Iterated forcing, category forcings, generic ultrapowers ... · of chapter 1 gives the basic facts...

Iterated forcing, category forcings, generic ultrapowers,

generic absoluteness

Matteo Viale,

Giorgio Audrito, Silvia Steila,

and the collaboration of Raphael Carroy.

List of Corrections

manca riferimento – M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Manca riferimento – M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Manca riferimento – M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Manca riferimento – Silvia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Manca riferimento – M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Aggiungere dimostrazione dell’ultimo punto – M . . . . . . . . . . . . . . . . . . 88Cercare su mathscinet uso di lottery preparation in articoli ed aggiungere qui

referenze principali su questo argomento – M . . . . . . . . . . . . . . . . . 116Aggiungere referenza – M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

What can be found in this book

The book contains:

• A self-contained account of the theory of iterated forcing, as well as the main resultsregarding the preservation of (semi)properness through limit stages by means of(revised) countable support iterations.

• A compact self-contained account of the general theory of generic ultrapowers inducedby a (tower of) normal ideal(s). In particular how to force the existence of idealsinducing generic ultrapower with strong closure properties, and a self-contained proofof the presaturation of the stationary towers. The latter results are also used to givea complete presentation of Woodin’s generic absoluteness results for second ordernumber theory in the presence of large cardinals.

• A detailed account of the first author’s work on generic absoluteness results for (largefragments of) third order arithmetic in the presence of strong forcing axioms.

The book is divided in six parts. It grew out of a series of lecture notes the first authorprepared for two PhD courses, one on iterated forcings and semiproperness and the otheron generic ultrapowers embeddings and Woodin’s generic absoluteness for projective sets ofreals. Only the sixth part of the book contains original research material by the first author(and Aspero). The first two parts contain material which is by now well-established; thismakes an hard task (at least for us) the correct attribution of the main results. The third,fourth and fifth parts of the book present major results (at least in the authors’ opinion)mainly by Shelah (on semiproper iterations), Foreman, Magidor, Shelah (on forcing axioms,Martin’s maximum, and their consistency relative to supercompactness), Foreman, Magidor(on the general theory of generic ultrapowers), Woodin (on the stationary towers andon generic absoluteness for second order number theory). Clearly we’re not mentioninghere many of the contributors to the development of these topics. We hope for their

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comprehension, and we’ll try in the subsequent pages to report with the greatest care thework of any scholar whose results appear in this book.

The book is organized as follows:

1. Part I introduces the basic objects we will be dealing with, i.e. on the one handthe category of complete boolean algebras and complete homomorphisms betweenthem and on the other hand normal ideals and towers of normal ideals. The coreof chapter 1 gives the basic facts on Stone duality between the category of booleanalgebras and that of compact Hausdorff spaces as well as a categorial characterizationof completeness for an homomorpshim i : B→ C of boolean algebras in terms of theexistence of an adjoint map π : C → B (Theorem 1.4.1). This will be repeatedlyused in our analysis of iterated forcing, among other reasons because it gives a∆0-characterization (therefore absolute between transitive structures) of the propertyof being a complete homomorphism, (this is in sharp contrast with the fact thatbeing a complete boolean algebra is not an absolute property). Chapter 2 deals withthe notions of normal ideal I on a set X and of generalized stationarity. We proveresults of Burke showing that (a tower of) normal ideal(s) is the projection of thenonstationary ideal restricted to a stationary set. Other basic facts about normalityand stationary sets are proved.

2. Part II introduces the type of structures we will be studying: i.e. boolean valuedmodels of set theory of type V B with B a complete boolean algebra, and genericultrapowers of type Ult(V,G) with G a V -normal (tower of) filter(s). We aimed togive a unified presentation of these two distinct type of structures, hence we firstdevelop the basic theory of boolean valued models in chapter 3, introducing the keynotion of full B-model, then we prove Los Theorem for full B-models, as well as therules governing the forcing relation on these type of models. Our approach to thesematters is inspired by Hamkins and Seabold [27]. Chapter 4 gives a fast account ofthe basic theory of forcing for boolean valued models of set theory, recalling withsketchy proofs the main results needed in the sequel of the book. Chapter 5 presentsgeneric ultrapowers of type Ult(V,G) as quotients by certain type of ultrafiltersG ⊆ P (P (X)) of the full P (P (X))-model given by functions f : P (X) → V inV . In this manner we can give a unified treatment both of generic ultrapowerembeddings and of the ultrapower embeddings induced by standard large cardinals;here (and everywhere in the book we deal with these topics) we continue along thelines of Foreman’s chapter for the Handbook [18], Larson’s book on stationary towerforcing [30], and Foreman and Magidor’s [20]. We make a point to prove all the basicresults about elementary embeddings using minimal assumptions, so to be able touse them both when dealing with standard ultrapower embeddings given by largecardinals, or when dealing with generic ultrapower embeddings given by a V -genericfilter on P (P (X)) /I for some normal ideal I on X. Along the way we also deal withtowers of normal ideals. In the end we show how to describe standard large cardinalproperties such as hugeness or supercompactness by means of this technology. Acommon theme of this part of the book is to outline the common features shared bygeneric ultrapowers and boolean valued models of set theory. Many of the remainingparts of the book analyze which are the situations in which the two types of models(V [G] and Ult(V,G)) are very close to each other. A more general approach to genericultrapowers which encompasses as special cases both the towers of normal ideals(which are the focus of the present book) and the notion of generic extender (whichis the generic counterpart for strongness and superstrongness) has been devised by

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Audrito and Steila in [4], however we decided not to pursue it in the present book.

3. Part III deals with the general theory of iterated forcing, which is here developedin the framework of boolean algebras, fully exploiting all the results on these typeof objects gathered in Chapter 1. Chapter 6 deals with two-steps iterations, whileChapter 7 deals with iterations of limit length. Here and in chapter 10 we develop onDonder and Fuchs approach to iterated forcing [22]. In chapter 6 we also introducecategory forcings (i.e. any class forcing whose conditions are set-sized forcing notionsand which is ordered by (a subfamily of) the complete embeddings existing betweenits conditions). This concept will gain more and more importance in the sequel ofthe book, and will become the central topic of the last part of the book.

4. Part IV deals with forcing axioms, properness and semiproperness. Chapter 8 gives athorough analysis of different types of forcing axioms and of their mutual interactions:it is shown that the axiom of choice, Baire’s category theorem, and Shoenfield’sabsoluteness results can all be naturally seen as forcing axioms; stationary sets arealso used to give a different characterization of forcing axioms in terms of a strongform of the downward Lowenheim-Skolem theorem. Chapter 9 introduces propernessand semiproperness. We link these concepts to forcing axioms and give a topologicaland algebraic characterization of both of these properties. Chapter 10 gives the mainresults regarding (semi)proper iterations, mainly their preservation through limitstages.

5. Part V deals with stationary tower forcings and generic ultrapowers induced by(towers of) normal ideals. Chapter 11 presents the main results of Woodin regardingstationary towers (i.e. that they induce almost huge generic elementary embeddings)and a key result by Foreman (Theorem 11.3.1) regarding ideal forcings (i.e. forcingsof type P (P (X)) /I with I a normal ideal on X). Foreman’s theorem gives an exactcharacterization of which type of forcings can consistently become isomorphic toan ideal forcing; along the way it provides an informative description of the closureproperties of the generic ultrapower embedding induced by these ideal forcings.Chapter 12 proves one of Woodin’s main achievements: i.e the invariance of secondorder number theory in the presence of large cardinals axioms; specifically it isproved that the theory of the Chang model L(Ordω) is generically invariant if weassume the existence of class many supercompact cardinals. Chapter 13 provesthe consistency of Martin’s maximum relative to the existence of a supercompactcardinal. It next addresses an analysis of the category forcing whose conditionsare stationary set preserving complete boolean algebras and whose order relationis given by the complete homomorphisms between them. Among many things it isshown that (assuming class many supercompact cardinals) Martin’s maximum canbe formulated as the assertion that the class of presaturated towers is dense in thiscategory forcing. This shows that very strong forcing axioms can also be formulatedin the language of categories in terms of density properties of class partial orders.These two last chapters serve as a motivation for the last part of the book, where wewill look at suitable generalizations to third order number theory (and beyond) ofWoodin’s generic absoluteness results for second order number theory.

6. In part VI we develop a general theory of category forcings: in Chapter 14 it is shownthat for a variety of natural classes of forcings Γ closed under two-steps iterationsthe class partial order (Γ,≤Γ) is particularly well behaving (C ≤Γ B if and only ifthere exists a complete homomorphism of B into C with a generic quotient in Γ).

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This expands on [50] where just the case Γ = SSP is studied. For example if δ is alarge enough cardinal (Γ ∩ Vδ,≤Γ) ∈ Γ, and in case Γ = SSP and δ is supercompactwe also obtain that (Γ ∩ Vδ,≤Γ) forces Martin’s maximum. Also it is shown that anyforcing in Γ is absorbed as a complete subforcing into some (Γ ∩ Vδ,≤Γ) whenever δis large enough. Chapter 15 presents a new forcing axiom CFA(Γ) and proves thatthe forcings in Γ preserving CFA(Γ) cannot change the theory of L(OrdκΓ) for acardinal κΓ which is naturally attached to Γ. In case Γ is the class of stationary setpreserving forcings, CFA(Γ) is a natural strengthening of Martin’s maximum, andcan be formulated as a density property of the class forcing (Γ,≤Γ) which makesthe theory of L(Ordω1) generically invariant with respect to forcings preservingCFA(Γ). We can produce similar results for a variety of other classes Γ as well. Thisexpands the results of [3, 50], and generalizes to (a very large fragment of) third orderarithmetic Woodin’s generic absoluteness results. In our opinion these results givea sound a posteriori explanation of the success forcing axioms have met in settlinga variety of problems undecidable on the basis of ZFC. Several applications of themain theorems of these chapters will appear in [2].

How to read the book

Parts I and II are the backbones for all the other results of the book. They contain materialwhich is familiar to most of the potential readers. However we discourage the readers fromskipping them entirely, since the approach to forcing and iterations taken in the book is inmany respects different from what one encounters in most forcing books or papers in settheory: we made a point to exploit as much as we could the algebraic and categorial toolsone disposes of when dealing with the category of boolean algebras with homomorphisms;we believe that, despite the initial effort required for the reader, this approach pays off. Asafe road for the reader is to skim through the first five chapters and try to get a clearidea of the statements of the main results, omitting their proofs (when those are present).

Parts III and IV deal with iterated forcings. Chapter 8 only serves as a motivation forthe results in part VI.

Part V on stationary towers and ideal forcings can be read independently of parts IIIand IV. There will be need of the results of Part IV just in Chapter 13 for the proof of theconsistency of Martin’s maximum.

Part VI reposes on all the results presented in the first five parts of the book, andrequires the reader to have gained familiarity with the content of almost all of them.

Several selection of topics are possible, we list the following two:

Woodin’s generic absoluteness results for second order number theory:

Skim through parts I and II, then read sections 11.2.1, 11.4, and chapter 12.

Preservation theorems for limits of semiproper iterations:

Skim through parts I and II (omitting entirely chapter 5) and then read Parts IIIand IV. Section 6.3 and chapter 8 can be omitted. The consistency proof of Martin’smaximum is given in section 13.1 and can be read independently of the otherresults of Part V (provided the reader has some familiarity with the notion ofsupercompactness).

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Prerequisites

Strictly speaking the unique prerequisite the reader is required to have is a good familiaritywith the forcing method as presented in any of the by-now numerous textbooks on thistopic (such as [7, 28, 29] or the first author’s notes [54], which gives a presentation offorcing in line with the approach taken in this book). Some familiarity with large cardinalswill be of great help to understand the content of chapter 5 and of part V, even if completeproofs are given in the book for all results on ultrapowers by standard large cardinalswe will use. Some familiarity with the notion of (semi)properness and its variants, andespecially with some of its applications can also be of great thelp but is not strictly needed.

Who made what.

Part I is due to all four authors, and its current presentations owes much to the contributionof R. Carroy. The overall architecture of the book has also seen an important contributionof Carroy. Chapters 5, 6, 7, 9, 10, 11 are mostly due to G. Audrito and S. Steila. Parts ofthese chapters and the remainder of the book is due to M. Viale. The last part of the book(VI) develops on ideas of Viale and Aspero; a paper coauthored by them containining anumber of applications of the results of Part VI is in preparation [2]; due to the length ofproofs, we decided to move the general theory of category forcings to this book, and leaveits applications to the joint paper.

Some motivations to write the book

The driving motivation (at least for the first author) to embark in the researches presentedin this book has been the following: give a satisfactory sound explanation rooted inmathematical logic for the success forcing axioms have met in settling so many problemsformalizable in third order arithmetic. Most of these problems are otherwise undecidableon the basis of ZFC alone. Why forcing axioms have been so successful? We believe thatthe results of part VI provide a solid mathematical explanation: strong forcing axiomsyield as a natural byproduct strong generic absoluteness results for third order arithmetic.This occurs much in the same way as large cardinal axioms do for second order numbertheory. Therefore large cardinals and forcing axioms transform forcing from a tool to proveindependence results into a tool to prove theorems. Moreover the axiom of determinacyAD plays a crucial role in providing a new powerful tool to settle most of the questionson second order arithmetic which are independent from ZFC; likewise Martin’s maximumplays the same role for third order arithmetic as AD does for second order arithmetic. Wedo not want to dwelve further on this theme here; the motivated reader is referred to [53],the introductory parts of [3, 51, 52], and (for a more technical set of results) to the resultsof Chapter 8.

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Contents

0.1 Axiomatizations of Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . 10

I Boolean algebras and generalized stationarity 16

1 Basics on partial orders, topology, boolean algebras 181.1 Boolean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.1.1 Lattices and boolean algebras . . . . . . . . . . . . . . . . . . . . . . 181.1.2 Complete boolean algebras . . . . . . . . . . . . . . . . . . . . . . . 201.1.3 Stone spaces of boolean algebras . . . . . . . . . . . . . . . . . . . . 221.1.4 Boolean completion of a partial order . . . . . . . . . . . . . . . . . 23

1.2 Homomorphisms of boolean algebras and Stone duality . . . . . . . . . . . . 241.2.1 Adjoint pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.2.2 Quotient homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3 Appendix 1: Basics on orders and topology . . . . . . . . . . . . . . . . . . 321.3.1 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.3.2 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.4 Appendix 2: Fundamental properties of adjoint pairs . . . . . . . . . . . . . 35

2 Clubs and normal ideals 372.1 Generalized clubs and generalized stationarity . . . . . . . . . . . . . . . . . 37

2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.2 Generalized stationarity versus classical stationarity . . . . . . . . . 392.1.3 First properties of the non-stationary ideals . . . . . . . . . . . . . . 40

2.2 Skolem Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2.1 Fast Skolem functions . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Normal Fine vs Non Stationary . . . . . . . . . . . . . . . . . . . . . . . . . 442.3.1 Normal Fine Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3.2 Non-stationary ideals and normal fine ideals . . . . . . . . . . . . . . 462.3.3 Projection and Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4 Towers of normal fine ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

II Boolean valued models 54

3 Boolean Valued Models 563.1 Boolean valued models and boolean valued semantics . . . . . . . . . . . . . 563.2 Full boolean valued models and Los theorem . . . . . . . . . . . . . . . . . 60

3.2.1 Los theorem for full boolean valued models . . . . . . . . . . . . . . 603.2.2 Full B-models for a non complete boolean algebra B . . . . . . . . . 62

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3.2.3 Sufficient conditions for fullness . . . . . . . . . . . . . . . . . . . . . 633.3 Homomorphisms of boolean valued models . . . . . . . . . . . . . . . . . . . 65

4 Forcing 684.1 Boolean valued models for set theory . . . . . . . . . . . . . . . . . . . . . . 68

4.1.1 Embeddings and boolean valued models . . . . . . . . . . . . . . . . 714.2 Basic properties of forcing extensions . . . . . . . . . . . . . . . . . . . . . . 72

4.2.1 Preservation of regular cardinals in forcing extensions . . . . . . . . 72

4.2.2 Computing HV [G]λ in forcing extensions . . . . . . . . . . . . . . . . 74

4.3 Class forcing and set forcing with posets . . . . . . . . . . . . . . . . . . . . 754.3.1 V P versus V RO(P ) for set sized forcings P . . . . . . . . . . . . . . . 77

5 Generic ultrapowers 805.1 Normal (towers of) ultrapowers . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 V -normal ultrafilters and towers . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2.1 V -normal ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2.2 Towers of V -normal ultrafilters . . . . . . . . . . . . . . . . . . . . . 855.2.3 Generic elementary embeddings versus (towers of) V -normal ultrafilters 88

5.3 Normal (towers of) ultrafilters in forcing extensions . . . . . . . . . . . . . . 915.4 Large cardinals defined by means of normal ultrapowers . . . . . . . . . . . 93

III Iterations of forcing notions 97

6 Two-steps iterations 996.1 Two-steps iterations and generic quotients . . . . . . . . . . . . . . . . . . . 99

6.1.1 Two-steps iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.1.2 Generic quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.1.3 Equivalence of two-steps iterations and injective complete homomor-

phisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.1.4 Generic quotients of generic quotients, aka three-steps iterations . . 1086.1.5 Preservation of chain conditions under two steps iterations . . . . . 109

6.2 Definable classes of forcing notions closed under two-steps iterations . . . . 1106.3 Category forcings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7 Iteration systems 1167.1 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . 1167.2 Sufficient conditions for the equality of direct and inverse limit, and preser-

vation theorems for the < λ-cc . . . . . . . . . . . . . . . . . . . . . . . . . 1197.3 Generic quotients of iteration systems . . . . . . . . . . . . . . . . . . . . . 1227.4 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.4.1 Distinction between direct limits and full limits . . . . . . . . . . . . 1237.4.2 Direct limits may not preserve ω1 . . . . . . . . . . . . . . . . . . . . 125

7.5 Iterable classes of forcing notions . . . . . . . . . . . . . . . . . . . . . . . . 1267.5.1 Weakly iterable forcing classes . . . . . . . . . . . . . . . . . . . . . 1277.5.2 The lottery preparation forcings . . . . . . . . . . . . . . . . . . . . 128

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IV Forcing axioms, properness, semiproperness 130

8 Forcing axioms I 1328.1 The axiom of choice and Baire’s category theorem are forcing axioms . . . . 1328.2 Forcing axioms and stationarity I . . . . . . . . . . . . . . . . . . . . . . . . 1348.3 Ωκ is closed under two-steps iterations . . . . . . . . . . . . . . . . . . . . . 1378.4 Forcing axioms as Σ1-reflection properties . . . . . . . . . . . . . . . . . . . 1378.5 Which forcings can be in Ωκ? . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.6 Forcing axioms and stationarity II: MM++ . . . . . . . . . . . . . . . . . . . 1418.7 Forcing axioms and category forcings . . . . . . . . . . . . . . . . . . . . . . 143

9 Properness and semiproperness. 1459.1 Shelah’s properness and semiproperness . . . . . . . . . . . . . . . . . . . . 1459.2 Algebraic definition of properness and semiproperness . . . . . . . . . . . . 1519.3 Topological characterization of properness and semiproperness . . . . . . . 1549.4 FA+

ω1(Countably closed) implies SP = SSP . . . . . . . . . . . . . . . . . . . 155

9.5 A common framework for the classes of proper semiproper and stationaryset preserving forcings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

10 Proper and semiproper iterations 15910.1 Two-steps iterations of (semi)proper posets . . . . . . . . . . . . . . . . . . 159

10.1.1 Two-steps iterations of proper forcings . . . . . . . . . . . . . . . . . 16210.2 (Semi)proper iteration systems . . . . . . . . . . . . . . . . . . . . . . . . . 163

10.2.1 Iterations of proper forcings . . . . . . . . . . . . . . . . . . . . . . . 167

V Selfgeneric ultrapowers, generic absoluteness for L(Ordω), Martin’smaximum 169

11 Self-generic presaturated ideals and towers 17111.1 Self-generic towers and self-generic ideal forcings . . . . . . . . . . . . . . . 17211.2 Sufficient conditions to get (almost) huge generic elementary embeddings . 174

11.2.1 Characterization of presaturation for towers of normal ideals . . . . 17411.2.2 Sufficient condition granting the presaturation for normal ideal forcings176

11.3 Foreman’s duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17811.3.1 Forcing the existence of presaturated towers: a guiding example . . 185

11.4 Self-genericity and presaturation of the stationary tower forcings . . . . . . 18711.4.1 Basic properties of tower forcings concentrating on sets of order type

at most λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18711.4.2 The stationary towers are presaturated and self-generic . . . . . . . 18911.4.3 Other properties of stationary towers . . . . . . . . . . . . . . . . . . 193

12 Generic absoluteness for L([Ord]ω) 194

13 Forcing axioms II 20013.1 Consistency of MM++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20013.2 Selfgeneric towers, forcing axioms, and category forcings . . . . . . . . . . . 202

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VI Forcing Axioms and Category Forcings 208

14 Category forcings 21214.1 Definitions and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 21214.2 κ-suitable category forcings . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

14.2.1 Proper, semiproper and stationary set preserving forcings are ω1-suitable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

14.2.2 Γ-Freezeability and Γ-rigidity . . . . . . . . . . . . . . . . . . . . . . 22014.2.3 From freezeability to total rigidity . . . . . . . . . . . . . . . . . . . 22414.2.4 Key properties of Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

14.3 When is (Γ ∩ Vδ,≤Γ) a partial order in Γ? . . . . . . . . . . . . . . . . . . . 22614.3.1 Proof of Theorem 14.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . 228

14.4 The quotient of (Γ B)V by a V -generic G for a B ∈ ΓV is ΓV [G] . . . . . . 23014.5 Other properties of the class forcing Γ and of UΓ

δ . . . . . . . . . . . . . . . 23214.6 MM++ and the relation between the stationary towers and the category

forcing (SSP,≤SSP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23414.6.1 A rough analysis of the forcing axiom MM++ . . . . . . . . . . . . . 23414.6.2 Duality between SSP-forcings and stationary sets . . . . . . . . . . . 23614.6.3 SSP-superrigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

15 Category forcing axioms 23915.1 Γ-superrgidity and category forcing axioms . . . . . . . . . . . . . . . . . . 241

15.1.1 Basic properties of Γ-superrigid self-generic forcings . . . . . . . . . 24215.1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24315.1.3 Proof of Theorem 15.1.7 . . . . . . . . . . . . . . . . . . . . . . . . . 24515.1.4 Proof of Theorem 15.1.9 . . . . . . . . . . . . . . . . . . . . . . . . . 247

9

0.1 Axiomatizations of Set Theory

We assume throughout the platonistic stance that there is a definite mathematical entitycalled the universe of sets V and an ∈-relation holding between two sets a, b ∈ V if and onlyif a belongs to b. We also assume that it is meaningful to speak of the proper subclasses ofV and to consider the inclusion relation ⊆ holding between subclasses A,B of V if everyset belonging to A belongs to B. Set theory describes the mathematical properties of thisuniverse of sets and of its proper classes.

Throughout the book we will work in two distinct natural first order axiomatizations ofset theory: the standard ZFC axiom system and the Morse-Kelley axiom system MK withsets and classes. While most (if not all) the results in the book can be proved in ZFC, thelast part of the book presents a number of results which are naturally formulated in MKsince they concern the properties of certain (proper) classes of forcing notions. Even thoughall the classes we will be interested are definable in ZFC, it is convenient to formulateour results in MK in order to avoid the cumbersome verifications that all our argumentsinvolving proper classes can be carried in ZFC. Throughout the book, with the exceptionof the last part, the readers can choose whichever of the two axiom systems as the basetheory over which they prefer to formalize our results. In the last part we will explicitlyembrace MK as our base theory. Also it will be convenient in many cases to formulate ZFCin a language with classes, i.e. to resort to the NBG axiom system for set theory. Hencewe list below all these three axiom systems and recall that NBG is a conservative extensionof ZFC (all formulae not containing class quantifiers provable in NBG are also provable inZFC).

We will work in a language L = ∈,⊆,= with three binary relation symbols forequality, membership, and containment. We feel free to adopt standard shorthands in theset theory practice, such as:

• the use of restricted quantifiers ∀x ∈ y,∃x ∈ y,

• the use of defined predicates such as x = z ∈ y : φ(z, x1, . . . , xn) (which is ashorthand for a formula ψ(x, y, x1, . . . .xn) defined by means of φ(z, x1, . . . , xn)),

• the use of defined constants (such as ∅, ω, ω1, . . . ),

• the use of definable functions such as the rank function, etc....

• the fact that most of our reasonings about sets can be formalized in the above firstorder language.

The ZFC axioms for set theory

Letting L = ∈,⊆,=, we formalize ZFC as the following list of L formulae.

1. Extensionality for sets: Two sets are equal if they have the same elements:

∀x∀y(x ⊆ y ↔ ∀z ∈ x(z ∈ y))

and∀x∀y(x = y ↔ (x ⊆ y ∧ y ⊆ x)).

2. Pairing: For any sets x and y there is a set containing x, y:

∀x∀y∃z(x ∈ z ∧ y ∈ z)

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3. Infinity: There is an infinite set:

∃z∃y[y ∈ z ∧ ∀x(x ∈ z → ∃t[(t ∈ z) ∧ ∀w(w ∈ t↔ (w ∈ x ∨ w = x)))]]

4. Union: For every set a its union ∪a exists:

∀y∃z∀w(w ∈ z ↔ ∃x ∈ y w ∈ x).

5. Powerset: For every set a its powerset P (a) exists:

∀y∃z∀w(w ∈ z ↔ w ⊆ y).

6. Foundation: Every nonempty set has an ∈-minimal element:

∀y∃x(x ∈ y ∧ ∀z ∈ x¬z ∈ y)

7. ZFC-Comprehension: For any formula φ(x, x1, . . . , xn) with displayed free variables

∀x1, . . . xn∀x∃y∀z[z ∈ y ↔ (z ∈ x ∧ φ(z, x1, . . . , xn))].

8. Collection: For all formulae φ(x, y, x1, . . . , xn) with displayed free variables

∀x1 . . . ∀xn∀w(∀x ∈ w∃yφ(x, y, x1, . . . , xn)→ ∃z∀x ∈ w∃y ∈ zφ(x, y, x1, . . . , xn)).

9. Choice: There exists a well-ordering of any set.

ZF is the axiom system consisting of all the above axioms except the axiom of choice,ZF \ P is the axiom system consisting of all the above axioms except power set and choice.ZFC \ P is the axiom system consisting of all the above axioms except power set.

The Morse-Kelley Set Theory MK

We present the axioms of Morse-Kelley Set Theory MK, following the notation of [1]. Tosimplify the readability, we work in a two sorted language L2 extending the first orderlanguage L = ∈,⊆,= for set theory with variables for sets and classes: capital lettersX,Y, Z, . . . will range over classes and sets, small letters x, y, z, . . . will range just oversets. Each set is a class and the class of sets is definable in the L2 formulation of MK bythe formula φ(X) ≡ ∃Y X ∈ Y . Hence the reader can convert our presentation of MK in L2

in a standard first order axiomatization replacing all quantifiers ∀x occurring in a formulaφ of L2 by the string ∀x∃Y x ∈ Y → and ∃x by the string ∃x∃Y x ∈ Y ∧.

1. Extensionality: Two classes are equal if they have the same elements:

∀X∀Y (X ⊆ Y ↔ ∀z(z ∈ X → z ∈ Y ))

and∀X∀Y (X = Y ↔ (X ⊆ Y ∧ Y ⊆ X)).

2. Pairing: For any sets x and y there is a set x, y:

∀x∀y∃z∀w[w ∈ z ↔ (w = x ∨ w = y)]

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3. Infinity: There is an infinite set:

∃z∃y[y ∈ z ∧ ∀x(x ∈ z → ∃t∀w(w ∈ t↔ (w ∈ x ∨ w = x)))]

4. Union: For every class A its union ∪A exists:

∀Y ∃Z∀w(w ∈ Z ↔ ∃x ∈ Y w ∈ x).

5. Powerset: For every class A its powerset P (A) exists:

∀Y ∃Z∀w(w ∈ Z ↔ ∀x ∈ w x ∈ Y ).

6. Foundation: Every nonempty class has an ∈-minimal element:

∀Y ∃x(x ∈ Y ∧ ∀z ∈ x¬z ∈ Y )

7. MK-Comprehension: For any formula ϕ(x,X1, . . . , Xn)

∀X1, . . . Xn∃Y (Y = x : ϕ(x,X1, . . . , Xn)).

8. Collection: If a class R is a binary relation and a is a set contained in its domain,there is a set b such that for all u ∈ a there is v ∈ b with u R v. In particular (usingcomprehension) if F is a class function and a is a set

F [a] = y : ∃x ∈ a 〈x, y〉 ∈ F

is a set.

9. Global choice: There exists a global class well-ordering of the universe of sets.

Recall that a model with classes has the form 〈M, CM ,∈M 〉, where M is the collectionof sets in the model (and it is itself a model of ZFC), and CM is the collection of classes inthe model; if M is truly a transitive class contained in the universe V of sets (i.e. a ∈Mentails a ⊆ M and ∈M is the true ∈-relation restricted to M) each class X in CM is anelement or a subset of M . We often denote an MK model 〈M, CM 〉 just by CM and omitthe reference to its class of sets M (M can be recovered from CM as the class of elementsof CM satisfying ∃y x ∈ y). In some cases we denote the MK model 〈M, CM 〉 just by itsfamily of sets M if the intended meaning of CM is clear from the context or is not relevantfor our argument (i.e the argument is truly a ZFC argument in which there is no need torefer to proper classes).

The Godel-Bernays Set Theory NBG

NBG is obtained replacing from the MK-system the comprehension axiom to its weakerform:

NBG-Comprehension: For any formula ϕ(x,X1, . . . , Xn) in which quantifiersrange just over sets we have that

∀X1, . . . Xn∃Y (Y = x : ϕ(x,X1, . . . , Xn)).

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In our formulation of NBG, the axiom of global choice of MK is replaced with the standardaxiom of choice of ZFC.

V is for us the standard model of ZFC, in the following sense:

(V,∈,⊆,=) |= ZFC

according to the rules of Tarski semantics for L applied to the structure (V,∈,⊆,=) (weare here taking the liberty to consider also proper classes as the possible domain of a firstorder structure). In the same spirit V with its collection of classes C is the standard modelof MK (and of NBG), in the sense that:

(V, C,∈,⊆,=) |= MK

according to the rules of Tarski semantics for L2 applied to the structure (V, C,∈,⊆,=).

Set theoretic terminology

trcl(x), rank(x) denote respectively the transitive closure and the rank of a given set x.Vα is the set of x such that rank(x) < α and for a cardinal κ Hκ is the set of x such that|trcl(x)| < κ. We use P (x), [x]κ, [x]<κ to denote the powerset of x, the set of subsets ofx size κ, and the set of subsets of x of size less than κ. Given a set M , πM : M → ZMdenotes its Mostowski collapse map onto a transitive set ZM . The notation f : A→ B isimproperly used to denote partial functions in A×B, AB to denote the collection of allsuch (partial) functions, and f [A] to denote the pointwise image of A through f . We usesat for sequence concatenation and sax where x is not a sequence as a shorthand for sa〈x〉.We use t C s to denote that s = t (|t|−1). CH denotes the continuum hypothesis and c thecardinality of the continuum itself. We prefer the notation ωα instead of ℵα for cardinals.

For a L2-formula φ(~y, ~Z), ~y are the displayed free variables of set type of φ and ~Z arethe displayed free variables of class type of φ. We introduce the following Levy hierarchyfor formulae of L2 over a theory T :

• φ(~y, ~Z) is ∆00(T ) or Σ0

0(T ) or Π00(T ) if it is provably equivalent in T to a formula in

which all quantifiers are of type ∃y ∈ x, ∀y ∈ x.

• φ(~y, ~Z) is Σ0n+1(T ) if it is provably equivalent in T to a formula of type ∃xψ(x, ~y, ~Z)

with ψ(x, ~y, ~Z) Π0n(T ).

• φ(~y, ~Z) is Π0n+1(T ) if it is provably equivalent in T to a formula of type ∀xψ(x, ~y, ~Z)

with ψ(x, ~y, ~Z) Σ0n(T ).

• φ(~y, ~Z) is ∆0n(T ) if it is provably equivalent in T to a formula of type Π0

n(T ) and toa formula of type Σ0

n(T ).

• All Σ0n(T ) formulae will also be Σ1

0(T ), ∆10(T ), Π1

0(T ).

• φ(~y, ~Z) is Π1n+1(T ) if it is provably equivalent in T to a formula of type ∀Xψ(X, ~y, ~Z)

with ψ(X, ~y) Σ1n(T ).

• φ(~y, ~Z) is Σ1n+1(T ) if it is provably equivalent in T to a formula of type ∃Xψ(X, ~y, ~Z)

with ψ(X, ~y, ~Z) Π1n(T ).

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• φ(~y, ~Z) is ∆1n(T ) if it is provably equivalent in T to a formula of type Π1

n(T ) and toa formula of type Σ1

n(T ).

• Qmn (T ) denotes any among the family of formulae Σmn (T ), Πm

n (T ), ∆mn (T ).

• If T consists of the logical axioms, we just write that φ is Qin rather than Qin(T ).

• We omit the upper index of the classes Σ0n(T ), Π0

n(T ), ∆0n(T ) for formulae φ(~x) of

this type whose displayed free variables are just of set-type and the Qn(T )-formulaestand for the formulae with just set-type free variables of type Σn(T ), Πn(T ), ∆n(T ).

• Given M ⊆ V transitive and sets a1 . . . , ak ∈M , a formula

ψ(x1, . . . , xi, a1, . . . , ak)

with just set-type free variables is Qn(M) in parameters a1, . . . , ak if for some Qn-formula

φ(x1, . . . , xi, y1, . . . , yk)

we have that for all b1, . . . , bi ∈M

M |= ψ(b1, . . . , bi, a1, . . . , ak)

if and only ifM |= φ(b1, . . . , bi, a1, . . . , ak).

• Given sets a1 . . . , ak ∈ V , a formula

ψ(x1, . . . , xi, a1, . . . , ak)

with just set-type free variables is Qn(T ) in parameters a1, . . . , ak if it is Qn(M) inparameters a1, . . . , ak for all transitive sets M which are models of T with a1, . . . , an ∈M .

We use 〈M,CM 〉 ≺in 〈N,CN 〉 to denote that 〈M,CM ,∈〉 is a Σin-elementary substructure

of 〈N,CN ,∈〉.We will often encounter the following scenario: we have transitive classes V,M ⊆W

which are all models of ZF and an elementary embedding j : V →M (i.e. it preserves truthof all formulae with no free variables and parameters in V ). To avoid ambiguities in theintended meaning of the above, we always assume the following: W comes in pair with itsfamily of classes C so that 〈W, C〉 is a model of NBG, j, V,M,W ∈ C, j is elementary justwith respect to formulae φ(~x) with just set-type bounded variables and set type parametersoccurring in them.

We denote by crit(j) (the critical point of j) the least ordinal moved by j (if such anordinal exists). In general j might not have a critical point, however we will see that in allcases of interest, j has a critical point unless it is the identity.

Notations

We will have to deal with objects ranging over many different domains. For this reason wewill try to stick to the following conventions:

• X, Y are either sets or classes;

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• X<ω denote the set of finte sequences on X;

• x, y and z are elements of X or Y ;

• s, t are elements of X<ω and Y <ω;

• Z is an element in P (X) or P (Y );

• S, T are elements of P (P (X));

• I, J are ideals on P (X) or P (Y );

• I is the dual filter of I;

• I+ is the set of positive elements of I.

• I is a tower of ideals.

• πM : M −→ ZM = πM [M ] ⊆ V is the Mostowsky’s collapse of the structure (M, ε),and jM : ZM →M denotes its inverse.

• The first letter of the greek letters such as α, β, γ denote ordinals, while lettersoccurring later in the alphabet such as κ, λ, δ, θ denote cardinals.

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Part I

Boolean algebras and generalizedstationarity

16

This part introduces the two types of mathematical entities which will be the bricksover which we develop the main body of this book: boolean algebras and normal ideals.The first chapter outlines the algebraic and topological properties of boolean algebrasrelevant for us, while the second gives a detailed account of the properties of stationarysets and of normal ideals.

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Chapter 1

Basics on partial orders, topology,boolean algebras

We recall some basic algebraic properties of (complete) boolean algebras and (complete)homomorphisms between them. Among other things we briefly recall Stone duality betweenthis category and that of compact Hausdorff spaces, we show that any partial order admitsa unique boolean completion up to isomorphism, and we give a ∆0-characterization ofthe notion of complete homomorphism. Useful definitions about orders and topology aregathered in an appendix to this chapter. The reference text for unexplained details andmissing proofs for results in the first two sections is [54, Chapter 2]; reference texts on thematerial of this chapter are [8, 23, 33, 43], in particular:

• our treatment of Stone duality for complete boolean algebras follows loosely Balcarand Simon’s appendix on general topology in the third volume of [33],

• we are inspired by [8, Chapters 1, 6] in our presentation of the basic properties ofhomomorphisms of bolean algebras.

1.1 Boolean algebras

We assume the reader is familiar with the basic terminolgy and facts on partial orders andtopological space, else we invite him to skim through the appendix of this chapter 1.3. Wealso assume familiarity with the basic properties of rings and ring homomorphisms (elsewe refer the reader to any basic undergraduate text in algebra).

1.1.1 Lattices and boolean algebras

A join-semilattice (P,≤) is a partial order such that every pair of elements (x, y) of Padmits a unique least upper bound denoted by x ∨ y, the join of x and y.

Dually, a partial order (P,≤) is a meet-semilattice when any two elements x and y inP have a unique greatest lower bound denoted by x ∧ y, the meet of x and y.

A partial order (P,≤) is a lattice if it is both a join-semilattice and a meet-semilattice.A lattice (P,≤) is bounded if it has a greatest element 1P and a least element 0P which

satisfy 0P ≤ x ≤ 1P for every x in P .A lattice (P,≤) is distributive if for all x, y and z in P we have

x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z).

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Let (P,≤) be a bounded lattice. A complement of an element a ∈ P is an elementb ∈ P such that a ∨ b = 1P and a ∧ b = 0P .

Remark 1.1.1. In a distributive lattice, if a has a complement it is unique. In this case wedenote by ¬a the complement of a.

A lattice is complemented if it is bounded and every element has a complement.A lattice (P,≤) is complete if every subset X = xi : i ∈ I of P has a meet (or infimum)∧

i∈I xi and a join (or supremum)∨i∈I xi.

Notice that if X = ∅, then∧∅ = 1 and

∨∅ = 0, so a complete lattice is always

bounded.A boolean algebra B is a bounded complemented distributive lattice. To define a boolean

algebra we consider a signature given by a sextuple (0, 1,∨,∧,¬,≤).

Example 1.1.2. Given a (non-empty) set X and a topology τ on X:

• let 0, 1, ∨, ∧, ¬ and ≤ be respectively ∅, X,∪,∩,¬ and ⊆, then the power set P (X)of X is a boolean algebra and is a complete lattice;

• the family τ and the family of closed sets τ c are bounded distributive sublattices ofP (X) (with the same operations we have on P (X));

• the family CLOP(X, τ) of clopen set of τ is a boolean subalgebra of P (X);

• the partial order (τ,⊆) has suprema for all of its subsets;

• the partial order (τ c,⊆) has infima for all of its subsets.

It is often convenient to introduce further operations on a boolean algebra. For examplegiven a and b elements of a boolean algebra a \ b = a ∧ ¬b, and a∆b = (a \ b) ∨ (b \ a).Notice that if B is P (X), these turn out to be the natural set theoretic operations onsubsets of X.

We denote by B+ the positive elements of B. We often consider B+ when referring toB as an order, otherwise some definitions could indeed become trivial.

We can also gather the formal definition of a bounded complemented distributive latticein an equational characterization of a boolean algebra:

Lemma 1.1.3. Let (B,∧,∨,¬, 0, 1) be a sextuple consisting of a set B, two total binaryoperations ∧ and ∨, a total unary operation ¬ on B and two elements 0 and 1 of B.(B,∧,∨,¬, 0, 1) is a boolean algebra if and only if the following hold:

a ∨ (b ∨ c) = (a ∨ b) ∨ c associativity

a ∧ (b ∧ c) = (a ∧ b) ∧ ca ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) distributivity

a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)a ∨ b = b ∨ a commutativity

a ∧ b = b ∧ aa ∨ 0 = a identity

a ∧ 1 = a

a ∨ ¬a = 1 complements

a ∧ ¬a = 0

On a boolean algebra (B, 0, 1,∨,∧,¬,≤) we can define an order relation a ≤ b given bya ∧ b = a (or equivalently a ∨ b = b).

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Ideals and filters on boolean algebrasThe following also holds:

Lemma 1.1.4. (B,∧,∨,¬, 0, 1) is a boolean algebra if and only if letting a∆b = (a ∨ b) ∧¬(a ∧ b) we have that (B,∧,∆, 0, 1) satisfies the axiom of a multiplicatively idempotentcommutative ring with ∆ interpreting the sum and ∧ the multiplication.

Definition 1.1.5. Let B be a boolean algebra, H ⊆ B. Its dual H = ¬b : b ∈ H.

Fact 1.1.6. Let B be a boolean algebra. I ⊆ B is an ideal on B seen as a commutative ringif and only if I is a filter on B+ seen as a partial order1.

By the above Lemma and Fact, we get the following:

Lemma 1.1.7. For any boolean algebra B and ideal I on B, B/I is a boolean algebra whoseelements are the equivalence classes [a]I = b ∈ B : a∆b ∈ I with operations inherited byB: ¬[a]I = [¬a]I , [a]I ∧ [b]I = [a ∧ b]I . . .

Proof. 〈B,∆,∧, 0B, 1B〉 is a commutative ring with idempotent multiplication, hence so isits quotient B/I modulo any ideal I on B.

1.1.2 Complete boolean algebras

If B is also complete (i.e. it admits suprema and infima with respect to all of its subsets),then it is a complete boolean algebra, or cba for short, we denote by

∨A the suprema (in

the sens of the order on B) of a subset A of a boolean algebra B (if it exists), and by∧A

its infima (if it exists).Given a boolean algebra B, a subset X of B+ is dense if it is dense in the poset (B+,≤)

(i.e. for any b ∈ B+ there is a ≤ b in X). A boolean algebra B is atomless if the order(B+,≤) is atomless (i.e. with no minimal elements under ≤). A boolean algebra is atomicif the set of minimal elements (i.e. atoms) of (B+,≤) is dense in (B+,≤).

Given a boolean algebra B, and some b ∈ B+, the boolean algebra B b is given bya ∈ B : a ≤B b, with the operations inherited from B. The top element of B b is b.

We need the following property of complete boolean algebras:

Fact 1.1.8. Assume B is a complete boolean algebra and X ⊆ B. Then∨X = 1B iff

X ∩ B+ is a predense2 subset of (B+,≤) in the sense of the order.More generally for any dense set D ⊇ B+ and any a ∈ B+ a =

∨q ∈ D : q ≤B a

In particular a complete boolean algebra B can be split in the disjoint sum of an atomicboolean algebra and of an atomless boolean algebra. I.e. there is c ∈ B such that B ¬c isatomless, and B c is atomic.

Preservation of completeness under quotientsRemark that the class of complete boolean algebras is closed under products and

complete subalgebras, but in general it is not closed under quotients (for example P (N) iscomplete, while P (N) /Fin is not, Fin being the ideal of finite subsets of N).

1See 1.3.1 for the definition of filter on a partial order.2See 1.3.1 for the definition of predensity.

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Definition 1.1.9. Let B be a boolean algebra and κ a cardinal.B is κ-complete (respectively < κ-complete) if

∨B J (the suprema of J under the order

relation on B) exists for all J ∈ [B]κ (respectively J ∈ [B]<κ).B is complete if it is |B|-completeLet I ⊆ B an ideal. I is κ-complete (respectively < κ-complete) if

∨B J ∈ I for all

J ∈ [I]κ (respectively J ∈ [I]<κ).

Lemma 1.1.10. Assume κ ≥ λ, B is κ-complete and I ⊆ B is a λ-complete ideal. ThenB/I is λ-complete. Hence if B and I are both < δ-complete, so is B/I .

Proof. Given [aξ]I : ξ < λ ⊆ B/I , a =∨ξ<λ aξ ∈ B exists since B is κ-complete. [a]I

is the suprema of [aξ]I : ξ < λ: if [c]I ≥ [aξ]I for all ξ, we get that dξ = aξ \ c ∈ Ifor all ξ ≤ λ and a \ c ≤ d =

∨B dξ : ξ ≤ λ (d exists by κ-completeness of B), hence

[c]I ≥ [a]I .

Complete boolean algebras of regular open setsEvery complete boolean algebra can be considered as the family of regular open sets of

some given topological space. The first step in this direction is to show that the regularopen sets of a given topological space have a natural structure of complete boolean algebra.

Definition 1.1.11. Let (X, τ) be a topological space. A ∈ τ is regular open if3 A =Reg (A). We denote by RO(X, τ) the collection of regular open sets in X with respect toτ . If no confusion can arise we write RO(X) instead of RO(X, τ).

Remark 1.1.12. Any clopen subset of a topological space is regular. Any open intervalof R with the usual topology is regular. If U and V are open regular then so is U ∩ V .Moreover any isolated point x ∈ X of a topological space X is such that x is clopen andthus regular.

We set 0 = ∅, 1 = X, U ≤ V if and only if U ⊆ V and we equip RO(X) with thefollowing operations:

U ∨ V =Reg (U ∪ V ) ,

U ∧ V =U ∩ V,∨i∈I

Ui =Reg

(⋃i∈I

Ui

),

∧i∈I

Ui =Reg

(⋂i∈I

Ui

),

¬U =X \ Cl (U) .

The following holds:

Theorem 1.1.13. Assume (X, τ) is a topological space. Then RO(X) with the operationsdefined above is a complete boolean algebra.

The following characterization of regular open sets is useful:

Lemma 1.1.14. Let (X, τ) be a topological space. For any open A ∈ τ we have:

Reg (A) = x ∈ X : ∃U ∈ τ open set containing x such that A ∩ U is dense in U .3Reg (A) is the interior of the closure of A, see 1.3.2.

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Hence if (P,≤) is a partial order, we get that for the forcing topology4 τP and someA ⊆ P

Reg (A) = q ∈ P : A ∩Nq is dense in Nq ,

since Nq is the smallest open neighborhood of q in τP .

Remark that if U is an open neighborhood of x witnessing that x ∈ Reg (A) any V ⊆ Uopen neighborhood of x is equally well a witness of x ∈ Reg (A), since A∩W = A∩U ∩Wis a dense open subset of W for any open W ⊆ U .

1.1.3 Stone spaces of boolean algebras

There is a natural functor that attaches to a boolean algebra the Stone space of itsultrafilters. These spaces turn out to be compact and 0-dimensional. Complete booleanalgebras are those whose Stone spaces have the property that their regular open sets areclopen.

Definition 1.1.15. Let B be a boolean algebra. G ⊂ B is a prefilter in B if and only iffor every a1, . . . , an ∈ G, a1 ∧ · · · ∧ an > 0B.A prefilter G is a filter if it contains all its finite meets and is upward closed.A filter G is an ultrafilter if it satisfies the additional condition:

∀a ∈ B, a ∈ G ∨ ¬a ∈ G.

A filter G is principal if a ∈ G for some atom a ∈ B.

We recall the following essential result:

Theorem 1.1.16 (Prime ideal theorem). Assume F is a prefilter on a boolean algebra B.Then F can be extended to an ultrafilter G on B.

Let B be a boolean algebra. We define:

St(B) = G ⊆ B : G is an ultrafilter;

τB is the topology on St(B) generated by 5Nb = G ∈ St(B) : b ∈ G : b ∈ B.

The topological space (St(B), τB) is the Stone space of B. We have:

1. for all b ∈ B, Nb ∩N¬b = ∅;

2. for all b ∈ B, Nb ∪N¬b = St(B);

3. for all b1, . . . , bn ∈ B, Nb1 ∩ . . . ∩Nbn = Nb1∧...∧bn ;

4. for all b1, . . . , bn ∈ B, Nb1 ∪ . . . ∪Nbn = Nb1∨...∨bn .

We outline the key properties of Stone spaces.

Theorem 1.1.17 (Stone duality for boolean algebras). Given a boolean algebra B, wehave that:

• (St(B), τB) is a Hausdorff 0-dimensional, compact topological space.

4The forcing topology on a partial order (P,≤) is the topology whose open sets are the downward closedsubsets of P , see 1.3 for details.

5I.e., the smallest topology that contains Nb : b ∈ B.

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• The map

φ :B→ CLOP(St(B))

b 7→ Nb

is an isomorphism, hence the clopen sets of τB are the sets Nb for b ∈ B, and form abasis for τB.

• There is a natural correspondence between open (closed) subsets of St(B) and ideals(filters) on B:

– U ⊆ St(B) is open if and only if

c ∈ B : Nc ⊆ U is an ideal on B

– F ⊆ St(B) is closed if and only if

c ∈ B : Nc ⊇ F is a filter on B

• G is an isolated point of St(B) if and only if G = Ga = b ∈ B : a ≤ b is a principalultrafilter generated by some atom a ∈ B.

It is possible to go the other way round, i.e. take a topological space and attach to it aboolean algebra, we already outlined one natural option assigning to (X, τ) the completeboolean algebra RO(X, τ). There is also another possibility, which assigns to (X, τ) thefamily of clopen subsets of X.

Proposition 1.1.18. Let (X, τ) be a 0-dimensional compact topological space. Then (X, τ)is homeomorphic to the Stone space of CLOP(X, τ) via the map

π : X −→ St(CLOP(X, τ))

x 7−→ Gx = U ∈ CLOP(X, τ) : x ∈ U ∈ St(CLOP(X, τ)).

For a given topology τ on X there are two natural boolean algebras we can attach to it:CLOP(X, τ) and RO(X, τ). Observe that CLOP(X, τ)+ is always contained in RO(X, τ)+

and that if τ is 0-dimensional, any open set contains a clopen set, hence CLOP(X, τ)+ is adense subset of RO(X, τ)+.

We give a necessary and sufficient condition to ensure that CLOP(X, τ) and RO(X, τ)coincide.

Proposition 1.1.19. Assume B is a boolean algebra. B is complete if and only if theregular open sets of St(B) overlap with the clopen subsets of St(B).

We say that a topological space (X, τ) is extremally disconnected if CLOP(X, τ) =RO(X, τ).

1.1.4 Boolean completion of a partial order

Every pre-order can be completed to a complete boolean algebra.

Theorem 1.1.20. Let (Q,≤Q) a pre-order. There exists a unique (up to isomorphism)cba B and a map j : Q→ B such that:

1. j preserves order and incompatibility (i.e. both a ≤Q b⇒ j(a) ≤B j(b) and a ⊥ b⇔j(a) ∧ j(b) = 0B hold).

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2. j[Q] is a dense subset of the partial order (B+,≤).

Notice that while B is unique, there can be many j : Q→ B which satisfy the aboverequirements.

Proof. Define:

j : Q −→ RO(Q)

p 7−→ Reg (↓ p) .

The map j defines a complete embedding of Q into RO(Q)+ with a dense image whichpreserves order and incompatibility. The uniqueness up to isomorphism of the booleancompletions of Q can also be established.

Remark 1.1.21. (Q,≤) is a separative pre-order if and only if the map j : Q→ RO(Q) ofTheorem 1.1.20 is an injection.

Notation 1.1.22. Given a partial order P we denote by RO(P ) the algebra of regularopen sets of the forcing topology on P . Given a topological space (X, τ) we denote byRO(X) (or RO(X, τ) in case confusion can arise) the algebra of regular open sets of τ .

1.2 Homomorphisms of boolean algebras and Stone duality

We introduce the type of homomorphisms between boolean algebras that will be of interestfor us and we analyze their basic properties.

1.2.1 Adjoint pairs

The terminology we adopt comes from category theory (i.e. adjoint functors for categorieswhich are partial orders), and from Stone duality which (as we just saw) establish thecorrespondence of the notion of homomorphism between boolean algebras with that ofcontinuous function between compact Hausdorff spaces. Much of what we say drawsfrom [8, Chapters 1, 6].

Definition 1.2.1 (Def. 1.3, Thm 1.2 [8]). Let P,Q be partial orders and i : P → Q,π : Q→ P be order preserving maps between them. The pair (i, π) forms an an adjointpair (or a Galois connection or a pair of residuated mappings) if for all p ∈ P and q ∈ Q

i(p) ≥ q if and only if π(q) ≤ p.

Proposition 1.2.2. Let P,Q be partial orders and i : P → Q, π : Q → P be such that(i, π) is an adjoint pair. It holds that:

1. i π(c) ≥ c for all c ∈ Q,

2. π is defined byπ(c) = inf

Pb ∈ P : i(b) ≥ c ,

3. i is defined byi(b) = sup

Qc ∈ Q : b ≥ π(c) ,

4. i π i = i, and π i π = π,

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5. if i is injective π i is the identity map on P .

Hence any order preserving map i : P → Q (π : Q→ P ) can have at most one π : Q→ P(i : P → Q) such that (i, π) is an adjoint pair.

Proof.

1. Since (i, π) is an adjoint pair we get that i(b) ≥ c if and only if b ≥ π(c), thereforei π(c) ≥ c if and only if π(c) ≤ π(c), which is clearly the case.

2. Observe that i π(c) ≥ c by the first item, hence π(c) is in the set on the right-handside. Moreover b ≥ π(c) if and only if i(b) ≥ c since (i, π) is an adjoint pair. Henceπ(c) is the minimum of the set on the right-hand side.

3. Observe that π i(b) ≤ b since i(b) ≤ i(b) trivially holds and (i, π) is an adjointpair, hence i(b) is in the set on the right-hand side. Moreover b ≥ π(c) if and only ifi(b) ≥ c since (i, π) is an adjoint pair. Therefore i(b) is the maximum of the set onthe right-hand side.

4. π(i(b)) ≤ b if and only if i(b) ≤ i(b) since (i, π) is an adjoint pair. Thereforei π i(b) ≤ i(b) since i is order preserving. i(π(i(b))) ≥ i(b) if and ony if π(i(b)) ≥π(i(b)) since (i, π) is an adjoint pair. Hence i π i(b) = i(b). The other assertion isproved exactly in the same vein.

5. If i is injective i π i(b) = i(b) if and only if π i(b) = b for all b ∈ P .

Notation 1.2.3. In case (i, π) forms an adjoint pair, π is the adjoint, the residual, theretraction, or the projection associated to i.

Definition 1.2.4. Let B,C be boolean algebras (not necessarily complete).A map k : B→ C is a homomorphism if it preserves the boolean operations, a κ-complete

homomorphism if it moreover maps predense subsets of size at most κ of B+ to predensesubsets of C+, a complete homomorphisms if it is |B|-complete, an isomorphism if it is abijective homomorphism.

i is a regular embedding if it admits an adjoint πi.

Remark 1.2.5. Assume B,C are κ-complete boolean algebras. i : B→ C is a κ-completehomomorphisms if and only if i[

∨BA] =

∨C i[A] for all A ⊂ B of size at most κ.

The following theorem links the notion of (complete injective) homomorphism to thatof (open surjective) continuous function and of adjoint pair:

Theorem 1.2.6. Let i : B→ C be an homomorphism of boolean algebras.Define

π∗i : St(C)→ St(B)

G 7→ i−1[G],

Then:

1. π∗i is continuous and closed;

2. i is injective if and only if π∗i is surjective, in which case π∗[Ni(b)] = Nb;

3. i is complete if π∗i is open.

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4. Assume i is injective. Then i has an adjoint πi if and only if π∗i is an open map, inwhich case:

• πi is defined by the identity Nπi(c) = π∗i [Nc].

• i is complete,

• πi(c) ∧ b = πi(c ∧ i(b)) for all b ∈ B and c ∈ C.

• πi is order and suprema preserving.

5. Assume i is an injective complete homomorphism. Then

i :RO(St(B))→ RO(St(C))

A 7→ Reg(⋃

Ni(b) : Nb ⊆ A)

is a regular embedding extending i (i.e. modulo the identification of B with CLOP(St(B))and C with CLOP(St(C)) i(Nb) = Ni(b)). Moreover i has an adjoint π if and only ifπ is the restriction to B of the adjoint π of i (i.e. modulo the identification of B withCLOP(St(B)), π B is the map Nc 7→ π∗i [Nc]).

Remark 1.2.7. Let B be the boolean algebra given by finite and cofinite subsets of ω andi : B → P (ω) be the inclusion map. It can be checked that i is a complete injectivehomomorphism and π∗i is not an open map6. In particular the notion of complete injectivehomomorphisms is slightly weaker than that of regular embedding. In any case in theremainder of the book we will just consider complete injective homomorphism which arealso regular embeddings.

Proof. Let π∗ denote π∗i in what follows. Observe that

π∗[Nc] = H : ∃G ∈ Nc (i[H] ⊆ G) = H ∈ St(B) : c ∧ i(d) > 0 for all d ∈ H

by the prime ideal theorem.

1. Immediate since π∗−1[Nb] = Ni(b) for all b ∈ B.

2. The surjectivity of π∗ follows by the prime ideal theorem:

If i is injective, i[H] is a prefilter, hence there is G ∈ St(B) containing it, andπ∗(G) = H.

If i is not injective and i(b) = 0 for some b > 0, we have that Nb ∩ ran(π∗) = ∅, henceπi is not surjective.

3. Assume i is not complete. Let D be a predense subset of B such that i[D] is notpredense in C. Fix c ∈ C such that c ∧ i(d) = 0C for all d ∈ D. Then

A =⋃d∈D

Nd

is a dense open subset of St(B) and π∗[Nc]∩A = ∅. Since St(B) \A is closed nowheredense, this gives that π∗[Nc] is closed (since St(B) is compact Hausdorff) and nowheredense, hence it is not open.

6This counterexample has been given by D. Monk.

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4. First assume π∗ is open. By 3 i is complete. Let us define πi : C→ B by the identityπ∗[Nc] = Nπi(c) for c ∈ C. πi is well defined since π∗ is an open and closed map,hence π∗[Nc] is always clopen. πi is immediately seen to be order preserving. πi isalso surjective: For b ∈ B

Nb = π∗[Ni(b)] = Nπi(i(b)).

Now we show the remaining properties of πi and π∗:

πi preserves suprema: Assume c =∨

CA; this occurs if and only ifNc = Reg (⋃Na : a ∈ A).

Now π∗ is open, hence π∗[⋃Na : a ∈ A] is open; it is also dense in π∗[Nc]

since π∗ is open surjective. Hence

Nπi(c) = π∗[Nc] = Reg(π∗[⋃Na : a ∈ A]

),

giving that πi(c) =∨

B πi[A].

πi is the adjoint of i: It suffices to prove that

πi(c) =∧b : i(b) ≥ c ,

by 1.2.2.

First we prove πi(c) ≤ b for all b such that i(b) ≥ c: Assume i(b) ≥ c. LetH ∈ π∗[Nc] = Nπi(c), this occurs if and only if i(d)∧ c > 0c for all d ∈ H, givingthat i(d)∧ i(b) ≥ i(d)∧c > 0C for all d ∈ H (since i(b) ≥ c), which occurs only ifb ∈ H; since this occurs for all H ∈ π∗[Nc] we have that π∗[Nc] ⊆ Nb, thereforeπi(c) ≤ b. To conclude it suffices to show that i(πi(c)) ≥ c. Assume not towardsa contradiction; then 0C < ¬(i(πi(c)) ∧ c = i(¬πi(c)) ∧ c. Hence there existsH ∈ Ni(¬πi(c)) with c ∈ H. Then on the one hand i−1[H] = π∗(H) ∈ π∗[Nc] =Nπi(c) by definition of πi(c), and on the other hand πi(c) 6∈ i−1[H] by definitionof H; we reached a contradiction.

πi(c) ∧ b = π(c ∧ i(b)) for all b ∈ B and c ∈ C:

Nπi(c∧i(b)) = H ∈ St(B) : c ∧ i(b) ∧ i(d) > 0 for all d ∈ H =

= H ∈ Nb : c ∧ i(d) > 0 for all d ∈ H =

= H ∈ St(B) : c ∧ i(d) > 0 for all d ∈ H ∩Nb =

= π∗[Nc] ∩Nb =

= Nπi(c) ∩Nb =

= Nπi(c)∧b.

On the other hand assume i has an adjoint πi = π. We will prove that π∗[Nc] = Nπ(c)

for any c ∈ C, this will grant that π∗ is an open map.

Towards this aim we establish a few algebraic properties of π.

We start showing that π preserves the ∨ operation: Since π(c) =∧b : i(b) ≥ c we

get that

π(c ∨ d) =∧e : i(e) ≥ c ∨ d =

∧b : i(b) ≥ c ∨

∧f : i(f) ≥ d = π(c) ∨ π(d).

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Now we prove that the identity

π(i(b) ∧ c) = b ∧ π(c)

holds for the pair (i, π), and any b ∈ B, c ∈ C:

For b ∈ B, c ∈ C, the following three equations hold:

π(c ∧ i(b)) ∨ π(c ∧ ¬i(b)) = π(c); (1.1)

(π(c) ∧ b) ∨ (π(c) ∧ ¬b) = π(c); (1.2)

(π(c) ∧ b) ∧ (π(c) ∧ ¬b) = 0B. (1.3)

Furthermore, by definition of π, we have

π(c ∧ i(b)) ≤ π(c) ∧ b; (1.4)

π(c ∧ ¬i(b)) = π(c ∧ i(¬b)) ≤ π(c) ∧ ¬b. (1.5)

By (1.4), (1.5), and (1.3) we get

π(c ∧ i(b)) ∧ π(c ∧ ¬i(b)) ≤ (π(c) ∧ b) ∧ (π(c) ∧ ¬b) = 0B.

Moreover, by (1.1) and (1.2),

π(c ∧ i(b)) ∨ π(c ∧ ¬i(b)) = π(c) = (π(c) ∧ b) ∨ (π(c) ∧ ¬b).

By the laws of boolean algebras, we conclude that

π(c ∧ i(b)) = π(c) ∧ b and π(c ∧ ¬i(b)) = π(c) ∧ ¬b.

Now we can prove that Nπi(c) = π∗[Nc] for all c ∈ C, this grants that π∗ is an openmap, hence i is complete, and all the required properties of π∗ and πi holds.

To this aim recall that π∗[Nc] = H ∈ St(B) : c ∧ i(d) > 0 for all d ∈ H. But wehave just shown that c ∧ i(d) > 0 if and only if π(c) ∧ d > 0. Hence H ∈ π∗[Nc] ifand only if H ∈ Nπ(c), as was to be proved.

5. Assume i is complete and injective. We first show the following:

For all A dense open subset of St(B)

B = π∗−1[A] =⋃

Ni(b) : Nb ⊆ A

is a dense open subset of St(C).

Clearly B is open since π∗ is continuous. Moreover B =⋃

Ni(b) : Nb ⊆ A

, and Ais an open dense subset of B if and only if b : Nb ⊆ A is a predense subset of B+.Since i is complete, i(b) : Nb ⊆ A is a predense subset of C+, which occurs if andonly if B is a dense open subset of St(C).

Now we prove that i is an homomorphism. For A regular open subset of St(B) wehave that

i(A) = Reg(⋃

Ni(b) : Nb ⊆ A)

andi(¬A) = Reg

(⋃Ni(b) : Nb ∩A = ∅

);

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therefore i(A)∪ i(¬A) is dense in St(C) since A∪¬A is dense open in St(B). On theother hand

i(A) ∩ i(B) = Reg(⋃

Ni(b) : Nb ⊆ A)∩ Reg

(⋃Ni(c) : Nc ⊆ B

)=

= Reg(⋃

Ni(b∧c) : Nb ⊆ A, Nc ⊆ B)

=

i(A ∩B).

These two identities suffice to prove that i is an homomorphism. Clearly i(Nb) = Ni(b),hence i extends i.

Let π(A) = Int (⋂B : i(B) ⊇ A). Then it is immediate to check that π is the

adjoint of i by 1.2.2. since

π(A) = Int(⋂B : i(B) ⊇ A

)= inf

RO(St(B))B : i(B) ⊇ A .

Therefore i is a regular embedding with adjoint π which extends i.

If i is a regular embedding, then π∗i [Nc] = Int (π∗i [Nc]) is clopen for all c ∈ C by 4.Assuming this is the case for all c ∈ C, we get that

πi(Nc) = Int(⋂

B ∈ RO(St(B)) : i(B) ⊇ Nc

)= Int (π∗i [Nc]) = π∗i [Nc].

Conversely if πi(Nc) is clopen for all c ∈ C we get that πi B is the adjoint of i, hencei is a regular embedding and πi(Nc) = π∗i [Nc] can be again obtained as above using4 for the regular embedding i.

For the sake of completeness we also outline that any continuous map between compactHausdorff spaces gives rise to an homomorphism between the boolan algebras of clopensets, which is also complete if the map is open:

Lemma 1.2.8. Assume X,Y are compact Hausdorff and f : X → Y is continuos, then:

if : CLOP(Y )→ CLOP(X)

U 7→ f−1[U ],

is an homomorphism, which is complete if f maps open sets to open sets, morever in thiscase πif (U) = f [U ] for all U clopen subset of X.

Proof. Left to the reader.

1.2.2 Quotient homomorphisms

The following definitions and results will be used to analyze the quotient homomorphismsinduced by V -generic filters, as well as the homomorphisms defining a tower of ideals.

Definition 1.2.9. Let k : B→ C an homomorphism of boolean algebras, I be an ideal onB, and J an ideal on C.

• J projects to I through k if k[I] ⊆ J and k[I+] ⊆ J+.

• J is the lift of I by k if J =↓ k[I].

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Proposition 1.2.10. Let k : B → C be a complete homomorphism of boolean algebras,π : C→ B be such that (k, π) forms an adjoint pair, I be an ideal on B, and J be an idealon C.


1. J projects to I through k if and only if π[J+] = I+ and π[J ] ⊇ I.

2. J is the lift of I by k if and only if π[J ] = I and π[J+] = I+. Hence if J is the liftof I by k, J projects to I through k.

Proof.

1. First assume J projects to I through k. We show that π[J+] = I+ and π[J ] ⊇ I.Clearly k[I] ⊆ J gives that π[J ] ⊇ π k[I] = I. Now J+, J partition C in twodisjoint pieces, and π is surjective, hence B = π[J+]∪ π[J ]. Assume π(c) ∈ I ∩ π[J+].Then c ≤ k(π(c)) ∈ J , since J projects to I through k, a contradiction. Thereforeπ[J+] ⊆ I+. Since k[I+] ⊆ J+ and I+ = π k[I+], we get that π[J+] = I+.

Conversely assume π[J+] = I+ and π[J ] ⊇ I. Then k[I+] = k π[J+] ⊆ J+, andk[I] ⊆ k π[J ] ⊆ J . Hence I projects to J through k.

2. Assume J =↓ k[I]. Then c ∈ J+ if and only if c 6≤ k(b) for all b ∈ I if and only ifπ(c) 6≤ b for all b ∈ I if and only if π(c) ∈ I+. Hence π[J+] = I+ and π[J ] = I. Theconverse is left to the reader.

Lemma 1.2.11. Assume k : B→ C is an homomorphism. Let I be an ideal on B and Jan ideal on C such that J projects to I through k. Then

k/I,J :B/I → C/J

[b]I 7→ [k(b)]J

is an injective homomorphism.Assume moreover that k has an adjoint π and that J is the lift of I by k. Then

π/I,J :C/J → B/I

[c]J 7→ [π(c)]I

is well defined and π/I,J is the adjoint of k/I,J , hence k/I,J is a regular embedding (eventhough B/I ,C/J need not be complete).

Proof. The condition k[I] ⊆ J grants that k/I,J is an homomorphism on the quotientalgebras, the condition k[I+] ∩ J = ∅ grants that k/I,J is injective.

To prove the second part of the proposition, we first observe that k/I,J([b]I) ≥ [c]J ifand only if c ∧ k(¬b) = c ∧ ¬k(b) ∈ J if and only if7 π(c) ∧ ¬b = π(c ∧ k(¬b)) ∈ I, if andonly if [π(c)]I ≤ [b]I .

Hence[π(c)]I = inf [b]I : k/I,J([b]I) ≥ [c]J .

This gives that π/I,J([c]J) = [π(c)]I is a well defined order-preserving map from C/J ontoB/I and also that (k/I,J , π/I,J) forms an adjoint pair.

7Here we are crucially using that J is the lift of I by k.

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Remark 1.2.12. In general the notion of canonical projection through k is weaker than thatof being the lift by k: For example if k : B→ C is injective but not surjective, H ∈ St(B),G ∈ St(C) are ultrafilters with k[H] ⊆ G, we have that G canonically projects to H, but Gis never the lift of H by k, since ↓ k[H] is not a prime ideal.

Proposition 1.2.13. Assume k : B → C is a κ-complete homomorphism of κ-completeboolean algebras. Let J be an ideal on C which projects to I ideal on B through k. AssumeI is κ-complete. Then k/I,J is κ-complete as well.

Proof. Assume A = [ai]I : i < κ ⊆ B/I . Let a =∨i<κ ai. Then [a]I is the supremum of

A: clearly it is an upper bound, and if [b]I ≥ [ai] for all i < κ we have that ai \ b = ci ∈ I,hence c =

∨i<κ ci ∈ I (by κ-completeness of I) and a \ b ≤ c. Similarly one checks that

[k(a)]J is the supremum in C/J of k/I,J [A].

Remark 1.2.14. Remark in sharp contrast, that while the property of being a completeboolean algebra is not absolute between transitive structures (in an outer model some newsubset of the boolean algebra may not have a supremum), the property of being a regularembedding is absolute (and regular embeddings are just a very slight strengthening of thenotion of complete injective homomorphism).

Notation 1.2.15. Let B be a complete boolean algebra, and let b ∈ B+. Then

B b= c ∈ B : c ≤ b,

and

Restb : B→ B bc 7→ c ∧ b.

is the restriction map from B to B b.Let i : B→ C be a complete homomorphism. We define

ker(i) =∨b ∈ B : i(b) = 0C

coker(i) = ¬ ker(i)

Remark 1.2.16. We can always factor a complete homomorphism i : B→ C as the restrictionmap from B to B coker(i) (which we can trivially check to be a complete and surjectivehomomorphism) composed with the regular embedding i Bcoker(i)

. This factorizationallows to generalize easily many results on regular embeddings to results on completehomomorphisms.

Later in these notes we will use the following Lemma to extend various results oni : B→ C to subalgebras of the form B πi(c), C c.

Lemma 1.2.17 (Restriction). Let i : B→ C be a regular embedding, c ∈ C, then

ic : B πi(c) → C cb 7→ i(b) ∧ c

is a regular embedding and its associated retraction is πic = πi (Cc).

Proof. First suppose that ic(b) = 0, then by Proposition 1.4.1.??,

0 = πi(ic(b)) = πi(i(b) ∧ c) = b ∧ πi(c) = b.

Hence ic is injective. Furthermore, for any d ≤ c,πic(d) =

∧b ≤ πi(c) : i(b) ∧ c ≥ d

=∧b ≤ πi(c) : i(b) ≥ d = πi(d),

concluding the proof.

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1.3 Appendix 1: Basics on orders and topology

We recall in this last section some basic facts and terminology on orders and topology wehave used in the previous parts of this chapter.

1.3.1 Orders

A pre-order, also called quasi-order or qo, is a set P equipped with a reflexive and transitivebinary relation denoted by ≤P . An antisymmetric qo is a partial order, or poset, or evenjust po. Every qo has an associated strict relation denoted by <P and defined by x <P yif and only if x ≤P y and y 6≤P x.

Remark that if P is a po then the strict relation <P is just ≤P \∆P , where ∆P standsfor the diagonal in P 2. Remark also that this is far from being true in any qo, since forinstance the total relation P 2 on P is a qo.

In a clear context we write ≤ instead of ≤P .When x ≤ y holds we say that x is below y. When x is either below or above y, we

say that x and y are comparable. An order (P,≤) is total when any two elements arecomparable.

Let (P,≤) be a qo.We say that two elements x, y in P are compatible and we write x||y if there is z ∈ P

such that both z ≤ x and z ≤ y hold. Otherwise x and y are incompatible, which is denotedby x ⊥ y.

A ⊆ P has c as an upper bound if c ≥ b for all b ∈ A. A has supremum a ∈ P if it is aleast upper bound: i.e. a is an upper bound for A and c ≥ a any c upper bound for A.Notice that if P is a partial order the upper bound of A is unique if it exists.

We define the notions of lower bound and infimum dually reversing the ≤-relationA partial order is complete if it admits suprema and infima for all its subsets.A subset D of P is dense in P if for all x in P there is some y in D below x, it is

predense if its downward closure

↓ D = q : ∃x ∈ D, q ≤ x

is dense.We say that (P,≤) is separative if for all x and y in P , if x is not below y then there is

some z below x that is incompatible with y. Formally,

∀x ∈ P ∀y ∈ P ( x 6≤ y → ∃z ≤ x(z ⊥ y)) .

We say that a separative (P,≤) is atomless if it does not have minimal elements, in thefollowing strong sense: given any x in P there are elements y ⊥ z of P strictly below x.

A map i : P → Q between partial orders is a morphism if it preserves the orderrelation, an embedding if it preserves the order and the incompatibility relations, a completeembedding if it maps predense subsets of P in predense subsets of Q. Remark that acomplete embedding need not be injective, natural examples of non-injective completeembeddings are given by the natural dense embedding A 7→ Reg (A) of the partial order(τ \ ∅ ,⊆) given by a topology τ on some space X into the complete boolean algebra ofregular open sets of τ .

Definition 1.3.1. Let P be a partial order.G ⊆ P is a filter on P if it is upward closed and any two elements p, q of G have a

common refinement r ≤ p, q in G.

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A ⊆ P is a prefilter if any finite subset of A has a common refinement in P .Dually an ideal I on P is a downward closed subset of P such that any two elements

p, q of I are bounded above by some r ≥ p, q also in I.Given a family X of subsets of P G ⊆ P is X -generic if G ∩X 6= ∅ for all X ∈ X .

Lemma 1.3.2. Let P be a partial order. Assume X is a countable family of predensesubsets of P . Then there is a filter G on P which is X -generic.

1.3.2 Topological spaces

A topology on a given set X is a family τ ⊆ P (X) that contains ∅ and X and that is closedunder arbitrary unions and finite intersections. We call (X, τ) a topological space.

Remark that (τ,⊆) is a complete partial order for any topological space (X, τ).The elements of τ are the open sets for the topology τ . Complements of open sets

are called closed sets, we denote by τ c the family of closed sets (the family of closed setsof a topological space is closed under arbitrary intersections and finite unions). When aset A is both open and closed, we call it a clopen set of τ and we denote this family byCLOP(X, τ).

A basis σ for a topological space (X, τ) is a subfamily of τ with the property that everyopen set in τ can be written as an union of elements of σ. We say that τ is generated by σ.

We say that U ⊂ X is a neighborhood of some x ∈ X if x ∈ U .A Hausdorff space (X, τ) is a topological space (X, τ) in which any two distinct points

x and y can be separated by two open sets U and V in τ , that is x is in U , y is in V andU and V are disjoint.

A 0-dimensional space (X, τ) is a topological space (X, τ) whose clopen sets form abasis.

Given a topological space (X, τ) and an arbitrary subset A of X, we denote by Cl (A)(the closure of A) the smallest closed set containing A. We denote by Int (A) (the interiorof A) the biggest open set contained in A. An open set A is regular open if A = Int (Cl (A)).For any A ⊆ X Reg (A) = Int (Cl (A)) denotes the regularization of the set A.

Given B ⊆ A, B is dense in A if Cl (B) = Cl (A). Remark that if B is dense in A andC ⊆ A is open, then B ∩C is dense in C. B is nowhere dense in A if A \Cl (B) is a densesubset of A.

A map f : X → Y between topological spaces (X, τ) and (Y, σ) is:

• continuous if the preimage by f of any open set of Y is open,

• open if the (direct) image of an open set of X is open in Y ,

• closed if the (direct) image of a closed set of X is closed in Y ,

• a projection if it is an open and continuous surjection,

• a homeomorphism if it is an open and continuous bijection.

Given a topological space (X, τ) and Y ⊆ X the restriction of τ to Y is given by thefamily A ∩ Y : A ∈ τ and is a topology on Y .

Topology of partial ordersA pre-order can always be equipped with a topological structure. Let (P,≤) a pre-

order. For each X ⊆ P ↓ X = q ∈ P : ∃ a ∈ X q ≤ a denote its downward closure,↑ X = q ∈ P : ∃ a ∈ X a ≤ q its upward closure and for each p ∈ P we let

Np :=↓ p .

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The sets Np form a basis of a topology τP on P , which we call the forcing topology for lackof a better terminology. We remark the following:

• The open sets of P in this topology are the downward closed subsets of P withrespect to the order ≤ (dually the closed sets in τ cP are the upward closed subsets ofP ).

• For any p ∈ P , Np is the smallest open set to which p belongs.

• A subset D of P is dense in the sense of the order iff it is dense in P with respect tothe forcing topology.

• The family of open sets of this forcing topology is closed under arbitrary intersections,since the family of downward closed subsets of P has this property. In particular theorder topologies are always complete and distributive sublattices of P (P ) (see thenext subsection 1.1 for a definition of complete and distributive lattice).

Remark 1.3.3. This topology is not to be confused with the one induced by a linear order.For example the family of open sets for the forcing topology induced by the linear order(R, <) is given by the intervals of the form (−∞, a) or (−∞, a] as a ranges in R. We areinterested in order topologies for orders which are not linear. For any pre-order (P,≤)containing p 6= q with p ≤ q the induced forcing topology is not Hausdorff: p ∈ U for anyopen neighborhood of q, since p ∈ Nq.

Examples of the kind of partial orders we will focus on are given by (τ \ ∅ ,⊆), whereτ is a topology on some space X with no isolated points.

Example 1.3.4. The simplest example of a non-separative partial order is given by thestandard topology τ on R. Consider the partial order (τ \ ∅ ,⊆), this is an atomlessnon-separative partial order: First of all notice that open sets A and B in this partial orderare incompatible iff they have empty intersections. Next notice that any non-empty openset contains two disjoint non-empty proper open subsets, showing that (τ \ ∅ ,⊆) is anatomless partial order. Finally to see that (τ \ ∅ ,⊆) is not separative, let A 6= B be suchthat A ∩B is a dense subset of both (for example A = (0, 1) ∪ (1, 4), B = (0, π) ∪ (π, 4)).We have that any V ⊆ A contains an U ⊆ B and conversely, thus giving that neitherA ≤ B, nor B ≤ A, but also that no non-empty open subset of A is incompatible with B.

Example 1.3.5. The partial order (2<ω,⊇) is the standard example of a separativeatomless partial order: s ⊥ t iff s ∪ t is not a function and s||t iff s ∪ t = s or s ∪ t = t.

These two partial orders give rise to isomorphic boolean completions, cfr.: Theo-rem 1.1.20.

Remark 1.3.6. Let (X, τ) be a topological space. Then:

• D ⊂ X is a dense and open iff σD = O ∈ τ : O ⊆ D is a dense subset of the partialorder (τ \ ∅,⊆) which is open in the forcing topology induced by ⊆ on τ \ ∅.

• σ ⊆ τ \ ∅ is predense in the partial order (τ \ ∅,⊆) iff ∪σ = Dσ is an open densesubset of X with respect to the topology τ .

Product topologiesLet I be a set of indexes and for all i ∈ I, let (Xi, τi) be a topological space and

X =∏i∈I Xi be the cartesian product of the sets Xi. The product topology τ on X is the

weakest topology making all the projections maps πi : f 7→ f(i) continuous. It is generatedby the family of sets of the form

∏i∈I Ai, where each Ai is open in Xi and Ai 6= Xi only

for finitely many i.

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CompactnessA topological space (X, τ) is compact if any of the following equivalent conditions are

met:

• every family F of closed sets with the finite intersection property8 has a non-emptyintersection.

• Every open covering of X has a finite subcovering.

We recall that for an Hausdorff space X a subset Y which is compact in the relativetopology inhereted from X is closed. This yields the following fundamental property ofcontinuous functions between compact Hausdorff spaces:

Assume f : X → Y is continuous with X, Y compact Hausdorff. Then f is aclosed map.

We emphasize the following two statements:

• We focus on either Hausdorff compact spaces or on forcing topologies.

• We often interplay between the topological notion of density and the notion of densesubset of a partial order.

1.4 Appendix 2: Fundamental properties of adjoint pairs

Due to the prominent role adjoint pairs will play in our analysis of iterated forcing (seechapters 6, 7, 9, 10), we collect below the fundamental properties of adjoint pairs.

Theorem 1.4.1. Let i : P → Q and π : Q→ P be maps between partial orders (P,<P ),(Q,< Q) such that π is the adjoint of i. Then:

(A) i(p) =∨q : π(q) ≤ p;

(B) π(q) =∧p : i(p) ≥ q;

(C) i π(c) ≥ c for all c ∈ Q;

(D) i π i = i, and π i π = π;

(E) if i is injective π i is the identity map on P ;

(F) the assertion “i : P → Q has as adjoint π” is a ∆0-property, hence it is absolutebetween transitive structures to which i, P,Q belong.

Assume moreover that P = B+ and Q = C+ for complete boolean algebras B,C. Then(letting i(0B) = 0C and π(0C) = 0B):

(i) i is a complete injective homomorphism;

(ii) π is a surjective, order and suprema preserving map with ker(π) = 0C;

(iii) π(c) ∧ b = π(c ∧ i(b)) for all b ∈ B and c ∈ C;

8F has the finite intersection property if any finite subfamily of F has a non-empty intersection. Afamily A of subsets of X such that

⋃A = X is a covering of X.

35

(iv) moreover for complete boolean algebras B and C, i : B→ C is a complete homomor-phism if and only if it has an adjoint π defined by B.

Finally if I is an ideal on B and J =↓ i[I] is the lift of I to C via i:

i/I,J :B/I → C/J (1.6)

[b]I 7→ [i(b)]J

π/I,J :C/J → B/I (1.7)

[c]J 7→ [π(c)]I

are well defined and π/I,J is the adjoint of k/I,J , hence k/I,J is a regular embedding (eventhough B/I ,C/J need not be complete).

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Chapter 2

Clubs and normal ideals

2.1 Generalized clubs and generalized stationarity

In this section we recall the properties of generalized stationarity. Reference texts are[28], [30, Chapter 2]. The notion of generalized stationary set is rooted in the downwardLowenheim-Skolem theorem for countable languages stating that:

Let 〈Ri : i ∈ I, fj : j ∈ J, ck : k ∈ K〉 be a countable first order signature, andM = 〈M,RMi : i ∈ I, fMj : j ∈ J, cMk : k ∈ K〉 a structure of large size (i.e. withdomain an uncountable set of size λ) for this signature. Given a subset X of Mof size κ with κ ≤ λ, there is an N ≺M such that X ⊆ N and |N | = |X|+ ℵ0.

In particular this theorem provides “many” elementary substructures of M: for examplefor any infinite X ⊆M we can “blow” X to a larger set of the same cardinality which isan elementary substructure of M .

The notion of generalized stationarity (and of normal fine ideal which we are goingto introduce later on) aims to give a precise meaning to the concept of “many”. Morespecifically the Lowenheim Skolem theorem gives us the possibility to find substructures ofM which maintain certain second order properties ofM which are not expressed by the firstorder theory ofM (specifically that of having size at least κ for some κ ≥ ω). Moreover thetheorem states that we can find many such structures of a fixed infinite cardinality κ < λ(by the theorem any subset of M of size κ can be a subset of an elementary substructure ofsize κ). The notion of generalized club gives us the tool to describe precisely what is meantby “many”: the basic idea being that a subset of P (M) is large if it contains “almostall” elementary substructures of M. The notion of generalized stationarity captures theconcept that the set of elementary substructures ofM which maintain certain second orderproperties of M (for example that of being a model of uncountable size) is always “nonnegligible”.

The standard proof of this theorem produces a Skolem function f : M<ω → M andshows that any superset N of X closed under f (i.e. such that f [N<ω] ⊆ N) is anelementary substructure of M. It appears that the notion of Skolem function is moremanageable from a combinatorial point of view than that of first order structure. For thisreason in the development of the notion of generalized stationarity one is led to focus onSkolem functions on a set M , which for us are just functions f : M<ω →M , rather thanon first order structures with domain M .

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2.1.1 Definitions

Definition 2.1.1. Let X be a non-empty set. A set C is a club on P (X) if there is afunction f : X<ω → X such that

C ⊇ Cf =Y ∈ P (X) : f [Y ]<ω ⊆ Y

.

Given a function f : X<ω → X we write Cf to denote the club defined from f .

Definition 2.1.2. A set S is stationary on P (X) if it intersects every club on P (X).

Example 2.1.3. The set X is always stationary since every club contains X. AlsoP (X) \ X and [X]κ are stationary for any infinite κ ≤ |X| (following the proof of thewell-known downwards Lowhenheim-Skolem Theorem). Notice that every element of aclub C must contain fC(∅), a fixed element of X. Moreover Ca = Z ⊆ X : a ∈ Z is aclub as witnessed by the constant function ca : X<ω → X, s 7→ a for all a ∈ X. Finally ifX = xn : n ∈ ω is countable X is a club (choose g(s) = x|s| for s ∈ X<ω). Hence thenotion of stationarity for a set S ⊆ P (X) is non-trivial (i.e. different from that of club)only if X is uncountable.

Remark 2.1.4. The reference to the support set X for clubs or stationary sets may beomitted, since every set S can be club or stationary only on

⋃S (otherwise S is disjoint

from Ca for some a /∈ ∪S).

There is a key property of stationary sets worth to mention. Given any first-orderstructure M in a countable language with domain M , we can define a Skolem functionfM : M<ω →M (i.e., a function coding solutions for all existential first-order formulas overM). The set C of all elementary submodels of M contains a club (the one correspondingto fM ). Henceforth, every set S stationary on X must contain an elementary submodel ofany first-order structure in a countable language with domain X. Moreover the followingproperty of clubs will be repeatedly used throughout these notes:

Fact 2.1.5. Assume M ≺ Hθ and f : X<ω → X in M . Then M ∩X ∈ Cf .

Proof. f ∈ M ≺ Hθ grants that f(s) ∈ M ∩X for any s ∈ X<ω ∩M = (X ∩M)<ω (thelatter equality holds since M ≺ Hθ is closed under finite sequences). The conclusion followsimmediately.

Definition 2.1.6. A set S is subset modulo club of T , in symbols S ⊆∗ T , if⋃S =

⋃T =

X and there is a club C on X such that S ∩ C ⊆ T ∩ C. Similarly, a set S is equivalentmodulo club to T , in symbols S =∗ T , if S ⊆∗ T ∧ T ⊆∗ S.

Definition 2.1.7. Let X be a set and let A ⊆ P (P (X)).The dual of A is the set A given by P (X) \ S : S ∈ A.I ⊆ P (P (X)) is an ideal with support X if it is an ideal on the complete boolean

algebra P (P (X)).F ⊆ P (P (X)) is a filter with support X if it is the dual of an ideal with support X.If I is an ideal I+ denotes the I-positive sets, i.e. P (P (X)) \ I.

Remark 2.1.8. I is an ideal with support X if and only if I is a filter.˘A = A.

Definition 2.1.9. Let I ⊆ P (P (X)) be an ideal and let S ∈ I+. We define

I S= I ∩ P (S) .

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Remark 2.1.10. Observe the natural isomorphism [A]IS 7→ [A∩S]I between P (P (X)) /IS∼=

A ∈ P (P (X)) : A ⊆ S /I , where I S is the ideal generated by I∪P (X) \ S, i.e. thesmallest ideal containing I ∪ P (X) \ S.

Definition 2.1.11. The non-stationary ideal on X is

NSX = S ∈ P (P (X)) : ∃f Cf ∩ S = ∅ .

Analogously the club filter on X is

CFX = C ⊂ P (P (X)) : ∃f C ⊇ Cf .

By definition for any element in S ∈ NSX there exists a function f such that Cf ∩S = ∅,we will write that f is a witnessing function for S.

Remark 2.1.12. If |X| = |Y |, then P (X) and P (Y ) are isomorphic and so are CFX andCFY (or NSX and NSY ): hence we can suppose X ∈ Ord or X ⊇ ω1, or X transitive ifneeded.

Lemma 2.1.13. CFX is a σ-complete filter on P (X), and the stationary sets are exactlythe CFX-positive sets.

Proof. CFX is closed under supersets by definition. Given a family of clubs Ci, i < ω,let fi be the function corresponding to the club Ci. Let π : ω → ω2 be a surjection,with components π1 and π2, such that π2(n) ≤ n. Define g : X<ω → X to be g(s) =fπ1(|s|)(s π2(|s|)). It is easy to verify that Cg ⊆

⋂i<ω Ci.

2.1.2 Generalized stationarity versus classical stationarity

The generalized notion of club and stationary set we just introduced is closely related tothe usual one defined for subsets of (regular) cardinals (see [28, Chapter 8]).

Lemma 2.1.14. C ⊆ ω1 is a club in the classical sense ([28, Def. 8.1]) if and only ifC ∪ ω1 is a club. S ⊆ ω1 is stationary in the classical sense ([28, Def. 8.1]) if and onlyif it is stationary.

Proof. Let C ⊆ ω1 + 1 be a club in the generalized sense. Then C is closed: given anyα = supαi with f [αi]

<ω ⊆ αi, f [α]<ω =⋃i f [αi]

<ω ⊆⋃i αi = α. Furthermore, C is

unbounded: given any β0 < ω1, define a sequence βi by taking βi+1 = sup f [βi]<ω. Then

βω = supβi ∈ C.Let now C ⊆ ω1 be a club in the classical sense. Let C = cα : α < ω1 be an

enumeration of the club. For every α < ω1, let dαi : i < ω ⊆ cα+1 be a cofinal sequencein cα+1 (eventually constant), and let eαi : i < ω ⊆ α be an enumeration of α. Define fCto be fC((cα)n) = dαn, fC(0aαn) = eαn, and fC(s) = 0 otherwise. The sequence eαi forces allclosure points of fC to be ordinals, while the sequence dαi forces the ordinal closure pointsof fC being in C.

Lemma 2.1.15. If κ is a regular uncountable cardinal, C ⊆ κ contains a club in theclassical sense if and only if C ∪κ contains the ordinals of a club in the generalized sense.S ⊆ κ is stationary in the classical sense if and only if it is stationary in the generalizedsense.

Proof. If C is a club in the generalized sense, then C ∩ κ is closed and unbounded by thesame reasoning of Lemma 2.1.14. Let now C be a club in the classical sense, and definef : κ<ω → κ to be f(s) = min c ∈ C : sup s < c. Then Cf ∩ κ is exactly the set ofordinals in C ∪ κ that are limits within C.

39

Remark 2.1.16. If S is stationary in the generalized sense on ω1, then S ∩ ω1 is stationary(since ω1 + 1 is a club by Lemma 2.1.14), while this is not true for κ > ω1. In this case,P (κ) \ (κ + 1) is a stationary set: given any function f , the closure under f of ω1 iscountable, hence not an ordinal.

2.1.3 First properties of the non-stationary ideals

In this section we present two key results on the non-stationary ideal: Fodor’s Lemma andUlam’s Theorem. To state them we need to recall the definitions of diagonal union anddiagonal intersection.

Definition 2.1.17. Given a non-empty set X and S ⊆ P (X)\∅, f : S → X is regressiveif it is a choice function on S, i.e. if f(Z) ∈ Z for all Z ∈ S.

Definition 2.1.18. Let Sx : x ∈ X ⊆ P (P (X)). We define:

•aSx : x ∈ X = Z ⊆ X : ∀x ∈ Z(Z ∈ Sx) (diagonal intersection);

•`Sx : x ∈ X = Z ⊆ X : ∃x ∈ Z(Z ∈ Sx) (diagonal union).

Lemma 2.1.19 (Fodor). CFX is closed under diagonal intersection. Equivalently, everyfunction f : P (X) → X that is regressive on a CFX-positive set is constant on a CFX-positive set.

Proof. Given a family Ca, a ∈ X of clubs, with corresponding functions fa, let g(aas) =fa(s). It is easy to verify that Cg = ∆a∈XCa.

Assume by contradiction that f : P (X) → X is regressive (i.e. f(Y ) ∈ Y ) in aCFX -positive (i.e. stationary) set, and f−1 [a] is non-stationary for every a ∈ X. Then,for every a ∈ X there is a function ga : [X]<ω → X such that the club Cga is disjoint fromf−1 [a]. Without loss of generality, suppose that Cga ⊆ Ca = Y ⊆ X : a ∈ Y . As in thefirst part of the lemma, define g(aas) = ga(s). Then for every Z ∈ Cg and every a ∈ Z, Zis in Cga , hence is not in f−1 [a] (i.e., f(Z) 6= a). So f(Z) /∈ Z for any Z ∈ Cg, thereforeCg is a club disjoint with the stationary set on which f is regressive, a contradiction.

The above equivalence holds for any filter F . We postpone its proof to Lemma 2.3.3 inSection 2.3.

Remark 2.1.20. The generalized club filter is never < ω2-complete. Let Y ⊆ X besuch that |Y | = ω1, and Ca be the club corresponding to fa : [X]<ω → a; thenC =

⋂a∈Y Ca = Z ⊆ X : Y ⊆ Z is disjoint from the stationary set [X]ω, hence is not a

club.

Theorem 2.1.21 (Ulam). Let κ be an infinite cardinal. Then for every stationary setS ⊆ κ+, there exists a partition of S into κ+ many disjoint stationary sets.

Proof. For every β ∈ [κ, κ+), fix a bijection πβ : κ → β. For ξ < κ, α < κ+, define

Aξα = β < κ+ : πβ(ξ) = α (notice that β > α when α ∈ ran(πβ)). These sets form a(κ × κ+)-matrix, the Ulam Matrix, with the property that two sets in the same row or

column are always disjoint. Moreover, every row is a partition of⋃α<κ+ A

ξα = κ+, and

every column is a partition of⋃ξ<κA

ξα = κ+ \ (α+ 1).

Let S be a stationary subset of κ+. For every α < κ+, define fα : S \ (α + 1) → κ

by fα(β) = ξ if β ∈ Aξα. Since κ+ \ (α + 1) is a club in the classical sense, and hence isstationary in the generalized sense, every fα is regressive with a stationary domain. by

40

Fodor’s Lemma 2.1.19 there exists a ξα < κ such that f−1α [ξα] = Aξαα ∩ S is stationary.

Define g : κ+ → κ by g(α) = ξα, g is regressive on the stationary set κ+ \ κ; again byFodor’s Lemma 2.1.19, let ξ∗ < κ be such that g−1 [ξ∗] = T is stationary. Then, the rowξ∗ of the Ulam Matrix intersects S in a stationary set for stationary many columns T . SoS can be partitioned into S ∩Aξ

∗α for α ∈ T \ min(T ), and S \

⋃α∈T\min(T )A

ξ∗α .

Remark 2.1.22. In the proof of Theorem 2.1.21 we actually proved something more: theexistence of a Ulam Matrix, i.e. a κ× κ+-matrix such that every stationary set S ⊆ κ+ iscompatible (i.e., has stationary intersection) with stationary many elements of a certainrow.

2.2 Skolem Hulls

We want to find a very efficient proof of the downward Lowenheim-Skolem theorem whichgives the following:

Let L = 〈Ri : i ∈ I, fj : j ∈ J, ck : k ∈ K〉 be a first order countable signature,and M = 〈M,RMi : i ∈ I, fMj : j ∈ J, cMk : k ∈ K〉 a structure of large size (i.e.with domain an uncountable set of size λ) for this signature. Then we can findh Skolem function for M such that for any infinite X ⊆M , N = h[X<ω] ⊇ Xis already an elementary substructure of M .

The standard proof of the downward Lowenheim-Skolem theorem produces a functiong : M<ω → M such that, given any set X, letting X0 = X, Xn+1 = Xn ∪ g[X<ω

n ], andN =

⋃n∈ωXn, one gets that N ⊇ X is an elementary substructure of M of size |X|+ ℵ0.

We want to be able to get this N in just one step using h and setting N = h[X<ω] ratherthan in ω-steps using g. While this result appears to be rather technical, it plays a crucialrole in many of the basic results to follow on stationary sets and normal fine ideals. Thissection aims to give a detailed proof of how to construct such a function h. We willformulate our result just for very specific types of first order languages, however it is clearthat with suitable complications the same strategy will be able to define the required typeof Skolem functions for all types of countable first order signature. The reader may skipall proofs and just keep track of Proposition 2.2.3 to follow.

2.2.1 Fast Skolem functions

Let L be a first order language containing one and only one n-ary function symbol fn forall n ∈ ω. Given a set X, let LX stand for the expansion of L where we added a constantsymbol cx for each element of x.

Notation 2.2.1. When we write t(x0, . . . , xn) for a term in L with free variables (x0, . . . , xn),we really consider the free variables (x0, . . . , xn) as an ordered sequence rather than anon-ordered set.

More specifically, the notation t(x0, . . . , xn) stands for: t is a term of L, and (x0, . . . , xn)is an enumeration of the set of free variables occuring in t such that if i < j ≤ n holds,then the first occurence of the variable xi in the representation of t as a string of symbolsoccurs before that of xj . In particular, the sequence (x0, . . . , xn) has no repetitions. Whenno confusion can arise we write ~x and t(~x) instead of (x0, . . . , xn) and t(x0, . . . , xn).

Given a map g : X<ω → X, there is a natural interpretation of CT (LX), the set ofclosed terms (i.e. with no free variables) of LX , inside X given by the map ν : CT (LX)→ Xdefined as follows:

41

• ν(cx) = x for all x ∈ X,

• If t1, . . . , tn are closed terms of LX , then

ν(fn(t1, . . . , tn)) = g(ν(t1), . . . , ν(tn)).

In this way each term t(x0, . . . , xn) of L in the free variables (x0, . . . , xn) (and withoutconstant symbols!!) can be identified with a function which we denote by

t(x0, . . . , xn) : Xn+1 → X

s 7→ ν(t(x0/cs(0), . . . , xn/cs(n))).

First of all notice that any substitution t(x0/y0, . . . , xn/yn) such that yi 6= yj for all i 6= jwill yield the same function t(x0, . . . , xn). In particular we can enumerate the functionst : Xn → X given by these kind of terms as a family

tnk : n, k ∈ ω

such that tnk : Xn → X is induced by a given term tk(x0, . . . , xn−1) in n-free variables. Ifno confusion on the arity of tnk can arise we drop the superscript n denoting its arity andwe denote tnk just by tk.

Notice also the following trivial fact:

Fact 2.2.2. Assume t(x0, . . . , xm) is a term of L in the free variables (x0, . . . , xm) andeach ti(y

i0, . . . , y

ini) is a term in L in the free variables (yi0, . . . , y

ini) so that (yij 6= ylk) for

i 6= l and j, k arbitrary. The term

t′(y00, . . . , y

0n0, . . . , yi0, . . . , y

ini . . . , y

m0 , . . . , y

mnm) =

t(x0/t0(y00, . . . , y

0ni), . . . , xm−1/tm−1(ym−1

0 , . . . , ymnm−1))

gives a functiont′ : X

∑i≤m ni → X

with the property that

t(t0(s0), . . . , tm(sm)) = t′(sa0 . . .a sm).

We leave the proof of this fact to the masochistic reader.We now fix φ : ω → ω × ω surjective such that the preimage of any couple (i, j) is

infinite and contained in ω \ i+ 1, and we let φ(n) = (in, jn).Let hg : X<ω → X be defined by

hg(u) = ti|u|j|u|

(u i|u|).

Proposition 2.2.3. For any g : X<ω → X, hg : X<ω → X is such that:

1. Chg is contained in Cg,

2. for all infinite Z ⊂ X, hg[Z<ω] ∈ Chg ,

3. if Z ∈ Chg , hg[Z<ω] = Z.

Proof. We prove that the three items of the proposition are satisfied by g and hg = h asfollows:

42

1. Ch is contained in Cg: Assume Z ∈ Ch is infinite and s ∈ Z<ω has length n. Thenfn(x0, . . . , xn−1) is a term in L in the free variables (x0, . . . , xn−1) thus it gives afunction tnk : Xn → X for some k ∈ ω. This means that

g(s) = fn(x0, . . . , xn)(s) = tnk(s) = h(u)

for any u ⊇ s in Z<ω such that φ(|u|) = (n, k). The desired conclusion follows.

2. For all infinite Z ⊆ X, h[Z]<ω ∈ Ch:

WARNING: To avoid heavy notation in what follows we will systematically drop thesuperscript indexing the functions tnm we will use below, thus tnm will be denoted bytm, its arity will be clear from the context.

Pick s finite sequence with range in h[Z<ω]. Then

h(s) = ti|s|j|s|

(s i|s|) = tj|s|(s i|s|).

Now for all l < i|s| = m,

s(l) = h(ul) = ti|ul|j|ul|

(ul i|ul|) = tj|ul|(ul i|ul|).

We let i|ul| = nl for all l < m. So we get

h(s) = tj|s|(tj|u0|(u0 n0), . . . , tj|um−1|

(um−1 nm−1)).

In particular for each l < mtj|ul|

: Xnl → X

is the function given by a fixed term tl(yl0, . . . , y

lnl−1) and

tj|s| : Xm → X

is the function given by a particular term t′(x0, . . . , xm−1).

We can suppose that ylk 6= yhe for all l 6= h, k < nl, e < nh. Then we can let

t(y00, . . . , y

0n0, . . . , yl0, . . . , y

lnl, . . . , ym−1

0 , . . . , ym−1nm−1

) =

t′(x0/t0(y00, . . . , y

0nl−1), . . . , xm−1/tm−1(ym−1

0 , . . . , ym−1nm−1−1)).

By the above fact we get that

h(s) = t(u0 na0 . . .

a um−1 nm−1).

Observe that this latter sequence is in Z<ω. Let u denote this sequence and extendit to u′ so that i|u′| = |u| and

tj|u′| = t(y00, . . . , y

0n0, . . . , yl0, . . . , y

lnl, . . . , ym−1

0 , . . . , ym−1nm−1

).

Then h(s) = h(u′) ∈ h[Z<ω]. Since s ∈ h[Z<ω] was chosen arbitrarily, we are done.

43

2.3 Normal Fine vs Non Stationary

We now introduce normal fine ideals I ⊆ P (P (X)). These ideals are characterized by theproperty of being closed under diagonal unions indexed by X, or equivalently by the factthat regressive functions defined on I+ are constant on a set in I+. By Lemma 2.1.19 (andProposition 2.3.7 below), we have that the non-stationary ideal is normal fine, outlininga (somewhat at least on a first sight) surprising analogy between the analysis of firstorder elementary substructure (i.e. stationary sets) of some signature on a set X and theanalysis of the properties of choice functions on P (X) \ ∅ (i.e. the regressive functionson X). We will see that this analogy is not an accident, as all normal fine ideals can beseen as the projection of the non-stationary ideal restricted to a specific stationary set(Theorem 2.3.19). A detailed comparison between the concept of normality and that ofstationarity is carried in Subsection 2.3.2.

2.3.1 Normal Fine Ideals

Definition 2.3.1. Let X be a set and let I ⊆ P (P (X)).

1. I is normal if for any f : S → X such that S /∈ I and f is a choice function, thereexists x ∈ X such that Y ∈ S : f(Y ) = x ∈ I+.

2. I is fine if ∀x ∈ X Cx = Y ⊆ X : x ∈ Y ∈ I.

Definition 2.3.2. Let I be a normal fine ideal with support X, and f : S → X bea regressive function such that S /∈ I. x ∈ X is a pressing down constant for f ifY ∈ S : f(Y ) = x ∈ I+.

Lemma 2.3.3 (Fodor). Let I ⊆ P (P (X)) be an ideal, then the following are equivalent:

1. I is normal fine;

2. if Sx : x ∈ X ⊆ I then`Sx : x ∈ X ∈ I.

Proof.

1⇒ 2 Assume by contradiction 2 fails as witnessed by Sx : x ∈ X ⊆ I. Let S =`Sx : x ∈ X ∈ I+, and define

f : S −→ X

Z 7−→ x,

where x ∈ Z is such that Z ∈ Sx. Let x be the pressing down constant which existsby the normality of I. Then Y ∈ S : f(Y ) = x ⊆ Sx ∈ I. Contradiction.

2⇒ 1 Assume, for the sake of contradiction, f : S → X is a regressive function with S ∈ I+

such that ∀x ∈ X f−1[x] ∈ I. Let Sx = f−1[x] and set T =`Sx : x ∈ X. We

have S ⊆ T since

Z ∈ S =⇒ f(Z) = x for some x ∈ Z =⇒ Z ∈ Sx for some x ∈ Z =⇒ Z ∈ T.

Then T ∈ I+. Contradiction.

44

A normal fine ideal I on P (P (Y )) is not even ω1-complete in general (see Re-mark 2.1.20), nonetheless its quotient algebra is always |Y |-complete:

Proposition 2.3.4 (Proposition 2.22 [18]). Suppose that X ⊆ Y and I is a normal, fineideal on Y . Suppose that A = [Sx]I : x ∈ X ⊆ P (P (Y )) /I . Then in P (P (Y )) /I wehave ∨

A = [hSx : x ∈ X]I

and ∧A = [

iSx : x ∈ X]I .

Therefore P (P (Y )) /I is always |Y |-complete.

Proof. For any x ∈ X Sx ∩ Cx ⊆`Sx : x ∈ X. Therefore Sx ≤I

`Sx : x ∈ X, since

Cx ∈ I. Viceversa, we have to prove that if T ≤I`Sx : x ∈ X and T /∈ I, then there

exists x ∈ X such that T ∩Sx /∈ I. We can assume that T ⊆`Sx : x ∈ X, thus for each

Z ∈ T there is x(Z) ∈ Z such that Z ∈ Sx(Z). So the map

f : T −→ X

Z 7−→ x(Z)

is regressive. Since I is normal fine, there exists a positive set T0 ⊆ T and a fixed x ∈ Xsuch that for all Z ∈ T0, x(Z) = x. This implies that T0 ⊆ Sx and we are done.

Definition 2.3.5. Let I ⊆ P (P (X)) be an ideal.

• I is ω-fine if for any Z ⊆ X countable, Y ⊆ X : Z ⊆ Y ∈ I.

• I is ω-normal if ∀f : S → X<ω with S ∈ I+ and such that ∀Z ∈ S f(Z) ∈ Z<ω,there exists t ∈ X<ω such that z ∈ S : f(z) = t ∈ I+.

Proposition 2.3.6. Every normal fine ideal is ω-normal, ω-fine, and countably closed.

Proof. Let I ⊆ P (P (X)) be a normal fine ideal.

I is ω-fine. Let Z = xn : n ∈ ω. Suppose by contradiction that Y ⊆ X : Z * Y ∈ I+.Since Cx0 ∈ I we have S = Y ⊆ X : Z * Y ∩ Cx0 ∈ I+. Define the followingregressive function:

f : S −→ X

Y 7−→ xn,

where n is such that x0, . . . , xn ⊆ Y and xn+1 /∈ Y . Since f is regressive and I isnormal, let xm be the pressing down constant. Then f−1[xm] ∈ I+ and for anyY ∈ f−1[xm] xm+1 /∈ Y . This is a contradiction since I is fine.

I is countably closed. Let Z = xn : n ∈ ω be a subset of X. Assume Sn : n ∈ ω ⊆ Iand put Tn = Sn ∩ Cxn ∈ I. Then, since I is fine, we can define

Tx =

Tn if x = xn;

Cx otherwise.

ThenaTx : x ∈ X ∈ I, since I is normal fine. Moreover Y ⊆ X : Z ⊆ Y ∈ I

since I is ω-fine. ThereforeiTx : x ∈ X ∩ Y ⊆ X : Z ⊆ Y ∈ I .

45

Now we are done since:iTx : x ∈ X ∩ Y ⊆ X : Z ⊆ Y ⊆

⋂Tn : n ∈ ω ⊆

⋂Sn : n ∈ ω .

I is ω-normal. Given f : S → X<ω such that ∀Z ∈ S f(Z) ∈ Z<ω, define Sn =Z ∈ S : |h(Z)| = n. Since S ∈ I+ and I is countably closed there exists n suchthat Sn ∈ I+. Let us define Tm for any m ≤ n by induction. Put T0 = Sn. GivenTm for some m ≤ n− 1 define xm and Tm+1 as follows. Let

hm : Tm −→ X

z 7→ h(z)(m).

and let xm be the pressing down constant. Put Tm+1 = h−1m [xm] ∈ I+. Then

Tn ∈ I+ and for all Z ∈ Tn h(Z) = (x0, . . . , xn−1) = t. Hence we are done.

2.3.2 Non-stationary ideals and normal fine ideals

We will show that the nonstationary ideals form a directed, coinitial and cofinal family NSof normal fine ideals in the sense that:

• NSX is always normal fine for any |X| > 1, and NSY canonically projects1 to NSXfor every X ⊆ Y (NS consists of normal fine ideals and is directed),

• any normal fine ideal on X contains NSX (NS is coinitial in the class of normal fineideals),

• for every normal fine ideal I, there is an X and a stationary set S such that NSX Scanonically projects to I (NS is cofinal in the class of normal fine ideals).

We start showing that NSX is normal fine.

Proposition 2.3.7. NSX is a normal fine ideal if |X| > 1.

Proof. We must show three properties:

NSX is fine. Given x ∈ X, put

fx : X<ω −→ X

s 7−→ x.

Hence Cx = Cfx is a club and P (X) \ Cx ∈ NSX .

NSX is an ideal. Let S0, S1 ∈ NSX , then by definition there are f0, f1 : X<ω → X suchthat Si ∩ Cfi = ∅. Fix x0 6= x1 ∈ X and define g : X<ω → X as follows:

g(xi ∗ s) =

fi(s) if i ∈ 0, 1 ;

x|s| mod 2 otherwise (i.e. if s(0) 6= x0, x1).

Then Cg ⊆ Cf0 ∩ Cf1 ∩ Cx0 ∩ Cx1 is disjoint from S0 ∪ S1.

1Notion to be defined below in Subsection 2.3.3

46

NSX is normal. We will prove that if Sx : x ∈ X ∈ NSX then

hSx : x ∈ X ∈ NSX .

Applying Fodor’s Lemma we obtain our thesis. Assume Sx : x ∈ X ∈ NSX andlet fx : X<ω → X be such that Sx ∩ Cfx = ∅. Define f : X<ω → X such thatf(x ∗ s) = fx(s).

First of all observe that Cf ⊆aCfx : x ∈ X. Assume Z ∈ Cf , then ∀s ∈ Z f(s) ∈

Z. Let s = x ∗ t, then f(s) = fx(t) where x ∈ Z and t ∈ Z<ω. This implies thatfx(t) ∈ Z for any t ∈ Z<ω, hence for any x ∈ Z, Z ∈ Cfx . Thus Z ∈

aCfx : x ∈ X.

Moreover Cf ⊆aCfx : x ∈ X implies Cf ∩

`Sx : x ∈ X = ∅, therefore

hSx : x ∈ X ∈ NSX .

We now show that normal fine ideals canonically project to each other, we need tointroduce the relevant definition:

2.3.3 Projection and Lifting

Definition 2.3.8. If S ⊆ P (P (X)) and Z ⊆ X ⊆ Y we define

• S ↓ Z = Z0 ∩ Z : Z0 ∈ S;

• S ↑ Y = Z0 ⊆ Y : Z0 ∩X ∈ S.

The following is trivial but crucial (recall Def. 1.2.1 and 1.2.3):

Fact 2.3.9. Given X ⊆ Y the map iXY : S 7→ S ↑ Y defines a complete injectivehomomorphism of P (P (X)) into P (P (Y )) with associated projection πXY : T 7→ T ↓ X.

Notation 2.3.10. For sets S, T S ≤NS T is a short-hand for the assertion that forX = ∪S ∪ ∪T S ↑ X ∩ Cf ⊆ T ↑ X ∩ Cf for some f : X<ω → X.

Recall now Def.1.2.9.

Definition 2.3.11. Let X ⊆ Y be non-empty sets and let I ⊆ P (P (X)) and J ⊆P (P (Y )) be ideals.

• J canonically projects to I if it projects to I through iXY , i.e. if

S ∈ I ⇐⇒ S ↑ Y ∈ J

for any S ∈ P (P (X)), or equivalently if and only if iXY [I] ⊆ J and iXY [I+]∩ J = ∅.

• J is the canonical lift of I if it is the lift of I through iXY , i.e. if J =↓ iXY [I].

Lemma 2.3.12. Let X ⊆ Y be non-empty sets and let I ⊆ P (P (X)) and J ⊆ P (P (Y ))be ideals. Then the following are equivalent:

1. ∀S ∈ P (P (X)) (S ∈ I =⇒ S ↑ Y ∈ J);

2. ∀T ∈ P (P (Y )) (T /∈ J =⇒ T ↓ X /∈ I).

47

Proof. First prove that 1 implies 2. Assume T /∈ J and that T ↓ X ∈ I, by 1 we have(T ↓ X) ↑ Y ∈ J , but this is a contradiction since (T ↓ X) ↑ Y ⊇ T and J is an ideal.Now assume that 2 holds in order to prove 1. Let S ∈ I, if S ↑ Y /∈ J , then by 2, we have(S ↑ Y ) ↓ X /∈ I, but this is a contradiction since S = (S ↑ Y ) ↓ X and I is an ideal.

Definition 2.3.13. Let X ⊆ Y be non-empty sets. Given a function f : X<ω → X wedefine

f ↑ Y : Y <ω −→ Y

s 7−→

f(s) if s ∈ X<ω

s(0) otherwise.

Given a function g : Y <ω → Y we define

g ↓ X : X<ω −→ X

s 7−→

g(s) if g(s) ∈ Xx0 otherwise,

where x0 is a fixed element of X.

Proposition 2.3.14. Assume X ⊆ Y are non-empty sets. Then NSY canonically projectsto NSX .

Proof.

S ∈ NSX =⇒ S ↑ Y ∈ NSY . Assume h : X<ω → X is such that S ∩ Ch is empty. ThenS ↑ Y ∩ Ch↑Y is also empty.

S /∈ NSX =⇒ S ↑ Y /∈ NSY . Assume S /∈ NSX and let g : Y <ω → Y . Find h : Y <ω → Ysuch that Ch ⊆ Cg and h[Z<ω] ∈ Ch for any infinite Z ⊆ Y as in Proposition 2.2.3.Since S /∈ NSX there exists Z ∈ Ch↓X ∩ S. Observe that h[Z<ω] ∈ Ch and

h[Z<ω] ∩X = h ↓ X[Z<ω] = Z ∈ S.

Thereforeh[Z<ω] ∈ Ch ∩ (S ↑ Y ) ⊆ Cg ∩ (S ↑ Y ).

We now argue that NSX is the smallest fine normal fine ideal on X:

Proposition 2.3.15. Let I ⊆ P (P (X)) be a normal, fine ideal. Then I ⊇ NSX .

Proof. Assume, for the sake of contradiction, that S ∈ I+ and S ∩ Cg = ∅ for someg : X<ω → X. Then for any Z ∈ S there exists t ∈ Z<ω such that g(t) /∈ Z. Define

h : S −→ X<ω

Z 7−→ t ∈ Z<ω such that g(t) /∈ Z.

By Proposition 2.3.6 I is ω-normal, hence there exists t ∈ X<ω such that T = h−1[t] ∈ I+.Therefore T ⊆ Z ∈ S : g(t) /∈ Z, hence

T ∩ Cg(t) = ∅ ∈ I+.

Contradiction.

48

We now turn to the proof that the family NS is cofinal with respect to the class ofnormal fine ideals.

Lemma 2.3.16 (Proposition 3.44 [18]). Assume that I ⊆ P (P (X)) is a normal fine idealand let g : H<ω

θ → Hθ with X ∈ Hθ. There exists h : H<ωθ → Hθ such that:

• Ch ⊆ Cg, Ch ↓ X = Ch↓X , and any Z ∈ Ch is infinite.

• The set T consisting of all Z ∈ Ch↓X such that h[Z<ω] ∩ X = Z /∈ D for allD ∈ h[Z<ω] ∩ I is in the filter I.

Proof. Fix h : H<ωθ → Hθ witnessing Proposition 2.2.3 relative to g. Clearly h satisfies the

first part of the proposition. We are left to prove the second part of the proposition: Firstof all notice that all Z ∈ Ch↓X are infinite, and such that Z = h[Z<ω] ∩X.

Assume towards a contradiction that

Ch↓X \ T = S =Z ∈ Ch↓X : ∃h(s) ∈ h[Z<ω] ∩ I(Z ∈ h(s))

∈ I+.

Define

f : S −→ X<ω

Z 7−→ sZ ,

where sZ ∈ Z<ω is a witness that Z ∈ S (i.e. is such that h(sZ) ∈ I and Z 6∈ h(sZ)). Byω-normality and fineness, there exists s ∈ X<ω such that

Z ∈ S : f(z) = s = Z ∈ S : s ⊆ Z ∧ Z ∈ h(s) ∈ I+.

Now h(s) ∈ I andZ ∈ S : s ⊆ Z ∧ Z ∈ h(s) ⊆ h(s) ∈ I.

This is a contradiction.

Notation 2.3.17. Given an ideal I ⊆ P (P (X)) and θ large enough so that I ∈ Hθ, define

SHθ(I) =i

T ↑ Hθ : T ∈ I.

We remark the following immediate consequence of the definition of SHθ(I):

Property 2.3.18. For all θ > 2|I|

M ≺ Hθ : I ∈M and ∀A ∈ I ∩M,M ∩X /∈ A ∗=i

T ↑ Hθ : T ∈ I.

Moreover for all 2|I| < θ < λ

SHθ(I) ↑ Hλ∗= SHλ(I).

In particular in view of the above property we denote by S(I) the set(s) SHθ(I) forsome (all) large enough θ.

Theorem 2.3.19 (Burke). Assume I ⊆ P (P (X)) is a normal fine ideal and θ is largeenough so that P (P (X)) ∈ Hθ. Then SHθ(I) is stationary and I is the projection ofNSHθ SHθ (I).

Proof. We prove both properties of SHθ(I) = S(I) as follows:

49

S(I) is stationary. Let g : H<ωθ → Hθ and let h and T be the witnesses of Lemma 2.3.16

applied to g. Then for any Z ∈ T we have

h[Z<ω] ∈ Ch ∩ S(I) ⊆ Cg ∩ S(I).

NSHθ S(I) projects to I. Let T ∈ I, we show that T ↑ Hθ ∈ NSHθ S(I): Let C =M ≺ Hθ : T ∈M, hence by Property 2.3.18:

(P (X) \ T ) ↑ Hθ ⊇ M ≺ Hθ : M ∩X /∈ T ⊇ S(I) ∩ C.

Since C is a club and T ↑ Hθ ∩ S(I) ∩ C = ∅, T ↑ Hθ ∩ S(I) is non-stationary.

On the other hand, let S ∈ I+ and g : H<ωθ → Hθ. We can fix h : H<ω

θ → Hθ and Tas in Lemma 2.3.16 applied to g. W.l.o.g. (modifying h eventually) we can assumethat S ∈ h[Y <ω] for all Y ∈ Ch↓X . Since T ∈ I, S ∩ T ∈ I+, hence we can findZ ∈ S ∩ T , i.e. such that Z = h[Z<ω] ∩X /∈ A for all A ∈ I ∩ h[Z<ω]. Therefore

h[Z<ω] ∈ S ↑ Hθ ∩ S(I) ∩ Ch ⊆ S ↑ Hθ ∩ S(I) ∩ Cg.

The proof is completed.

2.4 Towers of normal fine ideals

We will now focus on towers of normal fine ideals.

Proposition 2.4.1. Assume X ⊆ Y and I, J are normal fine ideals with support respec-tively Y , X such that J projects to I. Then iXY /IJ is an injective homomorphism whichis |X|-complete.

Proof. iXY /IJ is an injective homomorphism. By Lemma 1.2.11.

iXY /IJ is |X|-complete. Given [Sx]I : x ∈ X we want to prove that

iXY /IJ(∨[Sx]I : x ∈ X) =

∨iXY /IJ([Sx]I) : x ∈ X .

By Proposition 2.3.4 and since I and J are normal fine and fine we have∨[Sx]I : x ∈ X = [

hSx : x ∈ X]I ,

and ∨[Sx ↑ Y ]J : x ∈ X = [

hSx ↑ Y : x ∈ X]J .

Therefore

iXY /IJ(∨[Sx]I : x ∈ X) = [

hSx : x ∈ X ↑ Y ]J

= [Z : ∃z ∈ Z ∩X : (Z ∩X ∈ Sx)]J = [hSx ↑ Y : x ∈ X]J

=∨[Sx ↑ Y ]J : x ∈ X =

∨iXY /IJ([Sx]I) : x ∈ X .

Observe that in general the homomorphism iXY /IJ is not complete.

Notation 2.4.2. Given X ⊆ Y , ideals I with support X and J with support Y such thatJ canonically projects to I, we will often improperly denote (to avoid an unnecessary heavynotation) the quotient homomorphism iXY /IJ by iXY loosing track of the reference to I andJ while still assuming these maps act with domain P (P (X)) /I and range P (P (Y )) /J .

50

Definition 2.4.3.I = IX : X ∈ Vδ

is a tower of normal fine ideals of height δ if IX ⊆ P (P (X)) is normal fine for any X ∈ Vδand IY canonically projects to IX if Y ⊇ X.

Notation 2.4.4. Given a tower of normal fine ideals I = IX : X ∈ Vδ, let

SVδ(I) =i

T ↑ Vδ : T ∈ IX , X ∈ Vδ.

Theorem 2.4.5 (Burke). Assume I = IX : X ∈ Vδ is a tower of normal fine ideals ofinaccessible height δ. Then SVδ(I) is stationary and IX is the projection of NSVδ SVδ (I)

for any X ∈ Vδ.

Proof.

SVδ(I) is stationary. Take any f : V <ωδ → Vδ. We look for Z ∈ Cf ∩ SVδ(I). Since δ is

inaccessible, there exists α < δ strong limit cardinal such that f Vα : V <ωα → Vα. Since

α is strong limit, X ∈ Vα entails that IX ∈ Vα as well, hence IX : X ∈ Vα ⊆ Vα.Moreover IVα canonically projects to IX ∈ Vα for all X ∈ Vα, therefore

SVα(I ∩ Vα) =i

T ↑ Vα : T ∈ IX , X ∈ Vα∈ IVα .

In particular (since IVα is normal fine) SVα(I ∩ Vα) /∈ NSVα . This implies that thereexists Z ⊆ Vα such that

Z ∈ SVα(I ∩ Vα) ∩ CfVα ⊆ SVδ(I) ∩ Cf .

NSVδ SVδ (I) projects to IX for any X ∈ Vδ. Fix X ∈ Vδ and let T ∈ P (X). We needto prove that T ↑ Vδ ∈ NSVδ SVδ (I) if and only if T ∈ IX .

Assume first T ∈ IX . Let C = M ≺ Vδ : T ∈M, then C is a club and

(P (X) \ T ) ↑ Vδ ⊇ SVδ(IX) ∩ C ⊇ SVδ(I) ∩ C.

Hence T ↑ Vδ ∩ SVδ(I) is non-stationary.

Assume now that T ∈ I+X . Let f : V <ω

δ → Vδ. As before we can find α < δstrong limit such that f Vα : V <ω

α → Vα and X ∈ Vα. Since T ∈ I+X , we have

T ↑ Vα ∈ I+Vα

. On the other hand we already showed that SVα(I ∩ Vα) ∈ IVα , hence

SVα(I ∩ Vα) ∩ T ↑ Vα ∈ I+Vα

is stationary. Therefore there exists Z such that

Z ∈ SVα(I ∩ Vα) ∩ T ↑ Vα ∩ CfVα ⊆ SVδ(I) ∩ T ↑ Vδ ∩ Cf .

This completes the proof.

We conclude giving a description of the tower forcings induced by a tower of normalfine ideals of strong limit height.

Definition 2.4.6. Let δ be a strong limit cardinal.Tδ denotes the boolean algebra obtained as the direct limit of the system of complete

homomorphisms iXY : P (P (X))→ P (P (Y )) : X ⊆ Y ∈ Z, i.e.:Tδ =

⋃X∈Vδ P (P (X)) /≡ where S ≡ T if and only if letting XS = ∪S and XT = ∪T

S ↑ (XS ∪XT ) = T ↑ (XS ∪XT ),

with operations:

51

• [S] ∧ [T ] = [S ↑ (XS ∪XT ) ∩ T ↑ (XS ∪XT )],

• [S] ∨ [T ] = [S ↑ (XS ∪XT ) ∪ T ↑ (XS ∪XT )],

• ¬[S] = [P (P (∪S)) \ S].

Lemma 2.4.7. Assume δ is inaccessible. Then (Tδ,∧,∨,¬, [∅], [P (P (0))]) is a < δ-complete boolean algebra.

Moreover assume I = IX : X ∈ Vδ is a tower of normal fine ideals. Then I =[S] : S ∈

⋃I is a < δ-complete ideal on Tδ, hence T Iδ = Tδ/I is a < δ-complete boolean

algebra.

Proof. We leave to the reader to check that Tδ is a boolean algebra. We show that Tδ is< δ-complete. Assume [Si] : i ∈ α ⊆ Tδ with α < δ. Let Xi = ∪Si Find α < ξ < δ suchthat Si ∈ Vξ for all i < α. We leave to the reader to check that [

⋃iXiVξ(Si) : i < α

] is

the exact upper bound of [Si] : i ∈ α.Similarly one can check that the ≤I order on Tδ admits suprema for all the < δ-sized

subsets of Tδ (we can use Burke’s theorems 2.3.19 and 2.4.5 to find an exact upper boundin I for any subset of I of size less than δ). Otherwise Lemma 2.3.4 in combination withthe proof of Theorem 2.4.5 can be used to compute the supremum in T Iδ of a family[Si]I : i < ξ ⊆ T Iδ for some ξ < δ.

Remark 2.4.8. Let I = IX : X ∈ Vδ be a tower of normal fine ideals of height aninaccessible cardinal δ. Let J =

⋃I and I = [S] ∈ Tδ : S ∈ J. Consider the partial order

(Vδ \ J,≤I) where S ≤I T if letting ∪S = X and ∪T = Y , we have that

(S ↑ X ∪ Y ) \ (T ↑ X ∪ Y ) ∈ IX∪Y .

We leave to the reader to check that the map S 7→ [S]I defines a dense embedding of theabove partial order in (Tδ/I)+.

To avoid confusion between the case of forcings induced by normal fine ideals andforcings induced by towers of normal fine ideals, and to simplify our notation, we adoptthe following conventions:

Notation 2.4.9. From now on, for the remainder of this book, we will just say I is anormal ideal on X to subsume that I is normal and fine ideal on X.

Let I = IX : X ∈ Vδ be a tower of normal ideals of height an inaccessible cardinal δand I be the ideal [S] : S ∈

⋃I on Tδ. We denote Tδ/I by T Iδ . [S]I for S ∈ Vδ denotes

a generic element of T Iδ . Moreover to avoid an exceedingly heavy notation (in case wefeel that this cannot generate misunderstandings) we identify T Iδ with the forcing notion(Vδ \

⋃I,≤I).

Remark 2.4.10. Let I be a tower of normal ideals of height δ with δ inaccessible.We obtain that [S]I ≤ [T ]I if and only if S ≤I T (where setting X =

⋃S and Y =

⋃T ,

S ⊆ P (X), T ⊆ P (Y ) are stationary and

S ↑ (Y ∪X) ⊆IY ∪X T ↑ (Y ∪X)).

Moreover the supremum and infimum of a family [Si]I : i < ξ ∈ T Iδ with ξ < cof(δ)V

52

are given respectively by the I-equivalence classes of the sets∧Si : i < ξ =

M ⊆ (

⋃Si : i < ξ) ∪ ξ : ∀i ∈M ∩ ξ,M ∩

⋃Si ∈ Si

=

iSi ↑ (ξ ∪

⋃Si : i < ξ) : i < ξ

,∨

Si : i < ξ =M ⊆ (

⋃Si : i < ξ) ∪ ξ : ∃i ∈M ∩ ξ,M ∩

⋃Si ∈ Si

=

hSi ↑ (ξ ∪

⋃Si : i < ξ) : i < ξ

.

A great deal of this book will investigate the forcing properties of boolean algebras oftype T Iδ or P (P (X)) /I with I normal or I a tower of normal ideals of height δ.

53

Part II

Boolean valued models

54

We introduce the type of first order structures we will be studying in the remainderof the book: boolean valued models of set theory of type V B with B a complete booleanalgebra, and generic ultrapowers of type Ult(V,G) with G a (tower of) V -normal filter(s).We aim to give a unified presentation of these two distinct type of structures, hence wefirst develop the basic theory of boolean valued models in chapter 3, where we introducethe key notion of full B-model, and we prove Los Theorem for full B-models, as well asthe rules governing the forcing relation on these type of models. Our approach to thesematters is inspired by Hamkins and Seabold [27]. Chapter 4 gives a fast account of thebasic theory of forcing for boolean valued models of set theory, recalling with sketchy proofsthe main results needed in the sequel of the book. Chapter 5 presents generic ultrapowersof type Ult(V,G) as quotients by certain type of ultrafilters G ⊆ P (P (X)) of the fullP (P (X))-model given by functions f : P (X)→ V in V .

55

Chapter 3

Boolean Valued Models

This chapter presents some basic properties of boolean valued models and homomorphismsbetween them. Detailed references for all results we mention without proofs are [47, 54] orthe more encyclopedic (and a bit out of date) [37].

1. In the first section we give the formal definition of boolean semantic for any first orderlanguage, and we present the soundness theorem for the semantic for the language ofset theory. The boolean valued semantic selects a given complete boolean algebraB and assigns to every statement φ a boolean value in B. The boolean operationswill reflect the behavior of the propositional connectives; it will require more careto give a meaning to atomic formulae and to quantifiers, and we need that B hasan high degree of completeness in order to be able to interpret quantifiers in ourboolean semantic. The standard Tarski semantics will be recovered when we choosethe boolean algebra 0, 1 as B.

2. The second section carves a bit more into the theory of B-valued modelsM and theirTarski quotient M/G induced by an ultrafilter G ∈ St(B). We state a necessary andsufficient condition (that of being a full model) on a B-valued model M which givesa complete control on how truth in M/G is determined by the topological propertiesof G as a point of St(B) via a Los theorem for full boolean valued models. We alsoprove a version of the forcing theorem relating the boolean value of a formula φ in aB-valued model M to the topological density of the family of G such that M/G |= φ.Next we give an exact characterization of the degree of completeness B must havewith respect to a given B-model M in order to grant that M is a full B-model towhich Los theorem applies. We also introduce Cohen’s forcing relation on a B-valuedmodel M and compare it to the B-valued semantic for M. Finally we show that theproperty of being a full B-valued model is preserved by passing to quotients.

3. The third section outlines the basic properties of homomorphisms between B-valuedmodels.

3.1 Boolean valued models and boolean valued semantics

In this section we give the formal definition of a boolean valued model for any first orderrelational language (i.e. a language containing no function symbols), and we introduce asound semantic for these languages. We limit ourselves to analyze relational languages toavoid some technicalities arising in the semantical interpretation of function symbols inboolean valued models.

56

Definition 3.1.1. Let L = Ri : i ∈ I, cj : j ∈ J be a language with no function symbols(a relational language in the sequel) and B a Boolean algebra. A B-valued model M for Lconsists of:

1. A non-empty set M . The elements of M are called names.

2. The Boolean value of the equality symbol. That is, a function

M2 −→ B

〈τ, σ〉 7−→ Jτ = σKMB

3. The interpretation of symbols in L. That is:

• for each n-ary relation symbol R ∈ L, a function

Mn −→ B

〈τ1, . . . , τn〉 7−→ JR(τ1, . . . , τn)KMB

• for each constant symbol c ∈ L, a name cM ∈M .

We require that the following conditions hold:

1. For all τ, σ, π ∈M ,

Jτ = τKMB = 1, (3.1)

Jτ = σKMB = Jσ = τKMB , (3.2)

Jτ = σKMB ∧ Jσ = πKMB ≤ Jτ = πKMB . (3.3)

2. If R ∈ L is an n-ary relation symbol, for all 〈τ1, . . . , τn〉, 〈σ1, . . . , σn〉 ∈Mn,(n∧i=1

Jτi = σiKMB

)∧ JR(τ1, . . . , τn)KMB ≤ JR(σ1, . . . , σn)KMB . (3.4)

Notation 3.1.2. We feel free to confuse a boolean structure M = 〈M,RMi : i ∈ I〉 withits domain M when no confusion can arise.

We define now the semantic of a boolean valued model: assume we have fixed anL-structure M, its Tarski semantic can be seen as a function that takes a L-statement ϕand assigns 1 or 0 to ϕ according to the fact thatM ϕ orM 6 ϕ. We want to generalizethis framework letting this evaluation function get its values inside any given booleanalgebra B. To simplify slightly our treatment, in the beginning we will assume that B iscomplete though this is not strictly necessary (in subsection 3.2.2 we characterize exactlythe amount of completeness for B and M which is needed to give a satisfactory B-valuedsemantics for M). We adopt the following strategy to define the semantic of a booleanvalued structure for L:

• Given M = 〈M,=M, RMi : i ∈ I, cMj : j ∈ J〉 B-valued model for a relationallanguage L = Ri : i ∈ I, cj : j ∈ J, we expand L to LM = L∪ ca : a ∈M addingconstant symbols for all elements of M so that ca is always assigned to a. In sucha way we can interpret in M formulae with constant symbols in the place of freevariables.

57

• FRV(L) denotes the set of free variables for the formulae of the language L, and anymap ν : FRV(L)→M is an assignment.

• Given an assignment ν, a free variable x, and b ∈M , νx/b denotes the assignment ν ′

such that ν ′(y) = ν(y) for all y 6= x in FRV(L) and such that ν ′(x) = b.

• If y = 〈y0, . . . , yn−1〉 is an n-tuple of free variables, ν(y) is a short-hand for 〈ν(y0), . . . , ν(yn−1)〉.

• If a = 〈a0, . . . , an−1〉 is an n-tuple of elements of M , ca is a short-hand for the n-tupleof constant symbols of LM 〈ca0 , . . . , can−1〉.

Definition 3.1.3. Let M = 〈M,=M, RMi : i ∈ I〉 be a B-valued model for the relationallanguage L = Ri : i ∈ I.

We evaluate all formulae of LM without free variables (but possibly with constantsymbols) as follows:

- JR(ca1 , . . . , can)KMB = RMi (a1, . . . , an).

- Jϕ ∧ ψKMB = JϕKMB ∧B JψKMB .

- J¬ϕKMB = ¬B JϕKMB .

- Jϕ→ ψKMB = ¬B JϕKMB ∨B JψKMB .

- J∃xϕ(x, ca)KMB =∨b∈M Jϕ(cb, ca)KMB .

- J∀xϕ(x, ca)KMB =∧b∈M Jϕ(cb, ca)KMB .

If φ(x1, . . . , xn) is a formula of LM with free variables x1, . . . , xn and ν is an assignment,

we let ν(φ(x1, . . . , xn)) =qφ(cν(x1), . . . , cν(xn))

yMB

.

To simplify notation we confuse from now on the constant symbol ca ∈ LM with itsintended interpretation a ∈M .

Remark 3.1.4. Some comments:

- In the definition of J∃xϕ(x, a)KMB and J∀xϕ(x, a)KMB we are (apparently) using thatB is complete.

- Clearly the definitions of Jφ ∨ ψKMB and Jφ→ ψKMB is redundant once we have defined

J¬ϕKMB and Jϕ ∧ ψKMB . Also J∀xϕ(x, a)KMB is redundant once we have defined J¬ϕKMBand J∃xϕ(x, y)KMB .

- If no confusion can arise, we avoid to put the superscript M and the subscript B inJφK.

- If B = 0, 1, the semantic we have just defined is the usual Tarski semantic for firstorder logic.

Now we outline that this semantic is sound and complete with respect to first ordercalculus.

Definition 3.1.5. A statement ϕ in the language L is valid in a B-valued model M for Land the boolean algebra B if JϕK = 1B. A theory T is valid in M if every axiom ϕ ∈ T isvalid.

58

Theorem 3.1.6 (Soundess and Completeness). Let L be a relational first order language.An L-formula ϕ is provable syntatically by an L-theory T if and only if for all cba Bν(ϕ) = 1B for every assignment ν : FRV(L)→M on a boolean valued model M for L inwhich T is valid.

Definition 3.1.7. Let B be a cba and let L = Ri : i ∈ I be a relational language whereRi is a mi-ary relation symbol for every i ∈ I. Suppose that M = (M,Ri : i ∈ I) is aB-model for L. Let J be an ideal on B and G be its dual filter. The quotient model M/Jis the B/J -valued model defined as follows:

- its domain M/J is the set [h]J : h ∈M where [h]J = f ∈M : Jf = hK ∈ G;

- JRi/J([f1]J , . . . , [fmi ]J)KB/J = [JRi(f1, . . . , fmi)KB]J for every i ∈ I.

We feel free to denote these quotient models as M/J or M/G.

We leave to the reader to check that these quotients are B/J -valued models.

Remark 3.1.8. In case the dual of J is an ultrafilter G, B/J = 2 and M/J is a standardTarski model for a first order language. In this case we say thatM/J is the Tarski quotientof M by G.

In general B-valued models are not extensional, i.e. there can be f 6= g ∈ M suchJf = gK = 1B, consider for example the case of L∞-functions on R, i.e. the essentiallybounded measurable functions. This can be viewed as a B-valued model for B the completeboolean algebra given by measurable sets modulo null sets: Jf = gK is the equivalenceclass modulo the null ideal of the measurable set on which the two functions agree. Twomeasurable functions f, g which disagree on a measure 0-set are such that Jf = gK = 1B.Nonetheless it is customary to identify these functions passing to the quotient structureL∞ = L∞/0B, whose equivalence classes are given by essentially bounded measurablefunctions agreeing modulo a null set, obtained by passing to quotient L∞ by the trivialideal 0B.

Definition 3.1.9. A B-valued model M with domain M is extensional if Jf = gKB = 1Bentails f = g for all f, g ∈M .

Remark 3.1.10. Given any B-valued modelM and any ideal I on B,M/I is an extensionalB/I -model, hence M/1B is an extensional B-model.

We have no reasons to believe that if a formula which is not quantifier free is true in aB-valued model M, then it is also true in M/G, for some G ∈ St(B). In general this isfalse, as the following example shows:

Example 3.1.11. Fix the language L = <,C consisting of two relation symbols, where< is binary and C is unary. Let B = RO(R) and Cω(R) denote the analytic functions withdomain R (i.e. those defined by a power series converging on all of R). Consider the B-valued model for the language L M = (Cω(R),=, <B, CB) with the following interpretationof the atomic formulae:

Jf = gK =Reg (x ∈ R : f(x) = g(x)),Jf <B gK =Reg (x ∈ R : f(x) < g(x)) ,

JCB(f)K =⋃

Reg (U : f U is constant) .

59

We leave to the reader to check that (Cω(R),=, <B, CB) is a B-valued model. Now, fix anyf ∈ Cω(R) and look at the formula φ(f) := ∃y (f < y ∧ C(y)).

J∃y (f <B y ∧ CB(y))K =∨

g∈Cω(R)

Jf <B g ∧ CB(g)K ≥

=∨a∈R

Jf <B caK ∧ JCB(ca)K ≥

where ca is the constant function ca(x) = a

=∨a∈R

Jf <B caK ≥

≥∨n∈Z

Jf < canK = where an = sup(f (n− 1, n))

=Reg

(⋃n∈Z

(n− 1, n)

)= R.

Therefore, we have that (Cω(R),=, <B, CB) |= φ(f) and in particular

(Cω(R),=, <B, CB) |= ∃y (idR < y ∧ C(y)),

where idR is the identity function x 7→ x.Now, consider F = (n,+∞) : n ∈ Z ⊆ RO(R). Since F satisfies the finite intersectionproperty (that is, F is closed under intersection of finite subsets), we can extend F to someG ∈ St(RO(R)). Consider the model given by the quotient M/G. The identity functionidR has the property that for any a ∈ R

J¬([idR]G <B [ca]G)K = (a,+∞) ∈ G.

For any analytic function f either JC(f)K = R or JC(f)K = ∅; moreover JC(f)K = R if andonly if f is constant. It follows that M/G |= ¬∃y (idR < y ∧ C(y)).

3.2 Full boolean valued models and Los theorem

Example 3.1.11 shows that quotients of boolean valued models may not preserve validityof formulae with quantifiers. To overcome this issue we are led to the definition of fullboolean valued models as those boolean valued models for which the above problem doesnot occur. We show that fullness characterizes the preservation of satisfiability in anyquotient and we give several examples of full boolean valued models.

3.2.1 Los theorem for full boolean valued models

Definition 3.2.1. Fix a language L and a complete boolean algebra B, a B-valued modelM for L is full if for every formula φ(x, y) and a ∈M |y|

J∃xφ(x, a)KMB = Jφ(b, a)KMBfor some b ∈M .

Theorem 3.2.2 ( Los theorem). Let B be a (complete)1 boolean algebra. Assume M is afull B-valued model. For any G ∈ St(B), f1, . . . , fn ∈M , and for all formulae φ(f1, . . . , fn)

M/G |= φ([f1]G, . . . , [fn]G) if and only if Jφ(f1, . . . , fn)KMB ∈ G.1In subsection 3.2.2 we will replace this assumption on the completeness of B with a weker condition

requiring the existence of suprema just for certain families of subsets of B determined by the B-valuedsemantics of M .

60

Proof. By induction on the complexity of φ(f1, . . . , fn). The case for φ atomic holdsby definition; propositional connectives are easily handled. Assume φ(f1, . . . , fn) =∃xψ(x, f1, . . . , fn), then

M/G |= ∃xψ(x, [f1]G, . . . , [fn]G) iff M/G |= ψ([h], [f1]G, . . . , [fn]G)

for some h ∈Miff Jψ(h, f1, . . . , fn)K ∈ G

for some h ∈Mwhich implies J∃xψ(x, f1 . . . , fn)K ∈ G

Moreover, since M is full, the viceversa also holds: pick h ∈M such that

J∃xψ(x, f1 . . . , fn)K = Jψ(h, f1 . . . , fn)K ,

we have that Jψ(h, f1 . . . , fn)K belongs to G if and only if J∃xψ(x, f1 . . . , fn)K does.

The following Lemma outlines a fundamental link between full B-valued models andthe topological properties of St(B).

Lemma 3.2.3 (Forcing lemma I). Let B be a (complete)2 boolean algebra. Let M be a fullB-model and G ∈ St(B). Then, for any formula φ the following statements are equivalent

1. JφK ≥ b.

2. Dφ = G ∈ St(B) :M/G |= φ is dense in Nb.

3. Dφ ⊇ Nb.


According to Lemma 3.2.3, if M is full, we can check that a formula φ is valid in Mby showing it is valid in M/G for densely-many G ∈ St(B).

Examples of full boolean valued models: standard ultraproducts We now sketchan argument to show that the familiar notion of ultraproduct of Tarski models is a specialcase of a quotient of a full boolean valued model.

Let X be a set. Then P (X) is an atomic complete boolean algebra. Notice that alltheorems proved so far applies equally well to atomic complete boolean algebras even ifin the examples we focused on atomless, complete boolean algebras. A key observation isthat x : x ∈ X is a maximal antichain and a dense open set in P (X)+. Now observethat St(P (X)) is the space of ultafilters on X and X can be identified inside St(P (X))as the open dense set Gx : x ∈ X where Gx is the principal utrafilter on P (X) given byall supersets of x. Another key observation is the following:

Fact 3.2.4. Let Mx : x ∈ X be a family of Tarski-models in the first order relationallanguage L each with domain Mx. Then N =

∏x∈XMx is the domain of a full P (X)-model

N letting for each n-ary relation symbol R ∈ L,

JR(f1, . . . , fn)KP(X) = x ∈ X : Mx |= R(f1(x), . . . , fn(x)) .2See the previous footnote regarding the degree of completeness of B needed for this Lemma.

61

Let G be any non-principal ultrafilter on X. Then, using the notation of the previousfact, N/G is the domain of the familiar ultraproduct of the family (Mx : x ∈ X) by G andthe usual Los Theorem for ultraproducts of Tarski models is the specialization to the caseof the full P (X)-valued model N of Theorem 3.2.2. Notice that in this special case, if theultraproduct is an ultrapower of a model M, the embedding a 7→ [ca]G (where ca(x) = afor all x ∈ X and a ∈M) is elementary. A good deal of our work in the remainder of thisbook will be to establish to what extent this is the case for other examples of full B-valuedmodels we will be looking at.

3.2.2 Full B-models for a non complete boolean algebra B

We have essentially used that B is a complete boolean algebra to assign a truth value toexistential formulae. Nonetheless in many cases we do not need that B is complete inorder to show that a certain B-valued model is full. It suffices that the model satisfies thefollowing definition:

Definition 3.2.5. Let B be a boolean algebra andM be a B-valued model for a relationallanguage L.M is a full B-valued model if for all formulae φ(x0, . . . , xn) and all a1, . . . , an ∈ M ,

there exists a b ∈M such that

Jφ(b, a1, . . . , an)K ≥B Jφ(c, a1, . . . , an)K

for any c ∈M , i.e.:

Jφ(b, a1, . . . , an)K =∨c∈M

Jφ(c, a1, . . . , an)K .

In order to check that this definition holds for a B-valued model M, one must proceedby induction on the complexity of the formula φ(x0, . . . , xn) to check that

Jφ(a0, a1, . . . , an)K ∈ B

is well defined for all a0, . . . , an ∈M . This is always the case for atomic formulae φ, sinceM is a B-valued model, and it holds for quantifier free formula by the rules of booleanconnectives. Assuming that

Jφ(a0, a1, . . . , an)K ∈ B

for all a0, . . . , an has been defined, one has to check whether

J∃xφ(x, a1, . . . , an)K ∈ B

can be defined asJφ(b, a1, . . . , an)K =

∨c∈M

Jφ(c, a1, . . . , an)K

for some b ∈M such that

Jφ(b, a1, . . . , an)K ≥B Jφ(c, a1, . . . , an)K

for any c ∈M .On the other hand once one has been able to check that M is a full B-valued model,

we automatically get that Los theorem 3.2.2 and the forcing lemma 3.2.3 holds forM evenif B is not complete, since the proof of these results used just the assumption that M is afull B-valued model, not the one that B is a complete boolean algebra.

Fullness is an absolute property:

62

Lemma 3.2.6. Assume V ⊆ W are transitive models of MK (or ZFC) and M is a fullB-valued model in V whose domain and relations are classes in V and in W . Then Mremains a full B-valued model also in W .

A comment is in order for the above statement: even if the domain M of M and allthe relations R : Mk → B used to define M in V are definable classes in V , in W we arenot considering the structure N obtained as the extension in W of the relations used in Vto define M (which could be quite different from M!), but the structure M itself. This ispossible if V is itself a definable class in W or if all the classes of the MK model V are alsoclasses of the MK model W .

Proof. To evaluate the formula Jφ(f1, . . . , fk)KMB = b one needs to check just quantifiersranging over the domain of M and the boolean algebra B. Hence the truth value assignedto this formula is the same in W and in V since the structures M and the boolean algebraB are the same in the two models.

Hence, once we have established the fullness of a boolean valued model in some transitivemodel V of ZFC (MK), its fullness is propagated to all the transitive outer models W of V .

3.2.3 Sufficient conditions for fullness

We show two methods to obtain full B-models:

• Quotients of full B-valued models remain full: this will be used to obtain that genericultrapowers are full, being Tarski quotients of full B-models.

• B-models with the mixing property are full: this will be used to argue that theboolean valued models of set theory of the form V B are full.

Lemma 3.2.7. Assume M is a full B-valued model and J is an ideal on B. Then M/Jremains a full B/J -valued model and is also extensional.

Proof. By Remark 3.1.10 M/J is an extensional B/J -valued model.We proceed by induction on the logical complexity of formulae to show that

Jφ([f1]J , . . . , [fn]J)KB/J = [Jφ(f1, . . . , fn)KB]J

for all formulae φ(x1, . . . , xn) and f1 . . . , fn ∈M . The thesis holds by definition for atomicformulae, and the induction is trivially checked for propositional connectives. In case ofquantifiers

J∃xφ(x, [f1]J , . . . , [fn]J)KB/J =

=∨B/J

Jφ([f ]J , [f1]J , . . . , [fn]J)KB/J : f ∈M

=

=∨B/J

[Jφ(f, f1, . . . , fn)KB]J : f ∈M .

Since M is full there exists g ∈M such that

J∃xφ(x, f1, . . . , fn)KB = Jφ(g, f1, . . . , fn)KB ,

hence Jφ(f, f1, . . . , fn)KB ≤ Jφ(g, f1, . . . , fn)KB for all f ∈M . This gives that

[Jφ(f, f1, . . . , fn)KB]J ≤ [Jφ(g, f1, . . . , fn)KB]J = Jφ([g]J , [f1]J , . . . , [fn]J)KB/J

63

for all f ∈M . Hence

J∃xφ(x, [f1]J , . . . , [fn]J)KB/J =∨B/J

[Jφ(f, f1, . . . , fn)KB]J : f ∈M ≤ Jφ([g]J , [f1]J , . . . , [fn]J)KB/J .

The other inequality is trivial.

Remark 3.2.8. Observe the following:

• For an ideal J on B, B/J may not be anymore a complete boolean algebra even if Bis (also B/J may be a complete boolean algebra even if B is not).

• Los theorem for full boolean valued models is the instantiation of the above Lemmato the case of J being a prime ideal on B.

• If M is a full P (X)-model, M/J is also full for any ideal J on P (X).

Another sufficent condition which guarantees fullness is the Mixing property.

Definition 3.2.9 (Mixing property manca riferimento – M). Let B be a complete booleanalgebra and M a B-valued model. M has the mixing property if for all τa : a ∈ A family ofelements of M indexed by an antichain A ⊆ B+, there exists τ ∈M such that Jτ = τaK ≥ afor all a ∈ A.

Lemma 3.2.10 (Maximum Principle [7, Lemma 1.27] ). Let B be a complete booleanalgebra and M a B-valued model with the mixing property. Then M is full.

We prove the Lemma. We first need the following piece of notation:

Notation 3.2.11. Let B be a boolean algebra and b ∈ B+. D ⊆ B+ is predense below b if

q ∧ b : q ∈ D ∪ ¬b

is predense in B+.

Proof.J∃xϕ(x, τ)K ≥ Jϕ(σ, τ)K

holds always. So we want to show that

Jϕ(σ, τ)K ≥ J∃xϕ(x, τ)K

for some σ ∈M. Letu0 = J∃xϕ(x, τ)K > 0B.

LetD =

u ∈ B+ : there is some σu ∈ V such that u ≤ Jϕ(σu, τ)K

.

D is dense and open below u0 in B+. Let A be a maximal antichain of D; clearly∨u : u ∈ A = u0.

Now we can appeal to the Mixing lemma to find σ ∈ M such that Jσ = σuK ≥ u for anyu ∈ A. Thus for each u ∈ A we have

u ≤ Jσ = σuK ∧ Jϕ(σu, τ)K ≤ Jϕ(σ, τ)K .

ThereforeJ∃xϕ(x, τ)K = u0 =

∨A ≤ Jϕ(σ, τ)K .

The proof is complete.

64

A standard example of a B-model (with B a cba) satisfying the mixing property is theboolean valued model for set theory V B given by Cohen’s forcing method. This will beshown in the next chapter.

Lemma 3.2.12 (Forcing lemma II). Let B be a (complete)3 boolean algebra. Given aB-valued model M for a relational language L with the mixing property, φ(x0, . . . , xn) aformula of the language L, a0, . . . , an ∈M , define:

b φ(a0, . . . , an) (to be read as b forces φ(a0, . . . , an))

iff b ≤ Jφ(a0, . . . , an)K.Then the following holds:

1. b φ iff the set of G ∈ St(B) such that M/G |= φ is dense in Nb,

2. b φ ∧ ψ iff b φ and b ψ,

3. b ¬φ iff c 6 φ for all c ≤ b,

4. b φ ∨ ψ iff the set of c ≤ b such that c φ or c ψ is dense below b in B+.

5. b ∃xφ(x) iff the set of c ≤ b such that c φ(σ) for some σ ∈M is dense below b.

6. For all G ∈ St(B) and all φ formulae with parameters in M and no free variable,M/G |= φ if and only if b φ for some b ∈ G.

7. For all φ formulae with parameters in M JφK =∨b : b φ.


3.3 Homomorphisms of boolean valued models

Definition 3.3.1. Let M be a B-valued model and N a C-valued model in the samerelational language L. Let

i : B→ C

be a morphism of boolean algebras and Φ ⊆ M × N a relation. The couple 〈i,Φ〉 is amorphism of boolean valued models if:

1. domΦ = M ;

2. given (τ1, σ1), (τ2, σ2) ∈ Φ:

i(Jτ1 = τ2KMB ) ≤ Jσ1 = σ2KNC ,

3. given R an n-ary relation symbol and (τ1, σ1), . . . , (τn, σn) ∈ Φ:

i(JR(τ1, . . . , τn)KMB ) ≤ JR(σ1, . . . , σn)KNC ,

An injective morphism is a morphism such that in 2 equality holds.An embedding of boolean valued models is an injective morphism such that in 3 equality

holds.An embedding 〈i,Φ〉 from M to N is called isomorphism of boolean valued models if

i is an isomorphism of boolean algebras and for every b ∈ N there is a a ∈M such that(a, b) ∈ Φ.

3See the previous footnote regarding the degree of completeness of B needed for this Lemma.

65

Notation 3.3.2. A boolean couple 〈B,M〉 is a pair given by a boolean algebra B and aB-valued model M .

Remark 3.3.3. 〈i,Φ〉 is a morphism between the boolean couples 〈B,M〉 and 〈C,N〉 if andonly if letting J be the trivial ideal 0C the map Φ′ defined by τ 7→ [σ]J for (τ, σ) ∈ Φ is afunction and is also a well defined morphism of the boolean couple 〈B,M〉 with the booleancouple 〈C,N/J〉: observe that if Φ is a morphism Φ′ is a function; given (τ, σ1), (τ, σ2) ∈ Φ,

1C = i(1B) = i(Jτ = τK) ≤ Jσ1 = σ2K

hence [σ1]J = [σ2]J , since N/J is extensional. We leave to the reader to check the rest.In particular morphisms between extensional boolean valued models are maps and not

just binary relations.

Definition 3.3.4. Suppose M is a B-valued model and N a C-valued model (both in thesame language L) such that B is a complete subalgebra of C, M ⊆ N , and

JR(τ1, . . . , τn)KMB = JR(τ1, . . . , τn)KNC

for all relation symbols R. Let IdM be the immersion of M into N . Then 〈IdB, IdM 〉 isan embedding of boolean valued models and N is said to be a boolean extension of M.

Proposition 3.3.5. Let M be a B-valued model and N a C-valued model in the samelanguage L. Assume 〈i,Φ〉 is an isomorphism of boolean valued models.

Then for any L-formula φ(x1, . . . , xn), and for every (τ1, σ1), . . . , (τn, σn) ∈ Φ we havethat:

i(Jφ(τ1, . . . , τn)KMB ) = Jφ(σ1, . . . , σn)KNC

Morphisms of boolean valued models are preserved by quotients.

Proposition 3.3.6. Let 〈B,M〉 and 〈C,N〉 be two boolean couples in the language L. LetF be a filter in B, G a filter on C and i : B→ C a morphism of boolean algebras such thatG ⊇ i[F ]. Then 〈B/F ,M/F 〉 and 〈C/G,N/G〉 are still boolean couples.

Assume now Φ ⊆M ×N is such that 〈i,Φ〉 is a morphism of boolean valued models.Let

i/F,G : B/F → C/G[b]F 7→ [i(b)]G

andΦ/F,G = (α, β) ∈M/F ×N/G : ∃σ ∈ α, τ ∈ β such that (σ, τ) ∈ Φ .

Then 〈i/F,G,Φ/F,G〉 is a morphism between the extensional boolean valued models M/Fand N/G. Moreover, if 〈i,Φ〉 is an injective morphism, embedding, or isomorphism ofboolean valued models, and i/F,G is injective, then 〈i/F,G,Φ/F,G〉 is respectively an injectivemorphism, embedding, or isomorphism of boolean valued models.

Proof. Given (α, β) ∈ Φ/F,G, we let σα ∈ M and τβ ∈ N be two elements such that(τα, σβ) ∈ Φ and α = [τα]F , β = [σβ]G.

1. Since dom(Φ) = M , it follows that Φ/F,G is everywhere defined.

2. Consider (α1, β1), (α2, β2) ∈ ΦF . Then

i/F,G(Jα1 = α2KB/F ) = i/F,G([Jτα1 = τα2K]F ) =

= [i(Jτα1 = τα2K)]G ≤ [Jτβ1 = τβ2K]G = Jβ1 = β2KC/G .

66

3. Let R be an n-ary relation symbol in L and (α1, β1), . . . , (αn, βn) ∈ Φ/F,G. Then

i/F,G(JR(α1, . . . , αn)KB/F ) = i/F,G([JR(τα1 , . . . , ταn)K]F ) = [i(JR(τα1 , . . . , ταn)K)]G ≤

≤ [JR(σβ1 , . . . , σβn)K]G = JR(β1, . . . , βn)KC/G .

It can be easily checked that whenever equality holds in 2-3 of Definition 3.3.1, equalityholds as well in the above equations. The proof is concluded.

67

Chapter 4

Forcing

This chapter sums up some general facts about forcing we need in the remainder of thesenotes. We assume the reader is already familiar with the standard development of forcingas done for example in [29]. Reference texts for this chapter are [7, 28, 29] or the notes[54].

We focus our analysis of the forcing method following the approach by means of booleanvalued models, the advantage being that we will make extensive use of the algebraicapparatus we developed so far. Nonetheless in some parts of this book (notably the last oncategory forcings), we cannot neglect the standard approach to forcing by means of posetstaken originally by Cohen (Kunen [29] is the reference text). The main reason being thatin case the forcing notions we consider are proper classes, their boolean completion maynot exist even as a proper class, hence for class forcings the unique reasonable approach isto generalize to the proper class setting the Cohen’s presentation of the forcing method viapartial orders. This is done for example in [1] and we will follow this approach referringthe reader to the paper for details and proofs.

We start developing forcing by means of boolean valued models using as base theoryZFC since this is notationally simpler. Next we show which slight changes one has toimplement to handle the case of forcing over models of MK. Finally we present the approachto class forcing by means of posets over models of MK and compare the two approaches.

4.1 Boolean valued models for set theory

We introduce the standard boolean valued models for set theory; it is practical to work inthe language L = ∈,⊆,= with three binary relation symbols. For the moment we workwith base theory ZFC.

Definition 4.1.1 (Manca riferimento – M [54]). Let 〈M,∈,=,⊆〉 be a model of ZFC withM transitive. Let B ∈M be a complete boolean algebra in 〈M,∈〉. The canonical B-valuedmodel 〈MB,=B,∈B,⊆B〉 for L is the following class definable inside M :

MB = a ∈M : a : MB → B is a partial function.

The boolean relation =B,∈B,⊆B are the following class functions RB : (MB)2 → B (forR ∈ ∈,⊆,=) definable inside M :

• ∈B (b0, b1) =rb0 ∈ b1

z

B=∨r

a = b0

z

B∧ b1(a) : a ∈ dom(b1)

,

• ⊆B (b0, b1) =rb0 ⊆ b1

z

B=∧¬b0(a) ∨

ra ∈ b1

z

B: a ∈ dom(b0)

,

68

• =B (b0, b1) =rb0 = b1

z

B=

rb0 ⊆ b1

z

B∧

rb1 ⊆ b0

z

B,

When the context is clear, we will omit indexes.

Remark 4.1.2. The definition of MB is a shorthand for a recursive definition by rank in M .Similarly b0 R b1 for b0, b1 ∈MB and R ∈ ∈,=,⊆ is defined by a recursion on the rankof (b0, b1).

It can be shown that MB has the mixing property, hence it is full and therefore Lostheorem holds for MB.

Lemma 4.1.3 (Mixing [7, Lemma 1.25] or [54]). Assume 〈M,∈,=,⊆〉 models ZFC andM is transitive. Let B be a complete boolean algebra in 〈M,∈〉. Then 〈MB,∈B,=B,⊆B〉has the mixing property.

We will also use the following variant of the mixing Lemma:

Lemma 4.1.4 (Mixing [7, Lemma 1.25] or [54]). Let B be a complete boolean algebrain V , and θ > |B| be regular. Then 〈HB

θ ,∈B,=B,⊆B〉 has the mixing property, whereHBθ = Hθ ∩ V B.

Proof. Run the proof of the mixing Lemma inside Hθ and check that this can be done onthe basis of the axioms of ZFC holding in Hθ.

Theorem 4.1.5 (Cohen-Solovay-Scott-Vopenka [7, Theorem 4.1]). Assume 〈M,∈,=,⊆〉models ZFC and M is transitive. Let B be a complete boolean algebra in M . Let G be anyultrafilter on B. Then

(MB/G,∈B /G,=B /G,⊆B /G)

is a Tarski model of ZFC.Moreover the following holds:

• If M is transitive and models all axioms of ZFC except the powerset axiom, then

(MB/G,∈B /G,=B /G,⊆B /G)

is a Tarski model of the same axioms.

• If M is transitive and models the replacement scheme for Σn-formulae, then

(MB/G,∈B /G,=B /G,⊆B /G)

is a Tarski model of extensionality and of the replacement scheme for Σn-formulae.

Definition 4.1.6. Assume 〈M,∈,=,⊆〉 models ZFC and M is transitive. Let B be acomplete boolean algebra in 〈M,∈,=,⊆〉.

G is an M -generic ultrafilter for B if it is an ultrafilter on B and G ∩ D 6= ∅ for allD ∈M predense subset of B.

Given G ultrafilter on B, define by recursion on the rank of b ∈MB

bG = aG : ∃p ∈ G 〈a, p〉 ∈ b.

M [G] = bG : b ∈MB.

69

Notation 4.1.7. For a complete boolean algebra B ∈ V , GB ∈ V B denotes the canonicalname for a V -generic filter for B, i.e.

GB = 〈b, b〉 : b ∈ B.

Also we let Jτ ∈ V K be a short hand for the boolean value∨Jτ = xK : x ∈ V .

Remark 4.1.8. The above notation needs a bit of explanation: it is well known that for anatomless boolean algebra B in V there cannot be a V -generic ultrafilter G ∈ St(B), sincefor all G ∈ St(B) B+ \G is open dense and disjoint from G.

Nonetheless the forcing statement

rG is a V -generic filter for B

z= b

makes perfectly sense as it is formalizable in first order logic by the sentence

r(∀x ∈ ˇPD(B)G ∩ x 6= ∅) ∧ (G is an ultrafilter on B)

z

where PD(B) ∈ V is the collection of predense subset of B and is a definable subset ofP (B). It can be checked that

rGB is a V -generic filter for B

z= 1B.

In view of the forcing theorem below, one can safely work under the assumption thatV -generic filters G for B exist and translate by means of the forcing theorem her/hisconclusions regarding the first order properties holding in V [G] to statements assertingthat in V the corresponding forcing statements have a certain positive boolean value.

Theorem 4.1.9 (Cohen’s forcing theorem [7, Lemma 4.11], [28, Theorem 14.6, Theorem14.29]). Assume 〈M,∈,=,⊆〉 models ZFC and M is transitive. Let B be a complete booleanalgebra in 〈M,∈,=,⊆〉, G be an M -generic filter for B. Then:

1. 〈MB[G],∈,⊆,=〉 is isomorphic to 〈MB/G,∈B /G,⊆B /G,=B〉 via the map whichsends bG to [b]G.

2. M [G] |= φ((b1)G, . . . , (bn)G) iffrφ(b1, . . . , bn

z∈ G.

3. M |= b ≤B

rφ(b1, . . . , bn)

ziff

M [G] |= φ((b1)G, . . . , (bn)G)

for all M -generic filters G for B such that b ∈ G.

Fact 4.1.10. Let 〈M,∈,=,⊆〉 be a model of ZFC with M transitive, B ∈M be a completeboolean algebra in M , G be an M -generic ultrafilter for B. Then

∧A ∈ G for any A ⊂ G

which belongs to M .

The following also holds:

Proposition 4.1.11. Assume 〈M,∈,=,⊆〉 models ZFC and M is transitive. Let B be acomplete boolean algebra in 〈M,∈,=,⊆〉. Then φ(x) ≡ (x ∈ MB) is a provably ∆1(M)-property in the parameter B. The same holds for φR(x, y, z) ≡ (Jx R yKB = z) with Ramong ∈,⊆,=.

70

More generally assume φ(x1, . . . , xl) is a Qi-formula for some i > 0.Then for all b1, . . . , bl ∈MB and b ∈ B

rφ(b1, . . . , bl)

z

B= b

is Qi(M) in parameters b1, . . . , bl,B, b.

Proof. The base case follows by a delicate inductive proof based on the very definition ofthe classes MB and RB for R among ∈,⊆,= inside M . To prove the remaining part of the

Proposition, observe that M |=r∃xφ(x, b1, . . . , bn)

z= b if and only if

M |= ∃σ ∈MBrφ(σ, b1, . . . , bn)

z= b.

An easy induction can now be carried to yield the desired conclusion.

4.1.1 Embeddings and boolean valued models

Complete homomorphisms of complete boolean algebras extend to natural ∆1-elementarymaps between the associated boolean valued models for set theory.

Proposition 4.1.12. Let i : B → C be a complete homomorphism. Define by recursionı : V B → V C by

ı(b)(ı(a)) = i b(a)

for all a ∈ dom(b) ∈ V B. Then the pair 〈i, i〉 is a boolean embedding of V B into V C andthe map ı is ∆1-elementary, i.e. for every ∆1 formula φ,

i(rφ(b1, . . . , bn)

z

B

)=

rφ(ı(b1), . . . , ı(bn))

z

C

Proof. We prove the result by induction on the complexity of φ. For atomic formulas ψ(either x = y or x ∈ y), we proceed by further induction on the rank of b1, b2.

i(rb1 ∈ b2

z

B

)= i(∨

b2(a) ∧rb1 = a

z

B: a ∈ dom(b2)

)=∨

i(b2(a)

)∧ i(rb1 = a

z

B

): a ∈ dom(b2)

=∨

i(b2(a)

)∧

rı(b1) = ı(a)

z

C: a ∈ dom(b2)

=

rı(b1) ∈ ı(b2)

z

C

i(rb1 ⊆ b2

z

B

)= i(∧

b1(a)→ra ∈ b2

z

B: a ∈ dom(b1)

)=∧

i(b1(a)

)→ i

(ra ∈ b2

z

B

): a ∈ dom(b1)

=∧

i(b1(a)

)→

rı(a) ∈ ı(b2)

z

C: a ∈ dom(b1)

=

rı(b1) ⊆ ı(b2)

z

C.

We used the inductive hypothesis in the last row of each case. Sincerb1 = b2

z=

rb1 ⊆ b2

z∧

rb2 ⊆ b1

z, the proof for ψ atomic is complete. In particular this gives that the

pair 〈i, i〉 is a boolean embedding of V B into V C.

71

For ψ quantifier-free formula, the proof is immediate: i is an embedding hence preserves∨, ¬. Suppose now that ψ = ∃x ∈ y φ is a ∆0 formula.

i(r∃x ∈ b1φ(x, b1, . . . , bn)

z

B

)=∨

i(b1(a)

)∧ i(rφ(a, b1, . . . , bn)

z

B

): a ∈ dom(b1)

=∨

i(b1(a)) ∧rφ(ı(a), ı(b1), . . . , ı(bn)

)zC

: a ∈ dom(b1)

=r∃x ∈ ı(b1) φ

(x, ı(b1), . . . , ı(bn)

)zC

Furthermore, if ψ = ∃x φ is a Σ1 formula, by the Maximum Principle there exists a

a ∈ V B such thatr∃xφ(x, b1, . . . , bn)

z

B=

rφ(a, b1, . . . , bn)

z

Bhence

i(r∃xφ(x, b1, . . . , bn)

z

B

)= i(rφ(a, b1, . . . , bn)

z

B

)=

rφ(ı(a), ı(b1), . . . , ı(bn)

)zC

≤r∃xφ

(x, ı(b1), . . . , ı(bn)

)zC

Thus, if φ is a ∆1 formula, φ and ¬φ are both Σ1, hence the above inequality holds andalso

i(rφ(b1, . . . , bn)

z

B

)= ¬i

(r¬φ(b1, . . . , bn)

z

B

)≥ ¬

r¬φ(ı(b1), . . . , ı(bn)

)zC

=rφ(ı(b1), . . . , ı(bn)

)zC,


Notation 4.1.13. In general all over these notes, for the sake of readability, we indicateB-names with their defining properties. Recurring examples of this behavior are thefollowing:

• If we have in V a collection bi : i ∈ I of B-names, we confuse bi : i ∈ I with aB-name b such that for all a ∈ V B

ra ∈ b

z=

r∃i ∈ I a = bi

z.

• If i : B→ C is a complete homomorphism, we denote by C/i[G] a B-name b such that

rb is the quotient of C modulo the ideal generated by the dual of i[GB]

z= 1B.

4.2 Basic properties of forcing extensions

4.2.1 Preservation of regular cardinals in forcing extensions

Definition 4.2.1. A partial order (Q,≤Q) is<δ-presaturated is for any family Aγ : γ < ξwith ξ < δ of maximal antichains

q ∈ Q+ : ∀γ < ξ |a ∈ Aγ : a and q are compatible < δ|

is open dense in (Q,≤Q).A boolean algebra B is <δ-presaturated if (B+,≤B) is <δ-presaturated.

72

Remark 4.2.2. Any partial order P is <ω-presaturated, any <δ-CC partial order is <δ-presaturated. Hence any complete boolean algebra B is < δ-presaturated, whenever itadmits a dense subset P of size less than δ. In particular RO(P ) is < θ-presaturated ifP ∈ Hθ.

Proposition 4.2.3. Let (Q,≤Q) in V be a partial order, and δ be a regular cardinal in V .TFAE:

1. Q is <δ-presaturated in V ;

2. RO(Q) preserves the regularity of δ in V (i.e.: Jδ is regularKRO(Q) = 1RO(Q)).

Proof. Let B = RO(Q). It suffices to prove the above equivalence for B.

(1)⇒(2). Assume thatrf : α→ δ

z= 1B for some α < δ. Then define for any γ < α

Aγ =aγβ =

rf(γ) = β

z: β < δ

.

By (1), for any q ∈ B+ there exists r ≤ q such that

|β : aγβ ∧ r > 0

| < δ

for any γ < α. Let

Xγ =β : aγβ ∧ r > 0

for any γ < α. Since δ is regular and α < δ, we have

|⋃Xγ : γ < α | < δ.

Let η = sup rank(Xγ) : γ < α < δ. Then for any γ < α

r =∨

aγβ ∧ r : β < δ

=∨

aγβ ∧ r : β ∈ Xγ

≤

≤∨

aγβ : β < η

=∨r

f(γ) = βz

: β < η

=rf(γ) < η

z.

Hencer ≤

∧rf(γ) < η

z: γ < α

=

rf [α] ⊆ η

z.

Therefore there is a dense open set of conditions r which force f to be not cofinal.

(2)⇒(1). Assume that Aγ : γ < α is a family of maximal antichains of B and put

Aγ =aγβ : β < δ

,

andf =

〈op(δ, β), aγβ〉 : β < δ, γ < α

,

where op(δ, β) is a canonical name for the pair 〈δ, β〉. Observe that

•rf(γ) = β

z= aγβ;

•rf : α→ δ

z= 1B.

73

By (2) f is forced to be bounded, i.e.:q : ∃η < δ

rf [α] ⊆ η

z≥ q

is open dense. We get

q ≤rf [α] ⊆ η

z=∧r

f(γ) < ηz

: γ < α.

Fix γ < α. Since rf(γ) < η

z= sup

rf(γ) = β

zβ < η

,

we haveq ≤

∨rf(γ) < β

z: β < η

=∨

aγβ : β < η.

Thus for any β ≥ η, q ∧ aγβ = 0. This means that

|aγβ : aγβ ∧ q > 0

| ≤ η < δ.

4.2.2 Computing HV [G]λ in forcing extensions

Lemma 4.2.4. Let λ be a regular cardinal in V and let B ⊆ Hλ be a < λ-presaturated cba.Assume G is V -generic for B. Then

Hλ[G] = xG : x ∈ V B ∩Hλ = HV [G]λ .

It is clear that the Lemma can be relativized to any (transitive) model of ZFC.

Proof. Since every element a of Hλ with λ regular is coded by a bounded subset of λ (i.e.a bounded subset of λ coding a binary relation whose transitive collapse is the transitiveclosure of a), and B preserves the regularity of λ, we can assume that every B-name for an

element of HV [G]λ is coded by a B-name for a function f : λ→ 2 such that f is allowed to

assume the value 1 only on a bounded subset of λ. In particular we let for any such f ,

Df =p ∈ B+ : ∃αp p ≤

rf−1[1] ⊆ αp

zand for all ξ < λ,

Eξ,f =p ∈ B : ∃i < 2 p ≤

rf(ξ) = i

zNotice that the above sets are open dense for any ξ, f , and also that p ∈ Df as witnessed

by αp implies that p ∈ Eξ,f for all ξ ≥ αp. In particular to decide the values of f belowany p ∈ Df we just need to consider the dense sets Eξ,f for ξ < αp.

Let p ∈ B+ be arbitrary, and let Aξ ⊆ Eξ,f ∩Df be maximal antichains for all ξ < αp.Since B preserves the regularity of λ, it is <λ presaturated; hence we can find q ≤ p suchthat q ∈ Df as witnessed by αp and

Bξ = r ∈ Aξ : r is compatible with q

has size less than λ for all ξ < αp. We can now use these antichains Bξ to cook up a namegq ∈ Hλ ∩ V B such that q forces that f = gq. By standard density arguments, the thesisfollows.

74

Bounding the cardinality of P (α) in forcing extensions We outline how the car-dinality of the powerset of some sets can be computed in a generic extension:

Lemma 4.2.5. Assume B is a cba and G is V -generic for B. Then every element ofP (α)V [G] is equal to τG for some τ ∈ V B with

τ =〈ξ, bξ〉 : ξ < α

.

Therefore the cardinality of P (α)V [G] is bounded by (|α||B|)V .

Proof. Any subset of α in V [G] is given by its characteristic function and the above set ofB-names describe all the possible characteristic functions in V [G] with domain α.

4.3 Class forcing and set forcing with posets

We start defining class-sized forcings models of MK. We briefly recall some definitions andgeneral facts about forcing with (set or class sized) posets in Morse-Kelley Set Theory,following closely [1]. When we restrict this approach to models of ZFC (i.e. we do notconsider classes) and to set sized forcings, it turns out to be the standard set-forcing overZFC, as presented for instance in [28, 29].

Definition 4.3.1. Let 〈M,CM 〉 be a model of MK. (P,≤P ) is a 〈M,CM 〉-forcing ifP,≤P∈ CM , ≤P is a partial ordering on P with greatest element 1P . G ⊆ P is an〈M,CM 〉-generic filter for P if

1. G is a filter (i.e upward closed and such that finitely many elements of G always havea common refinement in G with respect to ≤P ),

2. G ∩D 6= ∅ whenever D ⊆ P is dense and D ∈ CM .

When we assume M to be countable and transitive and CM to be countable as well,this yields that for each p ∈ P there exists G such that p ∈ G and G is 〈M,CM 〉-genericfor P .

Definition 4.3.2. Let 〈M,CM 〉 be a model of MK with M transitive. Define

• MP0 := ∅;

• MPα+1 :=

q : q is a subset of MP

α × P in M

;

• MPλ :=

⋃MPα : α ∈ Ord(M)

;

• MP =⋃α∈Ord(M)M

Pα ;

• CPM :=X ∈ C : X is a subclass of MP × P

.

• x := 〈y, 1P 〉 : y ∈ x for any x ∈M .

• X := 〈y, 1P 〉 : y ∈ X for any X ∈ C.

Remark 4.3.3. It is possible to check that MP is a class in CM (which is also the extensionof a formula in the parameter P ) and CPM is the extension of a formula in the parameterP . This can be done much as in the same fashion as one argue for set sized forcings intransitive models of ZFC. We refer the reader to [1] for the details.

75

Given a 〈M,CM 〉-generic filter G for P , we define the interpretations of set- andclass-names recursively:

Definition 4.3.4. Let 〈M,CM 〉 be a model of MK, (P,≤P ) a 〈M,CM 〉-forcing, andG ⊆ P an 〈M,CM 〉-generic filter for P .

• qG = rG : ∃p ∈ G(〈r, p〉 ∈ q) for all q ∈MP .

• XG =qG : ∃p ∈ G(〈q, p〉 ∈ X)

for all X ∈ CPM .

Definition 4.3.5.

(M,CM )[G] = (M [G], CM [G]) = (qG : q ∈MP

,XG : X ∈ CPM

).

Lemma 4.3.6 (Lemma 9 [1]). Let 〈M,CM 〉 be a model of MK with M transitive, andP ∈ CM be a poset with maximal element 1P . Then for all G ⊆ P 〈M,CM 〉-generic filtersfor P :

• ∀x ∈M(x ∈MP ∧ xG = x) and ∀X ∈ CM (X ∈MP ∧ XG = X).

• M ⊆M [G], CM ⊆ CM [G], G ∈ C[G].

• (M,CM )[G] is transitive and is contained in any transitive MK-model 〈N,C〉 suchthat M ⊆ N , CM ⊆ C, G ∈ C.

• OrdM [G] = OrdM .

• 〈M [G], CM [G]〉 is a model of extensionality and Class comprehension.

Remark 4.3.7. Notice that we do not assert that 〈M [G], CM [G]〉 is a model of MK. Thisis in general false, the problematic axioms to be checked in 〈M [G], CM [G]〉 being thereplacement axiom and the power-set axiom: for example if G is 〈M,CM 〉-generic forColl(ω,Ord)M , G makes all ordinals of M -countable in M [G], and all well orders in CM [G]countable as well, hence 〈M [G], CM [G]〉 is not a model of replacement.

Let us now introduce the external and internal forcing relation, we refer the reader to[1, 28, 29] for the proof of their equivalence and their exact formulation.

Definition 4.3.8 (External forcing relation). Let 〈M,CM 〉 be a model of MK withM transi-tive, and P ∈ CM be a poset with maximal element 1P . Let p ∈ P , ϕ(x1, . . . , xn, X1, . . . , Xn)be a formula with displayed free variables, q1, . . . , qm be P -set-names and X1, . . . , Xn beP -class-names.

p P ϕ(q1, . . . , qm, X1, . . . , Xn),

if and only if for any G ⊆ P which is an (M,CM )-generic filter for P with p ∈ G, we have

(M,C)[G] |= ϕ((q1)G, . . . , (qn)G, (X1)G, . . . , (Xn)G).

Lemma 4.3.9 (Definability Lemma — Lemma 11 [1]). Let 〈M,CM 〉 be a model of MKwith M transitive, and P ∈ CM be a poset with maximal element 1P . For any L2-formulaϕ(x1, . . . , xm, X1, . . . , Xn) with displayed free variables, the relation

p P ϕ(q1, . . . , qm, X1, . . . , Xn)

is definable in (M,CM ) with parameters p, q1, . . . , qm, X1, . . . , Xn, P .

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Lemma 4.3.10 (Truth Lemma — Lemma 12 [1]). Let 〈M,CM 〉 be a model of MK with Mtransitive, and P ∈ CM be a poset with maximal element 1P . Assume G is 〈M,CM 〉-genericfor P . Then

(M,C)[G] |= ϕ((q1)G, . . . , (qn)G, (X1)G, . . . , (Xn)G)

if and only if there is p ∈ G such that

p ϕ(q1, . . . , qm, X1, . . . , Xn)

We will need the following:

Definition 4.3.11 (Pretameness — Definition 20 [1]). Let 〈M,CM 〉 be a model of MKwith M transitive, and P ∈ CM be a poset with maximal element 1P .

P is pretame if for all families Ai : i ∈ a of maximal antichains of P definable in〈M,CM 〉 and indexed by an a ∈M , there is a dense set q of conditions in P such that forall i ∈ a

〈M,CM 〉 |= r ∈ Ai : r and q are compatible is a set.

Lemma 4.3.12 (Theorem 23 [1]). Let 〈M,CM 〉 be a model of MK with M transitive,and P ∈ CM be a poset with maximal element 1P . Assume P is pretame and G is〈M,CM 〉-generic for P . Then 〈M,CM 〉[G] models the Replacement axiom.

Remark 4.3.13. Pretameness of P corresponds to < Ord-presaturation of P (see Def. 4.2.1)and the Pretameness Lemma corresponds in the class partial order scenario to Proposi-tion 4.2.3. More specifically assume δ is inaccessible, P ⊆ Vδ is a class partial order inVδ+1, then Vδ+1 models that P is pretame according to Def. 4.3.11 if and only if V modelsthat P is < δ-presaturated according to Def. 4.2.1.

Theorem 4.3.14 (Cohen’s forcing Theorem — Theorem 23 [1]). Assume 〈M,CM 〉 is amodel of MK with M transitive, and P ∈M is a set-sized poset with maximal element 1P .Assume G is 〈M,CM 〉-generic for P . Then 〈M,CM 〉[G] models MK.

In the following, with abuse of notation, we write V P , V [G] to denote the genericextension of the standard MK-model (V, C) given by the class of all sets and the family ofall classes.

Remark 4.3.15. Restricting our attention to transitive models M of ZFC and to set sizedforcings P ∈M , the above results and definitions provides the usual definability and truthlemmas for the corresponding forcing relation as defined in [28, 29] relative to M .

More specifically one can prove the version of all of the above results in which onesystematically omit any reference to classes in the formulation of the relevant properties.We leave the details to the reader.

4.3.1 V P versus V RO(P ) for set sized forcings P

We now relate the presention of forcing for set sized posets in models of ZFC with theboolean valued approach to forcing outlined before. In certain cases (for example in thedefinition of the name β in the proof of Lemma 10.2.3) it is convenient to allow a name ato be a relation, as it occurs for the P -names in V P .

Definition 4.3.16 (Manca riferimento – M). Let V be a transitive model of ZFC, P bea partial order in V , and iP : P → RO(P ) be a dense embedding of P into its booleancompletion.

V P =a ∈ V : a ⊆ V P × P

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• ∈∗P (b0, b1) =rb0 ∈ b1

z∗P

=∨r

a = b0

z∗P∧ iP (b) : 〈a, b〉 ∈ b1)

,

• ⊆∗P (b0, b1) =rb0 ⊆ b1

z∗P

=∧¬iP (b) ∨

ra ∈ b1

z∗P

: 〈a, b〉 ∈ b0

,

• =∗P (b0, b1) =rb0 = b1

z∗P

=rb0 ⊆ b1

z∗B∧

rb1 ⊆ b0

z∗P

.

Notation 4.3.17. For any formula φ, we denote by JφK∗P the boolean value assigned to φby the above boolean valued model.

Remark 4.3.18. Once again the definition of the classes V P , =∗P , ⊆∗P , ∈∗P is a shorthandfor a recursive definition by rank, and (apparently) depends on the choice of iP . We willbriefly outline below that different choices of iP produce isomorphic boolean valued models,hence we will not bother to specify which iP is chosen to define 〈V P ,=∗P ,⊆∗P ,∈∗P 〉.

In case P is a set sized poset, the following theorem links the forcing relation p Pφ(q1, . . . , qn) defined in the previous section on V P to the boolean valued model

〈V P ,=∗P ,∈∗P ,⊆∗P 〉

defined above.

Theorem 4.3.19 (Manca riferimento – Silvia). Assume P ∈ V is a set-sized partial orderwith maximal element 1P . Let iP : P → RO(P ) in V be a dense embedding of P intoRO(P ) used to define (V P ,=∗P ,∈∗P ).

Then for any L2-formula φ(x1, . . . , xn) b1, . . . , bn ∈ V P ,

V |= p P φ(b1, . . . , bn) ⇐⇒ V |= iP (p) ≤rφ(b1, . . . , bn)

z∗P.


Theorem 4.3.20 (Manca riferimento – M). Assume (V,∈) is a transitive model of ZFCand P ∈ V is a partial order. Let B = RO(P )V ∈ V . Then V models that

〈V B,=B,∈B,⊆B〉

and〈V P ,=∗P ,∈∗P ,⊆∗P 〉

are isomorphic and full B-valued models for L = ∈,⊆,=.More precisely assume iP : P → B is the dense embedding of P into its boolean

completions used to define 〈V P ,=∗P ,∈∗P ,⊆∗P 〉, then the map

iP :V P → V B

fa =〈fb,

∨iP (p) : 〈b, p〉 ∈ a

〉 : 〈b, p〉 ∈ a

,

is the desired boolean isomorphism.

Remark 4.3.21. The above result can be relativized to any (transitive) model M which is amodel of ZFC. We leave the details to the reader.

Let us add the following observation: the notion of being a complete boolean algebra isnot absolute (for example if B in V is a cba and H is V -generic for a complete subalgebra ofB, then B is not anymore complete in V [H], since new maximal antichains of B without asup in V have been added). On the other hand, the notion of being a complete embeddingi : P → Q is absolute between transitive models:

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Lemma 4.3.22. Assume V ⊆ W are transitive models of some T extending ZFC. LetP,Q ∈ V be partial orders and i : P → Q an embedding. Then i is a complete embeddingin V if and only if it is a complete embedding in W .

In which case, let B = RO(P )V and Q = RO(Q)W . Then i lifts to a ∆1-preservinghomomorphism i : V P →WQ between the B-valued model V P and the Q-valued model WQ

(i is a definable class in W , if V is a class of the NBG model W ).

Proof. Clearly if i : P → Q is a complete embedding in W it remains so in V , hence onlyone direction of the above equivalence is non-trivial. To simplify matters assume P,Q areseparative, otherwise first pass to their boolean completions as computed in V . Assumei : P → Q is complete in V , then in V i extends to a regular embedding i : RO(P )→ RO(Q)with associated adjoint π. Then (i, π) remain an adjoint pair also in W , giving that iremains a complete embedding also in W (we are repeatedly using 1.4.1 in V and in W ).

Now let B = RO(P )V and Q = RO(Q)W and define i : V P → WQ by σ 7→〈i(τ), i(a)〉 : 〈τ, a〉 ∈ σ

. We leave to the reader to prove (by inductions on ranks of

σ, τ ∈ V B) thatV |= Jσ R τKB = b

if and only if

W |=ri(σ) R i(τ)

z

Q= i(b)

for R a relation among ∈,⊆,=.Similarly we can prove that

V |= J∃xφ(x, σ1, . . . , σn)KB ≥ p

entails thatW |=

r∃xφ(x, i(σ1), . . . , i(σn))

z

Q≥ i(p)

for φ a Σ0-formula.This suffices to prove that the boolean value of ∆1-properties is preserved when passing

from V P to WQ.

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Chapter 5

Generic ultrapowers

We introduce the second type of boolean valued model we will be interested in the remainderof this book, which are the boolean valued models giving raise to ultrapowers. Genericultrapowers of type Ult(V,G) are quotients by certain type of ultrafilters G ⊆ P (P (X))of the full P (P (X))-model given by functions f : P (X) → V in V . We aim to give aunified treatment both of generic ultrapower embeddings and of the ultrapower embeddingsinduced by standard large cardinals; here (and everywhere in the book we deal with thesetopics) we continue along the lines of Foreman’s chapter for the Handbook [18], Larson’sbook on stationary tower forcing [30], and Foreman and Magidor’s [20]. We make a pointto prove all the basic results about elementary embeddings using minimal assumptions, soto be able to use them both when dealing with standard ultrapower embeddings given bylarge cardinals, or when dealing with generic ultrapower embeddings given by a V -genericfilter on P (P (X)) /I for some normal ideal I on X. Along the way we also deal withtowers of normal ideals. In the end we show how to describe standard large cardinalproperties such as hugeness or supercompactness by means of this technology. A commontheme of this chapter is to outline the common features shared by generic ultrapowers andboolean valued models of set theory. Many of the remaining parts of the book analyzewhich are the situations in which the two types of models (V [G] and Ult(V,G)) are veryclose to each other. A more general approach to generic ultrapowers which encompasses asspecial cases both the towers of normal ideals (which are the focus of the present book)and the notion of generic extender (which is the generic counterpart for strongness andsuperstrongness) has been devised by Audrito and Steila in [4], however we decided not topursue it in the present book.

5.1 Normal (towers of) ultrapowers

Definition 5.1.1. Let X be a non-empty set

Ult(V,X) =f : P (X)V → V : f a function in V

,

Define for any relation R ⊆ V k

RX(f1, . . . , fk) = JR(f1, . . . , fk)KP(P(X)) = Y ⊆ X : R(f1(Y ), . . . , fk(Y )) .

Lemma 5.1.2. Let R1, . . . Rj be arbitrary ki-ary relations on V for i = 1, . . . , j. Then

〈Ult(V,X), RX1 , . . . , RXj 〉

is a full P (P (X))-valued model.

80

Proof. The proof is straightforward and is left to the reader.

By Lemma 3.2.7 we obtain the following:

Theorem 5.1.3 ( Los Theorem for ultrapowers). Assume I ⊆ P (P (X)) is an ideal. ThenUlt(V,X)/I is a full and extensional P (P (X))-valued model.

Moreover if I is an ultrafilter, the map jI : V → Ult(V,X)/I mapping a 7→ [ca]I iselementary (where ca : P (X)→ V is constant with value a).

Proof. The first part of the theorem is immediate by Lemma 3.2.7. The second part of thetheorem is a reformulation of the well known result that a first order structure elementarilyembeds in its ultrapowers via the diagonal embedding. See for more details 5.2.4 below.

Similarly we can handle directed systems of ultrapowers (recall Def. 2.4.6, 2.3.13):

Definition 5.1.4. Let δ be an inaccessible cardinal. Let

Ult(V, δ) =f : P (X)V → V : X ∈ Vδ, f a function in V

.

We endow Ult(V, δ) with the structure of a Tδ-valued model (recall Def. 2.4.6) as follows:given R ⊆ V k and f1, . . . , fk ∈ Ult(V, δ), let Y ∈ Vδ be such that dom(fj) ⊆ Y for allj = 1, . . . , k. Then

Rδ(f1, . . . , fk) = [Z ⊆ Y : R(f1 ↑ Z, . . . , fk ↑ Z)]

Lemma 5.1.5. Let R1, . . . Rj be arbitrary ki-ary relations on V for i = 1, . . . , j and δ bean inaccessible cardinal. Then

〈Ult(V, δ), Rδ1, . . . , Rδj〉

is a full Tδ-valued model.


Similarly to Theorem 5.1.3, we have the following:

Theorem 5.1.6 ( Los Theorem for directed systems of ultrapowers). Assume δ is inac-cessible and I is an ideal on Tδ. Then Ult(V, Tδ)/I is a full and extensional T -valuedmodel.

Moreover if I is an ultrafilter, the map jI : V → Ult(V, δ)/I mapping a 7→ [ca]I iselementary (where ca : P (∅)→ V is constant with value a).

In the remainder of this chapter we will analyze two specific cases of the above results,i.e. those induced respectively by a normal ideal on X or by a tower of normal ideals I ofheight an inaccessible δ. In these two cases, much more can be said about the propertiesof the maps jI .

Our main focus will be on the cases in which the duals of I and⋃I are (towers of)

normal ultrafilters. However this assumption is rather strong: in most cases there cannotbe interesting normal ultrafilters on X in V , much for the same reasons for which therecannot be V -generic ultrafilters for an atomless boolean algebra B ∈ V . Hence we arenaturally led to analyze the situation in which we have a pair of models V ⊆ W of ZFC(or MK), and we have in W a (tower of) V -normal ultrafilter(s) GX on X (respectively onVδ for some δ inaccessible in V ).

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All over this chapter we have the following scenario: we have transitive classes V,M ⊆Wwhich are all models of ZFC and an elementary embedding j : V → M (i.e. it preservestruth of all formulae with no free variables and parameters in V ). To avoid ambiguities inthe intended meaning of the above, we always assume the following: W comes in pair withits family of classes C so that 〈W, C〉 is a model of NBG, j, V,M,W ∈ C, j is elementary justwith respect to formulae φ(~x) with just set-type bounded variables and set type parametersoccurring in them.

5.2 V -normal ultrafilters and towers

We now focus on pairs of transitive models V ⊆W of ZFC (or MK) and we consider (towersof) ultrafilters G in W on the boolean algebra P (P (X))V (respectively on the booleanalgebra T Vδ for some δ inaccessible in V ).

5.2.1 V -normal ultrafilters

Definition 5.2.1. Let V ⊆ W be transitive models of ZFC (or MK), X ∈ V and G ⊆P (P (X))V with G ∈W . G is a V -normal ultrafilter on X if it is a fine filter on X, andfor any regressive

f : P (X)V −→ X

belonging to V there exists S ∈ G such that f [S] = x0 for some x0 ∈ X.

Remark 5.2.2. We focus on transitive models to simplify the discussion, however our resultscan be formulated in order to work for any pair V ⊆W of models of ZFC (or MK).

Notation 5.2.3. Assume V ⊆ W are transitive models of ZFC (or MK) and let G ∈ Wbe a V -normal ultrafilter on X. Define in W Ult(V,G) = Ult(V,X)/G.

Theorem 5.2.4 ( Los). Assume V ⊆ W are transitive models of ZFC (or MK) and letG ∈W be a V -normal ultrafilter on X. Then

Ult(V,G) ϕ([f1], . . . , [fn]) ⇐⇒ Z ⊆ X : V ϕ(f1(Z), . . . , fn(Z)) ∈ G.

Proof. Ult(V,X) is a full P (P (X))V -model in V . This assertion is absolute betweentransitive structures (see 3.2.6), hence Ult(V,X) is a full P (P (X))V -model in W as well.Now we conclude applying Lemma 3.2.7 in W to G and Ult(V,X).

Definition 5.2.5. Given I ⊆ P (P (X)) normal fine ideal on X, PI is the boolean algebraP (P (X)) /I.

Remark 5.2.6. Given I ⊆ P (P (X)) normal fine ideal on X the map S 7→ [S]I induces acomplete embedding of the partial order (P (P (X)) \ I,≤I) (with S ≤I T if S \ T ∈ I)into (P (P (X)) /I)

+ with a dense image. Hence we feel free when convenient to confuseelements S of P (P (X)) with their corresponding class [S]I and PI wih the partial order(P (P (X)) \ I,≤I).

Moreover since I is normal, the diagonal union`Sx : x ∈ X of a subset Sx : x ∈ X

of P (P (X)) /I of size at most |X| defines its suprema in PI , hence PI is always an|X|-complete boolean algebra.

Proposition 5.2.7. Let in V , I ⊆ P (P (X)) be a normal fine ideal on X. Let G be aV -generic filter for P (P (X)) /I . Then G = S ⊆ P (X) : [S]I ∈ G ∈ V [G] is a V -normalultrafilter on X.

In particular Ult(V,G) is always a definable class in V [G] and, letting B = RO(P (P (I))),G ∈ V B a B-name for G:

82

V |= [S]I ≤qφ([f1]G, . . . , [fn]G)

yB

if and only if

V |= [S]I = [M ∈ S : V |= φ(f1(M), . . . , fn(M))]I .

Proof. Let f : P (X)V −→ X be a regressive function such that f ∈ V . Put

Df =S ∈ I+ : f S is constant

.

Df is open dense: let f : P (X)V −→ X be regressive. Given T ∈ I+, f T : T −→ Xis still regressive; by normality of I there exists T ′ ⊆ T such that f T ′ is constant andT ′ ∈ I+. T ′ ∈ Df refines T in ≤I . It is immediate to check that Df is open.

The second part of the proposition is an immediate consequence of its first part, of LosTheorem for Ult(V,G), and of the forcing Theorem for V B and for Ult(V,X). We leavethe details to the reader.

Remark 5.2.8. A V -normal ultrafilter G on P (P (X)) /I with I a normal ideal, captures afamily of open dense sets of size equal to the cardinality of the family of all the open densesubsets of P (P (X)) /I, however G may avoid some open dense sets and thus fail to befully V -generic for P (P (X)) /I.

Definition 5.2.9. Given x ∈ Vθ, define

ρx = 〈Z, πZ(x)〉 : x ∈ Z ⊆ Vθ .

Observe that if we want ρx to be total we can define

ρx = 〈Z, πZ [x ∩ Z]〉 : Z ∈ P (Vθ) .

If β ∈ θ

ρβ : P (Vθ) −→ θ

Z 7−→ otp(Z ∩ β).

The following proposition sums up the extra information we can extract from theembedding jG (defined in 5.1.3) in case G is a V -normal ultrafilter on P (P (X)).

Proposition 5.2.10. Let G ⊆ P (P (Vθ)) be a V -normal ultrafilter. Then:

1. ∀α ≤ θ ([ρα]G,∈G) ∼= (α,∈). Hence any α ∈ θ is represented by ρα in Ult(V,G).

2. There exists an isomorphism between (θ,∈) and an initial segment of OrdUlt(V,G).

3. For any x ∈ Vθ, (trcl(x),∈) ∼= (trcl([ρx]),∈G). This implies:

(a) any x ∈ Vθ is represented in Ult(V,G) by ρx;

(b) Vθ can be naturally identified with a subset of VUlt(V,G)θ .

4. For any x ∈ Vθ, jG[x] = 〈Z,Z ∩ x〉 : Z ⊆ Vθ.

83

5. Assume θ > λ and

Pλ (Vθ) = X ⊆ Vθ : X ∩ λ ∈ λ ≥ otp(X ∩ Vθ)

is in G. Then crit(jG) = λ and jG(λ) ≥ θ.

6. S ∈ G if and only if [IdP(X)]G ∈G jG(S).

Proof.

1. By induction on α. Assume the thesis holds for any β < α. Let [f ]G ∈G [ρα]G andput

T = Z ⊆ Vθ : f(Z) ∈ ρα = otp(Z ∩ α) .Then T belongs to G. For any Z ∈ T let βZ ∈ α∩Z be such that otp(Z ∩βZ) = f(Z)and define

g : T −→ θ

Z 7−→ βZ .

Since g is regressive and G is V -normal, we can find β < α pressing down constantfor g such that

Z ⊆ Vθ : f(Z) = ρβ(Z) = Z ⊆ Vθ : βZ = β ∈ G.

Therefore [f ]G = [ρβ ]G for some β < α. It is now easy to check (using Los Theoremand the inductive assumptions) that the map β 7→ [ρβ]G is order preserving andsurjective between (α,∈) and ([ρα]G,∈G). The thesis follows.

2. By Los Theorem Ult(V,G) |= [ρα]G ∈ Ord. This shows that there exists an isomor-phism between θ and an initial segment of the class of ordinals of the ultrapowerUlt(V,G).

3. In order to prove the thesis we need the following:

• For any x, y ∈ Vθx ∈ y ⇐⇒ [ρx]G ∈ [ρy]G :

x ∈ y if and only if πZ(x) ∈ πZ(y) for any (some) Z ⊆ Vθ such that x, y ∈ Z.Since G is fine, the thesis follows.

•[f ]G ∈ [ρx]G ⇐⇒ ∃y ∈ x [f ]G = [ρy]G :

Assume there exists y ∈ x such that [f ]G = [ρy]G, then [f ]G ∈ [ρx]G. On theother hand, assume [f ]G ∈ [ρx]G let

T = Z ⊆ Vθ : f(Z) ∈ ρx(Z) = πZ(x), x ∈ Z ∈ G.

For any Z ∈ T let yZ ∈ x ∩ Z be such that πZ(yZ) = f(Z) and define

g : T −→ Vθ

Z 7−→ yZ .

Since g ∈ V is regressive and G is V -normal, we have a pressing down constanty for g. This means that

Z ⊆ Vθ : f(Z) = ρy(Z) = Z ⊆ Vθ : yZ = y ∈ G.

Hence [ρy]G = [f ]G.

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4. Let Idx be such that Idx(Z) = Z ∩ x. On the one hand we have for any Z ⊆ Vθ

y ∈ x ⇐⇒ cy(Z) = y ∈ Idx(Z) = Z ∩ x.

Therefore for any y ∈ x, jG(y) = [cy]G ∈G [Idx]G. On the other hand if [f ]G ∈ [Idx]Gwe have that T = Z : f(Z) ∈ Z ∩ x ∈ G and f is regressive on T . Let y ∈ x be thepressing down constant for f , then [f ]G = [cy]G.

5. First of all observe that ∀β < λ Z ≺ Vθ : β ⊆ Z ∈ G. As a matter of fact, since Gis V -normal and fine, ∀β ∈ λ Aβ = Z ≺ Vθ : β ∈ Z ∈ G. Moreover Pλ (Vθ) ∈ G,hence Z ≺ Vθ : β ⊆ Z ⊇ Pλ (Vθ) ∩Aβ ∈ G.

Now we want to show that the critical point is at least λ: Let β < λ, then Bβ =Z ≺ Vθ : β ⊆ Z ∈ G, and for any Z ∈ Bβ, we have cβ(Z) = β and ρβ(Z) =πZ(β) = β, since πZ(β) = πZ [β ∩ Z] = πZ [β] = β. Hence [cβ]G = [ρβ]G for anyβ < λ.

We will now see that the critical point is λ: Let f(Z) = Z ∩ λ for any Z ∈ Pλ (Vθ).Then

Z : f(Z) < cλ(Z) ⊇ Pλ (Vθ)∩Z ∈ P (Vθ) : λ ∈ Z ∈ G ⇐⇒ [f ]G < [cλ]G = jG(λ).

On the other hand for any α < λ, [f ]G > [cα]G, since for any Z ∈ Z ⊆ Vθ : α+ 1 ⊆ Z ∈G, f(Z) > α = cα(Z).

6. The thesis follows from the following observations.

• jG[Vθ] = [IdP(Vθ)]G. In fact for any Z ⊆ Vθ and x ∈ Z:

x ∈ Vθ ⇐⇒ cx(Z) ∈ IdP(Vθ)(Z),

hence [cx] = jG(x) ∈G [IdP(Vθ)]G. For the other inclusion if [f ]G ∈ [IdP(Vθ)]G let

T = Z : f(Z) ∈ Z .

which belongs to G. Let x be the pressing down constant for f , then [f ]G = [cx]Gand we are done.

• Assume S ∈ G. Then

S = Z ⊆ Vθ : Z ∈ S =Z ⊆ Vθ : IdP(Vθ)(Z) ∈ cS(Z)

∈ G

if and only if

[IdP(Vθ)]G ∈G [cS ]G.

5.2.2 Towers of V -normal ultrafilters

Definition 5.2.11. Let V ⊆ W be transitive models of ZFC (or MK) and δ be in V astrong limit cardinal. G ∈W is a tower of V -normal ultrafilters of height δ if G ⊆ (Vδ)

V

and

• GX = G ∩ P (P (X))V is a V -normal ultrafilter for any X ∈ Vδ.

85

• GY projects to GX iff X ⊆ Y .

Recall the regular embedding iXY : P (P (X)) → P (P (Y )) mapping S ⊆ P (X) →S ↑ Y for any X ⊆ Y .

Proposition 5.2.12. Assume X ⊆ Y . Then i−1XY [H] is a V -normal ultrafilter on X,

whenever H is a V -normal ultrafilter on Y .

Proof. Let f : P (X) → X be regressive, then f ↑ Y is regressive. By normality thereexists a pressing down constant x0 ∈ X such that

T = Z ⊆ Y : f ↑ Y (Z) = x0 ∈ H.

Then iXY ([T ↓ X]) ≥ [T ], which means that T ↓ X ∈ i−1XY [H] and f is constant on it.

Definition 5.2.13. Let V ⊆ W be models of ZFC (or MK) and G ∈ W be a tower ofV -normal ultrafilter on Vδ.

• The induced ultrapower is:

Ult(V,G) = Ult(V, δ)/G = [f ]G : f ∈ Ult(V, δ) ,

• The associated embedding jG is:

jG : V −→ Ult(V,G)

a 7−→ [ca],

where

ca : ∅ −→ V

∅ 7−→ a.

• For any X,Y ∈ Vδ we define the factor maps

kXY : Ult(V,GX) −→ Ult(V,GY )

[f ]GX 7−→ [f ↑ Y ]GY

kX : Ult(V,GX) −→ Ult(V,G)

[f ]GX 7−→ [f ]G.

In analogy with what has been done in the previous subsection, we have the following:

Proposition 5.2.14. Assume I ⊆ Vδ is a tower of normal ideals of height an inaccessibleδ. Let G be V -generic for (T Iδ )+ ∼= (Vδ \ ∪I,≤I). Then G ∈ V [G] is a tower of V -normal

ultrafilters with GX ⊇ IX for any X ∈ Vδ.In particular Ult(V,G) is always a definable class in V [G] and, letting B = RO(T Iδ ),

G ∈ V B a B-name for G:

V |= [S]I ≤qφ([f1]G, . . . , [fn]G)

yB

if and only if

V |= [S]I = [M ∈ S : V |= φ(f1(M), . . . , fn(M))]I .

86

Proof. We prove that if G is V -generic for T Iδ , GX is a V -normal ultrafilter for any X ∈ Vδ(which is clearly disjoint from IX). Given a regressive function f : P (X)→ X, let

Df = [S]I : ∃x0 ∈ X ⊆ ∪S ∀Z ∈ S(f(Z ∩X) = x0) .

It is enough to show that Df is open dense, the conclusion will then follow easily.

Df is dense: Let [S]I > [0]I . Take Y = X ∪⋃S and consider the regressive function

g : S ↑ Y −→ X

Z 7−→ f(Z ∩X).

By normality of IY , let T ≤IY S ↑ Y , T ∈ I+Y such that g is constant on T . Then

[T ]I ≤ [S]I = [S ↑ Y ]I and [T ]I ∈ Df .

Df is open. If T ∈ Df and S ≤I T let X =⋃S ∪

⋃T , then S ↑ X ≤IX T ↑ X and

[S]I = [S ↑ X]I . Now for all Z ∈ S ↑ X, Z ∈ T ↑ X, therefore f(Z ∩X) = x0 whichgives that [S]I = [S ↑ X]I ∈ Df .

The second part of the proposition is left to the reader.

Notation 5.2.15. Given x ∈ Vα we define

ρVαx : P (Vα) −→ Vα

Z 7−→ πZ(x).

Proposition 5.2.16. Let V ⊆W be models of ZFC (with W a model of NBG). Let G ⊆ Vδbe a tower of V -normal ultrafilters with G ∈W .

1. jG is definable in W with parameters G and Vδ (provided V is a (definable) class inW ).

2. Ult(V,G) |= φ([f0]G, . . . , [fn−1]G) if and only if there exists α < δ such that dom(fi) ∈Vα for any i ∈ n and

M ≺ Vα : V |= φ(f0 ↑ Vα(M), . . . , fn−1 ↑ Vα(M)) ∈ G.

3. For any x ∈ Vδ, x is represented by [ρVαx ]G in the ultrapower, where α < δ is suchthat x ∈ Vα.

4. Assume δ > θ > λ and

Pλ (Vθ) = X ⊆ Vθ : X ∩ λ ∈ λ ≥ otp(X ∩ Vθ)

is in G. Then crit(jG) = λ and jG(λ) ≥ θ.

5. T ∈ G if and only if jG[⋃T ] ∈ jG(T ).

6. crit(kVα) ≥ α and kX = kY kXY for any X ⊆ Y ∈ Vδ.

V

Ult(V,GX) Ult(V,GY )

Ult(V,G)

jGX

jGY

jG

kX

kXY

kY

87

Aggiungeredimostrazionedell’ultimopunto – M

Proof. The proof of all items except the last one is a straightforward variation of the casefor V -normal ultrafilters and is left as an instructive exercise to the reader.

– M

Notation 5.2.17. In the following we will refer to forcings of type PI with I a normalideal as ideal forcings, while forcings of type T Iδ will be referred to as tower forcings.

5.2.3 Generic elementary embeddings versus (towers of) V -normal ul-trafilters

We now show that generic elementary embeddings j : V →M between transitive models ofZFC which are classes of some NBG-model W ⊇ V,M and (towers of) V -normal ultrafiltersare two sides of the same coin: from one we can define the other and conversely.

Theorem 5.2.18. Assume W is a model of NBG and j, V,M are classes in W with V,Mtransitive models of ZFC and j : V →M elementary. For any X ∈ V such that j[X] ∈M ,

let GX,j =S ∈ P (P (X))V : j[X] ∈ j(S)

. The following holds:

1. j Ord is not the identity map, hence it has a least ordinal moved, its critical pointcrit(j), which is a regular cardinal in V . Moreover j HV

crit(j) is the identity.

2. If M ⊆ V , the critical point of j is a strongly inaccessible cardinal in V .

3. GX,j ∈W is a V -normal ultrafilter on X.

4. GX,j projects to GY,j for all Y ∈ P (X)V .

5. Let kGX,j : Ult(V,GX,j)→M be defined by [f ]GX,j 7→ j(f)(j[X]). Then:

(a) Ult(V,GX,j) is well founded, kGX,j is elementary, and, assuming X is transitive,crit(kGX,j ) ≥ rank(X);

(b) kGX,j jGX,j : x 7→ [cx]GX,j 7→ j(cx)(j[X]) is j;

(c) for Y ∈ P (X)V , kGY,jGX,j : [f ]GY,j 7→ [f ↑ X]GX,j is elementary and such thatkGY,j = kGX,j kGY,jGX,j .

6. Assume G ∈W is a V -normal ultrafilter on some X ∈ V with Ult(V,G) well-founded,and identify Ult(V,G) with its transitive collapse. Then GX,jG = G. Moreover1

Ult(V,G) =jG(f)(jG[X]) : f ∈ V, f : P (X)V → V

jGX,j = jG and kGX,j is the identity map.

7. Assume G = GX : X ∈ Vδ ∈ W is a tower V -normal ultrafilter on Vδ such thatUlt(V,∪G) is well-founded, and identify Ult(V,∪G) with its transitive collapse. ThenGX,jG = GX for all X ∈ Vδ. Moreover

Ult(V,∪G) =jG(f)(jG [X]) : f ∈ V, X ∈ Vδ, f : P (X)V → V

.

1More precisely: the left-hand class and the right-hand class of the equation have the same transitivecollapse.

88

V

Ult(V,GX,j) Ult(V,GY,j)

M

jGX,j

jGY,j

j

kGY,j

kXY,j

kGX,j

Remark 5.2.19. In view of 5.2.18(6), whenever Ult(V,G) is a well founded ultrapowerinduced by a V -normal ultrafilter G on some set X, we will identify Ult(V,G) with itstransitive collapse and freely represent its elements by [f ]G or jG(f)(jG[X]) according towhat is more convenient. Similarly we will handle ultrapowers given by towers of normalideals according to 5.2.18(7).

Proof.

1. If j is not the identity map, let a ∈ V be of least rank such that j(a) 6= a. Let Rbe a well order of a and let predR(b) denote the set of R-predecessors of b for eachb ∈ a. Since rank(b) < rank(a) for all b ∈ a, we get that

j[A] = j(〈u, v〉) : 〈u, v〉 ∈ A = 〈j(u), j(v)〉 : 〈u, v〉 ∈ A = 〈u, v〉 : 〈u, v〉 ∈ A = A

and similarly j[X] = X for any A ⊆ a2 and X ⊆ a.

Let b ∈ a be least such that (predR(b), R) 6= (predj(R)(b), j(R)) if such a b exists, and

X = predR(b), otherwise let X = a. In both cases we obtain that S = R∩X2 is a well-order on X such that j(S) 6= S, j(X) 6= X, and (predS(c), S) = (predj(S)(c), j(S))for all c ∈ X. Hence (X,S) is a proper initial segment of (j(X), j(S)). Now lettingα be the order type of (X,S) and β be the order type of (j(X), j(S)), we have thatα 6= β and β = j(α), by elementarity of j. Hence j has a critical point γ.

Assume γ = crit(j) is not regular in V . Let f : α → γ be a cofinal sequence in Vwith α < γ (i.e. sup(ran(f)) = γ). Then in M it holds that j(f) : j(α) = α→ j(γ)is also cofinal in j(γ) (i.e. M |= sup(ran(j(f))) = j(γ)). Since M ⊆ W are bothtransitive, sup(ran(j(f))) = j(γ) holds in W . Now for all ξ < α,

j(f)(ξ) = j(f)(j(ξ)) = j(f(ξ)) = f(ξ)

(since ξ, f(ξ) < γ = crit(j)), hence j(f) = f . This gives that

j(γ) = sup(ran(j(f))) = sup(ran(f)) = γ,

a contradiction.

Finally let a ∈ HVγ , we must show j(a) = a. trcl(a) can be coded by an element

f ∈ 2α for some limit α < γ. Notice that rank(f) ≤ α < γ. Hence it suffices to showthat j(f) = f for all f ∈ 2α and for all limit α < γ. In this case j(f) ∈ j(2α) =j(2)j(α) = 2α and

j(f) ⊇ j[f ] = j(〈η, i〉) : 〈η, i〉 ∈ f = 〈j(η), j(i)〉 : 〈η, i〉 ∈ f = 〈η, i〉 : 〈η, i〉 ∈ f = f,

since all pairs in f have as components ordinals less than crit(j). Hence j(f) ∈ 2α isa function extending f ∈ 2α, which occurs only if j(f) = f .

89

2. Assume M ⊆ V and let γ = crit(j).

First assume γ is not a limit cardinal. Then γ = ν+ with j(ν) = ν (since γ is regularby the previous item). Since ν+ = γ < j(γ) = (ν+)M we get that M |= |γ| ≤ ν.Since M ⊆ V , we get that γ is not a cardinal in V , contradicting the assumptionthat γ is regular in V .

Finally assume γ is not strong limit. Let α < γ be least such that |2α| ≥ γ. Let2α = fξ : ξ < η with η ≥ γ being the size of 2α in V . Let R ⊆ (2α)2 be the wellorder on 2α induced by the map 〈fξ : ξ < η〉. Then j(R) is a well order of 2α in M .Since M ⊆ V , j(R) ∈ V . Observe also that j(〈f, g〉) = 〈f, g〉 for all 〈f, g〉 ∈ R, sinceall such pairs have rank less than γ. Hence R = j[R] ⊆ j(R). But a total linearorder on 2α cannot have any proper super relation on 2α which is still a total linearorder. Since j(R), R are both total linear orders on 2α in V , they must coincide. Now〈fξ : ξ < η〉 is the enumeration of 2α in order type η according to R, and we havethat also M models that j(〈fξ : ξ < η〉) is the unique enumeration of j(2α) = 2α inorder type j(η) according to j(R). The unique such enumeration being 〈fξ : ξ < η〉(since R = j(R)), we have that j(〈fξ : ξ < η〉) = 〈fξ : ξ < η〉. But now j(fγ) is (byelementarity of j) the j(γ)th-element of 2α according to the well order j(R) of j(2α),i.e. j(fγ) = fj(γ) 6= fγ . On the other hand observe that j(f) = f for all f ∈ 2α sincerank(f) < γ for any such f . We reached a contradiction.

3. Clearly j[X] ∈ j(S) or j[X] ∈ j(P (X)\S), hence GX is ultra. Assume f : P (X)V →X is regressive and in V . Then j(f) : P (j(X))M → j(X) is regressive and inM . Hence j(f)(j[X]) = j(z) for some z ∈ X. By elementarity, we get thatZ ∈ P (X)V : f(Z) = z

∈ GX , since j[X] ∈ j(

Z ∈ P (X)V : f(Z) = z

). Hence

GX,j is a V -normal ultrafilter.

4. S ∈ GY if and only if j[Y ] = j(Y ) ∩X ∈ j(S) if and only if j[X] ∈ j(S) ↑ j(X) =j(S ↑ X) if and only if S ↑ X ∈ GX .

5.

(a) We have that:Ult(V,GX,j) |= φ([f1]GX,j , . . . , [fn]GX,j )

if and only if

S =Z ∈ P (X)V : V |= φ(f1(Z), . . . , fn(Z))

∈ GX,j

if and only if

j[X] ∈ j(S) =Z ∈ P (j(X))M : M |= φ(j(f1)(Z), . . . , j(fn)(Z))

if and only if

M |= φ(j(f1)(j[X]), . . . , j(fn)(j[X])).

Hence kX is elementary.

Ult(V,GX) is well-founded since kX is ∈X /G-preserving and identifies Ult(V,GX)with a sub-class of a well-founded class.

It remains to show that kX(α) = α for any α < rank(X). W.l.o.g. we can assumeX = β is an ordinal in V . By 5.2.10(1) α = [ρα]GX,j for any α < β. Hence

kX(α) = kX([ρα]GX,j ) = j(ρα)(j[X]) = otp(j[X]∩j(α)) = otp(j[X∩α]) = otp(X∩α) = α,

as was to be shown.

90

(b) We have that

kX kGX,j (x) = kX([cx]GX,j ) = j(cx)(j[X]) = cj(x)(j[X]) = j(x).

(c) Left to the reader (as in 5.2.16(6)).

6. S ∈ GX,jG if and only if jG[X] ∈ jG(S) if and only if S ∈ G. The rest is left to thereader.

7. Left to the reader.

In the remainder of this book we will focus just on generic ultrapower embeddingswhich define well founded ultrapowers, this notion therefore deserves a definition:

Definition 5.2.20. Let I ∈ V be a normal ideal on X. I is precipitous if PI has the follow-ing property: whenever H is V -generic for all PI the generic ultrapower Ult(V,H) definedin V [H] is well founded in V [H]. Equivalently: there is a ∆1-preserving homomorphism

between the boolean valued models (Ult(V,X)/I ,∈X /I ,=X /I) and (V P

I,∈PI ,=PI ).

Similarly we define I being a precipitous tower.

5.3 Normal (towers of) ultrafilters in forcing extensions

Definition 5.3.1. Let B be a complete boolean algebra and H ∈ V B be such that

rH is a V -normal ultrafilter on P (X)

z

B= 1B.

SetI(H) =

S ⊆ P (X) :

rS ∈ H

z

B= 0B

and

iH : PI(H) −→ B

S 7−→rS ∈ H

z.

Proposition 5.3.2. Let B be a complete boolean algebra and H ∈ V B be such that

rH is a V -normal ultrafilter on X

z

B= 1B.


1. I(H) is a normal ideal.

2. iH is an homomorphism between boolean algebras which is |X|-complete.

3. Moreover if the range of iH is dense in B, iH extends to an isomorphism of therespective boolean completions.

Proof. A straightforward simplification of the proof of Proposition 5.3.4 to follow.

Similarly

91

Definition 5.3.3. Let B be a complete boolean algebra and H ∈ V B be such that

rH is a V -normal tower of ultrafilters on Vδ

z

B= 1B.

SetIX(H) =

S ⊆ P (X) :

rS ∈ H

z

B= 0B

,

I(H) =IX(H) : X ∈ Vδ

,

iH : T I(H)δ −→ B

S 7−→rS ∈ H

z.

Proposition 5.3.4. Let B be a complete boolean algebra and H ∈ V B be such that

rH is a V -normal tower of ultrafilters on Vδ

z

B= 1B.


1. IX(H) is a normal ideal for all X ∈ Vδ.

2. If X ⊆ Y , then IY (H) projects on IX(H).

3. The map iH is an homomorphism between boolean algebras which is < cof(δ)-complete.

4. Moreover if the range of iH is dense in B, iH extends to an isomorphism of therespective boolean completions.

Proof.

1. IX(H) is an ideal. IfrS ∈ H

z= 0 and T ≤ S are stationary, then

rT ∈ H

z= 0.

Moreover ifrS ∪ T ∈ H

z= b > 0, let G V -generic be such that b ∈ G. If

H = valG(H), then S ∪ T ∈ H implies either S ∈ H or T ∈ H. Hence

rS ∈ H

z> 0 or

rT ∈ H

z> 0.

IX(H) is fine.rCx ∈ H

z= 1B for all x ∈ X, since H is fine with boolean value 1B.

IX(H) is normal. Let T /∈ IX(H) and f : T → X regressive. SincerT ∈ H

z> 0B,

let G be V -generic such that T ∈ H = valG(H). Then there exists x ∈ X such

that S = Z ∈ T : f(Z) = x ∈ H. This means thatrS ⊆ T ∧ S ∈ H

z> 0B.

Hence S ∈ IX(H)+. Now S ⊆ T iff JS ⊆ T K > 0B. Hence S ≤IX T witnessesthe pressing down property for f with respect to T .

2. Since H = valG(H) is V -normal for all V -generic filters G:

S ∈ IX ⇐⇒rS ∈ H

z= 0B ⇐⇒

rS ↑ Y ∈ H

z= 0B ⇐⇒ S ↑ Y ∈ IY .

92

3. iH is a homomorphism.

iH(¬S) =r¬S ∈ H

z= ¬

rS ∈ H

z= ¬iH(S).

iH(S ∨ T ) =rS ∨ T ∈ H

z=

rS ∈ H

z∨

rT ∈ H

z= iH(S) ∨ iH(T );

iH is <δ-complete. It is enough to show that for all Si : i < ξ ∈ Vδ∨B

rSi ∈ H

z: i < ξ

=

rhSi : i < ξ ∈ H

z.

Let G be V -generic withr`Si : i < ξ ∈ H

z∈ G. Then H = valG(H) is

V -normal with S =`Si : i < ξ ∈ H. Let in V f : S → ξ be given by

f(Z) = iZ for some iZ ∈ ξ ∩Z. Then f ∈ V is regressive on S ∈ H, hence thereexists i < ξ such that Ti = f−1[i] ∈ H. Now Si ⊇ Ti ↓ ∪Si ∈ H, giving thatrSi ∈ H

z∈ G. Since this occurs for all V -generic filters G

∨B

rSi ∈ H

z: i < ξ

=

rhSi : i ∈ ξ ∈ H

z.

4. Immediate, hence left to the reader.

5.4 Large cardinals defined by means of normal ultrapowers

We briefly recall the large cardinal notions we will use in the remainder of this book.

Definition 5.4.1.

• κ is Vλ-supercompact in V if there is a V -normal fine ultrafilter G ∈ V on λconcentrating on

Pκ (λ) = X ⊆ Vλ : X ∩ κ ∈ κ > |X| and for some α πX [X] = Vα .

• κ is supercompact in V if it is Vλ-supercompact in V for all λ.

• κ is huge in V if for some strongly inaccessible λ > κ there is a V -normal fineultrafilter G ∈ V on Vλ concentrating on

X ⊆ Vλ : πX [X] = Vκ and X ∩ κ ∈ κ = otp(X ∩ λ) .

• κ is 2-huge in V if for some strongly inaccessible cardinals λ > δ larger than κ thereis a V -normal fine ultrafilter G ∈ V on Vλ concentrating on

X ⊆ Vλ : X ∩ κ ∈ κ = otp(X ∩ δ) and πX [X] = Vδ .

• κ is 2-superhuge in V if for all η there exist strongly inaccessible cardinals λ > δabove η and a V -normal fine ultrafilter G ∈ V on Vλ concentrating on

X ⊆ Vλ : πX [X] = Vδ and X ∩ κ ∈ κ = otp(X ∩ δ) .

93

Theorem 5.4.2. The following holds for a cardinal κ:

1. For all strong limit cardinals λ κ is Vλ-supercompact if and only if there there existsj : V →M elementary such that M ⊆ V , crit(j) = κ and Mλ ⊆M .

2. κ is huge if and only if there there exists j : V →M elementary such that M ⊆ V ,crit(j) = κ and M j(κ) ⊆M . Moreover the hugeness of κ is witnessed by a V -normalfine ultrafilter G ∈ V on Vj(κ) concentrating on

X ⊆ Vj(κ) : πX [X] = Vκ and X ∩ κ ∈ κ = otp(X).

3. κ is 2-huge if and only if there there exists j : V →M elementary such that M ⊆ V ,crit(j) = κ and M j2(κ) ⊆ M . Moreover the 2-hugeness of κ is witnessed by aV -normal fine ultrafilter G ∈ V on j2(κ) concentrating on

X ⊆ Vj2(κ) : X ∩ κ ∈ κ = otp(X ∩ j(κ)) and πX [X] = Vj(κ)

.

4. Assume there exists j : Vλ → Vλ elementary such that j[Vλ] ∈ V with crit(j) = κ.Then

Vλ |= κ is 2-superhuge.

Proof.

1. Follows the same lines of the second item.

2. First assume there exists j : V →M elementary such that M ⊆ V , crit(j) = κ andM j(κ) ⊆ M . Standard arguments (i.e. 5.2.18(2) and the closure properties of M)yield that κ, j(κ) are both strongly inaccessible in V . This gives that j[Vj(κ)] ∈M .

Now let G =S ⊆ P

(Vj(κ)

): j[Vj(κ)] ∈ j(S)

. We leave to the reader to check

that G ∈ V is a V -normal ultrafilter on Vj(κ). Observe that j−1 implements anisomorphism of j[Vj(κ)] with Vj(κ), hence it is the Mostowski collapsing map ofj[Vj(κ)]. Therefore

S =X ⊆ Vj(κ) : X ∩ κ ∈ κ and πX [X] = Vκ

∈ G,

since j[Vj(κ)] ∈ j(S).

Next assume that G ∈ V is a V -normal fine ultrafilter on λ concentrating on

X ⊆ Vλ : X ∩ κ ∈ κ and πX [X] = Vκ

for some strongly inaccessible λ > κ. Then jG : V → Ult(V,G) has critical point κand jG(κ) ≥ λ by Proposition 5.2.10(5).

Now let fi : i ∈ Vλ ∈ Ult(V, κ). Consider the function g : P (Vλ) → V given byg(X) = fi(X) : i ∈ X. By fineness of G, Ult(V,G) models that [fi]G ∈Vλ /G [g]Gfor all i ∈ Vλ. By normality of G, if [h]G ∈Vλ /G [g]G, there exists i ∈ Vλ such that[h]G = [fi]G. Hence [g]G has as extension the family [fi]G : i ∈ Vλ in Ult(V,G)with respect to ∈Vλ /G.

Since G ∈ V , Ult(V,G) is a definable class in V ; hence the above shows that:

• ∈Vλ /G is a well founded extensional relation on Ult(V,G),

94

• letting M be the transitive collapse of Ult(V,G) with respect to ∈Vλ /G, wehave that Mλ ⊆M .

To conclude we just need to show (modulo the identification of Ult(V,G) with M)that jG(κ) = λ.

Observe that |Pκ (λ)|κ = λ, since λ is strongly inaccessible, hence

|jG(κ)| = |[f ]G : f : Pκ (λ)→ κ| ≤ λ.

We conclude that jG(κ) ≥ λ is an ordinal of size λ in V , hence its cofinality is atmost λ in V . But any sequence of ordinals in V cofinal in jG(κ) of order type atmost λ is in M , since Mλ ⊆M . Moreover M models that jG(κ) is a regular cardinal,we conclude that jG(κ) = λ.

3. Follows the same lines of the second item.

4. We get that jn : Vλ → Vλ is elementary with critical point the strongly inaccessibleκ for all n and also that λ = supn∈ω j

n(κ), hence item (3) applies to each η < κ tofind that j2n witnesses that κ is 2-huge as witnessed by a V -normal fine ultrafilterG ∈ V on

X ⊆ j2n(κ) : X ∩ κ ∈ κ = otp(X ∩ jn(κ)) and πX [X] = Vjn(κ)

with j2n(κ) > jn(κ) > η.

We also need the following characterizations of supercompactness, hugeness and 2-hugeness which is essentially due to Magidor:

Theorem 5.4.3 (Magidor [32]). The following holds:

• κ is supercompact if and only if for all α

M ≺ Vα : M ∩ κ ∈ κ and (M,∈) ∼= (Vη,∈) for some η

is stationary.

• κ is superhuge if and only if for all α exists δ > α such that

M ≺ Vδ : M ∩ κ ∈ κ and (M,∈) ∼= (Vκ,∈)

is stationary.

• κ is 2-superhuge if and only if for all α > κ there are λ > δ > α such that

M ≺ Vλ : M ∩ κ ∈ κ, (M ∩ Vδ,∈) ∼= (Vκ,∈) and (M,∈) ∼= (Vδ,∈)

is stationary.

Remark 5.4.4. Later in this book we will argue that certain elementary maps jG : V →Ult(V,G) defined in a forcing extension V [H] (with H V -generic for some B) of V aregenerically almost-huge (i.e. such that Ult(V,G)<j(crit(j)) ⊆ Ult(V,G) holds in V [H]). Theproofs will be much more involved then the ones occurring in the proof of 5.4.2. Themain issue being the following: assume [fα]G : α < λ ∈ V [H] is a subset of Ult(V,G),

95

we need to argue that this family is the extension in Ult(V,G) of some [g]G ∈ Ult(V,G).We encounter the following problem: most likely fα : α < λ 6∈ V . In this case we canjust argue that there is a family τα : α < λ ∈ V of B-names such that fα = (τα)G foreach α < λ. By the forcing theorem for each α < λ there is some pα ∈ H such that pαforces that fα = τα. However in order to run in V [H] the argument we sketched in theproof of 5.4.2 that [fα]G : α < λ is the extension of [g]G, we need that in H there is aunique p which decides simultaneously for all α that fα = τα. This is a very strong requestwhich is satisfied only in very specific circumstances. We will outline several occasions inwhich this scenario occurs.

96

Part III

Iterations of forcing notions

97

We develop a general theory of iterated forcing in the framework of boolean algebras,fully exploiting all the results on these type of objects gathered in Chapter 1. Chapter 6deals with two-steps iterations, while Chapter 7 deals with iterations of limit length. Hereand in chapter 10 we develop on Donder and Fuchs approach to iterated forcing [22]. Inchapter 6 we also introduce category forcings (i.e. any class forcing whose conditions areset-sized forcing notions and which is ordered by (a subfamily of) the complete embeddingsexisting between its conditions). This concept will gain more and more importance in thesequel of the book, and will become the central topic of the last part of the book.

98

Chapter 6

Two-steps iterations

Complete homomorphisms are the boolean algebraic counterpart of two-steps iterations,this will be spelled out in details in this chapter.

Section 6.1 develops the basic theory of two steps iterations both from the point ofview of forcing (i.e. viewing a two-steps iteration as a poset of type P ∗ Q with Q aV P -name for a poset, which is the usual approach at least in the set theory community),and from an algebraic point of view (i.e. viewing a two-steps iteration as a completeinjective homomorphism i : B→ C). We give both approaches and show their equivalence.We also show how to treat algebraically three steps iterations. In the last two sections ofthe chapter we deal with the basic properties of classes of complete boolean algebras Γclosed under two-steps iterations (section 6.2), we also study (section 6.3) the basic featuresof class forcings whose conditions are (certain classes) of complete boolean algebras Γordered by certain classes of complete homomorphisms →Θ between them. We also startto outline how the logical properties which define Γ and →Θ affects the combinatorialproperties of the class forcing (Γ,→Θ). The analysis of this type of class forcings will gainmore and more importance in the sequel of the book, and the last part of the book will beentirely devoted to this topic.

Our analysis is based on the properties of adjoint pairs collected in Thm. 1.4.1.

6.1 Two-steps iterations and generic quotients

We start outlining the relation existing between V -generic extensions for B and C incase there is a complete homomorphism i : B→ C.

Lemma 6.1.1. Assume M |= ZFC is transitive and let i : B→ C be a complete homomor-phism in M , D ⊂ B, E ⊂ C be predense sets also belonging to M . Then i[D] and πi[E]are predense (i.e. predense subsets are mapped into predense subsets). Moreover πi mapsM -generic filters in M -generic filters.

Proof. First, let c ∈ C be arbitrary. Since D is predense, there exists d ∈ D such thatd∧π(c) > 0. By Property 1.4.1.?? relativized to M , also i(d)∧c > 0 hence i[D] is predense.Finally, let b ∈ B be arbitrary. Since E is predense, there exists e ∈ E such that e∧ i(b) > 0.By Property 1.4.1.?? also πi(e) ∧ b > 0 hence πi[E] is predense.

We leave to the reader to check that πi[G] is an ultrafilter for any G ∈ St(C). Nowlet D be a predense subset of B in M and assume G is M -generic for C. We have thati[D] is predense, hence i[D] ∩ G 6= ∅ by the M -genericity of G. Fix c ∈ i[D] ∩ G, thenπi(c) ∈ D ∩ πi[G] concluding the proof.

99

Lemma 6.1.2. Let i : B → C be an homomorphism of boolean algebras. Then i is acomplete homomorphism if and only if for every V -generic filter G for C, i−1[G] is aV -generic filter for B.

Proof. If i is a complete homomorphism and G is a V -generic filter, then i−1[G] is trivially afilter. Furthermore, given D dense subset of B, i[D] is predense so there exists a c ∈ G∩i[D],hence i−1(c) ∈ i−1[G] ∩D.

Conversely, suppose that i is not complete, i.e. that there exists an A ⊆ B such thati(∨A) 6=

∨i[A] (in particular, necessarily i(

∨A) >

∨i[A]). Let d = i(

∨A) \

∨i[A], G

be a V -generic filter with d ∈ G. Then i−1[G] ∩ A = ∅, hence it is not V -generic belowi−1(d) =

∨A ∈ i−1[G], a contradiction.

The remainder of the section is organized as follows:

• In its second part we define two-steps iterations of the form B ∗ Q, where Q is aB-name for a complete boolean algebra, following Jech [28, chapter 16]. We studythe basic properties of the natural injective complete homomorphism of B into B ∗ Q.

• In its third part we study the properties of the generic quotients C/i[G] (wherei : B → C is a complete homomorphism in V and G is V -generic for B). We showthat if we have a commutative diagram of complete homomorphisms in V

B C0

C1

i0

i1j

and G is a V -generic filter for B, the map defined by j/G([c]i0[G]) = [j(c)]i1[G] is acomplete homomorphism in V [G]. We also show a converse of this property.

• In its fourth part we show that the two approaches are equivalent in the sense thati : B → C is a complete homomorphism iff C is isomorphic to B ∗ (C/i[GB]) and weprove a converse of the above factorization property when we start from B-names forcomplete homomorphisms k : C→ D.

6.1.1 Two-steps iterations

We present two-steps iterations following [28].

Definition 6.1.3. Let B be a complete boolean algebra, and C be a B-name for a completeboolean algebra. We denote by B ∗ C the boolean algebra defined in V whose elements are

the equivalence classes of B-names for elements of C (i.e. a ∈ V B such thatra ∈ C

z

B= 1B)

modulo the equivalence relation:

a ≈ b ⇔ra = b

z

B= 1,

with the following operations:

[d]≈ ∨B∗C [e]≈ = [f ]≈ ⇐⇒rd ∨C e = f

z

B= 1B;

¬B∗C[d]≈ = [e]≈

for any e such thatre = ¬Cd

z

B= 1B.

100

Literally speaking our definition of B ∗ C yields an object whose domain is a family ofproper classes of B-names. By means of Scott’s trick we can arrange so that B ∗ C is indeeda set. We leave the details to the reader. We denote elements of B ∗ C just by [b] omittingthe subscript ≈ if no confusion arises.

Lemma 6.1.4. Let B be a complete boolean algebra, and C be a B-name for a completeboolean algebra. Then B ∗ C is a complete boolean algebra and the maps iB∗C, πB∗C definedas

iB∗C : B → B ∗ Cb 7→ [db]≈

πB∗C : B ∗ C → B[c]≈ 7→ Jc > 0KB

where db ∈ V B is a B-name for an element of C such thatrdb = 1C

z

B= b and

rdb = 0C

z

B=

¬b, are an injective complete homomorphism with its associated adjoint.

Proof. We leave to the reader to verify that B ∗ C is a boolean algebra. We can also checkthat

[c] ≤ [a] ⇐⇒ Jc ∨ a = aK = 1B ⇐⇒ Jc ∧ a = cK = 1B ⇐⇒ Jc ≤ aK = 1B.

Observe that B ∗ C is also complete: if [dα] : α < δ ⊆ B ∗ C, let c be such thatrc =

∨dξ : ξ < δ

z= 1B. Then [c] ≥

∨[dξ] : ξ < δ

since for all α < δ

r∨dξ : ξ < δ

≥ dα

z= 1B.

Moreover if ra ≥ dα

z

B= 1B,

for all α < δ, then ∧ra ≥ dα

z

B: α < δ

= 1B.

We conclude that ra ≥

∨dξ : ξ < δ

z

B= 1B,

hence [a] ≥ [c], which gives that [c] =∨

[dα] : α < δ

.

Now we prove that iB∗C is an injective complete homomorphism and that πB∗C is itsassociated adjoint map.

• First of all a standard application of the mixing lemma to the maximal antichainb,¬b and the family of B-names 1C, 0C shows that for each b ∈ B there exists a

[db] ∈ B∗ C such thatrdb = 1

z= b and

rdb = 0

z= ¬b. Therefore iB∗C is well-defined.

• iB∗C preserves negation. Observe that ¬[db] = [d¬b]. In fact we have that

r(¬Cdb) = 1C

z=

rdb = 0C

z= ¬b

and similarly r(¬db) = 0

z=

rdb = 1

z= b;

therefore [¬db] = [d¬b]. We conclude that

iB∗C(¬b) = [d¬b] = ¬[db] = ¬iB∗C(b).

101

• iB∗C preserves joins. Consider bα ∈ B : α < δ. We have that

r∨dbα : α < ξ

= 0

z=∧α<ξ

rdbα = 0

z=∧α<ξ

¬bα = ¬(∨α<ξ

bα).

We have alsor∨

dbα : α < ξ

= 1z≤

r∨dbα : α < ξ

> 0

z=∨α<ξ

rdbα > 0

z

=∨α<ξ

rdbα = 1

z≤

r∨dbα : α < ξ

= 1

z;

hencer∨

dbα : α < ξ

= 1z

=∨α<ξ

rdbα = 1

z=∨α<ξ bα. Therefore

iB∗C

(∨bα

)=[∨

dbα : α < ξ]

=∨α<ξ

[iB∗C(bα)

].

• iB∗C is injective. If iB∗C(b) = iB∗C(b′) = [d], then b′ =rd = 1

z= b.

• We have to show that πiB∗C([c]) =qc > 0

y: by applying the definition of adjoint map

associated to iB∗C,

πiB∗C([c]) =∧b ∈ B : iB∗C(b) ≥ [c].

If b is such that iB∗C(b) ≥ [c], thenrdb ≥ c

z= 1 and we obtain

b =rdb = 1

z=

rdb > 0

z≥

qc > 0

y∧

rdb ≥ c

z=

qc > 0

y;

this gives the first inequality

πiB∗C([c]) ≥qc > 0

y.

In order to obtain the other one, letqc > 0

y= e, so that iB∗C(

qc > 0

y) = [de]. Then

on the one hand:

¬qc = 0

y=

qc > 0

y=

rde = 1

z≤

rc ≤ de

z.

On the other hand: qc = 0

y≤

rc ≤ de

z.

In particular since ¬qc = 0

y∨

qc = 0

y= 1B we get that

rc ≤ de

z= 1B,

and thus that [c] ≤ [de] = iB∗C(qc > 0

y), giving that

πiB∗C([c]) ≤qc > 0

y

as was to be shown.

102

When clear from the context, we feel free to omit the subscripts in iB∗C, πB∗C.

Remark 6.1.5. This definition is provably equivalent to Kunen’s two-steps iteration ofposets, i.e. RO(P ∗ Q) (as defined in [29]) is isomorphic to RO(P )∗RO(Q) as defined above.

We need in several occasions the following fact:

Fact 6.1.6. A = [cα]≈ : α ∈ λ is a maximal antichain in D = B ∗ C, if and only ifrcα : α ∈ λ is a maximal antichain in C

z= 1.

Proof. It is sufficient to observe the following:qcα ∧ cβ = 0

y= 1 ⇐⇒ [cα]≈ ∧ [cβ]≈ =

[0]≈ ;

t∨α<λ

cα = 1

|

= 1 ⇐⇒∨α<λ

[cα]≈ =

[∨α<λ

cα

]≈

=[1]≈ .

6.1.2 Generic quotients

We now outline the definition and properties of generic quotients.

Proposition 6.1.7. Let i : B → C be an injective complete homomorphism of completeboolean algebras and G be a V -generic filter for B. Then C/i[G], defined with abuse ofnotation as the quotient of C with the ideal dual to the filter generated by i[G], is a booleanalgebra in V [G].

Proof. Immediate, hence left to the reader.

Lemma 6.1.8. Let i : B→ C be an injective complete homomorphism, GB be the canonicalname for a generic filter for B, Q be a B-name for the boolean algebra C/i[GB]. Fix d B-name

for an element of Q. Then there exists a unique c ∈ C such thatrd = [c]i[GB]

z= 1B.

Proof. First, notice that the B-name for the dual of the filter generated by i[G] is I =〈c,¬πi(c)〉 : c ∈ C.

Uniqueness. Suppose that c0, c1 are such thatrd = [ck]I

z= 1B for k < 2. Then

q[c0]I = [c1]I

y= 1B, hence

rc04c1 ∈ I

z= ¬πi(c04c1) = 1B. This implies that

πi(c04c1) = 0B ⇒ c04c1 = 0C ⇒ c0 = c1.

Existence. Let A ⊂ B be a maximal antichain among the elements deciding that d =

[c]I for some c ∈ C. For every a ∈ A let ca be such that a ≤rd = [ca]I

z. Let∨

i(a) ∧ ca : a ∈ A = c ∈ C. Then

q[c]I = [ca]I

y=

rc4ca ∈ I

z= ¬πi(c4ca) ≥ ¬πi(i(¬a)) = a,

since c4ca ≤ ¬i(a) = i(¬(a)). Thus,rd = [c]I

z≥

rd = [ca]I

z∧

q[c]I = [ca]I

y≥ a ∧ a = a

The above inequality holds for any a ∈ A, sord = [c]I

z≥∨A = 1B concluding the

proof.

103

Proposition 6.1.9. Let i : B → C be an injective complete homomorphism of completeboolean algebras and G be a V -generic filter for B. Then C/i[G] is a complete booleanalgebra in V [G].

Proof. In what follows to simplify our notation we will denote C/i[G] by C/J where J is

the ideal which is dual to the filter generated by i[G], i.e. J = JG with

J = 〈c,¬π(c)〉 : c ∈ C .

By Proposition 6.1.7, we need only to prove that C/J is complete. Let cα : α < δ ∈ V B

be a set of B names for elements of C/J (i.e. cα : α < δ is a shorthand for a τ ∈ V B suchthat

qτ : δ → C/J

yB

= 1B. By Lemma 6.1.8 for each α < δ there exists a unique dα ∈ Csuch that q

cα = [dα]Jy

= 1B.

We have that d =∨α<δ dα ∈ C, since C is complete. Clearly d ≥ dα entails that

V [G] |= ∀α < δ [d]J ≥ [dα]J . To complete the proof it is enough to show that [d]J is the leastupper bound of [dα]J : α < δ in V [G]. Fix c ∈ C such that V [G] |= ∀α < δ [c]J ≥ [dα]J ,we must show that V [G] |= [c]J ≥ [d]J . Now

¬π(dα ∧ ¬c) =rdα ∧ ¬c ∈ J

z=

q[c]J ≥ [dα]J

y∈ G

for all α < δ. So π(dα ∧ ¬c) 6∈ G for all α < δ. Observe that π(dα ∧ ¬c) : α < δ ∈V . Since G is V -generic we get that G ∩ π(dα ∧ ¬c) : α < δ = ∅ if and only if∨π(dα ∧ ¬c) : α < δ = π(d∧¬c) 6∈ G. Hence [d]J ≤ [c]J holds in V [G]. This shows that

V [G] |= C/J is complete for all V -generic filters G.

The construction of generic quotients can be defined also for injective complete homo-morphisms:

Proposition 6.1.10. Let B, C0, C1 be complete boolean algebras, and let G be a V -genericfilter for B. Let i0, i1, j form a commutative diagram of injective complete homomorphismsof complete boolean algebras as in the following picture:

B C0

C1

i0

i1j

Then j/G : C0/G → C1/G defined by j/G([c]i0[G]) = [j(c)]i1[G] is a well-defined injectivecomplete homomorphism of complete boolean algebras in V [G] with associated adjoint mapπj/G defined by πj/G([c]i1[G]) = [πj(c)]i0[G].

Proof. Let I be the dual ideal of G. Ik the ideal obtained by the downward closure of ik[I]for k = 0, 1. Then πj [I1] = I0 and πj [I

+1 ] = I+

0 .By Lemma 1.2.11 applied in V [G], j/G is an injective complete homomorphism and

(j/G, πj/G) forms an adjoint pair, since it still holds in V [G] that (j, πj) is an adjoint pairand I1 =↓ k[I0] holds in V [G].

The proof is completed.

104

6.1.3 Equivalence of two-steps iterations and injective complete homo-morphisms

We are now ready to prove that two-steps iterations and injective complete homomorphismscapture the same concept.

Theorem 6.1.11. If i : B→ C is an injective complete homomorphism of complete booleanalgebras, then B ∗ (C/i[GB])

∼= C.

Conversely if Q ∈ V B is a B-name for a complete boolean algebra, and G is V -genericfor B

(B ∗ Q)/iB∗C[G]∼= QG.

Proof. Let

i∗ : C→ B ∗ C/Gc 7→

[[c]i[G]

]≈

We show that i∗ is a bijective complete homomorphism. Let

J = 〈c,¬πi(c)〉 : c ∈ C

be a B-name for the ideal dual to the filter generated by i[G].

i∗ is well-defined and injective:[[c]i[G]

]≈

=[[d]i[G]

]≈

if and only if

r[c]i[G] = [d]i[G]

z= 1B

if and only if rc∆d ∈ J

z= 1B

if and only if πi(c∆d) = 0B if and only if c∆d = 0C if and only if c = d.

i∗ is surjective: by Lemma 6.1.8.

i∗ is a complete homomorphism: By definition of two-steps iteration:

i∗(¬c) =[[¬c]i[G]

]≈

=[¬[c]i[G]

]≈

= ¬[[c]i[G]

]≈

= ¬i∗(c);

and

i∗(∨cα) =

[[∨cα]i[G]

]≈

=[∨

[cα]i[G]

]≈

=∨[

[cα]i[G]

]≈

=∨i∗(cα).

For the second part of the proposition define in V [G]:

i∗ : (B ∗ Q)/iB∗C[G] → QG

[[q]≈]G 7→ qG

We leave to the reader to check that the above map is an isomorphism.

Notation 6.1.12. From now on given a complete boolean algebra B ∈ V a B-name Q ∈ V B

for a complete boolean algebra and a V -generic filter G for B we feel free to identify in V [G]the complete boolean algebras QG and B ∗ Q/i[G] modulo the isomorphism [[q]≈]G 7→ qG.

In particular we will identify the B-name q for an element of Q with the B-name [[q]≈]GB

for the corresponding element of (B ∗ Q)/GB.

105

Proposition 6.1.13. Let C0, C1 be B-names for complete boolean algebras, and let kbe a B name for a complete homomorphism from C0 to C1. Then there is a completehomomorphism i : B ∗ C0 → B ∗ C1 such that

rk = i/GB

z= 1B.

Moreover if k is a B name for an injective homomorphism, i is injective.

The following picture assumes G is V -generic for B.

V [G] :

(C0)G

(C1)G

(k)G V :

B B ∗ C0

B ∗ C1

i

i0

i1V [G] :

(B ∗ C0)/i0[GB]

(B ∗ C1)/i1[GB]

(C0)G

(C1)G

(k)Gi/GB

∼=

∼=

Proof. Let

i : B ∗ C0 → B ∗ C1

[d]≈ 7→ [k(d)]≈.

Since k is a B-name for a complete (injective) homomorphism with boolean value 1B,we have that [

d]≈

= [e]≈ ⇐⇒rd = e

z= 1⇒

⇒rk(d) = k(e)

z= 1 ⇐⇒

[k(d)

]≈

=[k(e)

]≈

This shows that i is well defined (in case k is forced to be injective,rd = e

z= 1 ⇐⇒

rk(d) = k(e)

z= 1, giving that i is injective). We have that i is a complete homomorphism,

since

i (¬ [c]≈) = i ([¬c]≈) =[k (¬c)

]≈

=[¬k (c)

]≈

= ¬[k (c)

]≈

= ¬i ([c]≈) ;

and

i(∨

[cα]≈

)= i([∨

cα

]≈

)=[k(∨

cα

)]≈

=[∨

k (cα)]≈

=∨[

k (cα)]≈

=∨i ([cα]≈) .

Moreover assume G is V -generic for B, then kG = i/G: it suffices to chase the followingdiagram

B B ∗ C0

B ∗ C1

iB∗C0

iB∗C1

i

106

i/G([[c]≈]iB∗C0[G]) = [i ([c]≈)]iB∗C1

[G] =[[k (c)

]≈

]iB∗C1

[G].

Proposition 6.1.14. Let G be V -generic for B, I be its dual ideal, and ij : B → Cj becomplete homomorphisms for j = 0, 1.

Assume C0/i0[G] and C1/i1[G] are isomorphic complete boolean algebras in V [G].

V [G] :V :

B C0

C1

i0

i1

C0/i0[G]

C1/i1[G]

k ∼=

Then C0 i0(b) and C1 i1(b) are isomorphic in V for some b ∈ G and k ∼= l/G.

V [G] :

C0 i0(b)/i0[G]

C1 i1(b)/i1[G]

C0/i0[G]

C1/i1[G]

l/G ∼=

=

=

k ∼=

V :

B C0

C1

i0

i1

C0 i0(b)

C1 i1(b)

l ∼=

rest

rest

Proof. By the previous Proposition we know that Cj/ij [G] are isomorphic to B∗Q/G (where

Q is a B-name for C0/i0[G]) via maps kj = (kj)G for both j = 0, 1. Let bj ∈ B be the

boolean value that kj is an isomorphism for both j. Then b = b0 ∧ b1 ∈ G. Define in V a

map l : C0 i0(b)→ C1 i1(b) by the rule c0 7→ c1 iffrk0([c0]i0[G]) = k1([c1]i1[G])

z≥ b.

We leave to the reader to check that l is an isomorphism in V and k ∼= l/G in V [G].

We also have the following:

Proposition 6.1.15. Assume B,C are complete boolean algebras, G is V -generic for C,and H ∈ V [G] is V -generic for B. Let H ∈ V C be such that HG = H. Then there existsr ∈ G such that the map

iq,H :B→ C q

b 7→rb ∈ H

z

C∧ q

is a complete homomorphism for all 0C < q ≤C r.

Proof. Let A ∈ V be the collection of predense subsets of B ∈ V . Then H is a V -genericultrafilter on B is the forcing statement:

φ(H, B, A) := (H is a filter on B) ∧ ∀τ ∈ A (H ∩ τ 6= ∅)

Hencer =

rφ(H, B, A)

z

C∈ G

We leave to the reader to check that the map iq,H defined above is a complete homomorphismfor all 0C < q ≤C r.

107

Lemma 6.1.16. Let i : B → C be a homomorphism with B,C cbas. For all r ∈ C+ letir : B→ C r be defined by b 7→ i(b)∧ r. The following are equivalent for any G V -genericfor C:

1. i−1[G] = H is in V [G] a V -generic filter for B,

2. ir = ir,H for some r ∈ G and some H ∈ V C such that H = HG is a V -generic filterfor B in V [G].

3. ir is a complete homomorphism for some r ∈ G.

Proof.

• 1 =⇒ 2. Assume that 1 holds, let H = 〈b, i(b)〉 : b ∈ B, then HG = i−1[G]; hence

by Proposition 6.1.15 there exists r ∈ G such that ir,H defined by b 7→rb ∈ H

z∧ r

is a complete homomorphism. Now observe that

ir,H(b) =rb ∈ H

z

C∧ r = i(b) ∧ r,

where the last equality holds sincera ∈ H

z

C≥ i(a) for all a ∈ B, hence choosing

a = b and a = ¬b we get the desired equality for any b ∈ B. We conclude that 2holds.

• 2 =⇒ 3. Assume that ir = ir,H and letrφ(H, B, A)

z

C(∈ G) as in the proof

Proposition 6.1.15. Then for any q ≤ r∧rφ(H, B, A)

z

Cwe have that iq,H is complete.

• 3 =⇒ 1. Assume that 3 holds. Then i−1[q ∧ r : q ∈ G] is V -generic for B byProposition 6.1.2, therefore i−1[G] is also V -generic for B.

6.1.4 Generic quotients of generic quotients, aka three-steps iterations

We can also handle the generic quotient of a generic quotient.

Proposition 6.1.17. Let B, C0, C1 be complete boolean algebras, and let G be a V -genericfilter for B. Let i0, i1, j form a commutative diagram of injective complete homomorphismsof complete boolean algebras as in the following picture:

B C0

C1

i0

i1j

Assume K is a V -generic filter for C0 with i0[G] ⊆ K. Then:

• K =

[q]i0[G] ∈ C0 : q ∈ K

is V [G]-generic for C0/i0[G],

• V [K] = V [G][K],

• V [K] |= C1/j[K]∼= (C1/i1[G])/j/G[K] via the map [q]j[K] 7→ [[q]i1[G]]j/G[K].

Conversely assume G is V -generic for B and K is a V [G]-generic filter for C0/i0[G]. Then:

• K =q ∈ C0 : [q]i0[G] ∈ K

is V -generic for C0 with i0[G] ⊆ K,

108

• V [K] = V [G][K],

• V [K] |= C1/j[K]∼= (C1/i1[G])/j/G[K] via the map [q]j[K] 7→ [[q]i1[G]]j/G[K].

Proof. We identify C0∼= B ∗ C0/i0[G]. Let D ∈ V [G] be a dense subset of C0/i0[G]. Let D

be a name for D such thatrD is dense in C0/i0[G]

z

B= 1. Then, by Fact 6.1.6,

D :=

[[q]i0[G]]≈ :r

[q]i0[G] ∈ Dz

B= 1

is dense for C0 in V . Hence there exists [[q]i0[G]]≈ ∈ K ∩ D. By definition, we get thatq ∈ K ∩D holds in V [G]. Since D ∈ V [G] was chosen arbitrarily among the dense subsetsof C K is V [G]-generic for C0/i0[G].

Note that K ∼= G ∗ K :=

[[q]i0[G]]≈ ∈ G : [q]i0[G] ∈ K

, from which it follows thatV [K] = V [G][K].

Let j′ be defined by [q]j[K] 7→ [[q]i1[G]]j/G[K].Using standard facts from ring theory we get that the map [q]j[K] 7→ [[q]i1[G]]j/G[K]

implements in V [K] the isomorphism C1/j[K]∼= (C1/i1[G])/j/G[K]. It remains to argue that

K is V -generic for C0: Let D ∈ V be a dense subset of C0. Define D :=

[q]i0[G] : q ∈ D

.Once again, since C0

∼= B ∗ C0/i0[G], and by Fact 6.1.6, D is dense in V [G]. Since K isV [G]-generic it follows that there exists [q]i0[G] ∈ K ∩D. Hence q belongs to K ∩ D. ThusK is V -generic for C0.

6.1.5 Preservation of chain conditions under two steps iterations

We show that < λ-CC and < λ-presaturation are preserved under two steps iterations.

Lemma 6.1.18. Let λ be a regular cardinal and i : B → C an injective complete homo-morphism. TFAE:

1. C is < λ-CC.

2. B is < λ-CC andrC/i[GB] is < λ-CC

z

B= 1B.

Proof. Let π be the adjoint of i.

6.1.18(1)⇒6.1.18(2) It is immediate to argue that B is < λ-CC since the pointwiseimage of an antichain of B is an antichain of C. By Fact 6.1.6 f : γ → C/i[GB] is a

B-name for an antichain of C/i[GB] if and only if

[f(α)]≈ : α < γ

is an antichain in

B ∗ (C/i[GB]). The latter is isomorphic to C by Thm. 6.1.11. Therefore γ < λ.

6.1.18(2)⇒6.1.18(1) Since B is < λ-presaturated (by Remark 4.2.2),

qλ is a regular cardinal

yB

= 1B

by Proposition 4.2.3.

Towards a contradiction fix A = cα : α < λ maximal antichain of C in V . Noticethat for all G V -generic filter for B we have that [cα]G : α < λ is a maximalantichain of C/i[G] and therefore it has size < λ in V [G]. Hence there is some ηG < λsuch that [cα]G = 0C/i[G]

for all ηG ≤ α < λ holds in V [G].

Let η be a B-name for an ordinal such that whenever G is V -generic for B, ηG = ηG.Let X be the maximal antichain B defined by the property that for all b ∈ X there

109

is a unique ηb < λ such that Jη = ηbKB = b. Since B is < λ-CC, ηb : b ∈ X is abounded subset of λ (it has size less than λ and consists of ordinals below λ). Letη = sup ηb : b ∈ X < λ. Then for all G V -generic for B we get that [cα]G = 0C/i[G]

for all η ≤ α < λ, since ηG ≤ η. Hence cα = 0C for all α ≥ η (since cα > 0 iffπ(cα) > 0 iff [cα]G > 0C/i[G]

for some G V -generic filter for B). This gives the desiredcontradiction.

Lemma 6.1.19. Let λ be a regular cardinal and i : B → C an injective complete homo-morphism. TFAE:

1. C is < λ-presaturated.

2. B is < λ-presaturated andrC/i[GB] is < λ-presaturated

z

B= 1B.

Proof. A trivial application of Proposition 4.2.3. Hence left to the reader.

6.2 Definable classes of forcing notions closed under two-steps iterations

We need to be able to characterize in the forcing language when a class Γ of forcing notionsis closed under two step iterations.

Definition 6.2.1. Let Γ be a definable class of forcing notions as the extension of theformula φΓ(x, aΓ) in the set-parameter aΓ.

Let B,C be complete boolean algebras.

• A complete homomorphism i : B→ C is Γ-correct if1

rC/i[GB] ∈ Γ

z

B=

rφΓ(C/i[GB], aΓ)

z

B≥B coker(i).

• Assume C is a complete boolean algebra and G is V -generic for C. Let H ∈ V [G] be V -

generic for B. Let H ∈ V C be such that HG = H and q =rH is V -generic for B

z∈

G.

Define for r ≤C q

ir,H :B→ C r

b 7→rb ∈ H

z

C∧ r.

Then ir,H is a complete homomorphism such that H = i−1r,H

[G] by Lemma 6.1.15. H

is Γ-correct for B in V [G] if for some r ∈ G, r ≤C q, letting J be the ideal ↓ (ir,H [H])

V [H] |= C/J ∈ ΓV [H].

A definable class of forcings Γ is closed under two steps iterations whenever B ∈ Γ andi : B→ C is Γ-correct entail that C ∈ Γ as well.

1Notice that we do not require a priori neither B nor C to be in Γ, even if in what follows we are mostlyinterested in the case in which this occurs for both of them.

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Lemma 6.2.2. Let Γ be a class of forcings which is the extension of the formula φΓ(x, aΓ)in the set-parameter aΓ. Let i : B→ C be a complete homomorphism. For all r ∈ C+ letir : B→ C r be defined by b 7→ i(b)∧ r. The following are equivalent for any G V -genericfor C:

1. i−1[G] = H is in V [G] a Γ-correct V -generic filter for B,

2. ir = ir,H for some r ∈ G and some H ∈ V C such that H = HG is Γ-correct in V [G].

3. ir is Γ-correct for some r ∈ G.

4. V |= i(rC/i[GB] ∈ Γ

z

B) ∈ G.

Proof. Left to the reader. It is an almost self-evident reformulation of the notion ofΓ-correct homomorphism and Γ-correct filter. Use 6.1.16 and the fact that in all relevant

cases the hypothesis grant that the boolean valuerC/i[GB] ∈ Γ

z

B∈ H = (GB)H .

Lemma 6.2.3. Assume Γ is a definable class of forcings. Let B, C0, C1 be completeboolean algebras, and let G be any V -generic filter for B. Let i0, i1, j be Γ-correct completehomomorphisms in V forming a commutative diagram of injective complete homomorphismsof complete boolean algebras as in the following picture:

B C0

C1

i0

i1j

Then in V [G], j/G : C0/G → C1/G is still a ΓV [G]-correct complete homomorphism.

Proof. By Proposition 6.1.10 j/G is a complete homomorphism. Assume K1 is V [G]-generic for C1/G = C1/i1[G]. Then K1 =

q : [q]i1[G] ∈ K1

is V -generic for C1 and

V [G][K1] = V [K1]. Hence K0 = j−1[K1] is a Γ-correct V -generic filter for C0 in V [K1],since j is Γ-correct in V , i.e.

V [K0] |= C1/j[K0] ∈ ΓV [K0].

Let K0 =

[q]i0[G] : q ∈ K0

. Then in V [K0], G = i−1

0 [K0] and C1/j[K0] is isomorphicto (C1/i1[G])/j/G[K0] via the map [q]j[K0] 7→ [[q]i1[G]]j/G[K0] (see Proposition 6.1.17).

Hence K0 = j−1/G[K1] and V [G][K0] = V [K0]. We get that

V [G][K0] |= (C1/i1[G])/j/G[K0] ∈ ΓV [G][K0].

Hence K0 is in V [G][K1] a ΓV [G]-correct V [G]-generic filter for C1/i1[G]. Since this occurs

for all K1 V [G]-generic for C1/i1[G], we conclude that j/G is ΓV [G]-correct in V [G].

Remark 6.2.4. Note that B,C0,C1 may not be in Γ! The Lemma gives control just on thebehaviour of the generic quotient of C1 with respect to the complete homomorphism j, butdoes not give any information regarding whether C1 itself is or not in Γ.

We also need the following:

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Proposition 6.2.5. Assume Γ is a definable class of forcings. Let G be V -generic forsome complete boolean algebra B. Assume k : B→ R is a Γ-correct homomorphism in V ,h : R/k[G] → Q is a Γ-correct homomorphism in V [G].

B RV :k

R/k[G]

Q

V [G] :

h

Then there are in V :

• C ∈ Γ,

• a Γ-correct homomorphism l : B→ C,

• a Γ-correct homomorphism h : R→ C,

such that:

• Q is isomorphic to C/l[G] in V [G],

• h/G ∼= h (modulo the isomorphism of Q with C/l[G]) holds in V [G],

• h k = l holds in V ,

• 0C 6∈ l[G].

B R

C

V :k

hl

2 ∼= B/G R/k[G]

C/l[G] ∼= Q

V [G] :

h/G∼=h

k/G∼=Id

l/G∼=Id

Proof. By Proposition 6.1.13 and Theorem 6.1.11 there are complete homomorphismsl : B→ C and h : R→ C such that Q ∼= C/l[G] and h = h/G with l = h k.

It remains to argue that l and h are Γ-correct in V . To see that l is Γ-correct we

proceed as follows: V [G] models that C/l[G]∼= Q ∈ ΓV [G]. Hence

rC/l[GB] ∈ Γ

z

B= r ∈ G.

Refining (if necessary) C to C l(r) and l to ll(r), we can assume w.l.o.g. that l(r) = 1C(hence r ≥ coker(l)), and conclude that l : B→ C is Γ-correct in V while maintaining thatC/l[G]

∼= Q, moreover 0C 6∈ l[G] since C/l[G]∼= Q is a non-trivial complete boolean algebra

in V [G].We also want to argue that h is Γ-correct in V :

V [G] |= h is ΓV [G]-correct

and Q ∼= C/l[G] ∈ ΓV [G], hence whenever K is V [G]-generic for Q and H = h−1[K] we getthat

V [G][H] |= Q/h[H] ∈ ΓV [G][H].

Now (modulo the identification of Q with C/l[G] in V [G])

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• K =q ∈ C : [q]l[G] ∈ K

is V -generic for C,

• H =r ∈ R : [r]k[G] ∈ H

is V -generic for R,

• V [K] = V [G][K],

• V [G][H] = V [H],

• the map [q]h[H] 7→ [[q]l[G]]h[H] defined in V [H] is an isomorphism of C/h[H] with(C/l[G])/h[H].

Conversely whenever K is V -generic for C, let G = l−1[K], and H = h−1[K]. Then (modulothe identification of Q with C/l[G] in V [G]):

• K =

[q]l[G] : q ∈ K

is V [G]-generic for Q,

• H =

[q]k[G] : q ∈ H

is V [G]-generic for R/k[G],

• V [K] = V [G][K],

• V [G][H] = V [H],

• the map [q]h[H] 7→ [[q]l[G]]h[H] defined in V [H] is an isomorphism of C/h[H] with(C/l[G])/h[H].

Hence whenever K is V -generic for C,

V [H] |= C/h[H]∼= Q/h[H] ∈ ΓV [G][H].

We conclude that rC/h[GR] ∈ Γ

z

R= 1R,

as was to be shown.

Remark 6.2.6. Note that with the notation of the proof of the Lemma it may be the casethat neither B nor R nor C are in ΓV and still, l, h are both Γ-correct in V (as for theprevious proposition, the Lemma gives information just on the generic quotients of theboolean algebras and not on the algebras themselves).

Remark 6.2.7. Most of the results in this chapter can be generalized to complete (non-injective) homomorphisms i : B → C, applying the relevant Lemmas to the injectivehomomorphism i coker(i). We leave the details to the reader.

6.3 Category forcings

Assume Γ is a class of complete boolean algebras and Θ is a family of complete homomor-phisms between elements of Γ closed under composition and containing all identity maps.(Γ,→Θ) is the category whose objects are the complete boolean algebras in Γ and whosearrows are given by complete homomorphisms i : B→ Q in →Θ. We call embeddings inΘ, Θ-correct embeddings. Notice that these categories immediately give rise to naturalclass partial orders associated with them, partial orders whose elements are the completeboolean algebras in Γ and whose order relation is given by the arrows in Θ (i.e. B ≤Θ C ifthere exists i : C→ B in →Θ). We denote these class partial orders by (Γ,≤Θ).

Depending on the choice of Γ and Θ these partial orders can be trivial (as forcingnotions), for example:

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Remark 6.3.1. Assume Γ = Ωℵ0 is the class of all complete boolean algebras and Θ isthe class of all complete embeddings, then any two conditions in (Γ,≤Θ) are compatible,i.e. (Γ,≤Θ) is forcing equivalent to the trivial partial order. This is the case since for anypair of partial orders P,Q and X of size larger than 2|P |+|Q| there are complete injectivehomomorphisms of RO(P ) and RO(Q) into the boolean completion of Coll(ω,X) (see [30,Thm A.0.7] and its following remark). These embeddings witness the compatibility ofRO(P ) with RO(Q).

On the other hand these partial orders will in general be non-trivial (see for exampleFact 8.7.1).

Since we want to allow ourselves more freedom in the handling of our class forcings(Γ,≤Θ) we allow elements of the category Γ to be arbitrary partial orders2 in Γ andwe identify the arrows in →Θ between the objects P and Q in Γ to be the Θ-correcthomomorphisms between the boolean completions of P and Q. We will be mainly interestedin these two types of category forcings:

• (Γ,≤Ω) where Γ is a definable (in set parameters) class of forcing notions and Ω is theclass of all complete homomorphisms between the boolean completions of elementsof Γ;

• (Γ,≤Γ) where Γ is a definable class of complete boolean algebras closed under twostep iterations, products, and complete subalgebras, and B ≤Γ C if there is a Γ-correcthomomorphism i : C→ B.

Suprema in (Γ,≤Θ)

Notation 6.3.2. Let B be a collection of complete Boolean algebras. BB =∏B, the

lottery sum of the algebras in B, is the Boolean algebra obtained by the cartesian product ofthe respective Boolean algebras with pointwise operations. Remark that

∏B is complete.

The name lottery sum is justified by the intuition that forcing with∏B corresponds to

forcing with a “random” algebra in B: since the set of p ∈∏B that are 1 in one component

and 0 in all the others form a maximal antichain of∏B, every V -generic filter G for

∏B

concentrates only on a specific B ∈ B (determined by the generic filter).Whenever Γ is a class of complete boolean algebras closed under products, the lottery

sum defines a natural∨

operation of suprema on subsets of (Γ,≤Ω). Moreover if Γ isclosed under two steps iterations, the lottery sum defines a sup operation also on (Γ,≤Γ):

Proposition 6.3.3. Assume Γ is a class of complete boolean algebras closed under products,and let A = Bi : i < γ be a family of complete boolean algebras in Γ. Then the productalgebra BA =

∏i<γ Bi endowed with coordinatewise operations is the exact upper bound of

A in (Γ,≤Ω). Moreover if Γ is also a definable class closed under two step iterations BA isthe supremum of A also in (Γ,≤Γ).

Proof. The adjoint maps πi : BA → Bi mapping f ∈∏j<γ Bj to f(i) define (Γ-correct)

complete homomorphisms witnessing that BA ≥Ω Bi.Moreover if C ≥Ω Bi for all i < γ as witnessed by ki : C → Bi, the map k : C → BA

mapping c 7→ 〈ki(c) : i < γ〉 witnesses that C ≥Ω BA. If Γ is closed under two stepsiterations and lottery sums, and each of the ki is Γ-correct, then one can check that k isΓ-correct as well.

2 Specifically one of our main aims will be to show that for certain classes Γ of complete boolean algebras(Γ ∩ Vδ,≤Γ ∩Vδ) ∈ Γ. In general (Γ ∩ Vδ,≤Γ ∩Vδ) is a non-separative partial order.

114

Why this ordering on class partial orders?

Given a pair (Γ,Θ) as above, we can define two natural order relations ≤Θ and ≤∗Θ on Γ.The first one is given by complete homomorphisms i : B→ Q in Θ (which is the one wedescribed before) and the other given by complete and injective homomorphisms i : B→ Qin Θ. Both notion of orders are interesting and set theorists are used to focus on this secondstricter notion of order since it is the one suitable to develop a theory of iterated forcing.However in the present book we focus mostly on complete (but possibly non-injective)homomorphisms because this notion of ordering grants that whenever we add a V -genericfilter for a C ≤Θ B, we will also be adding a V -generic filter for B by 6.1.1 applied to thei : B → C witnessing C ≤Θ B. Moreover the ≤Θ will grant us that whenever B is putinto a V -generic filter for (Γ,≤Θ) for suitably well behaved category forcings (Γ,≤Θ), thisV -generic filter will also add a V -generic filter for B. To understand why this occurs weleave to the reader to check that the map k : B → Γ mapping b 7→ B b is well defined,order and sup-preserving, whenever Γ is closed under complete subalgebras and lotterysums (which is trivially the case for most of our choices of Γ). The critical issue is to checkwhether the above map can be incompatibility preserving and a main result we will achieveis that this is the case for many interesting classes Γ.

If we decided to order the family Γ using injective homomorphisms in Θ we would getthat a V -generic filter for this other category forcing defined according to this stricter notionof order will just give a directed system of partial orders in Γ with injective homomorphismsin Θ between them, without actually giving V -generic filters for the partial orders in thisdirected system.

We will come back to these issues in great details in the last part of this book.

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Chapter 7

Iteration systems

In this chapter we present iteration systems and some of their algebraic and forcingproperties. In order to develop the basic theory of iterations, along the chapter we consideronly injective complete homomorphisms, we will use repeatedly several of the properties ofthese maps collected in Thm 1.4.1.

The first four sections prove in full details the basic properties of iterations of limitlength, in particular the first section introduces the key definitions, section 7.2 provesBaumgartner’s theorem on the preservation of < λ-CC through limit stages and outlinesits most relevant consequences, section 7.3 shows how to handle the generic quotiens of aniteration, 7.4 gives several counterexamples to certain false conclusions on the properties ofiterations one may be tempted to conjecture.

The last section of the chapter outlines the basic requirement a class of forcing notionsΓ must have in order to grant that most of the iterations of posets in Γ do have a limitwhich is also in Γ. We also introduce the lottery preparation for a class of forcings Γ: itprovides a smart way to organize an iteration of posets in Γ which is enough flexible togrant that the limit forces a variety of forcing axioms (depending on the length of the limit).This technique is due to Hamkins [24] and has been extensively used in the literature to

Cercare sumathscinetuso di lotterypreparation inarticoli edaggiungere quireferenzeprincipali suquestoargomento –M

prove the consistency of a number of forcing axioms – M. We will employ it in Parts IVand V to obtain the consistency of Martin’s maximum.

7.1 Definitions and basic properties

Definition 7.1.1. F = iαβ : Bα → Bβ : α ≤ β < λ is a complete iteration system ofcomplete boolean algebras iff for all α ≤ β ≤ γ < λ:

1. Bα is a complete boolean algebra and iαα is the identity on it;

2. iαβ is an injective homomorphism with associated adjoint παβ;

3. iβγ iαβ = iαγ .

If γ < λ, we define F γ= iαβ : α ≤ β < γ.

Bα Bβ Bγiα,β iβ,γ

iα,γ

116

Definition 7.1.2. Let F be a complete iteration system of length λ. Then:

• The inverse limit of the iteration is

lim←−(F) =

f ∈

∏α<λ

Bα : ∀α∀β > α παβ f(β) = f(α)

and its elements are called threads.

• The direct limit is

lim−→(F) =f ∈ lim←−(F) : ∃α∀β > α f(β) = iαβ(f(α))

and its elements are called constant threads. The support of a constant thread supp(f)is the least α such that iαβ f(α) = f(β) for all β ≥ α.

• The revised countable support limit is1

limrcs

(F) =f ∈ lim←−(F) : f ∈ lim−→(F) ∨ ∃α f(α) Bα cof(λ) = ω

.

We can define on lim←−(F) a natural join operation (which nonetheless produce thesuprema of a family of threads only assuming certain nice properties of the family, whilegiving in most cases just an upper bound for the family):

Definition 7.1.3. Let A be any subset of lim←−(F). We define the pointwise supremum ofA as ∨

A = 〈∨f(α) : f ∈ A : α < λ〉.

The previous definition makes sense since by Proposition 1.4.1.ii∨A is a thread.

Remark 7.1.4. It must be noted that if A is an infinite subset of lim←−(F),∨A might not

be the least upper bound of A in RO(lim←−(F)), as shown in Example 7.4.1. A sufficientcondition on A for this to happen is given by Lemma 7.1.11 below.

Definition 7.1.5. Let F = iαβ : α ≤ β < λ be an iteration system. For all α < λ, wedefine iαλ as

iαλ : Bα → lim−→(F)

b 7→ 〈πβ,α(b) : β < α〉a〈iαβ(b) : α ≤ β < λ〉

and παλπαλ : lim←−(F) → Bα

f 7→ f(α)

When it is clear from the context, we will denote iαλ by iα and παλ by πα.

Fact 7.1.6. We may observe that:

1. lim−→(F) ⊆ RCS(F) ⊆ lim←−(F) are partial orders with the order relation given bypointwise comparison of threads.

2. Every thread in lim←−(F) is completely determined by its tail. Moreover every thread inlim−→(F) is entirely determined by the restriction to its support. Hence, given a threadf ∈ lim←−(F), for every α < λ f α determines a constant thread fα ∈ lim−→(F) suchthat f ≤lim←−(F) fα.

1This definition can be appreciated just by the reader familiar with forcing, see Chapter 4 for details.

117

3. It follows that for every α < β < λ, iαλ = iαβ iβλ.

4. iαλ can naturally be seen as a injective homomorphism of Bα in any of RO(lim−→(F)),RO(lim←−(F)), RO(RCS(F)). Moreover by Property 1.4.1.ii in all three cases παλ =πiα,λ P where P = lim−→(F), lim←−(F), RCS(F).

5. If F is an iteration of length λ, and g : cof(λ)→ λ is an increasing cofinal map, thenwe have the followings isomorphisms of partial orders:

lim−→(F) ∼= lim−→(ig(α)g(β) : α ≤ β < cof(λ));

lim←−(F) ∼= lim←−(ig(α)g(β) : α ≤ β < cof(λ));

RCS(F) ∼= RCS(ig(α)g(β) : α ≤ β < cof(λ)).

hence we will always assume w.l.o.g. that λ is a regular cardinal.

Definition 7.1.7. lim−→(F) inherits the structure of a boolean algebra with boolean opera-tions defined as follows:

• f ∧ g is the unique thread h whose support β is the max of the support of f and gand is such that h(β) = f(β) ∧ g(β),

• ¬f is the unique thread h whose support β is the support of f such that h(β) = ¬f(β).

Fact 7.1.8. Assume g ∈ lim←−(F) and h ∈ lim−→(F). g ∧ h, defined as the thread whereeventually all coordinates α are the pointwise meet of g(α) and h(α), is the infimum of gand h in lim←−(F).

Remark 7.1.9. In general lim−→(F) is not complete and RO(lim−→(F)) cannot be identifiedwith a complete subalgebra of RO(lim←−(F)) (i.e. lim−→(F) and lim←−(F) as forcing notions ingeneral share little in common), as shown in Example 7.4.1. However, RO(lim−→(F)) can beidentified with a subalgebra of lim←−(F) that is complete (even though it is not a completesubalgebra, see the following proposition).

Proposition 7.1.10. Let F = iαβ : α ≤ β < λ be an iteration system. Then RO(lim−→(F)) 'D =

f ∈ lim←−(F) : f =

∨ g ∈ lim−→(F) : g ≤ f

.

Proof. We represent RO(lim−→(F)) as the family of regular open sets of the Stone space oflim−→(F). We also identify a g ∈ lim−→(F) with its associated neighborhood Ng in the Stonespace of lim−→(F). The isomorphism associates to a regular open U ∈ RO(lim−→(F)) the thread

k(U) =∨U , with inverse

k−1(f) = Reg(Ng : g ∈ lim−→(F) and g ≤lim←−(F) f

.)

First, we prove that

k−1(f) =⋃

Nh : h ∈ lim−→(F) and h ≤lim←−(F) f

:

LetA =

⋃Nh : h ∈ lim−→(F) and h ≤lim←−(F) f

.

We show that k−1(f) = A. One inclusion is clear since k−1(f) = Reg (A) ⊇ A by definition.

118

For the other assume that g ∈ lim−→(F) \A. Then

Ng 6⊆Nh : h ∈ lim−→(F) and h ≤lim←−(F) f

.

In particular g lim←−(F) f , and this is witnessed by some α > supp(g), so that g(α) f(α).

Let h = iα(g(α) \ f(α)) > 0, then h ∈ lim−→(F) and for all h′ ≤lim−→(F) h, h′ ⊥ f , giving that

Nh ∩A is empty. Therefore g 6∈ Reg (A) by Lemma 1.1.14, since A∩Ng is not dense in Ng.

Now we prove that k−1 k(U) =Ng : g ∈ lim−→(F) and g ≤

∨U

= U :

Since∨U ≥ f for all f ∈ U , it follows that U ⊆ k−1 k(U). Furthermore, since U

is a regular open set, if g /∈ U , there exists a g′ ≤lim−→(F) g such that Ng′ ∩ U is empty,

by Lemma 1.1.14. Suppose towards a contradiction that there exist a g ∈ lim−→(F) such

that g ≤∨U with Ng′ ∩ U empty for some g′ ≤ g. Let α be the support of g, so that

g(α) ≤∨f(α) : f ∈ U. Then, there exists an f ∈ U such that f(α) is compatible with

g(α), hence f ∧ g > 0 and is in U (since U is open). Since f ∧ g ≤ g, this is a contradiction.It follows that k(U) ∈ D for every U ∈ RO(lim−→(F)).

Furthermore, k−1 is the inverse map of k since we already verified that k−1 k(U) = Uand for all f ∈ D, k k−1(f) = f by definition of D.

Finally, k and k−1 are order-preserving maps since U1 ⊆ U2 iff∨U1 ≤

∨U2.

As noted before, the notion of supremum in lim←−(F) may not coincide with the notionof pointwise supremum. However the following Lemma (which will be repeatedly used inour analysis of semiproper iterations) shows that in some cases pointwise suprema are truesuprema:

Lemma 7.1.11. Let F = iαβ : α ≤ β < λ be an iteration system and A ⊆ lim←−(F) be an

antichain such that παλ[A] is an antichain for some α < λ. Then∨A is the supremum of

the elements of A in RO(lim←−(F)).

Proof. Suppose by contradiction that∨A <

∨A in RO(lim←−(F)). Then there exists

g ∈ lim←−(F) such that 0 < g ≤ ¬∨A ∧

∨A. Let α < λ be such that παλ[A] is an antichain

and let f ∈ A be such that f(α) is compatible with g(α). Such an f exists becauseg(α) ≤

∨f(α) : f ∈ A. We show that g and f are compatible: Consider

h = 〈g(β) ∧ iα,β f(α) : α ≤ β < λ〉.

Then h ≤ g is a thread of lim←−(F) by 7.1.8. It only remains to prove that h(β) ≤ f(β) for eachβ ≥ α. We have that h(β) ≤ g(β) ≤ supt(β) : t ∈ A, and also that h(β) is incompatiblewith t(β) for all f 6= t ∈ A for all α ≤ β < λ, since h(α) = g(α) ∧ f(α) ≤ f(α) ⊥ t(α).Hence the only possibility is that

h(β) ≤∨t(β) : t ∈ A ∧

(¬∨t(β) : t ∈ A, t 6= f

)= f(β)

for all β ≥ α. Therefore g and f are compatible. Contradiction.

7.2 Sufficient conditions for the equality of direct and in-verse limit, and preservation theorems for the < λ-cc

Even though in general lim−→(F) is different from lim←−(F), in certain cases they happen tocoincide:

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Lemma 7.2.1. Let F = iαβ : α ≤ β < λ be an iteration system such that lim−→(F) is<λ-cc. Then lim←−(F) = lim−→(F) is a complete boolean algebra.

Proof. First, since every element of RO(lim−→(F)) is the supremum of an antichain in lim−→(F),since lim−→(F) is <λ-cc and since λ is regular, the supremum of such an antichain can becomputed in some Bα for α < λ hence RO(lim−→(F)) = lim−→(F).

Let f be in lim←−(F) \ lim−→(F). Since f is a non-constant thread, for all α < β we havethat iαβ(f(α)) ≥ f(β) and for all α there is an ordinal βα such that iαβα(f(α)) > f(βα).By restricting to a subset of λ w.l.o.g. we can suppose that f(β) < iαβ(f(α)) for all β > α.Hence iαλ(f(α)) : α < λ is a strictly descending sequence of length λ of elements inlim−→(F)+. From a descending sequence we can always define an antichain in lim−→(F) settingaα = iαλ(f(α)) ∧ ¬iα+1,λ(f(α+ 1)). Since lim−→(F) is <λ-cc, this antichain has to be of sizeless than λ. Hence aα = 0 for coboundedly many α, giving that f(α+ 1) = iα,α+1(f(α)),so that f ∈ lim−→(F), contradiction.

In the formulation and proof of the following theorem we use a standard result aboutclubs and stationary sets: Fodor’s Lemma 2.1.19. For details see Chapter 2.

Theorem 7.2.2 (Baumgartner). Let λ be a regular cardinal and F = iαβ : α ≤ β < λbe an iteration system such that Bα is <λ-cc for all α and S =

α : Bα ∼= RO(lim−→(F α))

is stationary. Then lim−→(F) is <λ-cc.

Proof. Let fα : α < λ ⊆ lim−→(F)+. We show that it is not an antichain:Let h : λ→ λ be such that h(α) > α, supp(fα). Let C ⊆ λ be the club of closure points

of h (i.e. such that for all α ∈ C, h[α] ⊆ α). For each α ∈ S find gα ∈ lim−→(F α)+ suchthat gα ≤α fα(α) in RO(lim−→(F α)+).

Define a regressive function

φ : S → λα 7→ supp(gα).

By Fodor’s Lemma find ξ ∈ λ, and T ⊂ S ∩ C stationary such that φ[T ] = ξ.Consider the set gα(ξ) : ξ ∈ T. There are two possibilities:

• | gα(ξ) : ξ ∈ T | < λ. In which case since gα : α ∈ T has size λ (gα 6= gβ becausethey have distinct domain α 6= β), we must have that for some α < β ∈ T gα(ξ) =gβ(ξ).

• | gα(ξ) : ξ ∈ T | = λ. In which case since Bξ is <λ-cc, there are α < β ∈ T ∩ Csuch that gα(ξ) ∧ gβ(ξ) > 0Bξ .

In either cases we can find α < β ∈ T such that gα(ξ) ∧ gβ(ξ) > 0Bξ .We will show that fα ∧ fβ > 0lim−→(F). Towards this aim (since for h0, h1 ∈ lim−→(F) the

support of h0 ∧ h1 is the maximum of the supports of h0, h1), it suffices to show that

fα(ν∗) ∧ fβ(ν∗) > 0Bν∗ for ν∗ = max h(α), h(β) ≥ max supp(fα), supp(fβ).

We will prove it showing that

πξν∗(fα(ν∗) ∧ fβ(ν∗)) > 0Bξ . (7.1)

This suffices since ker(πξν∗) =

0Bν∗

(by 1.4.1(ii)).So we turn to the proof of equation 7.1.

120

Proof. First of all we observe that ν∗ = h(β): since α < β ∈ C, supp(fα) ≤ h(α) < β ≤h(β).

Since fβ(β) ≥ gβ ∈ lim−→(F β),

fβ(ν) = πνβ f(β) ≥ πνβ(gβ) = gβ(ν)

for all ν < β.In particular we get that

fβ(ν) ≥ iξ,ν(gβ(ξ))

for all ξ ≤ ν < β.Thus for ξ ≤ η = supp(fα) < β,

fβ(η) ≥η iξ,η(gβ(ξ)).

This gives that

πη,ν∗(fα(ν∗) ∧ fβ(ν∗)) =

= πη,ν∗(iη,ν∗(fα(η)) ∧ fβ(ν∗)) =

= πη,ν∗(fβ(ν∗)) ∧ fα(η) =

= fβ(η) ∧ fα(η),

where we used repeatedly the fundamental identity 1.4.1(iii).Now observe that

πξ,η(fβ(η) ∧ fα(η)) ≥≥ πξ,η(iξ,η(gβ(ξ)) ∧ fα(η)) =

= gβ(ξ) ∧ πξ,η(fα(η)) =

= gβ(ξ) ∧ fα(ξ) ≥≥ gβ(ξ) ∧ gα(ξ),

again using repeatedly the fundamental identity 1.4.1(iii).Putting all things together we get that

πξν∗(fα(ν∗) ∧ fβ(ν∗)) =

= πξη πην∗(fα(ν∗) ∧ fβ(ν∗)) =

= πξη(fβ(η) ∧ fα(η)) ≥ gβ(ξ) ∧ gα(ξ) >

> 0Bξ .

Equation 7.1 is proved.

Therefore fα ∧ fβ ∈ lim−→F+, hence fα : α < λ is not an antichain.

Corollary 7.2.3. Let λ > γ be regular uncountable cardinals and F = iαβ : α ≤ β < λbe an iteration system such that Bα is <γ-cc for all α and S =

α : Bα ∼= RO(lim−→(F α))

is stationary. Then lim−→(F) is <γ-cc.

Proof. By the previous theorem lim−→(F) is <λ-cc and it is equal to lim←−(F). If A ⊆ lim−→(F)is an antichain, it has size < λ, hence the set of supports of its element is contained insome ξ < λ. Therefore for f, g ∈ A f(ξ) ∧ g(ξ) = 0Bξ and the map f 7→ f(ξ) is injective.Since Bξ is < γ-cc we get that A must have size less than γ.

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Fact 7.2.4. Let λ < γ be regular uncountable cardinals and F = iαβ : α ≤ β < λ be aniteration system such that Bα is <γ-cc for all α. Then lim−→(F) is <γ-cc.

Proof. If A is an antichain in lim−→(F) we get that

A =⋃α<λ

Aα

where Aα = f : supp(f) = α. Now Aα has size less than γ since f(α) : f ∈ A is anantichain in Bα and the map f 7→ f(α) is injective on Aα. The conclusion follows.

Summing up all the previous results we can prove the following:

Theorem 7.2.5. Let γ be a regular uncountable cardinal and F = iαβ : Bα → Bβ : α ≤ β < ηbe a complete iteration system of complete boolean algebras such that:

• B0 is <γ-cc;

•rBα+1/iαα+1[GBα ] is < γ-cc

z

Bα= 1Bα for all α < η;

• Bβ = RO(lim−→(F β)) for all limit β.

Then Bβ is <γ-cc for all β < η.

Proof. The details are left to the reader. The proof is by induction on β < η. All theprevious results allow to prove the relevant induction step for Bβ according to whether βis a successor (Lemma 6.1.18), has cofinality less than γ (Fact 7.2.4), or has cofinality atleast γ (Corollary 7.2.3).

Elaborating on this result one already has all the tools needed to prove the consistencyof Martin’s axiom. Since the aim of this book is to provide detailed proofs of axiomsstronger than Martin’s axiom (such as the proper forcing axiom or Martin’s maximum), weomit to include here an explicit proof of the consistency of Martin’s axiom and we proceedto develop further our general theory of iteration systems.

7.3 Generic quotients of iteration systems

The results on generic quotients of the previous section generalize without much effort toiteration systems. In the following we outline how this occurs.

Lemma 7.3.1. Let F = iαβ : Bα → Bβ : α ≤ β < λ be a complete iteration system ofcomplete boolean algebras, Gγ be a V -generic filter for Bγ. Then F/Gγ = iαβ/Gγ : γ <α ≤ β < λ is a complete iteration system in V [Gγ ].

Proof. Apply Proposition 6.1.10 repeatedly.

Lemma 7.3.2. Let F = iαβ : Bα → Bβ : α ≤ β < λ be a complete iteration system ofcomplete boolean algebras, Gα be the canonical name for a generic filter for Bα and f be aBα-name for an element of lim←−(F/Gα). Then there exists a unique g ∈ lim←−(F) such thatrf = [g]Gα

z= 1Bα.

Proof. We proceed applying Lemma 6.1.8 at every stage β > α.

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Existence. For every β > α, by hypothesis f(β) is a name for an element of the quotient

Bβ/iαβ [Gα]. Let g(β) be the unique element of Bβ such thatrf(β) = [g(β)]iαβ [Gα]

z=

1Bα . Then,

rf = [g]Gα

z=

r∀β ∈ λ f(β) = [g(β)]Gα

z

=∧r

f(β) = [g(β)]iαβ [Gα]

z: β ∈ λ

=∧

1Bα = 1Bα

Uniqueness. If g′ is such thatrf = [g′]Gα

z= 1Bα then for every β > α,

rf(β) = [g′(β)]iαβ [Gα]

z=

1Bα . Such an element is unique by Lemma 6.1.8, hence g′(β) = g(β) defined above,completing the proof.

g is a thread. Apply once again Proposition 6.1.10 to infer that παβ(g(β)) = g(α) for allβ > α.

7.4 Examples and counterexamples

We now examine some aspects of iterated systems by means of examples. In the firstone we will see that lim←−(F) may not be a complete boolean algebra, and that lim−→(F)and lim←−(F) as forcing notions share little in common. In the second one, we will justifythe introduction of RCS-limits showing that in many cases lim−→(F) collapses ω1 even if allfactors of the iteration are preserving ω1. This shows that in order to produce a limit ofan iteration system that preserves ω1, one needs to devise subtler notions of limits thanfull and direct limits. This motivates the results of chapters 9 and 10 where it is shownthat RCS-limits are a nice notion of limit, since RCS-iterations of semiproper posets aresemiproper and preserve ω1. The last iteration system also provides an example of aniteration in which the direct limit is taken stationarily often but lim←−(F) 6= RO(lim−→(F)).

7.4.1 Distinction between direct limits and full limits

We start showing that the inverse limit and direct limit of the same iteration system maybe completely unrelated as forcing notions.

Example 7.4.1. Let F0 = in,m : Bn → Bm : n < m < ω be an iteration system suchthat for all n ∈ ω 1Bn Bn+1/G 6= 2, and B0 is atomless and infinite.

Lemma 7.4.2. There exists tm ∈ lim−→(F0) for each m ∈ ω such that the followings hold:

1. tn+1 : n ∈ ω ⊆ lim−→(F0) is an antichain;

2.∨tn+1 : n ∈ ω = 1;

3. tn+1 : n ∈ ω is a maximal antichain in lim−→(F0);

4. there exists t ∈ lim←−(F0) such that for all n ∈ ω, t ⊥ tn+1.

Proof. Let a0 = 1B0 . Since 1Bn Bn+1/GBn6= 2, there exists an+1 ∈ V Bn such that

1Bn = J0 < an+1 < 1K. Let an+1 ∈ Bn+1 be such thatran+1 = [an+1]GBn

z= 1Bn , which

exists by Lemma 6.1.8. Then

πn,n+1(an+1) = J0 < an+1K = 1

123

andπn,n+1(¬an+1) = Jan+1 < 1K = 1.

For all n > 0, let

tn = 〈in,m(¬an) ∧∧il,m(al) : l < n : m ∈ ω,m > n〉.

First of all we have

πn,n+1(∧il,n+1(al) : l ≤ n+ 1)

= πn,n+1(in,n+1(∧il,n(al) : l < n+ 1) ∧ an+1)

=∧il,n(al) : l ≤ n ∧ πn,n+1(an+1) =

∧il,n(al) : l ≤ n .

This implies also that

tn+1(n) = πn,n+1(¬an+1 ∧∧il,n+1(al) : l < n+ 1) =

= πn,n+1(¬an+1) ∧∧il,n(al) : l ≤ n) =

∧il,n(al) : l ≤ n .

1. Observe that for all 0 < m < n ∈ ω, tn ⊥ tm. As a matter of fact:

tm(n) = im,n(¬am) ∧∧il,n(al) : l < m ≤ ¬im,n(am),

tn(n) = ¬an ∧∧il,n(al) : l < n ≤ im,n(am).

2. In order to prove∨tm : 0 < m ∈ ω = 1, we prove by induction on n that∨

tm(n) : 0 < m ≤ n+ 1 = 1Bn .

If n = 0 then t1(0) = π0,1(a1 ∧ i0,1(a0)) = π0,1(a1) = 1. Now assume that it holds forn. Observe that

tn+1(n+ 1) ∨ tn+2(n+ 1) = (¬an+1 ∧∧im,n+1(am) : m < n+ 1)

∨ (an+1 ∧∧im,n+1(am) : m < n+ 1 =

∧im,n+1(am) : m < n+ 1

= in,n+1(∧im,n(am) : m ≤ n) = in,n+1(tn+1(n)).

Therefore

in,n+1(∨tm(n) : 0 < m ≤ n) ∨ tn+1(n+ 1) ∨ tn+2(n+ 1) =

= in,n+1(∨tm(n) : 0 < m ≤ n) ∨ in,n+1(tn+1(n)) =

= in,n+1(∨tm(n) : 0 < m ≤ n+ 1)) = 1.

3. We can now show that tm : 0 < m ∈ ω is a maximal antichain in lim−→(F0): Assumes ∈ lim−→(F0). Let n be the support of s. Then, by the previous item, there existsm > 0 such that tm(n) ∧Bn s(n) > 0Bn . Hence for all j > n

0Bn < tm(n) ∧Bn s(n) = πnj(tm(j) ∧ inj(s)).

We conclude that tm and s are compatible in lim−→(F0), since their meet is a positivethread in lim−→(F0). Since s was chosen arbitrarily, we get our thesis.

124

4. Let t = 〈∧im,n(am) : m ≤ n : n ∈ ω〉. It is a thread since for all l < n:

πl,n(∧im,n(am) : m ≤ n) =

∧im,l(am) : m ≤ l .

Moreover we have that t ⊥ tn for all n ∈ ω \ 0, since tn(n) < ¬an and t(n+ 1) <in,n(an) = an.

Proposition 7.4.3. RO(lim−→(F0)) is not a complete subalgebra of RO(lim←−(F0)). Moreoverlim←−(F0) is not closed under suprema.

Proof. We follow the notation of Lemma 7.4.2. For each m ∈ ω tm+1 ∈ lim−→(F0) andtm+1 ⊥lim←−(F0) t for all m. Hence

∨RO(lim←−(F0))

tn+1 : n ∈ ω 6=∨tn+1 : n ∈ ω = 1,

since the left term is orthogonal to t in RO(lim←−(F0)). Since tn+1 : n ∈ ω is a maximal an-tichain of lim−→(F0) which is not maximal in RO(lim←−(F0)), this implies also that RO(lim−→(F0))is not a complete subalgebra of RO(lim←−(F0)). Moreover since it is easy to check that a

thread f ∈ lim←−(F0) is a majorant of a family A of threads in lim←−(F0) iff f ≥∨A, we also

get that lim←−(F0) is not closed under suprema of its subfamilies.

7.4.2 Direct limits may not preserve ω1

We now produce an example of an iteration system of length bigger than ω1 whose directlimit does not preserve ω1. We use the following forcing notion:

Definition 7.4.4. Let λ be a regular cardinal. Namba forcing Nm(λ) is the poset ofall perfect trees T ⊆ λ<ω (i.e. everbranching and such that for every t ∈ T , the setα < λ : taα ∈ T has cardinality either 1 or λ), ordered by reverse inclusion.

Recall that a forcing notion P is stationary set preserving if

qS is stationary

yRO(P )

= 1

for all S stationary subset of ω1 ∈ V (see Definition 2.1.2 and Lemma 2.1.14 for the notionof stationary subset of ω1). Recall also that stationary set preserving forcings do notcollapse ω1 (see Lemma 9.1.3). In the example below we use only the following well-knownproperties of Namba forcing (see [21, Theorem A.2], [28, Theorem 28.10], [42]):

Fact 7.4.5. Nm(λ) is stationary set preserving (and thus preserves ω1) and forces thecofinality of λ to become ω and its size to become ω1.

Example 7.4.6. Let F1 = iα,β : Bα → Bβ : α ≤ β < λ be an iteration system suchthat S =

α < λ : Bα = lim−→(F1 α)

is stationary, and suppose that B0 is the boolean

completion of the Namba forcing Nm(λ) and Bα+1/Gα is forced to have antichains ofuncountable size.

Proposition 7.4.7. lim−→(F1) is not a dense subset of lim←−(F1) and lim−→(F1) collapses ω1.

125

Proof. Let iα : Bα → lim−→(F1) be the canonical embedding of Bα into lim−→(F1). Let f ∈ V B0

be the canonical name for a cofinal function from ω to λ, and let Aα be a Bα-name for anantichain of size ω1 in Bα+1/Gα , with Aα = aαβ : β < ω1 the corresponding antichain ofsize ω1 in Bα+1 obtained by repeated application of Lemma 6.1.8. Then for all α, β we

have that

s0 <

[aαβ

]Gα

< 1

= 1Bα hence πα,α+1(aαβ) = πα,α+1(¬aαβ) = 1Bα .

Let t ∈ V B0 be a name for the thread in lim←−(F1/G0) defined by the requirement that

for all n < ω rt(f(n)) = [a

f(n)0 ]G0

z

B0

= 1B0 .

Let t ∈ lim←−(F1) be the canonical representative for t obtained from Lemma 7.3.2. Suppose

by contradiction that t ∈ RO(lim−→(F1))+, so that there exists an r ≤ t in lim−→(F1)+. Since fis a B0-name for a cofinal increasing function from ω to λ, r cannot decide in lim−→(F1) a

bound for the value of i0(f(n)) for cofinitely many n, else λ would be the supremum ofsuch bounds and would have countable cofinality in V . Let b ∈ B0, b ≤ r(0) be such that

b B0 f(n) = γ

with γ > supp(r) and n large enough so that r cannot bound the value of i0(f(n)).Then t ∧ i0(b) ≤ iγ(aγ0) but r ∧ i0(b) cannot be below iγ(aγ0) since it has support

smaller than γ (and so is compatible with ¬aγ0 , that is an element that projects to 1Bγ ), acontradiction which shows that t is not refined by any element of lim−→(F1)+.

For the second part of the thesis, let G0 be V -generic for B0, f = fG0 . Let iα/G0 denotethe canonical embedding of Bα/G0 into lim−→(F1/G0).

Define g to be a lim−→(F1/G0)-name in V [G0] for a function from ω to ω1 as follows:

qg(n) = β

y= if(n)/G0([a

f(n)β ]G0).

Then g is forced to be a lim−→(F1/G0)-name for a surjective map from ω to ω1, since forevery t ∈ lim−→(F1/G0) and β ∈ ω1 we can find an n such that f(n) > supp(t) so that

[t]G0 ∧ if(n)/G0([af(n)β ]G0)

is positive and forces β to be in the range of g. Thus, lim−→(F1/G0) collapses ω1 to ω forevery G0 V -generic for B0. Since lim−→(F1) = B0 ∗ lim−→(F1/G0

) the same holds for lim−→(F1),

as witnessed by the following lim−→(F1)-name h for a function

rh(n) = β

z

RO(lim−→(F1))=∨

i0

(rf(n) = α

z

B0

)∧ aαβ : α ∈ λ

,

completing the proof.

7.5 Iterable classes of forcing notions

We now introduce natural sufficient conditions granting that a class of forcing notionsΓ has nice iteration properties. Specifically we want to address the following problem:Assume Γ is a definable class of forcing notions and

F = iαβ : Bα → Bβ : α ≤ β < λ

is an iteration system of cbas in Γ. When F admits a limit in Γ?In this section we isolate sufficient conditions granting that this is most often the case

if Γ satisfies them. In chapter 10 we show that the classes of proper, semiproper andstationary set preserving forcings satisfy these conditions.

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7.5.1 Weakly iterable forcing classes

We introduce the notion of weakly iterable forcing notions.

Notation 7.5.1. Let Γ be class of forcing notions definable as the extension of the formulaφΓ(x, aΓ) in the set-parameter aΓ and closed under two-steps iterations.

Γlim denotes the (definable) class of complete iteration systems F = iαβ : Bα → Bβ : α ≤ β < λwith iαβ such that

rBβ/iαβ [GBα ] ∈ Γ

z

Bα=

rφΓ(Bβ/iαβ [GBα ], aΓ)

z

Bα= 1Bα

for all α ≤ β < λ.

Definition 7.5.2. Let T be a theory extending NBG, Γ be a definable class of completeBoolean algebras in T , Σ : Γlim → Γlim be a definable class function in T , γ be a definablecardinal in T .

• An iteration system F = Bη : η < α ∈ Γlim of length α follows Σ if and only if forall β < α even2, F (β+1)= Σ(F β).

• Σ is a weak iteration strategy for Γ if and only if we can prove in T that forevery F = Bη : η < α of length α which follows Σ, Σ(F) has length α + 1 andF = Σ(F) α.

• Σ is a γ-weak iteration strategy for Γ if in addition Σ(F) = lim−→F whenever F =Bη : η < α and cof(α) = γ or cof(α) = α > γ, |B| for all B in F .

Definition 7.5.3. Let T be a theory extending NBG by a recursive set of axioms, Γ adefinable class of complete Boolean algebras, Σ : Γlim → Γlim a definable class function, γa definable cardinal.

Γ is γ-weakly iterable through Σ in T iff we can prove in T that:

• Γ is closed under two-step iterations and set-sized lottery sums;

• Σ is a γ-weak iteration strategy for Γ;

• 〈Γ,Σ〉 as computed in Vκ+1 is equal to 〈Γ ∩ Vκ,Σ ∩ Vκ〉 whenever κ is inaccessibleand Vκ+1 |= T .3

Γ is weakly iterable iff it is γ-weakly iterable through Σ for some γ,Σ.

Intuitively, Γ is weakly iterable if and only if there is a sufficiently nice strategy forchoosing limits in Γ for iterations of indefinite length in Γlim. We remark that the latterdefinition (for a T ⊇ NBG) is not related to a specific model V of T , and requires that theabove properties are provable in T , and hence hold for every T -model M : for example, ifT = MK they must hold in every Vκ+1 where κ is inaccessible. We feel free to omit thereference to T when clear from the context, and in particular when T = MK or NBG.

Many notable classes Γ are ωi-weakly iterable for some i = 0, 1:

• Ω (the class of all forcings) is ω-weakly iterable using a strategy Σ which takes finitesupport limits at limit stages and is the identity elsewhere.

2We remark that every limit ordinal is even.3Since κ is inaccessible, this statement is equivalent to Vκ+1 |= T \ NBG which is recursive.

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• The class SP of semiproper forcings is ω1-weakly iterable using a strategy Σ whichtakes revised countable support limits, and chooses Bα = Bα−1 ∗ Coll(ω1, |Bα−1|)at even successor stages α. We will prove this in full details in chapter 10. Bysimplifications of the argument for semiproperness, or as a corollary of this result,one can also prove that:

– the class of axiom-A forcing notions or of proper forcing notions are ω1-weaklyiterable using a strategy Σ which takes countable support limits at limit stagesand is the identity elsewhere.

– Locally <κ+-cc4 and <κ-closed are κ-weakly iterable using a strategy Σ whichtakes <κ-sized support limits at limit stages and is the identity elsewhere.

– The class of stationary set preserving forcings SSP is ω1-weakly iterable assumingthe existence of a proper class of supercompact cardinals, using a strategy Σwhich takes revised countable support limits and chooses Bα = Bα−1 ∗ C, whereC forces SP = SSP and collapses |Bα−1| to have size ω1 (details will be given inchapter 10).

7.5.2 The lottery preparation forcings

The definition of weak iterability for a definable class of forcing notions Γ provides theright conditions to carry out the lottery iteration PΓ,f

κ with respect to a partial functionf : κ → κ where κ is an inaccessible cardinal. The lottery iteration has been studiedextensively by Hamkins [24] and is one of the main tools to obtain the consistency offorcing axioms. We will employ these type of iterations in Parts IV and V to obtain theconsistency of Martin’s maximum and some of its variants and strengthenings.

Definition 7.5.4. Let Γ be γ-weakly iterable through Σ and f : κ → κ be a partial

function. Define Fξ =PΓ,fα : α < ξ

by recursion on ξ ≤ κ+ 1 as:

1. F0 = ∅ is the empty iteration system;

2. Fξ+1 = Σ(Fξ) if ξ is even;

3. Fξ+2 has PΓ,fξ+1 = PΓ,f

ξ if ξ + 1 is odd and f(ξ) is undefined;

4. Fξ+2 has PΓ,fξ+1 = PΓ,f

ξ ∗ C otherwise, where C is a PΓ,fξ -name for the lottery sum (as

computed in V PΓ,fξ ) of all complete Boolean algebras in Γ of rank less than f(ξ), i.e.,

a PΓ,fξ -name for

∏(Γ ∩ Vf(ξ)

).

We say that PΓ,fκ is the lottery iteration of Γ relative to f .

Proposition 7.5.5. Let T be a theory extending NBG by a recursive set of axioms, Γ beγ-weakly iterable through Σ, f : κ→ κ be a partial function with κ > γ inaccessible cardinalsuch that Vκ+1 |= T . Then:

1. PΓ,fκ exists and is in Γ;

2. PΓ,fκ is a <κ-cc complete boolean algebra and for all α < κ,

q2α ≤ κ

yPΓ,fκ

= 1PΓ,fκ

;

3. PΓ,fκ is definable in Vκ+1 using the class parameter f ;

4B is locally <κ-cc if it is the lottery sum of <κ-cc complete Boolean algebras.

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4. Let g : λ → λ with λ inaccessible be such that f = g κ, Vλ |= T . Then PΓ,gλ

absorbs every forcing in Γ ∩ Vg(κ) as computed in V PΓ,fκ . That is, for every B in

(Γ ∩ Vg(κ))V P

Γ,fκ , there is a condition p ∈ PΓ,g

λ such that PΓ,gλ p≤Γ PΓ,f

κ ∗ B.

Proof.

1. Follows from Σ being a weak iteration strategy for α < κ even, and from Γ beingclosed under two-step iterations and lottery sums for α odd.

2. Since Σ ∩ Vκ is Σ as computed in Vκ+1, we can prove by induction on α < κ that∣∣∣PΓ,fα

∣∣∣ < κ, hence PΓ,fα is <κ-cc for all α < κ. Furthermore, by definition of γ-weak

iteration strategy the set of ordinals:

S =α < κ : PΓ,f

α = lim−→ (Fκ α)⊇ α < κ : cof(α) = γ

is stationary in κ. It follows by Baumgartner’s Theorem 7.2.2 and Lemma 7.2.1that lim−→Fκ = lim←−Fκ is <κ-cc and a complete boolean algebra. Since Σ is a γ-weak

iteration strategy and κ = cof(κ) > γ,∣∣∣PΓ,fα

∣∣∣ for all α < κ, PΓ,fκ = Σ(Fκ) = lim−→Fκ

concluding the proof of the first part of the statement.

For the second part, given α < κ let x be a PΓ,fκ -name for a subset of α. Then x is

decided by α < κ antichains of size 2 (see Lemma 4.2.5), hence x = ıβ(y)5 for some

y ∈ V PΓ,fβ , β < κ.

Since∣∣∣PΓ,fβ

∣∣∣ < κ and κ is inaccessible, there are less than κ-many names for subsets

of α in V PΓ,fβ . Thus there are at most κ-many names for subsets of α in V PΓ,f

κ .

3. Straigthforward, given that 〈Γ ∩ Vκ,Σ ∩ Vκ〉 is 〈Γ,Σ〉 as computed in Vκ+1.

4. Let g : λ → λ and λ inaccessible be such that f = g κ and Vλ+1 |= T . Since〈Γ ∩ Vκ,Σ ∩ Vκ〉 is 〈Γ,Σ〉 as computed in Vκ+1 and the same holds for λ, lettingF = PΓ,g

α : α < λ, we have that

F κ= PΓ,gα : α < κ = PΓ,f

α : α < κ.

Hence PΓ,fκ = PΓ,g

κ . Furthermore any B in (Γ ∩ Vg(κ))V P

Γ,fκ is forced by PΓ,f

κ to be the

restriction of C =∏(

Γ ∩ Vg(κ)

)V PΓ,fκ

to a suitable condition q applying Definition

7.5.4.(4) in V PΓ,gλ (q is forced by PΓ,f

κ to be 1 in the component of C correspondingto B and 0 in all the other components of C). Hence we can find p ∈ PΓ,g

λ such that

PΓ,gκ+1 (pκ+1)

∼= PΓ,fκ ∗ B. Therefore,

PΓ,fκ ∗ B ∼= PΓ,g

κ+1 (pκ+1) ≥Γ PΓ,gλ p .

5Recall that ıβ is the natural embedding from VP

Γ,fβ to V PΓ,f

κ (see Proposition 4.1.12).

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Part IV

Forcing axioms, properness,semiproperness

130

Part IV deals with forcing axioms, properness and semiproperness. Chapter 8 gives athorough analysis of different types of forcing axioms and of their mutual interactions: itis shown that the axiom of choice, Baire’s category theorem, and Shoenfield’s absolutenessresults can all be naturally seen as forcing axioms; stationary sets are also used to givea different characterization of forcing axioms in terms of a strong form of the downwardLowenheim-Skolem theorem. Chapter 9 introduces properness and semiproperness. We linkthese concepts to that of forcing axioms and give a topological and algebraic characterizationof both of these properties. Chapter 10 gives the main results regarding (semi)properiterations, mainly their preservation through limit stages.

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Chapter 8

Forcing axioms I

Forcing is well-known as a versatile tool for proving consistency results. The axiom ofchoice and Baire’s category theorm are useful non-constructive principles which greatlysimplify (or in some cases are unavoidable assumptions for) the development of manymathematical theories. In this section we outline on the one hand how Baire’s categorytheorem and the axiom of choice can be regarded as specific instances of forcing axioms,and on the other hand how (by means of forcing axioms) one can transform forcing in anon-constructive tool to prove theorems by means of variations of Levy’s absoluteness andShoenfield’s absoluteness results.

Notation 8.0.1. Let P be a partial order and κ a cardinal.PD(P ) is the collection of predense subsets of P .PD(P, κ) is the collection of predense subsets of RO(P ) of size at most κ.

Definition 8.0.2. Let κ be a cardinal and (P,≤) be a partial order.

FAκ(P ) ≡ For all families Dα : α < κ of predense subsets of P , there is a filter G on Pmeeting all these predense sets.

Given a class Γ of partial orders FAκ(Γ) holds if FAκ(P ) holds for all P ∈ Γ.

Notation 8.0.3. Ωκ denotes the class of all forcings P such that FAκ(P ), Ω stands forΩℵ0 , the class of all posets (in view of Lemma 8.1.3 below).

8.1 The axiom of choice and Baire’s category theorem areforcing axioms

The axiom of choice AC and Baire’s category theorem BCT are non-constuctive principleswhich play a prominent role in the development of many fields of abstract mathematics.Standard formulations of the axiom of choice and of Baire’s category theorem are thefollowing:

Definition 8.1.1. AC ≡∏i∈I Ai is non-empty for all families of non empty sets Ai : i ∈ I,

i.e. there is a choice function f : I →⋃i∈I Ai such that f(i) ∈ Ai for all i ∈ I.

Theorem 8.1.2. BCT0 ≡ For all compact Hausdorff spaces (X, τ) and all countablefamilies An : n ∈ N of dense open subsets of X,

⋂n∈NAn is non-empty.

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There are large numbers of equivalent formulations of the axiom of choice and it maycome as a surprise that one of these is a natural generalization of Baire’s category theoremand naturally leads to the notion of forcing axiom.

A simple proof of the Baire Category Theorem is given by a basic enumeration argument(which however needs some amount of the axiom of choice to be carried):

Lemma 8.1.3. BCT1 ≡ Let (P,≤) be a partial order and Dn : n ∈ N be a family ofpredense subsets of P . Then there is a filter G ⊆ P meeting all the sets Dn.

Proof. Build by induction a decreasing chain pn : n ∈ N with pn ∈ ↓Dn and pn+1 ≤ pnfor all n. Let G = ↑ pn : n ∈ N. Then G is a filter and meets all the Dn.

Baire’s category theorem can be proved from the above Lemma (without any use ofthe axiom of choice) as follows:

Proof of BCT0 from BCT1. Given a compact Hausdorff space (X, τ) and a family of denseopen sets Dn : n ∈ N of X, consider the partial order (τ \ ∅ ,⊆) and the familyEn = A ∈ τ : Cl (A) ⊆ Dn. Then it is easily checked that each En is dense open in thepartial order (τ \ ∅ ,⊆). By Lemma 8.1.3, we can find a filter G ⊆ τ \ ∅ meeting allthe sets En. This gives that for all A1, . . . An ∈ G

Cl (A1) ∩ . . . ∩ Cl (An) ⊇ A1 ∩ . . . ∩An ⊇ B 6= ∅

for some B ∈ G (where Cl (A) is the closure of A ⊆ X in the topology τ .) By thecompactness of (X, τ), ⋂

Cl (A) : A ∈ G 6= ∅.

Any point in this intersection belongs to the intersection of all the open sets Dn.

Remark the interplay between the order topology on the partial order (τ \ ∅ ,⊆) andthe compact topology τ on X. Modulo the prime ideal theorem (a weak form of the axiomof choice), BCT1 can also be proved from BCT0.

It is less well-known that the axiom of choice has also an equivalent formulation assertingthe existence of filters on posets meeting sufficiently many dense sets.

Definition 8.1.4. Let λ be a cardinal. A partial order (P,≤) is < λ-closed if everydecreasing chain Pα : α < γ indexed by some γ < λ has a lower bound in P .

A cba B is < λ-closed if B+ contains a dense subset P such that (P,≤B) is < λ-closed.Γλ denotes the class of < λ-closed posets.

It is almost immediate to check that Γℵ0 is the class of all posets, and that BCT1 statesthat FAℵ0(Γℵ0) holds. The following formulation of the axiom of choice in terms of forcingaxioms has been handed to me by Todorcevic, I’m not aware of any published reference.In what follows, let ZF denote the standard first order axiomatization of set theory in thefirst order language ∈,= (excluding the axiom of choice) and ZFC denote ZF+ the firstorder formalization of the axiom of choice.

Theorem 8.1.5. The axiom of choice AC is equivalent (over the theory ZF) to the assertionthat FAκ(Γκ) holds for all regular cardinals κ.

We sketch a proof of Theorem 8.1.5, the interested reader can find a full proof in[36, Chapter 3, Section 2] (see the following hyperlink: Tesi-Parente). First of all, it isconvenient to prove 8.1.5 using a different equivalent formulation of the axiom of choice.

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http://www.logicatorino.altervista.org/matteo_viale/thesis-parente.pdf

Definition 8.1.6. Let κ be an infinite cardinal. The principle of dependent choices DCκstates the following:

For every non-empty set X and every function F : X<κ → P (X) \ ∅, there existsg : κ→ X such that g(α) ∈ F (g α) for all α < κ.

Lemma 8.1.7. AC is equivalent to ∀κDCκ modulo ZF.

The reader can find a proof in [36, Theorem 3.2.3]. We prove the Theorem assumingthe Lemma:

Proof of Theorem 8.1.5. We prove by induction on κ that DCκ is equivalent to FAκ(Γκ)over the theory ZF + ∀λ < κDCλ. We sketch the ideas for the case κ-regular1:

Assume DCκ; we prove (in ZF) that FAκ(Γκ) holds. Let (P,≤) be a <κ-closed partiallyordered set, and Dα : α < κ ⊆ P (P ) a family of predense subsets of P .

Given a sequence 〈pβ : β < α〉 call ξ~p the least ξ such that 〈pβ : ξ ≤ β < α〉 is adecreasing chain if such a ξ exists, and fix ξ~p = α otherwise. Notice that when the lengthα of ~p is successor then ξ~p < α.

We now define a function F : P<κ → P (P )\∅ as follows: given α < κ and a sequence~p ∈ P<κ,

F (~p) =

p0 if ξ~p = αd ∈ ↓Dα : d ≤ pβ for all ξ~p ≤ β < α

otherwise.

The latter set is non-empty since (P,≤) is <κ-closed, α < κ, and Dα is predense. By DCκ,we find g : κ→ P such that g(α) ∈ F (g α) for all α < κ. An easy induction shows thatfor all α the sequence g α is decreasing, so g(α) ∈ ↓Dα for all α < κ. Then

G = p ∈ P : there exists α < κ such that g(α) ≤ p

is a filter on P , such that G ∩Dβ 6= ∅ for all β < κ.Conversely, assume FAκ(Γκ), we prove (in ZF) that DCκ holds.Let X be a non-empty set and F : X<κ → P (X) \ ∅. Define the partially ordered set

P =s ∈ X<κ : for all α ∈ dom(s), s(α) ∈ F (s α)

,

with s ≤ t if and only if t ⊆ s. Let λ < κ and let s0 ≥ s1 ≥ · · · ≥ sα ≥ . . . , for α < λ, be achain in P . Then

⋃α<λ sα is clearly a lower bound for the chain. Since κ is regular, we

have⋃α<λ sα ∈ P and so P is <κ-closed. For every α < κ, define

Dα = s ∈ P : α ∈ dom(s) ,

and note that Dα is dense in P . Using FAκ(Γκ), there exists a filter G ⊂ P such thatG∩Dα 6= ∅ for all α < κ. Then g =

⋃G is a function g : κ→ X such that g(α) ∈ F (g α)

for all α < κ.

8.2 Forcing axioms and stationarity I

In an informal sense, assuming the forcing axiom for a broad class of posets suggests that anumber of different forcing has already been done in our model of set theory. This intuitiveinsight is reflected into the following equivalence.

1In this case the assumption ZF + ∀λ < κDCλ is not needed, but all the relevant ideas in the proof ofthe equivalence are already present.

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Definition 8.2.1. Let M be a set P a poset, B be a boolean algebra. G is M -generic forP if G ∩M ∩D is non-empty for all D ∈M dense subset of P . G is M -generic for B if itis M -generic for B+.

Theorem 8.2.2. Let B be a cba and κ be a cardinal. The following are equivalent:

1. FAκ(B).

2. For some cardinal θ with θ > 2|B|, κ, there exists an M ≺ H(θ), |M | = κ, B ∈ M ,κ ⊂M and a filter G M -generic for B.

3. The following set is stationary:

S =N ≺ H(|B|+) : κ ⊂ N ∧ |N | = κ ∧ ∃G filter N -generic for B

.

Proof. We prove these equivalences as follows:

1⇒2: Suppose that FAκ(B) holds and let M ≺ H(θ) be such that B ∈M , κ ⊂M , |M | = κ.There are at most κ dense subsets of B in M , hence by FAκ(B) there is a filter Gmeeting all those sets. However, G might not be M -generic since for some D ∈M ,the intersection G ∩D might be disjoint from M . Define:

N =x ∈ H(θ) : ∃τ ∈M ∩ V B Jτ = xK ∈ G

Clearly, N contains M (hence contains κ), moreover |N | ≤

∣∣M ∩ V B∣∣ = κ, since every

τ ∈M can be evaluated in at most one way by the elements of the filter G. To provethat N ≺ H(θ), let ∃xφ(x, a1, . . . , an) be any formula with parameters a1, . . . , an ∈ Nwhich holds in V . Let τi ∈MB, qi ∈ G be such that qi = Jτi = aiK ∈ G for all i < n.Define

Qφ = Jφ(x, τ1, . . . , τn)K : x ∈ Hθ ,this set is definable in M hence Qφ ∈M . Furthermore, Qφ ∩G is not empty sinceit contains q =

∧i<n qi ∈ G. By the mixing Lemma applied in H(θ), we can find

τ ∈ Hθ ∩ V B such that:

H(θ) ∨Qφ = Jφ(τ, τ1, . . . , τn)K =

∨x∈Hθ

(Jτ = xK ∧ Jφ(x, τ1, . . . , τn)K)

Hence, since M ≺ Hθ and τ1, . . . , τn ∈M ,

M ∃τ∨Qφ = Jφ(τ, τ1, . . . , τn)K =

∨x∈Hθ

Jτ = xK ∧ Jφ(x, τ1, . . . , τn)K)

Fix such a τ ∈M . Since the set Jx = τK : x ∈ H(θ) is a predense set definable inM , there is an a ∈ H(θ) such that q′ = Jτ = aK ∈ G. Then q′, τ witness that a ∈ N ,hence the original formula ∃xφ(x, a1, . . . , an) holds in N as witnessed by a.

Finally, we need to check that G is N -generic for B. Let D ∈ N be a dense subset ofB, and D ∈M be such that

1B =rD is dense ∧ D ∈ V

z

(recall that Jτ ∈ V K is a short hand for the boolean value∨Jτ = xK : x ∈ V ), and

q =rD = D

z∈ G. Since 1B =

rD ∩ G 6= ∅

z, by the fullness lemma, there exists a

τ ∈ H(θ) such that 1B =rτ ∈ D ∩ G

z. By elementarity there is such a τ also in M .

Let q′ ∈ G below q decide the value of τ , q′ ≤ Jτ = pK. Since q′ forces that p ∈ G, weget that q′ ≤ p, hence p ∈ G ∩D ∩N .

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2⇒3: Let M , G be as in the hypothesis of the theorem, and fix a collection D = 〈Dα :α < κ〉 of dense subsets of B. Define:

S =N ≺ H(|B|+) : κ ⊂ N ∧ |N | = κ ∧ ∃G filter N -generic for B

Note that S is definable in M , hence S ∈ M . Furthermore, since B ∈ M so doesH(|B|+), therefore M ∩ H(|B|+) ≺ H(|B|+) and M ∩ H(|B|+) is in S. Given anyCf ∈ M club on H(|B|+), since f ∈ M , we have that M ∩ H(|B|+) ∈ Cf . HenceHθ S ∩ Cf 6= ∅, and by elementarity, the same holds for M . Thus, S is stationaryin M and again by elementarity S is stationary also in V .

3⇒1: Given D ∈ [PD(B)]≤κ, find N ∈ S such that D ∈ N . Since κ ⊂ N and D has size κ,Dα ∈ N for every α < κ. Hence any N -generic filter G for B meets all the dense setsin D, verifying FAκ(B) for this collection.

Remark 8.2.3. A comment on the notion of M -genericity for an M ≺ Hθ is in order.Assume θ is inaccessible, so that Hθ |= ZFC. If M is transitive and G is M -generic forsome B ∈M , we have a clear idea of what is M [G] and we know that there is a naturalisomorphism between MB/G and M [G] mapping [τ ]G 7→ τG for τ ∈MB.

If M is not transitive but G is V -generic (but possibly not M -generic), we still have aclear definition of

M [G] =τG : τ ∈ V B ∩M

,

and the above map [τ ]G 7→ τG for τ ∈M ∩ V B is still an isomorphism definable in V [G].It remains to analyze the case in which M is not transitive and G is M -generic, but

not V -generic for B. Then:

• MB/G is the Tarski model with elements

[τ ]G =σ ∈M ∩ V B : Jσ = τK ∈ G

and the Los Theorem holds for MB/G.

• The definition of M [G] =τG : τ ∈M ∩ V B

with

τG =σG : τ(σ) ∈ G, σ ∈M ∩ V B

gives a transitive set which does not contain M .

• The definition of M [G] as

M [G] =τG : τ ∈ V B ∩M

makes sense only if G is V -generic: assume G is not V -generic, then it is most likelythe case that V B/G is not even well founded (for example this occurs if B is Cohen’sforcing and G ∈ V see [27]), hence the definition of

V [G] =τG : τ ∈ V B

with τG = σG : τ(σ) ∈ G still makes sense, but the forcing theorem for V [G] fails.So does the isomorphism between V B/G and V [G] (being the former ill-founded andthe latter transitive). Nonetheless the forcing theorem for V B/G holds.

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It is now not clear whether it has any meaning to look at M [G] and what its definitionshould be.

These difficulties can be resolved passing to the transitive collapse of M : let πM : M →N be the Mostowski’s collapsing map. Then H = πM [G] is N -generic for πM (B) = Q, thedefinition of N [H] =

τH : τ ∈ NQ

makes sense since N is transitive and H is N -generic

for Q; the maps [τ ]G 7→ [πM (τ)]H 7→ πM (τ)H implement isomorphisms of MB/G withNQ/H with N [H].

Moreover now N [H] ∈ V is a transitive model of ZFC, by the forcing theorem appliedto N,H. It is interesting to see how much resemblance do N [H] and V maintain: beingboth transitive, certainly N [H] is ∆1-elementary in V .

It is possible to formulate forcing axioms such as FA++(B) which are natural strength-enings of FAω1(B) requiring the existence of M -generic filters G for B such that the degreeof elementarity between πM [M ][πM [G]] and V is more than just ∆1-elementarity. We willaddress this issue in the latter section of this chapter.

8.3 Ωκ is closed under two-steps iterations

Lemma 8.3.1. For any cardinal κ the class Ωκ is closed under two-step iterations.

Proof. Assume FAκ(B) holds and B FAκ(Q) holds. Given Dα : α < κ family of predensesubsets of B ∗ Q, find M ≺ Hθ such that κ ⊆M with Dα : α < κ ∈M and G M -genericfor B. This is possible by 8.2.2 applied to B. Let πM : M → NM = N be the transitivecollapse of M , B = πM (B), Q = πM (Q), H = πM [G], Q = Q

H, Eα = πM (Dα),

Fα = q ∈ Q : ∃(p, q) ∈ Eα with p ∈ H and qH = q

for all α < κ. Then H is N -generic for B and N is transitive, hence N [H] is a genericextension of N . Moreover

N [H] |= FAκ(Q)

andN [H] |= Fα : α < κ is a family of predense subsets of Q

by the forcing theorem applied to N,H,B for the above statements, given that N models

thatrFAκ(Q)

z

B= 1B and N models that B forces:

Fα =q ∈ Q : ∃(p, q) ∈ Eα with p ∈ GB

defines a dense subset of Q.

Hence there is K ∈ N [H] filter on Q meeting all the predense sets Fα.Then

(p, q) : p ∈ G, πM (q) ∈ K

is a filter on B ∗ Q meeting all the predense sets Dα for all α < κ.

8.4 Forcing axioms as Σ1-reflection properties

The work of Bagaria [5] shows that forcing axioms entail strong forms of reflection2.Actually we will expand on these topics in the last part of the book which will show that

2Stavi and Vaananen [44] argued that generic absoluteness results can by themselves be seen as formsof forcing axioms; a line pursued further mainly by Hamkins (see among others [25, 26], the latter withJohnstone); the authors of this book contributed as well with [3].

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generic absoluteness results stem out as natural consequences of strong forcing axioms. Thissection essentially draws from [5]. We will use the notation M ≺n N to mean M ≺Σn N(or equivalently M ≺Πn N , M ≺∆n+1 N). We first recall the following lemma:

Lemma 8.4.1 (Levi’s Absoluteness). Let κ > ω be a cardinal. Then H(κ) ≺1 V .

Proof. Given any Σ1 formula φ = ∃x ψ(x, p1, . . . , pn) with parameters p1, . . . , pn in H(κ),if V ¬φ also H(κ) ¬φ since H(κ) ⊆ V and ψ is ∆0 hence absolute for transitive models.Suppose now that V φ, so there exists a q such that V ψ(q, p1, . . . , pn). Let λ be largeenough so that q ∈ H(λ). By the downward Lowenheim Skolem Theorem there exists anM ≺ H(λ) such that q ∈M , trcl(pi) ⊆M for all i < n, and |M | = ω ∪

∣∣⋃i<n trcl(pi)

∣∣ < κ.Let N be the Mostowski Collapse of M , with πM : M → N the corresponding isomorphism.Notice that πM (pi) = pi for all i < n. Since H(λ) ψ(q, p1, . . . , pn), the same holds for M ,hence N ψ(π(q), p1, . . . , pn). Since N is transitive of cardinality less than κ, N ⊆ H(κ)so π(q) ∈ H(κ) and H(κ) φ.

Lemma 8.4.2 (Cohen’s Absoluteness). Let T be any theory extending ZFC, and φ beany Σ1 formula with a parameter p such that T ` p ⊆ ω. Then T ` φ(p) if and only ifT ` ∃B (1B φ(p)).

Proof. The left to right implication is trivial (choosing the cba 2). For the reverseimplication, suppose that V |= T , hence V ∃B (1B = Jφ(p)K). Let B be any such cbain V and θ be such that p,B ∈ Vθ and Vθ satisfies a finite fragment of T large enough toprove basic ZFC facts and 1B = Jφ(p)K. Let M ≺ Vθ be countable with p,B ∈M and N beits transitive collapse. Then N (1Q = Jφ(p)K) where Q = π(B). Let G be N -generic for Q(G exists since N is countable), so that N [G] φ(p). Since φ is Σ1, φ is upward absolutefor transitive models, hence V φ(p). The thesis follows by completeness of first-orderlogic (we can run this argument in any model of T other than V ).

Cohen’s Absoluteness Lemma can be generalized to the case p ⊆ κ for any cardinal κ.

Lemma 8.4.3 (Generalized Cohen’s Absoluteness). Let T be any theory extending ZFC,κ be a cardinal, φ be a Σ1 formula with a parameter p such that T ` p ⊆ κ. Then T ` φ(p)if and only if T ` ∃B (1B φ(p) ∧ FAκ(B)).

Proof. The forward implication is trivial; the converse implication follows the proof ofLemma 8.4.2. Given p, B such that 1B = Jφ(p)K and FAκ(B) holds, by Theorem 8.2.2, letM ≺ H(θ) and G be such that |M | = κ, B ∈M , κ ⊂M and G is a filter M -generic for B.Since there are stationarily many such M , we can assume that p ∈M . Let πM : M → Nbe the transitive collapse map of M , then H = πM [G] is N -generic for Q = π[B] andp ⊆ κ ⊆M is not moved by π so that N [H] φ(p). Since φ is Σ1, φ is upward absolutefor transitive models, hence V φ(p).

Corollary 8.4.4. Assume κ is regular, B is a complete boolean algebra and FAκ(B) holds.Then κ is regular in V [G] for any V -generic filter G.

Proof. κ is a regular cardinal is a Π1-property expressible in Hκ+ . We conclude byLemma 8.4.3.

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8.5 Which forcings can be in Ωκ?

In view of the above results it becomes interesting to understand for each cardinal κ whatis the extent of the class Ωκ, for we can then use elements of this class to force the truth ofcertain Σ1-facts. Baire’s category theorem asserts that Ω = Ωℵ0 = Γℵ0 is the class of allforcing notions. The axiom of choice shows that Ωκ ⊇ Γκ (recall that the latter is the classof < κ-closed forcings). Shelah has outlined a clear drawing line which isolates a large classof posets P for which FAκ(P ) provably fails. Once again the notion of stationarity plays acrucial role in defining this class:

Definition 8.5.1. Let κ be a regular infinite cardinal. SSP(κ) is the class of completeboolean algebras B such that for all S ⊆ P (κ) stationary we have that

JS is stationaryKB = 1B.

Following the standard usage SSP(ω1) = SSP is the class of stationary set preservingposets.

Notation 8.5.2. Let Γ be a class of posets. A poset P is locally-Γ if and only if thereexists a p ∈ P such that P p = q ∈ P : q ≤ p is in Γ.

Theorem 8.5.3 (Shelah). Assume FAκ(B) holds and P (κ) admits a club subset of size κ.Then B is locally-SSP(κ).

Remark 8.5.4. Notice that the assumption of the lemma holds for all successor cardinal λ+

such that λ+ = 2λ. This is the case since C = [λ+]≤λ ∪ λ+ is a club subset of P (λ+) ofsize λ+. C is a club since λ+ is never a Jonsson cardinal, hence there is f : (λ+)<ω → λ+

such that the unique X ∈ Cf of size λ+ is λ+ by a classical result of Shelah [40].

Proof. Assume B is not locally-SSP(κ), let g : κ<ω → κ be such that Cg is a club subset ofP (κ) of size κ. Then there is a maximal antichain A of B such that for all b ∈ B there issome Sb stationary subset of Cg and fb ∈ V B B-name for a function from κ<ω → κ suchthat r

Cfb ⊆ Cg and Sb ∩ Cfb = ∅z

B= b

By the fullness Lemma we can find f ∈ V B such thatrfb = f

z

B≥ b for all b ∈ A.

Set for all s ∈ κ<ω, b ∈ A and Z ∈ Sb ∩ Cg

Ds =p ∈ B : ∃α

rf(s) = α

z

B≥ p

FZ,b =p ∈ B b : there are α 6∈ Z and s ∈ Z<ω such that

rf(s) = α

z

B≥ p.

We leave to the reader to check that Ds is open dense in B for all s ∈ κ<ω, and thatFZ,b is open dense in B b for all b ∈ A and Z ∈ Sb ∩ Cg. Now set for each Z ∈ CgaZ =

∨b ∈ A : Z ∈ Sb and

FZ = B ¬aZ ∪ FZ,b : b ∈ A,Z ∈ Sb

We leave to the reader to check that FZ is open dense for all Z ∈ Cg. Remark that thefamily

↓ A ∪ FZ : Z ∈ Cg ∪Ds : s ∈ κ<ω

is a family of size κ of dense open subsets of B.

139

Suppose by way of contradiction that FAκ(B) holds, and let G ∈ V be a filter thatintersects A, and all the Ds, FZ for s ∈ κ<ω and Z ∈ Cg. Hence there is a unique b ∈ Asuch that b ∈ G. Remark that f = fG : κ<ω → κ defined by fG(s) = α if

rf(s) = α

z

B∈ G

is a well defined total function, since G ∩Ds is non-empty for all s ∈ κ<ω.Find Z ∈ Sb ∩ Cf ∩ Cg, which exists since Sb is stationary. Then f(s) ∈ Z for all

s ∈ Z<ω. However G ∩ FZ 6= ∅ and b ∈ G, entail that G ∩ FZ,b 6= ∅. Hence there is α 6∈ Zand s ∈ Z such that

rf(s) = α

z

B∈ G. This gives that fG(s) = α 6∈ Z, contradicting our

assumption that Z ∈ S ∩ Cf .

Lemma 8.5.5. Stationary set preserving forcings preserve ω1.

Proof. If the Lemma fails there is B ∈ SSP and G V -generic for B in which (ω1)V is acountable ordinal. Remark that in this case

(ω1)V

is a club subset of P

((ω1)V

)in V [G]

(see the example right after 2.1.2). Hence all stationary subset of (ω1)V are no longerstationary in V [G] since they are disjoint from the club

(ω1)V

. A contradiction.

A main result of Foreman, Magidor, Shelah [16] is that B being SSP(ω1) can consistentlybe a sufficient condition to assert FAω1(B). They proved the following remarkable theorem:

Theorem 8.5.6 (Foreman, Magidor, Shelah). Let Martin’s maximum be the assertion thatFAω1(B) holds for all B ∈ SSP(ω1). Then Martin’s maximum is consistent relative to theexistence of a supercompact cardinal.

Martin’s maximum (and its variants) has given set theorists and mathematicians a verypowerful tool to obtain independence results: for any given mathematical problem we aremost likely able to compute its (possibly different) solutions in the constructible universeL and in models of Martin’s maximum. Actually much of the motivation of this bookcomes from the search for a sound explanation of the success this axiom has met. Martin’smaximum settles basic problems in cardinal arithmetic like the size of the continuumand the singular cardinal problem (see among others the works of Foreman, Magidor,Shelah [16], Velickovic [48], Todorcevic [46], Moore [34], Caicedo and Velickovic [9], andthe first author [49]), as well as combinatorially complicated ones like the basis problemfor uncountable linear orders (see Moore’s result [35] which extends previous work ofBaumgartner [6], Shelah [39], Todorcevic [45], and others). Interesting problems originatingfrom other fields of mathematics and apparently unrelated to set theory have also beensettled appealing to Martin’s maximum, as it is the case (to cite two of the most prominentexamples) for Shelah’s results [38] on Whitehead’s problem in group theory and Farah’sresult [15] on the non-existence of outer automorphisms of the Calkin algebra in operatoralgebra, which expands on previous works by Steprans, Shelah, Velickovic and others.

We will give a proof of a stronger version of Theorem 8.5.6 in section 13.1. For themoment we have (provably in ZFC) the following inclusions:

Γκ ⊆ Ωκ

for all cardinals κ andΩκ ⊆ SSP(κ)

for all cardinals κ such that P (κ) admits a club subset of size κ, with equality of the firsttwo for κ = ℵ0 and the consistent equality of the last two for κ = ℵ1.

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8.6 Forcing axioms and stationarity II: MM++

Definition 8.6.1. Let P be a partial order FA++ω1

(P ) holds if for all family Dα : α < ω1of dense subsets of P and for all family

Sα : α < ω1

⊆ V P of P -names for a stationary

subset of ω1 (i.e. such thatrSα ⊆ ω1 is stationary

z= 1RO(P )), there is a filter G on P

meeting all the Dα and such that

Sξ =ξ < ω1 : ∃p ∈ Gp P ξ ∈ Sα

is stationary for all α < ω1.

FA++ω1

(Γ) holds if FA++ω1

(P ) holds for all P ∈ Γ. MM++ stands for FA++ω1

(SSP).

Definition 8.6.2. For a regular cardinal λ, let Pλ(V ) be the class of sets Z such that|Z| < λ Z ∩ λ ∈ λ.

(a): Let θ be a regular cardinal, and M ≺ Hθ (so that M |= ZFC \ power-set axiom) bein Pω2(V ). Given B ∈ SSP ∩Hθ, let:

• πM : M → ZM be the transitive collapse of M ,

• Q = πM (B),ZQM = τ ∈ ZM : ZM |= τ is a Q-name .

G ultrafilter on B ∩M is (M,SSP)-correct if letting H = πM [G], we have that:

(i): G is M -generic for B. Equivalently H is ZM -generic for Q, hence

ZM [H] =τH : τ ∈ ZQ

M

is a transitive model of ZFC \ power-set axiom (with τH = σH : τ(σ) ∈ Hfor all τ ∈ ZQ

M ).

(ii): G is (M,SSP)-correct for B: For all S ∈ ZM [H]

V |= S is a stationary subset of ω1

if and only ifZM [H] |= S is a stationary subset of ω1;

i.e. G interprets correctly the B-names for stationary subsets of ω1 in M .

(b): For any B ∈ SSP and λ > |B| we let

T SSPB,λ = M ∈ Pω2(Hλ) : B ∈M and there exists an (M,SSP)-correct generic filter for B

and TB = T SSPB,|B|+ .

Remark 8.6.3. Notice that TB is stationary if and only if T SSPB,λ is stationary for some (any)

λ ≥ |B|+, 2ω1 : for all B-names τ for a stationary subset of ω1 there is σ ∈ H|B|++2ω1 such

that Jσ = τK = 1B (τ is essentially described by ω1-many partitions of B in two pieces,these partitions all belong to H|B|++2ω1 , use them to find σ).

Theorem 8.6.4. Let B be a cba and κ be a cardinal. The following are equivalent:

1. FA++ω1

(B).

141

2. For some cardinal θ with θ > 2|B|, κ, there exists an M ≺ H(θ), |M | = ω1, B ∈M ,ω1 ⊂M and an (M,SSP)-correct generic filter for B.

3. TB is stationary.

Proof. The proof is almost identical to the proof of 8.2.2. We sketch 1 implies 2 and leavethe rest to the reader.

Suppose that FA++ω1

(B) holds and let M ≺ H(θ) be such that B ∈M , ω1 ⊂M , |M | = ω1.

There are at most ω1-many dense subsets of B in M and at most ω1-many B-names S ∈Mfor stationary subsets of ω1 in M . By FA++

ω1(B) there is a filter G meeting all the dense sets

and evaluating each S as a stationary subset of ω1. However, G might not be M -genericsince for some D ∈M , the intersection G ∩D might be disjoint from M . Define:

N =x ∈ H(θ) : ∃τ ∈M ∩ V B ∃q ∈ G Jτ = xK ∈ G

Clearly, N contains M (hence contains ω1), moreover |N | ≤

∣∣M ∩ V B∣∣ = ω1, since every

τ ∈M can be evaluated in at most one way by the elements of the filter G. We can argueas in 8.2.2 that G is N -generic and N ≺ H(θ). Now we show the following:

Claim 1. For all T ∈ N B-name for a stationary subset of ω1, there is S ∈M such thatrS = T

z∈ G and

rS ⊆ ω1 is stationary

z= 1B.

Proof. Suppose T ∈ N is a B-name for a stationary subset of ω1, then for some τ ∈M ∩V B,rτ = ˇT

z∈ G. The following set is therefore in M :

D =A ∈ Hθ : bA =

rτ = ˇA

z> 0B and

rA ⊆ ω1 is stationary

z= 1B

.

Moreover T ∈ D. By the mixing Lemma applied in M ≺ Hθ find S0 ∈M such that

bA ≤rS0 = A

z

for all A ∈ D. Notice that rτ = ˇT

z= bT ∈ G.

HencerT = S0

z≥ bT =

rτ = ˇT

z∈ G, as was to be shown. Notice also that

rS0 ⊆ ω1 is stationary

z=∨D = q.

Let S ∈ M be such thatrS = S0

z= q and

rS = A

z= ¬q for some A ∈ M stationary

subset of ω1. Then S works.

Finally let πN : N → ZN be the transitive collapse of N , H = πN [G], Q = πN (B).Then H is ZN -generic for Q. We must show that any S ∈ ZN [H] which is stationaryaccording to ZN [H] is really stationary in V .

Now fix U ∈ ZQN such that S = UH , hence

q =rU ⊆ ω1 is stationary

z∈ H.

142

By the Mixing Lemma applied in ZN there is V ∈ ZQN such that

rV = U

z= q and

rV = A

z= ¬q for some A ∈ ZN stationary subset of ω1. Hence

rV ⊆ ω1 is stationary

z= 1Q

andrV = U

z≥ q ∈ H. Now V = πN (T ) for some T ∈ N such that

rT ⊆ ω1 is stationary

z= 1B.

By the Claim there is S ∈M such that

rS ⊆ ω1 is stationary

z= 1B.

andrS = T

z∈ G. Therefore

S = UH = VH =

=

α < ω1 :

rα ∈ V

z

Q∈ H

=

=α < ω1 :

rα ∈ T

z

B∈ G

=

=α < ω1 :

rα ∈ S

z

B∈ G

is stationary, since G evaluates S as a stationary subset of ω1.

8.7 Forcing axioms and category forcings

We give some more information on certain type of category forcings and relate these classforcings to the class partial order given by stationary sets, specifically we show that:

• (SSP,≤Ω), and (SSP,≤SSP) are non-trivial class forcing.

• Assuming Martin’s maximum or MM++, there is a natural order and sup preservingmap of these class forcings into the class boolean algebra given by stationary setsconcentraing on Pω2 (V ) ordered by the ≤NS-relation.

As we progress in our analysis of forcing axioms and tower forcings in the next chapters,we will outline nicer and nicer properties of this correspondence, for example we will show(see 13.2.7, 13.2.8) that the natural embedding of (SSP,≤Ω) and (SSP,≤SSP) into thestationary sets concentrating on Pω2 (V ) ordered by the ≤NS-relation is also incompatibilitypreserving in the presence of large cardinals.

Incompatibility in (SSP,Ω)

Fact 8.7.1. (SSP,≤Ω) is non-trivial.

Proof. Observe that if P is Namba forcing on ℵ2 and Q is Coll(ω1, ω2), then RO(P ),RO(Q)are incompatible conditions in (SSP,Ω): If R ≤SSP RO(P ),RO(Q), we would have that if

H is V -generic for R, ωV [H]1 = ω1 (since R ∈ SSP, see 8.5.5) and there are G,K ∈ V [H]

143

V -generic filters for P and Q respectively (since R ≤Ω RO(P ),RO(Q)). G allows to definein V [H] a sequence cofinal in ωV2 of type ω while K allows to define in V [H] a sequencecofinal in ωV2 of type (ω1)V . These two facts entail that V [H] models that cof(ωV1 ) = ω

contradicting the assumption that ωV [H]1 = ω1.

We now bring forward for the specific case of stationary set preserving forcings a dualityrelating certain categories of forcing notions with the category of normal ideals. The lastchapter of the book will develop the property of this duality in its full generality.

Notation 8.7.2. We denote by T λ be the quasi order on V given by S ≤T λ T if lettingX = ∪S ∪ ∪T we have that S ↑ X \ T ↑ X has non-stationary intersection with Pλ (X).

Lemma 8.7.3. Let for each cardinal λ and B ∈ Ωλ SB,λ be the set of M ∈ Pκ+

(H|B|+

)admitting an M -generic filter for B.

The map

Iκ :Ωκ → T κ+

B 7→ SB,κ

is order preserving and maps set sized suprema to set sized suprema.

Proof. Assume i : C→ B witnesses that B ≤Ω C. Let θ be large enough so that i,B,C ∈ Hθ.Let M ∈ SB,κ ↑ Hθ with i,B,C ∈M . Let H be M -generic for B.

Then i−1[H] is M -generic for C. Therefore (SC,κ ↑ Hθ) ∩ Pκ+ (Hθ) contains (SB,κ ↑Hθ) ∩ Pκ+ (Hθ) modulo a club, hence SB,κ ≤T κ SC,κ.

It is also immediate to check that for any set-sized family A of complete booleanalgebras,

SBA,κ =∨B∈A

SB,κ.

In the case we assume MM++ we can say even more:

Lemma 8.7.4. Assume MM++ The map

Iω1 :SSP→ T ω2

B 7→ TB (Recall Def. 8.6.2)

is ≤SSP-preserving and maps set sized suprema to set sized suprema.

Proof. Similar to the previous one: Assume i : C → B witnesses that B ≤SSP C. Let θbe large enough so that i,B,C ∈ Hθ. Let M ∈ TB ↑ Hθ with i,B,C ∈ M . Let H be(SSP,M)-correct for B. Then i−1[H] is M -generic for C. We leave to the reader to checkthat it is also (SSP,M)-correct for C. As before we conclude that TB ≤T κ TC.

We leave to the reader to check the preservation of set-sized suprema.

In section 13.2 we will show that in the presence of class many supercompact cardinalsthese maps are also incompatibility preserving.

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Chapter 9

Properness and semiproperness.

Shelah has isolated two very interesting classes which greatly affects the combinatorics ofP (ω1) and provide a useful insight on the properties of sets of ordinals of size ω1 in modelsof set theory. These are the classes of proper and semiproper forcings. In the remainder ofthis chapter we give a thorough presentation of their basic features and we link some ofthese results to what we have already outlined for the classes Γω1 , Ωω1 , SSP.

9.1 Shelah’s properness and semiproperness

We draw all the results of this section from [41].

Definition 9.1.1. Let X be a transitive set. N is an X-end extension of M if M ⊆ Nand M ∩X = N ∩X.

Assume ∪S ⊇ X and S,C ⊆ P (∪S) with C stationary. S is (X,C)-semistationary if

N ∈ C : N is an X-end extension of some M ∈ S

is a stationary subset of C.

S is semistationary if it is (ω1, [∪S]ℵ0)-semistationary.

Definition 9.1.2. Let B be a complete boolean algebra.

B is proper if for all S stationary subset of [∪S]ℵ0 we have that

qS is stationary

yB

= 1B.

B is semiproper if for all semistationary set S we have that

qS is semistationary

yB

= 1B.

A partial order P is (semi)proper if so is RO(P ).

Recall that a poset P has the CCC (countable chain condition) if all its antichains arecountable.

Lemma 9.1.3. The following holds:

• Countably closed posets and CCC posets are proper.

• Proper posets are semiproper.

145

• Semiproper posets are stationary set preserving.

• Stationary set preserving forcings preserve ω1.

Proof. We prove all implications as follows:

P CCC or countably closed entails P is proper: To argue that P is proper it suf-fices to show that:

P forces that Cf ∩ S is non-empty for any S ∈ V stationary set consisting

of countable sets and f ∈ V P P -name for a function from ∪S<ω → ∪S.

First assume P is CCC. Fix S ∈ V , f ∈ V P as above and pick θ large enough andM ≺ Hθ with f ∈M and Z = M ∩ ∪S ∈ S, which is possible since S is stationary.Now observe that for each s ∈ ∪S<ω we have that f(s) is a P -name for an elementof Z and is equal with boolean value 1RO(P ) to a P -name of the form

τs = 〈ap, p〉 : p ∈ A

with A a maximal antichain of P . Since P is CCC, A is countable.

Now M ≺ Hθ entails that for each s ∈M ∩ Z<ω there is

τs = 〈ap, p〉 : p ∈ A

as above and in M . Since A is countable, As = ap : p ∈ A is also countable andis in M . Hence As = ap : p ∈ A ⊆ M ∩ Z. Let G be V -generic for P . ThenfG(s) = (τs)G ∈ As ⊆ M ∩ Z. Hence Z is fG-closed in V [G]. Since this occurs for

all V -generic filters G, we conclude thatrZ ∈ S ∩ Cf

z

RO(P )= 1RO(P ).

Now assume P is countably closed. Given S ∈ V , f ∈ V P as above, and p ∈ P findagain θ large enough and M ≺ Hθ countable with P, p, f ∈M and Z = M ∩ ∪S ∈ S.Now let sn : n ∈ ω enumerate Z<ω. Build inside M a decreasing chain pn : n ∈ ωsuch that

• p0 = p,

• pn+1 ∈M refines pn and decides the value of f(sn).

By induction each pn ∈ M . Since pn+1 ∈ M and M ≺ Hθ, applying the TarskiVaught criterion to the formula with parameters in M

∃x (pn+1 P f(sn) = x),

we get that pn+1 decides that the value of f(sn) is an ordinal in M ∩∪S for all n ∈ ω.Since P is countably closed in V , we can find q ∈ P which refines all the pn for alln ∈ ω.

Then q forces that f [Z<ω] ⊆ Z, hence q forces that S ∩ Cf 6= ∅. Since the argumentcan be repeated below any p ∈ P we conclude that there is a dense set of q ∈ Pwhich force that S ∩ Cf 6= ∅. Hence

rS ∩ Cf 6= ∅

z

RO(P )= 1RO(P ).

146

B proper implies B semiproper: Assume B is proper in V . It suffices to show that anyS ∈ V semistationary in V remains semistationary in V [G].

Remark that an S ⊆ [X]ℵ0 with ∪S = X is semistationary in V if and only if

S =N ∈ [X]ℵ0 : ∃M ∈ S N ⊇M and N ∩ ω1 = M ∩ ω1

is stationary in V , and S ⊆ [X]ℵ0 is semistationary in V [G] if and only if

SV [G] =N ∈ ([X]ℵ0)V [G] : ∃M ∈ S N ⊇M and N ∩ ω1 = M ∩ ω1

is stationary in V [G].

Remark also that S ⊆ SV [G].

Now assume G is V -generic for B. Then S is stationary in V [G] (since B is proper),hence so is SV [G] being a superset of S. Since G is an arbitrarily chosen V -genericfilter for B, we conclude by the forcing theorem.

B semiproper implies B is stationary set preserving: This is an immediate conse-quence of the following observation (which is left to the reader):

S is a stationary subset of ω1 if and only if S is a semistationary subset ofω1.

Stationary set preserving forcings preserve ω1: Proved in 8.5.5.

The above results brings us the following picture regarding classes of forcing notions:

Remark 9.1.4. Let CCC, Γω1 , PR, SP, SSP, Ωω1 denote respectively the class of CCC,countably closed, proper, semiproper, stationary set preserving, and posets P satisfyingFAℵ1(P ). The following holds in ZFC:

CCC

<ω1-closed Ωℵ1

proper SP SSP⊆

⊆

⊇

⊆ ⊆

⊆

The work of Foreman, Magidor, Shelah shows that the equality

SP = SSP = Ωω1

is consistently possible and holds assuming Martin’s maximum (we will prove it in sec-tion 13.1).

There are several equivalent characterizations of properness and semiproperness, andwe will need the following:

Definition 9.1.5. (Shelah) Let P be a partial order, and fix M ≺ Hθ.

• q is an M -generic condition for P iff for every α ∈ V P ∩M such that 1P α ∈ θ,

q α ∈ M.

147

• q is an M -semigeneric condition for P iff for every α ∈ V P∩M such that 1P α < ω1,

q α < M ∩ ω1.

Notation 9.1.6. Assume P is a partial order, X is a set, G is a V -generic filter for P .

X[G] =σG : σ ∈ X ∩ V P

Remark 9.1.7. Note that whenever M ≺ Hθ and q ∈ P is M -(semi)generic for P , so is anyrefinement of q. Moreover q is M -(semi)generic for P if and only if for all G V -generic forP with q ∈M , we have that M [G] ∩ θ = M ∩ θ (M [G] ∩ ω1 = M ∩ ω1).

Fact 9.1.8. Let P be a partial order and θ > |P | be a cardinal. For all countable M ≺ Hθ

with P ∈M and for all q ∈ P the following holds:

(a) q is M -semigeneric for P if and only if it is M ∩H|P |++ω2-semigeneric for P .

(b) q is M -generic for P if and only if it is M ∩H|RO(P )|+-generic for P .

Proof. This is a consequence of the fact that all predense subsets of P are in H|P |+ ⊆H|RO(P )|+ , H|P |++ω2

. In particular any P -name τ for an ordinal is equal with boolean value1RO(P ) to one of the form

〈αp, p〉 : p ∈ Awith A a maximal antichain of P and αp an ordinal for each p ∈ A. Hence all P -namesfor ordinals below (ω1)V are equal to one which belongs to H|P |++ω2

. This immediatelyentails that for all countable M ≺ Hθ, q ∈ P is M -semigeneric for P if and only if it isM ∩H|P |++ω2

-semigeneric for P .It is slightly more delicate to argue that q is M -generic for P if and only if it is

M ∩H|RO(P )|+-generic for P .The reader can work out the details of this equivalence using the results given in

section 9.2: essentially one argues that q is M -generic for P if and only if

q ≤∧

RO(P )

∨RO(P )

(A ∩M) : A ∈M is a maximal antichain of P

(see in particular Propositions 9.2.2, 9.2.6). Since this is proved in full details there, wedecide to omit the proof here.

Notation 9.1.9. For each partial order P and θ such that P ∈ Hθ

S(P, θ) =M ∈ [Hθ]

ℵ0 : for all p ∈ P ∩M there is q ≤ p M -generic for P

and

S(P, θ, ω1) =M ∈ [Hθ]

ℵ0 : for all p ∈ P ∩M there is q ≤ p M -semigeneric for P.

Fact 9.1.10. For a partial order P and for all θ ≤ κ with P ∈ Hθ we have that

S(P, θ) ↑ Hκ ∩ [Hκ]ℵ0 =NS S(P, κ),

S(P, θ, ω1) ↑ Hκ ∩ [Hκ]ℵ0 =NS S(P, κ, ω1),

andS(P, κ) ↓ Hθ =NS S(P, θ),

S(P, κ, ω1) ↓ Hθ =NS S(P, θ, ω1).

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Proof. These are immediate consequences of Fact 9.1.8 above.

Lemma 9.1.11. Assume (P,≤) is a partial order The following holds:

• P is proper if and only if S(P, θ) =NS [Hθ]ℵ0 for some (any) θ > |RO(P )|.

• P is semiproper if and only if S(P, θ, ω1) =NS [Hθ]ℵ0 for some (any) θ > |P |+ ω1.

Proof. We start with a proof of the first item:

⇐: Assume P is not proper and let S witness this fact. Pick θ such that P (P ) , S ∈ Hθ. Itsuffices to show that there is some p ∈ P and a stationary set of countable M ≺ Hθ

which do not have an M -generic condition refining p.

First of all w.l.o.g. we can assume that ∪S is a cardinal κ smaller than θ. Since S isnon-stationary in V [G] for some G V -generic for P , we can find p ∈ G such that

JS is not stationaryK ≥ p

Hence we can find a P -name f for a function from κ<ω → κ such that for all Z ∈ Srf [Z<ω] 6= Z

z

B≥ p.

For each Z ∈ S we can pick a P -name s for an element of Z<ω such that

rf(s) 6∈ Z

z

B≥ p.

Now assume towards a contradiction that there is a club C of countable M ≺ Hθ

with p, f , P ∈ M , each of them admitting an M -generic condition qM refining p.Pick M ∈ (S ↑ Hθ) ∩ C and G V -generic for P with qM ∈ G. Let Z = M ∩ ∪S andfind r ∈ G, s ∈ Z<ω, α 6∈ Z such that r forces that s = s and that f(s) = α. Nowobserve that the P -name f(s) for an ordinal in κ belongs to M and r forces thatf(s) ∈ κ \M . Hence qM cannot be M -generic, contradicting our assumptions.

⇒: Assume that for some θ > 2|P | there is a stationary set of countable models M ≺ Hθ

and some pM ∈ P ∩M such that no refinement of pM is M -generic for P .

By pressing down (in V ) on this stationary set, we can find S ∈ V stationary set(in V ) of countable models M ≺ Hθ and a fixed p ∈ P such that for all M ∈ S norefinement of p is M -generic for P .

It is enough to argue that S ∈ V is not anymore stationary in V [G], whenever G is aV -generic filter for P with p ∈ G. Towards a contradictiona assume S is stationaryin V [G] for some G as above. Since no q ∈ G is M -generic for P for no M ∈ S, wehave that M [G] ∩ θ 6= M ∩ θ for all M ∈ S. Hence for each M ∈ S we can findαM ∈M ∩ V P P -name for an ordinal, such that (αM )G ∈ θ \M . By pressing down(in V [G]) on S (which is possible, since we are assuming that S is stationary in V [G])we can find a stationary (in V [G]) T ⊆ S and α ∈ V P such that αM = α for allM ∈ T and (α)G = α 6∈ M for all M ∈ T . Pick M ∈ T with α ∈ M (which ispossible since T is stationary in V [G]). Then M 3 α = (αM )G 6∈M , a contradiction.This concludes the proof of the left to right implication.

We now prove the second item:

149

⇐: Assume P is not semiproper and let S witness this fact with ∪S = X ⊇ ω1. Hence

SV [G] =N ∈ ([X]ℵ0)V [G] : ∃M ∈ S such that N ω1-end extends M

is non-stationary in V [G] for some G V -generic for P . Hence so is

S =N ∈ [X]ℵ0 : ∃M ∈ S such that N ω1-end extends M

,

since S ⊆ SV [G]. On the other hand S is stationary in V since S is semistationary inV .

Let S be a P -name such that SG = SV [G].

Pick θ such that P (P ) , S, S ∈ Hθ. It suffices to show that there is some p ∈ P and astationary set of countable M ≺ Hθ which do not have an M -semigeneric conditionrefining p.

Since SV [G] is non-stationary in V [G] for some G V -generic for P , we can findf : κ<ω → κ in V [G] such that Cf ∩ SV [G] is empty and Y ⊆ f [Y <ω] ∈ Cf for

all Y ∈ P (X)V [G] (by further refining an f ∈ V [G] such that Cf ∩ SV [G] is emptyeventually applying 2.2.3 in V [G] to f).

This gives that f [Z<ω] ∩ (ω1)V 6= Z ∩ (ω1)V for all Z ∈ S, else f [Z<ω] ∈ SV [G] ∩ Cf ,contradicting our assumptions on f .

Hence we can find p ∈ G and f ∈ V P such that:

• fG = f ,

• for all Z ∈ S rf [Z<ω] ∩ ω1 6= Z ∩ (ω1)V

z

B≥ p,

• for all Z ∈ S rZ ⊆ f [Z<ω] which is f -closed

z

B≥ p.

Working now in V , for each Z ∈ S we can pick in V a P -name s for an element ofZ<ω such that r

f(s) ∈ (ω1)V \ Zz

B≥ p.

Now assume towards a contradiction that there is a club C of countable M ≺ Hθ

with p, f , P ∈M , each of them admitting an M -semigeneric condition qM refiningp. Pick M ∈ (S ↑ Hθ) ∩ C (which is possible cince C is a club in [Hθ]

ℵ0 and S isstationary in V ) and G V -generic for P with qM ∈ G. Let Z = M ∩ X ∈ S andfind r ∈ G, s ∈ Z<ω, α ∈ (ω1)V \ Z such that r forces that s = s and that f(s) = α.Now observe that the P -name f(s) for an ordinal in (ω1)V belongs to M and rforces that f(s) ∈ (ω1)V \M . Hence qM cannot be M -semigeneric, contradicting ourassumptions.

⇒: Assume that for some θ > 2|P | there is a stationary set of countable models M ≺ Hθ

with some pM ∈ P ∩M such that no refinement of pM is M -semigeneric for P .

By pressing down (in V ) on this stationary set, we can find S ∈ V stationary set(in V ) of countable models M ≺ Hθ and a fixed p ∈ P such that for all M ∈ S norefinement of p is M -semigeneric for P .

150

It is enough to argue that S ∈ V is not anymore semistationary in V [G], wheneverG is a V -generic filter for P with p ∈ G. Towards a contradiction assume S issemistationary in V [G] for some G as above. Hence

SV [G] =N ∈ ([(Hθ)

V ]ℵ0)V [G] : ∃M ∈ S such that N (ω1)V -end extends M

is stationary in V [G].

Since no q ∈ G is M -semigeneric for P for any M ∈ S, we have that

M [G] ∩ (ω1)V 6= M ∩ (ω1)V

for all M ∈ S. On the other hand we claim the following:

Claim 2. In V [G] it holds that

(M ∩HVθ )[G] =

σG : σ ∈M ∩ V B

= M

for a club of M ≺ HV [G]θ .

Proof. One inclusion is clear for any M ≺ HV [G]θ . For the other, pick M ≺ H

V [G]θ+

with P,HVθ ∈M . Then M ≺ HV [G]

θ+ models that HV [G]θ = HV

θ [G] by 4.2.4 applied in

HV [G]θ+ to HV

θ , HV [G]θ and elementarity of M . Hence the thesis.

Now pick N ∈ SV [G] with N ≺ HV [G]θ and N = (N ∩HV

θ )[G]. Hence there existsM ∈ S such that N ∩ (ω1)V = M ∩ (ω1)V and N ⊇M . Therefore

M ∩ (ω1)V = N ∩ (ω1)V = (N ∩HVθ )[G] ∩ (ω1)V ⊇M [G] ∩ (ω1)V ⊇M ∩ (ω1)V ,

contradicting our assumption that

M [G] ∩ (ω1)V 6= M ∩ (ω1)V

for all M ∈ S.

9.2 Algebraic definition of properness and semiproperness

We now introduce another equivalent definition of (semi)properness which is more convenientfor our treatment of iterated forcing in Chapter 10.

Definition 9.2.1. Let B be a complete boolean algebra, PD(B, ω1) is the collection ofpredense subsets of B of size at most ω1.

Fix M ≺ Hθ countable for some θ > |B|. The boolean value

gen(B,M) =∧∨

(D ∩M) : D ∈ PD(B) ∩M

is the degree of genericity of M with respect to B.The boolean value

sg(B,M) =∧∨

(D ∩M) : D ∈ PD(B, ω1) ∩M

is the degree of semigenericity of M with respect to B.

151

Most of the proofs to follow are modular and apply equally well to the boolean valuesgen(B,M) and sg(B,M). Hence we prove all results just for the case of sg and leave tothe reader the corresponding proof for gen.

The next results show that the degree of (semi)genericity can be also calculated frommaximal antichains, and behaves well with respect to the restriction operation.

Proposition 9.2.2. Let B, M be as in the previous definition, and let A(B) be the collectionof maximal antichains of B, A(B, ω1) be the collection of maximal antichains of B of sizeat most ω1. Then

g(B,M) =∧∨

(A ∩M) : A ∈ A(B) ∩M

sg(B,M) =∧∨

(A ∩M) : A ∈ A(B, ω1) ∩M

Proof. Since A(B, ω1) ⊆ PD(B, ω1), the inequality

sg(B,M) ≤∧∨

(A ∩M) : A ∈ A(B, ω1) ∩M

is trivial. Conversely, if D = bα : α < ω1 ∈ PD(B, ω1) ∩M , define

AD =aα = bα ∧ ¬

∨bβ : β < α : α < ω1

By elementarity, since D ∈ M , also AD is in M . It is straightforward to verify that ADis an antichain, and since

∨AD =

∨D = 1, it is also maximal. Moreover, since aα ≤ bα

we have that∨AD ∩M ≤

∨D ∩M . Thus, for any D ∈ PD(B, ω1) ∩M , we have that∧

∨

(A ∩M) : A ∈ A(B, ω1) ∩M ≤∨D ∩M hence∧∨

(A ∩M) : A ∈ A(B, ω1) ∩M≤ sg(B,M)

The thesis follows.

Proposition 9.2.3. Let B be a complete boolean algebra and M ≺ Hθ for some θ > |B|.Then for all b ∈M ∩ B

sg(B b,M) = sg(B,M) ∧ b.

andgen(B b,M) = gen(B,M) ∧ b.

Proof. Observe that if A is a maximal antichain in B, then A ∧ b = a ∧ b : a ∈ Ais a maximal antichain in B b. Moreover for each maximal antichain Ab in B b ∩M ,A = Ab ∪ ¬b is a maximal antichain in B ∩M . Therefore

sg(B,M) ∧ b =∧∨

(A ∩M) ∧ b =∧∨

((A ∧ b) ∩M) = sg(B b,M).

We are now ready to introduce the algebraic definition of semiproperness and propernessfor complete boolean algebras and regular embeddings.

Definition 9.2.4. Let B be a complete boolean algebra, S be a stationary set on Hθ withθ > |B| and S ⊆ [Hθ]

ℵ0 .B is S-SP if for club many M ∈ S whenever b is in B∩M , we have that sg(B,M)∧b > 0B.

152

i : B→ C is S-SP if B is S-SP and for club many M ∈ S, whenever c is in C ∩M wehave that

π(c ∧ sg(C,M)) = π(c) ∧ sg(B,M).

B is S-proper if for club many M ∈ S whenever b is in B∩M , we have that gen(B,M)∧b > 0B.

i : B → C is S-proper if B is S-proper and for club many M ∈ S, whenever c is inC ∩M we have that

π(c ∧ gen(C,M)) = π(c) ∧ gen(B,M).

The previous definitions can be reformulated with a well-known trick in the followingform.

Proposition 9.2.5. B is S-SP iff for every ν θ regular, M ≺ Hν with B, S ∈ M andM ∩Hθ ∈ S we have that sg(B,M) ∧ b > 0 ∀b ∈ B ∩M .

i : B→ C is S-SP iff B is S-SP and for every ν θ regular, M ≺ Hν with i, S ∈ Mand M ∩Hθ ∈ S

π(c ∧ sg(C,M)) = π(c) ∧ sg(B,M)

∀c ∈ C ∩M .Similarly B is S-proper iff for every ν θ regular, M ≺ Hν with B, S ∈ M and

M ∩Hθ ∈ S we have that gen(B,M) ∧ b > 0 ∀b ∈ B ∩M .i : B → C is S-proper iff B is S-proper and for every ν θ regular, M ≺ Hν with

i, S ∈M and M ∩Hθ ∈ S

π(c ∧ gen(C,M)) = π(c) ∧ gen(B,M)

∀c ∈ C ∩M .

Proof. First, suppose that B, i : B→ C satisfy the above conditions. Then C = M ∩Hθ :M ≺ Hν , B, S ∈M is a club (since it is the projection of a club), and witnesses that B,i : B→ C are S-SP.

Conversely, suppose that B, i : B → C are S-SP and fix ν θ regular and M ≺ Hν

with B, S ∈ M , M ∩Hθ ∈ S. Since the sentence that B, i : B → C are S-SP is entirelycomputable in Hν and M ≺ Hν , there exists a club C ∈M such that ∪C = Hθ witnessingthat B, i : B→ C are S-SP. Furthermore, M models that C is a club hence M ∩Hθ ∈ C(see Fact 2.1.5) and sg(B,M) ∧ b > 0, π(c ∧ sg(C,M)) = π(c) ∧ sg(B,M) holds for anyb ∈ B∩M , c ∈ C∩M since C witnesses that B, i : B→ C are S-SP and M∩Hθ ∈ S∩C.

We may observe that if i : B→ C is S-SP, then C is S-SP. As a matter of fact c ∈ C∩Mis such that sg(C,M) ∧ c = 0 iff

0 = π(c ∧ sg(C,M)) = π(c) ∧ sg(B,M),

this contradicts the assumption that B is S-SP.Definition 9.2.1 is equivalent to the original Shelah’s notion of semiproperness and

properness in case S is a club of countable models. We now spell out the details of thisequivalence.

Proposition 9.2.6. Let B be a complete boolean algebra, and fix M ≺ Hθ countable. Then

sg(B,M) =∨q ∈ B : q is an M -semigeneric condition for B

andgen(B,M) =

∨q ∈ B : q is an M -generic condition for B

153

Proof. Given A = aβ : β < ω1 ∈ A(B), define αA = 〈γ, aβ〉 : γ < β < ω1. It isstraightforward to check that JαA < ω1K =

∨qαA = β

y: β < ω1

=∨aβ : β < ω1 = 1.

Conversely, given α ∈ V B∩M such that Jα < ω1K = 1, defineAα =aβ =

qα = β

y: β < ω1

.

It is straightforward to check that Aα ∈ A(B).Suppose now that q is an M -semigeneric condition, and fix an arbitrary A ∈ A(B) ∩M .

Then αA ∈M and JαA < ω1K = 1, hence

q ≤rαA < ˇ(M ∩ ω1)

z=∨q

αA = βy

: β ∈M ∩ ω1

=∨aβ : β ∈M ∩ ω1 =

∨A ∩M

It follows that q ≤∧∨

(A ∩M) : A ∈ A(B) ∩M = sg(B,M), hence

sg(B,M) ≥∨q ∈ B : q is a M -semigeneric condition

Finally, we show that sg(B,M) is a M -semigeneric condition itself. Fix an arbitraryα ∈ V B ∩M such that 1B α < ω1, and let Aα ∈ Aω1(B) be as above. Since α ∈M , alsoAα ∈M . Moreover,

rα < ˇ(M ∩ ω1)

z=∨q

α = βy

: β ∈M ∩ ω1

=∨aβ : β ∈M ∩ ω1 =

∨Aα ∩M ≥ sg(B,M)


Corollary 9.2.7. Let P be a partial order, then P is semiproper (respectively proper) ifand only if RO(P ) is C-SP (respectively C-proper) for some C club subset of [Hθ]

ℵ0 andθ > 2|P |.

Proof. First, suppose that P is semiproper in the sense of Shelah as witnessed by a club Cof countable elementary submodels of some Hθ with θ > 2|P |. Fix M ∈ C, b ∈ RO(P )∩M .Since P is dense in RO(P ), there exists a p ∈ P ∩M , p ≤ b, and by semiproperness thereexists a q ∈ P , q ≤ p ≤ b that is M -semigeneric. Then q > 0 and by Proposition 9.2.6,q ≤ sg(RO(P ),M). Hence sg(RO(P ),M) ∧ b ≥ q > 0.

Finally, suppose that RO(P ) is C-SP for some club of countable models C as above,and fix M ∈ C, p ∈ P ∩ M . Since P is dense in RO(P ), there exists a q ∈ P , q ≤sg(RO(P ),M) ∧ p. Then q is an M -semigeneric condition, since q ≤ sg(RO(P ),M) andthe set of semigeneric conditions is open.

9.3 Topological characterization of properness and semiproper-ness

An equivalent definition of semiproperness and properness can be stated also in topologicalterms, as a Baire Category property. Let B be a complete boolean algebra and St(B) be thespace of its ultrafilters defined in 1.1.3. The Baire Category Theorem states that given anyfamily of maximal antichains An : n ∈ ω of B, then

⋂n∈ω

⋃Na : a ∈ An is comeager

in St(B), so˚⋂

n∈ω

⋃Na : a ∈ An = St(B).

154

Now, let M ≺ Hθ, B ∈M , then if An : n ∈ ω is a subset of the set of the maximalantichains of B ∈M , the classical construction of an M -generic filter shows that⋂

n∈ω

(⋃Na : a ∈ An ∩M

)6= ∅.

However this does not guarantee that⋂n∈ω

(⋃Na : a ∈ An ∩M

)is comeager on some Nb in V,

i.e. that the above set has the Baire property in St(B). This latter requirement is exactlythe request that B is proper.

Proposition 9.3.1. B is proper if and only if ∀M ≺ Hθ with B ∈M , M countable

XM =⋂⋃

Na : a ∈ A ∩M : A ∈M maximal antichain of B

is such that for all c ∈M ∩ B there is b ∈ B with XM a comeager set on Nb ∩Nc.

Proof. As a matter of fact

∀c ∈M ∩ B ∃b(Nb ⊆ XM ∩Nc)

⇐⇒ ∀c ∈M ∩ B∃b ≤∧∨

(A ∩M) : A ∈M maximal antichain on B∧ c.

Proposition 9.3.2. B is semiproper if and only if ∀M ≺ Hθ with B ∈M , M countable

XM =⋂⋃

Na : a ∈ A ∩M : A ∈M maximal antichain of B, |A| = ω1

is such that for all c ∈M ∩ B there is b ∈ B with XM a comeager set on Nb ∧Nc.

9.4 FA+ω1

(Countably closed) implies SP = SSP

We will need the results of this section to prove the consistency of MM++.

Definition 9.4.1. FA+ω1

(P ) holds if for all family Dα : α < ω1 of dense subsets of P and

for all S ∈ V P P -name for a stationary subset of ω1 there is a filter G on P meeting all

the Dα and such thatα : ∃p ∈ Gp P α ∈ S

is stationary.

FA+ω1

(Γ) holds if FA+ω1

(P ) holds for all P ∈ Γ.

Theorem 9.4.2. Assume FA+ω1

(Γω1), where Γω1 is the class of posets admitting a countablyclosed dense subset. Then SP = SSP.

Proof. The inclusion SP ⊆ SSP holds in ZFC. Assume the other inclusion fails and let Qbe in SSP \ SP. Let θ be such that Q ∈ Hθ and let

S =M ∈ [Hθ]

ℵ0 : ∃qM ∈M ∩Q such that sg(RO(Q),M) ∧ qM = 0RO(Q)

.

Since Q 6∈ SP, S is stationary. By pressing down on S we can find q ∈ Q and T ⊆ Sstationary such that qM = q for all M ∈ T .

Consider the poset P given by continuous chains Mα : α < β of countable elementarysub-models of Hθ (i.e. such that Mγ =

⋃Mα : α < γ for all γ < β limit) ordered by

reverse inclusion. Clearly P is countably closed. Let

T = 〈α, p〉 : p = Mξ : ξ < β such that β > α and Mα ∈ T

155

Claim 3. P forces that T is a stationary subset of ω1.

Proof. Fix C P -name for a club subset of ω1. Given p ∈ P , find a countable M ≺ Hν

with ν >> θ with C, T , P ∈ M and M ∩Hθ ∈ T . Let M = xn : n ∈ ω and M ∩ ω1 =αn : n ∈ ω. Build inside M a decreasing chain pn : n ∈ ω such that p0 = p, for someβ ∈ (αn,M ∩ ω1) p2n+1 β ∈ C, for some N ∈ p2n+2 xn ∈ N and otp(p2n+2) > αn(i.e. p2n+2 = Mα : α < ν for some ν > αn). We leave to the reader to check thatq =

⋃pn : n ∈ ω ∪ M ∩Hθ ∈ P , q = Mξ : ξ ≤M ∩ ω1 with MM∩ω1 = M ∩Hθ, and

q forces that M ∩ ω1 ∈ C ∩ T .

By FA+ω1

(P ), find G filter on P such that G ∩Dα 6= ∅ for all α ∈ ω1 (where Dα is thedense subset of P given by p = Mξ : ξ < β with β ≥ α) and

T0 =α < ω1 : ∃p ∈ Gp P α ∈ T

is stationary. Let

Mα : α < ω1 = ∪G

and M =⋃Mα : α < ω1. Then M ≺ Hθ has size ω1 and contains ω1, Mα : α < ω1 is

a club subset of [M ]ℵ0 , and Mα : α ∈ T0 is a stationary subset of [M ]ℵ0 . Notice thatα ∈ T0 if and only if Mα ∈ T , giving that sg(RO(Q),Mα) ∧ q = 0RO(Q) for all α ∈ T0.

Now assume H is V -generic for Q with q ∈ H. Since Q ∈ SSP, we get that ωV [H]1 = ωV1

and T0 remains a stationary subset of ω1 in V [H]. Let

M [H] =σH : σ ∈M ∩ V P

and Mα[H] =

σH : σ ∈Mα ∩ V P

. Then in V [H] it holds that

M [H] =⋃Mα[H] : α < ω1 ,

Mα[H] : α < ω1 is a club subset of [M [H]]ℵ0 and Mα[H] : α ∈ T0 is a stationary subsetof [M [H]]ℵ0 .

Sinceω1 =

⋃Mα[H] ∩ ω1 : α < ω1 = Mα ∩ ω1 : α < ω1 ,

we can find in V [H] a club C of α < ω1 such that

Mα[H] ∩ ω1 = Mα ∩ ω1 = α.

Hence in V [H] we can find α ∈ T0 ∩ C. Therefore

V [H] |= Mα[H] ∩ ω1 = Mα ∩ ω1

i.e. sg(RO(Q),Mα) ∈ H. Since q ∈ G, this gives that sg(RO(Q),Mα) ∧ q ∈ H. Butα ∈ T0 if and only if Mα ∈ T , giving that sg(RO(Q),Mα) ∧ q = 0RO(Q). We reached acontradiction.

156

9.5 A common framework for the classes of proper semiproperand stationary set preserving forcings

The following is a curiosity, which does not have any specific application other thanoutlining some common properties shared by proper, semiproper and SSP-forcings.

Definition 9.5.1. Let P (x) be a property definable by means of a formula with parametersin one free variable ranging over sets. ΓP is the class of complete boolean algebras B suchthat

P (S)⇔qP (S)

yB

= 1B

for all S ∈ V .

We will be mainly interested in classes ΓP defined by a Π1-property ∀yφP (x, y, aP ),aP ⊆ κ for some cardinal κ, φP (x, y, z) a ∆0-formula in free variables for sets x, y, z, and∀y a quantifier ranging only over sets (and not on proper classes).

Notation 9.5.2. We write that i : B → C is P -correct to specify that it is ΓP -correct,B ≤P Q to specify that B ≤ΓP Q, and we let for Π1 properties P (x) ∀yφP (x, y, aP ) be theΠ1-formula whose extension in V is the class ΓP .

There is a tight interaction between the properties of a class of forcings Γ and thetheory T ⊇ ZFC in which we analyze this class. For example in our analysis of ΓP , we arenaturally led to work with theories T which extend ZFC but which are not preserved by allset sized forcings. For example this occurs for T = ZFC + ω1 is a regular cardinal whichis not preserved by Coll(ω, ω1), but is preserved by all SSP(ω1) forcings.

Remark 9.5.3. The class SP of semiproper complete boolean algebras, the class PR ofproper complete boolean algebras, and the class SSP of stationary set preserving forcingsare all of the form ΓP with P a Π1-definable property in the parameter ω1:

PR: Let ∀yφPR(x, y, ω1) be the formula stating that x is stationary and consists of countablesets, i.e.:

φPR(x, y, ω1) ≡ “ (∪x ⊇ ω1) ∧ (∀w ∈ x w is countable1) ∧ [(y : (∪x)<ω →∪x is a function) → (∃w ∈ x y[w<ω] ⊆ w)]”

SP: Let ∀yφSP(x, y, ω1) be the formula stating that x is semistationary and consists ofcountable sets, i.e.:

φSP(x, y, ω1) ≡ “(∪x ⊇ ω1) ∧ (∀w ∈ x w is countable) ∧ [(y : (∪x)<ω → ∪xis a function) → (∃w ∈ x w ⊆ y[w<ω] ∧ y[w<ω] ∩ ω1 = w ∩ ω1)]”

SSP: Let ∀yφSSP(x, y, ω1) be the formula stating that x is a stationary subset of ω1:

φSP(x, y, ω1) ≡ “(y is a club subset of ω1) → (∃w ∈ x ∩ y)”

SSP(κ): this is the class ΓP given by posets B which preserve stationary subsets of P (κ)and is defined by the property P (x) ≡ ∀yφP (x, y, κ) with φP given by

1This is the ∆1-property (over the theory ZFC) in the parameter ω1 given by the Σ1-formula

φ0(w) ≡ ∃α ∈ ω1∃f : α→ w bijective

and the Π1-formulaφ1(w) ≡ ∀f : ω1 → w f is not bijective.

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φP (x, y, κ) ≡ “(∀w ∈ x |w| < κ) ∧ [(y : κ<ω → κ is a function) →(∃w ∈ x y[w<ω] ⊆ w)]”

φPR(x, y, ω1), φSP(x, y, ω1), φSSP(x, y, ω1), φSSP(x, y, κ) are Π1-properties over the theoryZFC and the definition of properness (semiproperness, stationary set preserving) given bythese formulae witness that these classes are of the form ΓP with P (x) Π1-definable in theparameter ω1 (or κ for the last one).

The property of B being proper or semiproper is provably ∆2 in ZFC. B is (semi)properif and only if it satisfies:

• The Σ2-statement in the parameters B, ω

There is2 Hθ with B ∈ Hθ and C club subset of Hθ such that for allcountable M ∈ C with B ∈ M and M ≺ Hθ there exists r ∈ B which isM -(semi)generic and is compatible with all q ∈ B ∩M .

• The Π2-statement in the parameters B, ω

For all Hθ with P (B) ∈ Hθ, there is C club subset of H|B|+ such that forall countable M ∈ C with B ∈M and M ≺ H|B|+ there exists r ∈ B whichis M -(semi)generic and is compatible with all q ∈ B ∩M .

Observe that both statements hold in Vδ for some inaccessible δ if and only if they hold inV . Using this characterization of (semi)properness in a generic extension of B, one cancheck that i : B→ C is P -correct for the corresponding property P if and only if Vδ modelsthat i : B→ C is P -correct for any inaccessible δ to which i,B,C belong.

Also B being a stationary set preserving forcing is expressible by a ∆2-property inparameters B, ω which is absolute between Vδ and V for any inaccessible δ to which Bbelongs: checking whether B is stationary set preserving requires to test the preservationof stationary subsets of ω1 by quantifying just over B-names belonging to H|B|++ω2

. Weleave the details to the reader.

Fact 9.5.4. Assume P (x) is a Π1-property definable in the parameter aP ⊆ κ by theΠ1-formula ∀yφP (x, y, aP ). Then ZFC + ap ⊆ κ, κ is regular proves that ΓP is closedunder two step iterations, lottery sums and preimages of complete homomorphisms.


2The statement X = Hθ for some θ is Π1: it is the conjunction of the ∆1-properties φ0(X) ≡X istransitive and φ1(X) ≡ 〈X,∈〉 |= ZFC \ P and of the Π1-property φ2(X) ≡ ∀Z ∈ X∀Y (Y ⊆ Z ↔ Y ∈ X).

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Chapter 10

Proper and semiproper iterations

In this chapter we prove that iterations of (semi)proper boolean algebras are (semi)proper.We will use these results to infer the consistency of MM++ in Section 13.1 and to analyzecategory forcings in the last part of the book.

In the first part of the chapter we examine the case of two-step iterations, in the secondpart we focus on the limit case. Since the proof of the relevant results for properness iseasier, we just give detailed formulations and proofs for the semiproper case.

10.1 Two-steps iterations of (semi)proper posets

The notion of being S-SP can change when we move to a generic extension: for example, Scan be no longer stationary. In order to recover the “stationarity” in V [G] of an S whichis stationary in V , we are led to the following definitions:

Notation 10.1.1. Given a complete boolean algebra B ∈ V , a V -generic ultrafilter G onB, and a set M ⊆ V in V [G],we let

M [G] = aG : a ∈M ∩ V B.

Definition 10.1.2. Let S be a subset of P (Hθ), B ∈ Hθ be a complete boolean algebra,and G a V -generic filter for B. We define

S(G) = M [G] : B ∈M ∈ S.

Fact 10.1.3. Let S be a stationary set on Hθ, B ∈ Hθ be a complete boolean algebra, andG be a V -generic filter for B. Then S(G) is stationary in V [G].

Proof. Let f ∈ V B be such that fG : (HV [G]θ )<ω → H

V [G]θ . By 4.2.4 we can assume that

f ∈ HVθ+ . Let M ≺ HV

θ+ be such that M ∩HVθ ∈ S, and B, f , Hθ ∈M . Then fG ∈M [G],

hence M [G] ∩HV [G]θ ∈ CfG .

Claim 4.M [G] ∩HV [G]

θ = (M ∩HVθ )[G]

Proof. The right to left inclusion is clear. For the left to right inclusion, let σG ∈M [G] ∩HV [G]

θ for some σ ∈ M . By 4.2.4 applied in HVθ+ there exists τ ∈ HV

θ ∩ V B suchthat Jτ = σK = 1B. By elementarity of M , we can suppose that τ ∈ M as well. HenceτG = σG and τ ∈M ∩HV

θ .

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Thus M [G] ∩HV [G]θ ∈ S(G) ∩ CfG . The thesis follows.

Proposition 10.1.4. Let S be a stationary set of countable sets. Let B be an S-SPcomplete boolean algebra, and let C be such that

rC is S(G)-SP

z

B= 1B,

then D = B ∗ C and iB∗C are S-SP.

Proof. We start to verify that i = iB∗C is S-SP. Fix θ large enough Let C ∈ V B ∩Hθ be

the club that witnessesrC is S(G)-SP

z

B= 1, and let M ∈ S ↑ Hθ be countable and such

that C ∈M and M ≺ Hθ: this guarantees that V [G] M [G] ∩HV [G]θ ∈ CG.

We first prove that π(sg(D,M)) = sg(B,M). By Lemma 6.1.1 we obtain sg(D,M) ≤i(sg(B,M)), hence

π(sg(D,M)) ≤ (sg(B,M)).

Now we have to prove π(sg(D,M)) ≥ (sg(B,M)). Let G be V -generic for B withsg(B,M) ∈ G and C = CG = D/G, then, by the S-semiproperness of B,

V [G] M ∩ ω1 = M [G] ∩ ω1.

Therefore, by Lemma 6.1.6, in V [G] we get that

[sg(D,M)]i[G] =[∧∨

A ∩M : A ∈ PD(D, ω1) ∩M

]i[G] =

=∧

[∨A ∩M ]i[G] : A ∈ PD(D, ω1) ∩M

=

=∧∨

A′ ∩M [G] : A′ ∈ PD(C, ω1) ∩M [G]

=

= sg(C,M [G]).

Since this occurs for all V -generic G with sg(B,M) ∈ G, we can conclude that

r[sg(D,M)]i[GBs]

= sg(C,M [GB])z

B≥ sg(B,M).

Therefore

sg(B,M)∧r

[sg(D,M)]i[GB] > 0z

B= sg(B,M)∧

rsg(C,M [GB]) > 0

z

B= sg(B,M), (10.1)

using the S(GB)-semiproperness of C in V [GB]. Thus,

π(sg(D,M)) =r

[sg(D,M)]i[GB] > 0z≥ sg(B,M).

Finally, by Lemma 9.2.3 and 1.2.17, repeating the proof for B π([c]) and D [c] (that are atwo-step iteration of S-SP boolean algebras) we obtain that

π(sg(D,M) ∧ [c]) = sg(B,M) ∧ π([c])

hence i is S-SP. Finally, assume [c] ∈ D ∩M is incompatible with sg(D,M). Then

π(0) = π(sg(D,M) ∧ [c]) = sg(B,M) ∧ π([c]) = 0.

This implies π([c]) = 0 and [c] = 0 since B is S-SP, completing the proof that D is S-SP.

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Lemma 10.1.5. Let B, C0, C1 be S-SP complete boolean algebras, and let G be anyV -generic filter for B. Let i0, i1, j form a commutative diagram of regular embeddings asin the following picture:

B C0

C1

i0

i1j

Moreover assume that C0/i0[G] is S(G)-SP in V [G] and

rC1/j[GC0

] is S(GC0)-SPz

C0

= 1C0 .

Then in V [G], j/G : C0/G → C1/G is an S(G)-SP embedding.

Proof. Let G be V -generic for B. Pick K V [G]-generic for C0/G. Then we can let

H = c ∈ C0 : [c]G ∈ K,

and we get thatK = H/G = [c]G : c ∈ H.

Moreover H is V -generic for C0, V [H] = V [G][H/G] and in V [H] we have that S(H) =S(G)(H/G). Since this latter equality holds for whichever choice of K we make, this givesthat in V [G] it holds that j/G : C0/G → C1/G is a map such that

r(C1/G)/j/G[GC0/G

] is S(G)(GC0/G)-SPz

C0/G= 1C0/G .

So, by applying Proposition 10.1.4 in V [G], j/G is S(G)-SP in V [G].

Lemma 10.1.6. Let B, C0, C1 be complete boolean algebras. Assume B is S-SP and letG be any V -generic filter for B. For any n ∈ 2, let in;B → Cn be an injective completehomomorphism and let k be such that

rk : C0/i0[G] → C1/i1[G] is S(G)-SP

z

B= 1B

Then there exists a S-SP regular embedding j : C0 → C1 in V , such thatrk = j/GB

z= 1B.

B C0

C1

i0

i1j

C0/i0[GB]

C0/i1[GB]

jGB∼= kGB

Proof. By Proposition 6.1.13

j : B ∗ C0/i0[GB] → B ∗ C1/i1[GB]

[d]≈ 7→ [k(d)]≈.

is an injective complete homomorphism such thatrk = j/GB

z= 1B. By hypothesis

rC0/i0[G] is S(G)-SP

z

B= 1B

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Hence by Proposition 10.1.4, C0 is S-SP in V . We have to show that for any countableM ≺ Hθ such that M ∩ ∪S ∈ S and [c]≈ ∈ C1 ∩M :

πj([c]≈ ∧ sg(C1,M)) = πj([c]≈) ∧ sg(C0,M).

By Fact 6.1.6 for any n ∈ 2

sg(Cn,M) = [sg(Cn/in[GB],M [G])]≈.

Moreover by unfolding the definitions:

πj([c]≈) =∧

[d]≈ ∈ C0 : j([d]≈) ≥ [c]≈

=∧

[d]≈ ∈ C0 :rk(d) ≥ c

z= 1B

=[∧

d ∈ C0/i0[GB] : k(d) ≥ c]≈

= [πk(c)]≈.

Hence, since k is S(GB)-SP:

πj([c]≈ ∧ sg(C1,M)) = [πk(c ∧ sg(C1/i1[GB],M [GB]))]≈

= [πk(c) ∧ sg(C0/i0[GB],M [GB])))]≈

= πj([c]≈) ∧ sg(C0,M).

10.1.1 Two-steps iterations of proper forcings

Similarly we can handle the proper case:

Proposition 10.1.7. Let S be a stationary set of countable sets. Let B be an S-propercomplete boolean algebra, and let C be such that

rC is S(G)-proper

z

B= 1B,

then D = B ∗ C and iB∗C are S-proper.

Lemma 10.1.8. Let B, C0, C1 be S-proper complete boolean algebras, and let G be anyV -generic filter for B. Let i0, i1, j form a commutative diagram of regular embeddings asin the following picture:

B C0

C1

i0

i1j

Moreover assume that C0/i0[G] is S(G)-SP in V [G] and

rC1/j[GC0

] is S(GC0)-SPz

C0

= 1C0 .

Then in V [G], j/G : C0/G → C1/G is an S(G)-proper embedding.

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Lemma 10.1.9. Let B, C0, C1 be complete boolean algebras. Assume B is S-proper inV and let G be any V -generic filter for B. For any n ∈ 2, let in;B→ Cn be an injectivecomplete homomorphism and let k be such that

rk : C0/i0[G] → C1/i1[G] is S(G)-proper

z

B= 1B

Then there exists a S-proper regular embedding j : C0 → C1 in V , such thatrk = j/GB

z=

1B.

B C0

C1

i0

i1j

C0/i0[GB]

C0/i1[GB]

jGB∼= kGB

10.2 (Semi)proper iteration systems

The limit case needs a slightly different approach depending on the length of the iteration.We start with some general lemmas, then we will proceed to examine the different casesone by one.

Definition 10.2.1. Let S be a stationary set concentrating on countable sets. An iterationsystem F = iαβ : α ≤ β < λ is S-SP iff iαβ is S-SP for all α ≤ β < λ.

An iteration system F = iαβ : α ≤ β < λ is S-proper iff iαβ is S-proper for allα ≤ β < λ.

An iteration system F = iαβ : α ≤ β < λ is RCS iff for all α < λ limit ordinal wehave Bα = RO(RCS(F α)).

Fact 10.2.2. Let S be a stationary set concentrating on countable sets. Let F = iαβ :α ≤ β < λ be an S-SP (respectively S-proper) iteration system, f be in lim←−(F). Then

F f= (iαβ)f(β) : Bα f(α)→ Bβ f(β): α ≤ β < λ

is an S-SP (respectively S-proper) iteration system and its associated adjoints are therestriction of the original adjoints.


Lemma 10.2.3. Let F = iαβ : Bα → Bβ : α ≤ β < λ be an RCS and S-SP iterationsystem with S stationary on [Hθ]

ω. Let M be in S, g ∈ M be any condition in RCS(F),α ∈M be an RCS(F)-name for a countable ordinal, δ ∈M be an ordinal smaller than λ.

Then

• there exists a condition g′ ∈ RCS(F) ∩M below g with g′(δ) = g(δ) such that

g′ ∧ iδ(sg(Bδ,M))

forces that α < M ∩ ω1.

• If λ = ω1, then we can choose g′ ∈ M so that the support of g′ ∧ iδ(sg(Bδ,M)) iscontained in M ∩ ω1.

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Proof. Let D ∈M be the set of conditions in RCS(F) deciding the value of α (D is opendense by the forcing theorem):

D = f ∈ RCS(F) : ∃β < ω1 f α = β.

Consider the set πδ[D g] (which is open dense below g(δ) in Bδ by Lemma 6.1.1) and fixA ⊆ Bδ a maximal antichain in M contained in it, so that

∨A = g(δ). Let φ : A→ D g

be a map in M such that πδ(φ(a)) = a for every a ∈ A, and define g′ ∈ RCS(F) ∩M byg′ =

∨φ[A]. Observe that g′(δ) = g(δ) and g′ ≤ g by definition of pointwise supremum.

Moreover∨φ[A] is really the supremum of φ[A] in RO(lim←−(F)) by Lemma 7.1.11 (thus it

is the supremum in RO(RCS(F)) as well).Now we can define a name1 β ∈ V Bδ ∩M as:

β =〈γ, a〉 : a ∈ A, φ(a) RCS(F) α > γ

so that for any a ∈ A, a Bδ β = ξ iff φ(a) RCS(F) α = ξ. It follows that

rıδ(β) = α

z≥∨

φ[A] = g′. Moreover, sg(Bδ,M) ≤rβ < ˇM ∩ ω1

zand is compatible with g′(δ) ∈ M

(since Bδ is S-SP), so thatqα < ˇM ∩ ω1

y≥ g′ ∧ iδ(sg(Bδ,M)).

This proves the first conclusion in the Lemma.Assume now that λ = ω1, then RCS(F) = lim−→(F) and we can define a name γ ∈ V Bδ∩M

for a countable ordinal setting:

γ = 〈η, a〉 : a ∈ A, η < supp(φ(a)) .

Notice that γ is defined in such a way that for all β < ω1

Jγ = βKBδ =∨a ∈ A : supp(φ(a)) = β.

In particular this gives that:

iδ(Jγ < βK) ∧ g′ =

= iδ(∨a ∈ A : supp(φ(a)) < β) ∧

∨φ(a) : a ∈ A =

=∨φ(a) : a ∈ A, supp(φ(a)) < β.

Now observe that

g′ ∧ iδ(sg(Bδ,M)) =∨φ(a) ∧ iδ(sg(Bδ,M)) : a ∈ A.

Since sg(Bδ,M) ≤qγ < ˇM ∩ ω1

yBδ

, we get that:

g′ ∧ iδ(sg(Bδ,M)) =

= g′ ∧ iδ(sg(Bδ,M)) ∧ iδ(Jγ < M ∩ ω1K) =

=∨φ(a) : a ∈ A, supp(φ(a)) < M ∩ ω1 ∧ iδ(sg(Bδ,M)).

It is now immediate to check that this latter element of lim−→(F) has support contained inM ∩ ω1 as required.

1Literally speaking this is not a Bδ-name according to Definition 4.1.1, it is nonetheless a B+δ -name

considering B+δ as a partial order according to Definition 4.3.2. Since V Bδ and V B+

δ are isomorphic B-valuedmodels, the ambiguity can be resolved.

164

Lemma 10.2.4. Let F = inm : n ≤ m < ω be an S-SP iteration system with Sstationary on [Hθ]

ω. Then lim←−(F) and the corresponding inω are S-SP.

Proof. By Proposition 9.2.5, any countable M ≺ Hν with ν > θ, F , S ∈M , M ∩Hθ ∈ S,witnesses the semiproperness of every inm.

We need to show that for every f ∈ lim←−(F) ∩M , n < ω,

πnω(sg(RO(lim←−(F)),M) ∧ f) = sg(Bn,M) ∧ f(n)

this would also imply that RO(lim←−(F))is S-SP by the same reasoning of the proof of Lemma10.1.4. Without loss of generality, we can assume that n = 0 and by Lemma 9.2.3 and10.2.2 we can also assume that f = 1. Thus it is sufficient to prove that

π0ω(sg(RO(lim←−(F)),M)) = sg(B0,M)

Let αn : n ∈ ω be an enumeration of the lim←−(F)-names in M for countable ordinals.Let g0 = 1lim←−(F), gn+1 be obtained from gn, αn, n as in Lemma 10.2.3, so that

qαn < ˇM ∩ ω1

ylim←−(F)

≥ gn+1 ∧ in(sg(Bn,M))

and gn+1(n) = gn(n). Consider now the sequence g(n) = gn(n)∧ sg(Bn,M). This sequenceis a thread since in,n+1 is S-SP and gn(n) ∈M for every n, hence

πn,n+1(sg(Bn+1,M) ∧ gn+1(n+ 1)) = sg(Bn,M) ∧ πn,n+1(gn+1(n+ 1))

and πn,n+1(gn+1(n+1)) = gn+1(n) = gn(n) by Lemma 10.2.3. Furthermore, for every n ∈ ω,g ≤ gn since the sequence gn is decreasing, and g ≤ in(sg(Bn,M)) since g(n) ≤ sg(Bn,M).It follows that g forces that

qαn < ˇM ∩ ω1

ylim←−(F)

for every n, thus g ≤ sg(RO(lim←−(F)),M)

by Lemma 9.2.6. Then,

π0(sg(RO(lim←−(F)),M)) ≥ g(0) = g0(0) ∧ sg(B0,M) = sg(B0,M)

and the opposite inequality is trivial, completing the proof.

Lemma 10.2.5. Let F = iαβ : Bα → Bβ : α ≤ β < ω1 be an RCS and S-SP iterationsystem with S stationary on [Hθ]

ω. Then lim−→(F) and the corresponding iαω1 are S-SP.

Proof. The proof follows the same pattern of the previous Lemma 10.2.4. By Proposition9.2.5, any countable M ≺ Hν with ν > θ, F , S ∈ M , M ∩ Hθ ∈ S, witnesses thesemiproperness of every iαβ with α, β ∈M ∩ ω1.

As before, by Lemma 9.2.3 and 10.2.2 we only need to show that

π0(sg(RO(lim−→(F)),M)) ≥ sg(B0,M),

the other inequality being trivial. Let 〈δn : n ∈ ω〉 be an increasing sequence of ordinalssuch that δ0 = 0 and supn δn = δ = M ∩ ω1, and αn : n ∈ ω be an enumeration of thelim−→(F)-names in M for countable ordinals. Let g0 = 1lim←−(F), gn+1 be obtained from gn, αn,

δn as in Lemma 10.2.3, so that

qαn < ˇM ∩ ω1

y≥ gn+1 ∧ iδn(sg(Bδn ,M))

and the support of gn+1 ∧ iδn(sg(Bδn ,M)) is contained in M ∩ ω1. Consider now thesequence g(δn) = gn(δn) ∧ sg(Bδn ,M). As before, this sequence induces a thread on F δ,

165

so that g ∈ Bδ since F is an RCS-iteration, δ has countable cofinality and thus we cannaturally identify lim←−(F δ) as a dense subset of Bδ. Moreover we can also check that iδ(g)is a thread in lim−→(F) with support δ such that iδ(g)(α) = g(α) for all α < δ.

Since by Lemma 10.2.3

supp(gn+1 ∧ iδn(sg(Bδn ,M))) ≤ δ,

the relation iδ(g) ≤ gn+1 ∧ iδn(sg(Bδn ,M)) holds pointwise on all ω1, hence iδ(g) forcesthat

qαn < ˇM ∩ ω1

yfor every n. Thus, iδ(g) ≤ sg(RO(lim−→(F)),M) by Lemma 9.2.6 and

π0(sg(RO(lim−→(F)),M)) ≥ g(0) = g0(0) ∧ sg(B0,M) = sg(B0,M)

as required. The proof is completed.

Lemma 10.2.6. Let F = iαβ : Bα → Bβ : α ≤ β < λ be an RCS and S-SP iterationsystem with S stationary on [Hθ]

ω such that lim−→(F) is <λ-cc. Then lim−→(F) and thecorresponding iαλ are S-SP.

Proof. The proof follows the same pattern of the previous Lemmas 10.2.4 and 10.2.5. ByProposition 9.2.5, any countable M ≺ Hν with ν > θ, F , S ∈M , M ∩Hθ ∈ S, witnessesthe semiproperness of every iαβ with α, β ∈M ∩ λ.

As before, by Lemma 9.2.3 and 10.2.2 we only need to show that

π0(sg(RO(lim−→(F)),M)) ≥ sg(B0,M).

Let 〈δn : n ∈ ω〉 be an increasing sequence of ordinals such that δ0 = 0 and supn δn = δ =sup(M ∩ λ), and αn : n ∈ ω be an enumeration of the lim−→(F)-names in M for countableordinals. Let g0 = 1lim←−(F), gn+1 be obtained from gn, αn, δn as in Lemma 10.2.3, so that

qαn < ˇM ∩ ω1

y≥ gn+1 ∧ iδn(sg(Bδn ,M)).

Since lim−→(F) is <λ-cc by Theorem 7.2.1 we have that lim←−(F) = RO(lim−→(F)) = lim−→(F),so every gn is in lim−→(F) ∩M hence M has to model gn to be eventually constant, thussupp(gn) < δ. Then the sequence g(δn) = gn(δn) ∧ sg(Bδn ,M) induces a thread onF δ (hence g ∈ Bδ = RO(lim←−(F δ)) by the countable cofinality of δ) and iδ(g) ≤gn+1 ∧ iδn(sg(Bδn ,M)) for every n, so that iδ(g) ≤ sg(RO(lim−→(F)),M) by Lemma 9.2.6and

π0(sg(RO(lim−→(F)),M)) ≥ g(0) = g0(0) ∧ sg(B0,M) = sg(B0,M)

as required.

Theorem 10.2.7. Let F = iαβ : Bα → Bβ : α ≤ β < λ be an RCS and S-SP iterationsystem with S stationary on [Hθ]

ω, such that for all α < β < λ,

rBβ/iαβ[Gα] is S(Gα)-SP

z= 1Bα

and for all α there is a β > α such that Bβ |Bα| ≤ ω1. Then RCS(F) and thecorresponding iαλ are S-SP.

Proof. First, suppose that for all α we have that |Bα| < λ. Then, by Theorem 7.2.2, lim−→(F)is <λ-cc and RCS(F) = lim−→(F) hence by Lemma 10.2.6 we have the thesis.

Now suppose that there is an α such that |Bα| ≥ λ. Then by hypothesis there is aβ > α such that Bβ |Bα| ≤ ω1, thus Bβ cof λ ≤ ω1. So by Lemma 10.1.5 F/Gβ is a

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Bβ-name for an S(Gβ)-SP iteration system that is equivalent to a system of length ω or ω1

hence its limit is S(Gβ)-SP by Lemma 10.2.4 or Lemma 10.2.5 applied in V Bβ . Finally,RCS(F) can always be factored as a two-step iteration of Bβ and RCS(F/Gβ ), hence by

Proposition 10.1.4 we have the thesis.

Theorem 10.2.8. The class SP of semiproper forcings is ω1-weakly iterable in NBG + ω1

is a regular cardinal.

Proof. Let Σ : SPlim → SPlim be defined as follows:

1. If F ∈ SPlim is an iteration system of limit length, Σ(F) = F ∪ RCS(F),

2. If F = iαβ : Bα → Bβ : α ≤ β ≤ γ ∈ SPlim is an iteration system of odd successor

length γ + 1, Σ(F) = F ∪Bγ ∗ ˙Coll(ω1,Bγ)

.

Lemmas 10.2.4, 10.2.5, 10.2.6 and Theorem 10.2.7 show that Σ(F) ∈ SPlim for all F ∈ SPlim,and also that this is a provable statement in NBG + ω1 is a regular cardinal.

Now we run through the items of Def. 7.5.3:

• We know already that NBG proves that SP is closed under lottery sums, two-stepsiterations and complete subalgebras. This give that for any inaccessible δ, Vδ+1

proves these properties of SP, since Vδ+1 |= NBG.

• We have already proved that Σ is an ω1-weak iteration strategy for SP in NBG + ω1

is a regular cardinal.

• It is also clear ty the definition of Σ and of SP that 〈SP,Σ〉 as computed in Vκ+1 isequal to 〈SP ∩ Vκ,Σ ∩ Vκ〉 whenever κ is inaccessible.

Hence SP is ω1-weakly iterable through Σ in NBG + ω1 is a regular cardinal.

10.2.1 Iterations of proper forcings

Similarly we can handle proper forcings:

Lemma 10.2.9. Let F = iαβ : Bα → Bβ : α ≤ β < λ be an RCS and S-proper iterationsystem with S stationary on [Hθ]

ω. Let M be in S, g ∈ M be any condition in RCS(F),α ∈M be an RCS(F)-name for a countable ordinal, δ ∈M be an ordinal smaller than λ.

Then

• there exists a condition g′ ∈ RCS(F) ∩M below g with g′(δ) = g(δ) such that

g′ ∧ iδ(gen(Bδ,M))

forces that α < M ∩ ω1.

• If λ = ω1, then we can choose g′ ∈M so that the support of g′ ∧ iδ(gen(Bδ,M)) iscontained in M ∩ ω1.

Lemma 10.2.10. Let F = inm : n ≤ m < ω be an S-proper iteration system with Sstationary on [Hθ]

ω. Then lim←−(F) and the corresponding inω are S-proper.

Lemma 10.2.11. Let F = iαβ : Bα → Bβ : α ≤ β < ω1 be an RCS and S-proper iter-ation system with S stationary on [Hθ]

ω. Then lim−→(F) and the corresponding iαω1 areS-proper.

167

Lemma 10.2.12. Let F = iαβ : Bα → Bβ : α ≤ β < λ be an RCS and S-proper iterationsystem with S stationary on [Hθ]

ω such that lim−→(F) is <λ-cc. Then lim−→(F) and thecorresponding iαλ are S-proper.

Theorem 10.2.13. Let F = iαβ : Bα → Bβ : α ≤ β < λ be an RCS and S-proper itera-tion system with S stationary on [Hθ]

ω, such that for all α < β < λ,

rBβ/iαβ[Gα] is S(Gα)-proper

z= 1Bα

and for all α there is a β > α such that Bβ |Bα| ≤ ω1. Then RCS(F) and thecorresponding iαλ are S-proper.

Theorem 10.2.14. The class PR of proper forcings is ω1-weakly iterable in NBG + ω1 isa regular cardinal.

Remark 10.2.15. There is a great simplification in the properties of RCS-iteration of properforcings: its RCS-limit is always a full limit at stages of countable cofinality and a directlimit elsewhere. This holds because proper forcings have the countable covering property,see below, hence the intermediate proper forcings in an iteration of length a regularuncountable λ cannot change λ to become of countable cofinality, which is the only case inwhich the RCS-limit of an iteration system of length a regular uncountable λ can differfrom its direct limit.

Fact 10.2.16. Assume B is a proper forcing and G is V -generic for B. Then any countableset of ordinals in V [G] is contained in some set which is countable in V .

Proof. Assume σG is a countable set of ordinals in V [G]. Let C ∈ V be the stationary setof countable elementary sub-models of some Hθ with σ ∈M . Then C remains stationaryin V [G]. Hence if M ∈ C, M [G] ∩ θ = M ∩ θ. Since σG ∈M [G] is countable in V [G], wehave that σG ⊆M [G] ∩ θ = M ∩ θ ∈ V .

168

Part V

Selfgeneric ultrapowers, genericabsoluteness for L(Ordω), Martin’s

maximum

169

Part V deals with stationary tower forcings and generic ultrapowers induced by (towersof) normal ideals. Chapter 11 presents the main results of Woodin regarding stationarytowers (i.e. that they induce almost huge generic elementary embeddings) and a key resultby Foreman (Theorem 11.3.1) regarding ideal forcings (i.e. forcings of type P (P (X)) /Iwith I a normal ideal on X). Foreman’s theorem gives an exact characterization of whichtype of forcings can consistently become isomorphic to an ideal forcing; along the wayit provides an informative description of the closure properties of the generic ultrapowerembedding induced by these ideal forcings. Chapter 12 proves one of Woodin’s mainachievement: i.e the invariance of second order number theory in the presence of largecardinals axioms; specifically it is proved that the theory of the Chang model L(Ordω)is generically invariant if we assume the existence of class many supercompact cardinals.Chapter 13 proves the consistency of Martin’s maximum relative to the existence ofa supercompact cardinal. It next addresses an analysis of the category forcing whoseconditions are stationary set preserving complete boolean algebras and whose order relationis given by the complete homomorphisms between them. Among many things it is shownthat (assuming class many supercompact cardinals) Martin’s maximum can be formulatedas the assertion that the class of presaturated towers is dense in this category forcing. Thisshows that very strong forcing axioms can also be formulated in the language of categoriesin terms of density properties of class partial orders. These two last chapters serve as amotivation for the last part of the book, where we will look at suitable generalizationsto third order number theory (and beyond) of Woodin’s generic absoluteness results forsecond order number theory.

170

Chapter 11

Self-generic presaturated idealsand towers

In this chapter we analyze the kind of (towers of) normal ideals we are interested. Theirkey property is that they allow to define almost-huge generic embedding with small criticalpoint (as small as ω1, ω2), this allows to transfer many of the reflection arguments on thestructure Hκ one can use when dealing with standard large cardinal properties of κ toanalyze and study the combinatorics of Hℵ1 or Hℵ2 (or of Hθ for any θ which is the criticalpoint of a generic elementary embedding). The use of this type of generic elementaryembeddings is by now a standard technique, the terminology “generic embedding” was firstintroduced in Foreman’s [17]. We will denote the forcing notions induced by these (towersof) normal ideals as the self-generic forcings. The terminology will become transparentlater on. It turns out that there is a deep interaction between forcing axioms, self-genericforcings and generic absoluteness results: first of all these tower forcings are universalamong the class of forcings satisfying certain type of forcing axioms (for example assumingMartin’s maximum any forcing notion which is stationary set preserving is absorbed by apresaturated tower). The combination of this universality property with the fact that theself-generic forcings induce almost huge generic elementary embedding with small criticalpoints produce a variety of generic absoluteness results: in this part of the book we willprove Woodin’s generic absoluteness results for second order number theory, in the lastpart of the book we will produce a number of generic absoluteness results regarding thetheory with parameters of Hω2 or more generally of the Chang models L(Ordκ) with κ aregular cardinal.

Definition 11.0.1. Let V ⊆W be transitive models of ZFC (or MK) with V,M transitiveclasses in W . An elementary embedding j : V →M is:

• generically λ-supercompact if Mλ ⊆M holds in W . If W = V we just say that j isλ-supercompact.

• generically almost huge if crit(j) = δ and M<j(δ) ⊆ M holds in W . If W = V wejust say that j is almost huge.

• generically huge if crit(j) = δ and M j(δ) ⊆ M holds in W . If W = V we just saythat j is huge.

• δ is λ-supercompact in V if it is the critical point of a λ-supercompact j : V →M ⊆ V .

• δ is supercompact in V if it is the critical point of a λ-supercompact j : V →M ⊆ Vfor all λ.

171

• δ is (almost) huge in V if it is the critical point of an (almost) huge j : V →M ⊆ V .

• δ is superhuge in V if for all λ it is the critical point of an huge j : V →M ⊆ V withj(δ) > λ.

The remainder of the chapter will show how to generate a variety of (generically) almosthuge embeddings by means of tower or ideal forcings.

First of all we introduce the notion of self-generic tower and ideal forcings. Next weshow that the stationary towers of height a supercompact cardinal are self-generic andpresaturated.

11.1 Self-generic towers and self-generic ideal forcings

Notation 11.1.1. For a set Z, GZ = S ∈ Z : Z ∩ ∪S ∈ S.

Definition 11.1.2. Let M ≺ Hθ for some θ.

• For some X ∈ M , H ⊆ M ∩ P (P (X)) is an M -normal ultrafilter on X if for allregressive f : S → X with f ∈ M and S ∈ H, there is T ∈ H such that f T isconstant.

• For some δ ∈ M , H ⊆ M ∩ Vδ is an M -normal tower of ultrafilter if HX = H ∩P (P (X)) is an M -normal ultrafilter on X for all X ∈M ∩ Vδ.

Proposition 11.1.3. For any cardinal δ, and M ≺ H+δ , GM ∩ Vδ is an M -normal tower

of ultrafilters on M .

Proof. GM is a filter. Let T, S ∈ GM , then M ∩⋃S ∈ S and M ∩

⋃T ∈ T . Let

X =⋃S ∪

⋃T ∈M . Then

M ∩X ∈ S ↑ X ∩ T ↑ X = S ∧ T.

GM is ultra. Let T ∈M with X = ∪T , then

M ∩X ∈ T ∨M ∩X ∈ P (X) \ T.

GM is M-normal. Assume f : S →⋃S = X is regressive with S ∈ GM and f ∈M , put

Tx = N ∈ S : f(N) = x .

We have Tx : x ∈ X ∈M and

M |= S =NS

hTx : x ∈ X

(since the latter statement holds in Hδ+ and M ≺ Hδ+). In particular there is a clubCf : X<ω → X in M such that M ∩X ∈ Cf ∩ S and

S ∩ Cf =hTx : x ∈ X ∩ Cf .

Assume by contradiction that M ∩X /∈ Tx for any x ∈M ∩X then

M ∩X /∈hTx : x ∈ X ∩ Cf = S ∩ Cf .

A contradiction.

172

Can we render GM ∩ Vδ (or GM ∩ P (P (Vδ))) an M -generic ultrafilter for a towerforcing induced by a tower of height δ (or for an ideal forcing concentrating on boundedsubsets of Vδ)? We introduce the notion of self-genericity as a mean to describe towers andideal forcings for which this can occur.

Definition 11.1.4. Let I be a tower of normal ideals of height an inaccessible δ. LetT = T Iδ be the induced tower forcing.

• Given some transitive sets X ⊇ Vδ+1, M ⊆ X is I-self-generic if for all D ∈ Mwhich are predense subsets of T there exists S ∈ GM ∩ Vδ with [S]I ∈ D (i.e.[S]I : S ∈ GM ∩ Vδ is M -generic for T ).

• SGI is the set of M ≺ Hδ+ which are I-self-generic.

• I is a self-generic tower of ideals if SGI ∧ T is stationary for all [T ]I ∈ T +.

Similarly

Definition 11.1.5. Let I be a normal ideal on P (X). Let PI = P (P (X)) /I be theinduced ideal forcing.

• Given some transitive sets Y 3 P (P (X)), M ⊆ Y is PI -self-generic if for all D ∈Mwhich are predense subsets of PI there exists S ∈ GM ∩ P (P (X)) with [S]I ∈ D.

• SGI is the set of M ≺ H|P(P(X))|+ which are PI -self-generic.

• I is self-generic if SGI ∧ T is stationary for all [T ]I ∈ (PI)+.

Self-generic towers and ideals produce the following improvement of Burke’s theo-rems 2.3.19 and 2.4.5.

Proposition 11.1.6. Let I be a self-generic ideal on P (X) and θ > |P (P (X)) |. Then

SGI ≤NS

∧I =NS M ≺ Hθ : M ∩X 6∈ S for all S ∈M ∩ I .

Similarly, let I be a self-generic tower of ideals of height an inaccessible δ and θ > |Vδ|.Then

SGI ≤NS

∧I =NS M ≺ Hθ : M ∩X 6∈ S for all S ∈M ∩ ∪I .


Definition 11.1.7. We say that B is a self-generic forcing if it is the boolean completioneither of T IB for a self-generic tower IB or of PIB for a self-generic ideal IB.

Lemma 11.1.8. Assume B,C are self-generic tower forcings withSGIB ∈ ∪I+

C Then the map [T ]IB 7→ [SGIB ∧T ]IC extends to a complete homomorphismof B into C [SGIB ]IC.

Remark 11.1.9. We leave to the reader to formulate the corresponding proposition in caseIB or IC are self-generic ideals.

Proof. An instructive exercise for the reader.

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11.2 Sufficient conditions to get (almost) huge generic ele-mentary embeddings

We start outlining sufficient conditions granting that (a tower of) normal ideal(s) inducesan almost huge elementary embedding. In case of towers of normal ideals we get a necessaryand sufficient condition. In case of normal ideal we just get a sufficient condition for almosthugeness and another one for hugeness and beyond.

11.2.1 Characterization of presaturation for towers of normal ideals

Proposition 11.2.1. Let I be a tower of normal ideals of height an inaccessible δ whichconcentrates on

X : otp(X ∩ δ) ≤ γ+, and γ ⊆ X

for some cardinal γ < δ. The following are equivalent:

1. For any G V -generic for T Iδ

V [G] |= Ult(V,G)<δ ⊆ Ult(V,G).

2. RO(T Iδ ) is a <δ presaturated complete boolean algebra.

Proof. (1)⇒(2). By Proposition 4.2.3 it is enough to show that T Iδ preserves the regularityof δ. Let G be V -generic for T Iδ and jG : V → Ult(V,G) be the derived embedding.By (1) Ult(V,G) is < δ-closed thus it is well founded, so we can identify it withits transitive collapse. Moreover we have at the same time that crit(jG) = γ+

(by Prop. 5.2.16(4)) and that jG(γ+) ≥ δ. Suppose towards a contradiction thatjG(γ+) > δ. First of all observe that the family of functions f : P (Vα)→ γ+ in Vas α ranges on all ordinals less than δ suffice to represent all elements of Ult(V,G)-ordinals below jG(γ+). Next observe that this family belongs to V and has size δin V , in particular jG(γ+) < (δ+)V , thus it is an ordinal whose cofinality in V isat most δ. Now observe that V [G] models that |δ| = γ since Ult(V,G) ⊆ V [G] andUlt(V,G) models |δ| = γ, since δ < jG(γ+), which is the successor of γ in Ult(V,G).In particular V [G] models that jG(γ+) has cofinality at most γ. We conclude that

Ult(V,G) |= cof(jG(γ+)) ≤ γ,

since Ult(V,G) is < δ-closed in V [G] and γ < δ. This contradicts the regularity ofjG(γ+) in Ult(V,G).

This shows that δ = jG(γ+) is a regular cardinal in Ult(V,G). Now any subset ofδ of size less than δ in V [G] belongs to Ult(V,G) (since Ult(V,G) is < δ-closed inV [G]) and Ult(V,G) models this set to be bounded (since δ is a regular cardinal inUlt(V,G)). Hence this set is bounded also in V [G], which gives that δ is regular inV [G], as was to be shown.

(2) =⇒ (1). Let K be V -generic for T Iδ and let fi : i < ξ ∈ V [K] be a set of functionsfi : P (Xi)→ V in V with Xi ∈ Vδ for all i < ξ indexed by some ξ < δ. It is enoughto show the following:

Claim 5. There exists h : X → V in V such that for all [g]K ∈ Ult(V,K), [g]K ∈ [h]Kif and only if for some i < ξ [g]K = [fi]K .

174

Letrf : ξ → V

zbe a T Iδ -name for fi : i < ξ.

For any i < ξ let

Di =

[S]I : [S]I ≤rf(i) = fS,i

zfor some fS,i

,

and let Xi ⊆ Di be a maximal antichain in T Iδ . By assumption (2) (and by Prop. 4.2.3)there exists [T ]I ∈ K such that for any i < ξ,

| [S ∧ T ]I : [S]I ∈ Xi and [S ∧ T ]I > 0T | < δ.

Then Yi = [S ∧ T ]I : [S]I ∈ Xi is a maximal antichain in T Iδ below [T ]I . Hencefor any i < ξ,

∨Yi = [T ]I . Therefore

∧∨Yi : i < ξ = [T ]I .

On the other hand (since ξ < δ and δ is inaccessible) we can also observe that∧∨Yi : i < ξ

= [T ∗]I ,

where for some η < δ large enough

T ∗ =ih

S ∧ T : [S]I ∈ Xi : i < ξ

=

=M ≺ Vη : ∀i ∈M ∩ ξ,∃ [S]I ∈ Xi ∩M (M ∩

⋃S ∈ S)

,

since the size of each Yi is smaller than δ and ξ ∈ Vδ.Hence [T ]I = [T ∗]I . Define

h : T ∗ −→ Ordξ

M 7−→fS,i(M ∩ ∪S) : i ∈M ∩ ξ, [S]I ∈M ∩Xi, and (M ∩

⋃S ∈ S)

,

define h(M) arbitrarily for all other M ∈ P (Vη) .

We show the following:

Claim 6. [g]K ∈ [h]K if and only if for some i < ξ, [g]K = [fi]K .

Proof. We prove the claim as follows:

[fi]K ∈ [h]K for all i < ξ: Yi is a predense antichain below [T ]I for all i < η, henceK ∩ Yi = [Si ∧ T ]I for some unique [Si]I ∈ Xi for all i < η. Thus [Si]I ≤rf(i) = fSi,i

zand [Si]I ∈ K. This gives that fSi,i = fi for all i < η, since f is a

T -name for fi : i < η. Recall that [T ∗]I ∈ K, since [T ∗]I = [T ]I ∈ K. Hence[Si ∧T ∗]I ∈ K for all i < ξ. By definition of h, for any M ∈ Si ∧T ∗ with i ∈M

fi(M) = fS,i(M) ∈ h(M) =

= fS,j(M) : M ∩ ∪S ∈ S for the unique [S ∧ T ]I ∈ Yj ∩M, j ∈M ∩ η .

In particular we get that [fi]K ∈ [h]K for all i < ξ.

175

[g]K ∈ [h]K implies [g]K = [fi]K for some i < ξ: Assume [g]K ∈ [h]K with g :P (X) → V in V and (w.l.o.g. enlarging X if necessary) X ⊇ ∪T ∗. Weget that

M ∈ P (X) : g(M) ∈ h(M ∩ ∪T ∗) = T ∗∗ and [T ∗∗]I ∈ K.

This gives (in V ) that for each M in the above set there is some i ∈M ∩ ξ anda unique S ∈M such that [S ∧ T ]I ∈M ∩ Yi and g(M) = fS,i(M). By pressingdown on [T ∗∗]I ∈ K (since K is V -normal), we can find i < ξ such that

[M ∈ T ∗∗ : g(M) = fS,i(M ∩ ∪T ∗)]I ∈ K,

giving that [fi]K = [fS,i]K = [g]K .

The Claim is proved.

The Proposition is proved.

11.2.2 Sufficient condition granting the presaturation for normal idealforcings

It is not clear whether an analogue of this proposition can be proved for ideal forcings at δfor some inaccessible δ. For our purposes the following suffices:

Proposition 11.2.2. Let δ be a regular cardinal and I be a normal ideal on a set X of sizeδ such that PI is < δ-presaturated. Let H be V -generic for PI and K = S : [S]I ∈ H.Then Ult(V,K)<δ ⊆ Ult(V,K) holds in V [H].

Proof. W.l.o.g. we can suppose that X = δ = |X|. Fix (fi : i < γ) ∈ V [K] = V [H] suchthat fi : P (δ)→ V in V for some γ < δ. It suffices to find some g : P (δ)→ V in V suchthat [h]K ∈ [g]K if and only if [h]K = [fi]K for some i < γ.

Let f ∈ V PI be a PI -name for a function such that fH(i) = fi for all i < γ and f(i) isalways forced by PI to be a function f : P (δ)→ V in V .

For each i < γ, fix Ai maximal antichain in PI in V such that for each [S]I ∈ Ai [S]Iforces that f(i) = fS,i for some fS,i : P (δ)→ V in V . Since PI is < δ-presaturated in V ,we get that there is a dense open set D of conditions in PI such that for all [T ]I ∈ D

[S]I ∈ Ai : [S ∧ T ]I > 0PI =[Siη]I

: η ∈ γi

has size γi < δ for all i < γ. Hence for each [T ]I ∈ D

[T ]I ≤∧PI

∨PIAi : i ∈ γ

=[ih

Sij : j ∈ γi

: i < γ]

I

by Proposition 2.3.4. Therefore for each [T ]I ∈ D

T ∗ =ih

Sij : j ∈ γi

: i < γ

=M ⊆ δ : for all i ∈M ∩ γ there is η ∈M ∩ γi with M ∈ Siη

is in V and such that T ∗ ≥I T . Since D ∩H is non empty, let [T ]I ∈ H ∩D and

[S]I ∈ Ai : [S ∧ T ]I > 0PI =[Siη]I

: η ∈ γi

176

for all i < γ. Let T ∗ be defined as above as

ihSij : j ∈ γi

: i < γ

and g : T ∗ → V be defined by

g(M) = fS,i(M) : i ∈M ∩ γ, M ∈ S, S ∈ Ai

for M ∈ T ∗, g(M) = 0 otherwise. Then g ∈ V . Finally, since T ≤I T ∗ and [T ]I ∈ H,T ∗ ∈ K, and we get that

[h]K ∈ [g]K

if and only if (since T ∗ ∈ K)

M ∈ T ∗ : h(M) = fi(M) for some i ∈M ∩ γ ∈ K

if and only if for some i ∈ γ (since K is a V -normal ultrafilter)

M ∈ T ∗ : h(M) = fi(M) ∈ K

if and only if [h]K = [fi]K .The Proposition is proved.

The following proposition will grant sufficient conditions to get huge (or even morethan huge) generic elementary embeddings.

Proposition 11.2.3. Let δ be a regular cardinal and I be a normal ideal on a set X of sizeδ such that PI is γ-CC for some γ < δ. Then PI is a complete boolean algebra. Moreoverlet H be V -generic for PI and K = S : [S]I ∈ H. Then Ult(V,K)δ ⊆ Ult(V,K) holds inV [H].

Proof. W.l.o.g. we can suppose that X = δ = |X|. PI is automatically a complete booleanalgebra since it is γ-CC and δ-complete for some δ > γ.

Now fix (fi : i < δ) ∈ V [K] = V [H] such that fi : P (δ)→ V in V . It suffices to findsome g : P (δ)→ V in V such that [h]K ∈ [g]K if and only if [h]K = [fi]K for some i < δ.

Let f ∈ V PI be a PI -name for a function such that fH(i) = fi for all i < δ and f(i) isalways forced by PI to be a function f : P (δ)→ V in V .

For each i < δ, fix Ai maximal antichain in PI in V such that for each [S]I ∈ Ai [S]Iforces that f(i) = fS,i for some fS,i : P (δ)→ V in V . Since PI is γ-CC in V , we get that|Ai| ≤ γ for all i < δ. Hence

1PI =∧PI

∨PIAi : i ∈ δ

=[ih

Sij : j ∈ γ

: i < δ]

I

by Proposition 2.3.4, since PI is a δ-complete boolean algebra whose sups and infs arecomputed by means of diagonal unions and intersections. Therefore

T ∗ =ih

Sij : j ∈ γ

: i < δ

=M ⊆ δ : for all i ∈M ∩ δ there is η ∈M ∩ γ with M ∈ Siη

is in V and such that [T ∗]I = 1PI . Let g : T ∗ → V be defined by

g(M) = fS,i(M) : i ∈M ∩ δ, M ∈ S, S ∈ Ai

177

for M ∈ T ∗, g(M) = 0 otherwise. Then g ∈ V . Finally, since [T ∗] = 1PI , T∗ ∈ K, and we

get that[h]K ∈ [g]K

if and only if (since T ∗ ∈ K)

M ∈ T ∗ : h(M) = fi(M) for some i ∈M ∩ δ ∈ K

if and only if for some i ∈ δ (since K is a V -normal ultrafilter)

M ∈ T ∗ : h(M) = fi(M) ∈ K

if and only if [h]K = [fi]K .The Proposition is proved.

11.3 Foreman’s duality theorem

Now we focus on ideal forcings and we give a variety of methods to produce consistentlyself-generic ideal forcings with strong closure properties. They are all based on a remarkableresult of Foreman [19, 18], which (for the sake of avoiding some piece of heavy notation)is presented here in a weaker form. Foreman shows that all precipitous ideals I (recallDef. 5.2.20) in the ground model can be naturally lifted to precipitous ideals I in anygeneric extension by a forcing P which interacts nicely with PI . Roughly Foreman’s dualitytheorem states the following: Assume j : V → M is the ultrapower embedding inducedby some precipitous ideal I in V . Assume further that j P : P → j(P ) is a completehomomorphism. Then, whenever H is V -generic for P , I extends in V [H] to a precipitousideal I such that P I is forcing equivalent to j(P )/H in V [H]. There are lenghty details tobe addressed when I ∈ V is not a V -normal ultrafilter in V (in which case j exists only in ageneric extension by PI and the definition of j(P ) can occur only in this generic extension).Since we won’t use this more general case of Foreman’s duality, we limit ourselves toaddress the case in which I ∈ V is the dual of a V -normal ultrafilter inducing a wellfounded ultrapower. Other remarkable properties of this correspondence are the following:On the one hand, it gives a neat characterization of which definitions for a forcing notionP ∈ V can become in some generic extension of V the definitions of an ideal forcing notionof type P I . On the other hand, the nicer P and I interacts, the better I behaves in thegeneric extension by P . For example we produce as easy corollaries of Foreman’s dualitytheorem examples of ideals I on Vλ such that P I is a self-generic ideal forcing, a completeboolean algebra, a < λ-CC poset, an ideal forcing inducing λ-closed generic ultrapowerembeddings, etc. . . We will extensively use the results of this section in the last part of thisbook. The literature which exploit this duality theorem (even in other less elegant forms,prior to Foreman’s polished formulation we present below) is vast, Cumming’s [14] andForeman’s [18] articles in the Handbook of set theory are useful source of references andgive detailed proofs of many of the most interesting applications, for more recent uses ofthese techniques see for example [10, 11, 12, 13].

Theorem 11.3.1 (Foreman duality theorem: Thm. 17 [19]). Assume G ∈ V is a normalultrafilter on P (P (Vλ)) and j = jG : V → Ult(V,G) = M ⊆ V is its induced elementaryembedding with Ult(V,G) = M well-founded and identified with its transitive collapse.

Assume (P,≤) ∈ V is a partial order such that:

• j(P ) ⊆ λ,

178

• for some q ∈ j(P ), letting Q = RO(j(P ))M and RO(P )M = B, the map

iP,G = i :P 7→ Q+

p 7→ j(p) ∧Q q

extends to a regular embedding of B into Q q.

Then the following holds for any V -generic filter H for P :

1. For any K which is a V [H]-generic filter for Q/i[H], let K =q ∈ Q : [q]i[H] ∈ K

.

Then:

(a) K is a V -generic filter for Q, V [K] = V [H][K], and j extends in V [K] to

jK :V [H]→M [K]

τH 7→ j(τ)K

which is also elementary. Moreover jK V = j.

(b) In1 V [K]

LK =S ∈ P (P (Vλ))V [H] : j[Vλ] ∈ jK(S)

is a V [H]-normal ultrafilter on Vλ and LK ∩ V = G. Moreover M [K] =Ult(V [H], LK) and jK is the ultrapower embedding induced by LK .

V Ult(V,G) = M V

V [H] Ult(V [H], LK) = M [K] V [K] V [H][K]

j

jK

⊆

⊆ =

⊆

2. Let in V [H] L be a Q/i[H]-name such that LK = LK for any K V [H]-generic forQ/i[H]. Then V [H] models that

I(H,G,P ) = I =S ∈ P (P (Vλ))V [H] :

rS ∈ L

z= 0Q

is a normal ideal on Vλ and

iH,G,P :P (P (Vλ))V [H] /I → RO(Q/i[H])V [H]

[S]I 7→rS ∈ L

z

extends to an isomorphism of the respective boolean completions.

3. LetJ(H,G,P ) = J =

S ∈ P (P (Vλ[H]))V [H] :

rS ↓ Vλ ∈ L

z= 0Q

Then P I and P J are isomorphic via the map [S]I 7→ [S ↑ Vλ[H]]J , and J is the liftof I via the map iVλ,Vλ[H] : T 7→ T ↑ Vλ[H].

Proof.

1In what follows Vλ stand for the elements of V of rank less than λ.

179

1. Assume K is a V [H]-generic filter for Q/i[H], and K =q ∈ Q : [q]i[H] ∈ K

.

(a) Clearly K is V -generic for Q and V [H][K] = V [K] (by 6.1.17). By Lemma 4.3.22applied to the identity map on j(P ) q, we can identify MQ with a ∆1-elementaryRO(j(P ) q)V -valued submodel of V j(P )q .

Moreover the map i belongs to M . Hence M models that i is a regular embeddingof B into Q and M models that j(B) is a complete boolean algebra with Q =j(B) q (even though nor j(B), nor Q, nor B might be cbas in V ).

We get that the following key identity holds in M for any A ⊆ B:

q ∧j(B)

∨j(B)

j(A) = q ∧j(B) j(∨B

A) = i(∨B

A) =∨Q

i[A]. (11.1)

The first of the above equality holds by elementarity of j, the second by definitionof i, while the last holds by our assumptions that i is a complete homomorphism.

We have the following:

Claim 7. Set for all formulae φ and τ1, . . . , τn ∈ V B

Jφ(j(τ1), . . . , j(τn))KQ = j(Jφ(τ1, . . . , τn)KB) ∧ q.

ThenJφ(j(τ1), . . . , j(τn))KQ = i(Jφ(τ1, . . . , τn)KB)

for all formulae φ and τ1, . . . , τn ∈ V B.

Proof. First of all notice that j(Jφ(τ1, . . . , τn)KB) ∧ q is a well defined element ofQ since M |= ZFC.

We have that

M |= j(Jφ(τ1, . . . , τn)KB) = Jφ(j(τ1), . . . , j(τn))Kj(B)

by elementarity of j. Since i is a complete homomorphism, we have that∨Q

i(p) : p ≤ Jφ(τ1, . . . , τn)KB = i(Jφ(τ1, . . . , τn)KB).

By equation 11.1 and the forcing theorem applied to B in V and j(B) in M , weget:

i(Jφ(τ1, . . . , τn)KB) =

=∨Q

i[r ∈ P : r ≤ Jφ(τ1, . . . , τn)KB] =

= q ∧j(B)

∨j(B)

r ∈ j(P ) : r ≤ Jφ(j(τ1), . . . , j(τn))Kj(B)

=

= q ∧j(B) Jφ(j(τ1), . . . , j(τn))Kj(B) =

= Jφ(j(τ1), . . . , j(τn))KQ .

The thesis follows.

180

The hypotheses grants that j[H] ⊆↑ i[H] ⊆ K. By the above facts we get that

Jφ(τ1, . . . , τn)KB ∈ H iff Jφ(j(τ1), . . . , j(τn))KQ = i(Jφ(τ1, . . . , τn)KB) ∈ K

for all formulas φ and τ1, . . . , τn ∈ V B. This immediately gives that jK iselementary and well defined.

(b) Clearly j[λ] = [IdP(λ)]G ∈ Ult(V,G) = M ⊆M [K]. Hence (by 5.2.18(3) appliedto jK), LK ∈ V [K] is a V [H]-normal ultrafilter on λ.

We must show that M [K] = Ult(V [H], LK), and also that jK = jLK .

We first prove thatM [K] ⊆ Ult(V [H], LK) :

It suffices to prove that M ⊆ Ult(V [H], LK) and also that K ∈ Ult(V [H], LK).First of all:

Claim 8. The map [f ]G 7→ [f ]LK is the inclusion map of Ult(V,G) = M intoUlt(V [H], LK), hence M ⊆ Ult(V [H], LK).

Proof. We leave to the reader to check that it is enough to prove that for anyf ∈ V and h ∈ V [H], [h]LK ∈ [f ]LK if and only if h S= g S for some S ∈ LKand g ∈ V .

So assume h ∈ V [H] and f ∈ V are such that f : P (λ)V → V , h : P (λ)V → V ,and [h]LK ∈ [f ]LK . By 5.2.18(6) applied to LK and V [H]

[h]LK ∈ [f ]LK if and only if jK(h)(j[λ]) ∈ j(f)(j[λ]).

Hence jK(h)(j[λ]) ∈ j(f)(j[λ]). But

Ult(V,G) = M = j(g)(j[λ]) : g ∈ V, g : P (λ)→ V

by 5.2.18(6) applied to G and V . Therefore j(f)(j[λ]) ⊆M = Ult(V,G). Hencej(g)(j[λ]) = jK(h)(j[λ]) for some g : P (λ)→ V with g ∈ V . This gives that

S =Z ∈ P (λ)V : h(Z) = g(Z)

∈ LK ,

i.e. g S= h S for some g ∈ V and S ∈ LK as was to be shown.

Now let in V [H]

k :P (λ)V → V [H]

Z 7→ πZ [Z] ∩H

Claim 9. [k]LK = K.

Proof. Notice that jK(H) = K. Therefore in M [K]

jK(k) =〈Z, πZ [Z] ∩K〉 : Z ∈ P (j(λ))M

.

Moreover in M [K] we have that πj[λ][j[λ]] ∩K = λ ∩K = K. Hence

[k]LK = jK(k)(j[λ]) = πj[λ][j[λ]] ∩K = K.

181

This gives that K = [k]LK ∈ Ult(V [H], LK) ⊇ M . We conclude that M [K] ⊆Ult(V [H], LK).

Now assume

k : Ult(V [H], LK)→M [K]

[f ]LK 7→ jK(f)(j[λ])

is not the identity map. Then it must have a non-trivial critical point η (by5.2.18(1). Assume η = [f ]LK . Since η ∈ M , we can assume f ∈ V by what wehave already shown. This gives that

k([f ]LK ) = jK(f)(j[λ]) = j(f)(j[λ]) = [f ]G = [f ]LK

by what we have already shown. Therefore k(η) = η, contradicting that η ismoved by k. Hence k must be the identity, giving that M [K] = Ult(V [H], LK)and also that jK = jLK (the latter again by 5.2.18(6)).

2. iH,G,P is an injective homomorphism by 5.3.2. We must prove that it has a denseimage to conclude (again by 5.3.2). We show that j(P )/i[H] is in the range of iH,G,P .We define for p ∈ j(P )

Sp =Z ∈ P (λ)V : p ∈ Z, πZ(p) ∈ H

.

It suffices to prove the following:

Claim 10. For all K V [H]-generic for Q/i[H], letting K =q : [q]i[H] ∈ K

, we

have thatSp ∈ LK if and only if p ∈ K if and only if [p]i[H] ∈ K.

Assume the claim holds. Then iH,G,P ([Sp]I) = [p]i[H] for all p ∈ j(P ). It remains toprove the claim.

Proof. Observe that

jK(Sp) =Z ∈ P (j(λ))M : πZ(j(p)) ∈ jK(H)

=Z ∈ P (j(λ))M : πZ(j(p)) ∈ K

.

Now πj[λ](j(p)) = p. Hence p ∈ K if and only if j[λ] ∈ j(Sp) if and only ifSp ∈ LK .

3. Observe that S ∈ J if and only if for all K V -generic for Q/i[H] jK [Vλ[H]] ∈ jK(S)if and only if for all K V -generic for Q/i[H] j[Vλ] ∈ jK(S ↓ Vλ) if and only ifS ↓ Vλ ∈ I. Therefore J is the lift of I via iVλ,Vλ[H]. Moreover the above also showsthat [S]J 7→ [S ↓ Vλ[H]]I is injective: if [S]J = [T ]J and K is V -generic for Q/i[H]

jK [Vλ[H]] ∈ jK(S) if and only if jK [Vλ[H]] ∈ jK(T ) and j[Vλ] ∈ jK(S ↓ Vλ) if andonly if j[Vλ] ∈ jK(T ↓ Vλ). Hence [S ↓ Vλ[H]]I = [T ↓ Vλ[H]]I . This gives thatπVλ,Vλ[H]/I,J must be an isomorphism of P J with P I .

The proof of the theorem is completed in all its parts.

Lemma 11.3.2 (Preservation Lemma for almost huge embeddings). Let G ∈ V be a normalultrafilter on some λ > ω such that Ult(V,G)<λ ⊆ Ult(V,G) with jG : V → Ult(V,G) theinduced ultrapower embedding.

Let P be a forcing notion such that for some q ∈ j(P ):

182

• Q = RO(jG(P ) q)Ult(V,G) is < λ-presaturated in V ,

• the map p 7→ jG(p) ∧Q q can be extended (in Ult(V,G) and in V ) to a regularembedding i∗ : RO(P )→ Q (even though Q may not be complete in V ).

Let H be V -generic for RO(P ). Then in V [H]:

• Q/jG[H] is < λ-presaturated, and is forcing equivalent to the ideal forcing P I where Iis the normal ideal I(H,G,P ).

• Whenever K is V [H]-generic for P I , Ult(V [H],K) is < λ-closed in V [H][K].

Proof. Immediate by 11.3.1, and 11.2.2.

A basic outcome of Foreman’s duality is the following Lifting Lemma first isolatedby Levy and Solovay when they realized that small forcings do not affect large cardinalproperties [31].

Lemma 11.3.3 (Lifting Lemma I). Assume j : V →M ⊆ V is an elementary embeddingwith δ = crit(j) and M ⊆ V a transitive structure. Let P ∈ Vδ be a forcing notion.

Assume G is V -generic for RO(P ), then in V [G] the map

j : V [G]→M [G]

given by τG 7→ j(τ)H is elementary.Moreover if j is a supercompact, almost huge, huge embedding in V , so is j in V [G].

Proof. Left to the reader (else assuming j = jG for some V -normal ultrafilter G ∈ V onsome X ⊇ Vδ with crit jG = δ and G, apply Theorem 11.3.1 to P = j(P ) ∈ Vδ and thenLemma 11.3.2).

Lemma 11.3.4 (Preservation Lemma for huge embeddings). Let G ∈ V be a normalultrafilter on some Vλ for some uncountable cardinal λ > ω such that Ult(V,G)λ ⊆ Ult(V,G)with jG : V → Ult(V,G) the induced ultrapower embedding.

Let P be a forcing notion such that for some q ∈ j(P ):

• Q = RO(jG(P ) q)Ult(V,G) is γ-CC in V for some γ < λ;

• the map p 7→ jG(p) ∧Q q can be extended (in Ult(V,G) and in V ) to a regularembedding i∗ : RO(P )→ Q (even though Q may not be complete in V ).

Let H be V -generic for RO(P ). Then in V [H]:

• Q/jG[H] is a γ-CC complete boolean algebra, and is forcing equivalent to the ideal

forcing P I , where I is the normal ideal I(H,G,P ).

• P I is a complete boolean algebra and, whenever K is V [H]-generic for P I , Ult(V [H],K)is λ-closed in V [H][K].

Proof. P I is always a λ-complete boolean algebra; being also γ-CC, we conclude that P Iis a complete boolean algebra. P I induces a λ-closed generic ultrapower by 11.2.3.

We will need some bits of the notion of decisive ideal introduced by Foreman. Sincewe do not need to explore this notion at length we just present the Lemma capturing theproperties of ideals projecting to one another which are relevant for us.

183

Lemma 11.3.5 (Projection Lemma for more than huge embedding). Assume λ > ν ≥γ > δ are inaccessible and G ∈ V is a V -normal ultrafilter concentrating on

X ⊆ Vλ : (X,∈) ∼= (Vγ ,∈), X ∩ δ ∈ δ, otp(X ∩ λ) = γ, otp(X ∩ ν) = δ .

1. Let j = jG : V → Ult(V,G) be the embedding induced by G. Then crit(j) = δ,j(δ) = ν, and Ult(V,G)λ ⊆ Ult(V,G).

2. Furthermore let G0 = πVν ,Vλ [G]. Then G0 ∈ V is a V -normal ultrafilter concentratingon

X ⊆ Vν : (X,∈) ∼= (Vδ,∈), X ∩ δ ∈ δ .

and jG0 : V → Ult(V,G0) is a huge embedding such that jG = k jG0, wherek : [f ]G0 7→ [f ↑ Vλ]G is also elementary between Ult(V,G0) and Ult(V,G).

V Ult(V,G)

Ult(V,G0)

jG

jG0

k V

⊆

⊆

3. Finally assume that P ⊆ Vδ is < δ-presaturated. Then j(P ) = jG(P ) = jG0(P ) ⊆ Vνis < ν-presaturated and ν-CC both in Ult(V,G), Ult(V,G0), and V . Assume furtherthat for some q ∈ RO(j(P ))Ult(V,G0), the map

i : p 7→ p ∧RO(j(P ))Ult(V,G0) q = j(p) ∧RO(j(P ))Ult(V,G0) q

extends to a complete homomorphism of RO(P ) into Q = RO(j(P ))Ult(V,G0) q.

Let:

• H be V -generic for P ,

• I be the ideal induced on P (P (Vλ[H]))V [H] by Foreman’s duality theorem appliedto P,G,H,

• I0 be the ideal induced on P (P (Vν [H]))V [H] by Foreman’s duality theoremapplied to P,G0, H.

Then in V [H] it holds that:

(a) PI is a complete boolean algebra and induces a λ-closed generic ultrapowerembedding.

(b) PI0 is self-generic and induces a < ν-closed generic ultrapower embedding.

(c) PI ∼= Q/i[H]∼= PI0.

(d) The map iVν ,Vλ : [S]I0 7→ [S ↑ Vλ]I implements in V [H] the isomorphism betweenPI0 and PI obtained by the composition of the natural isomorphisms of PI andPI0 with Q/i[H] given by Foreman’s duality theorem.

The following picture assumes that G is V [H]-generic for Q/i[H], K =q ∈ Q : [q]H ∈ G

is V -generic for Q, K (respectively K0) is the V [H]-generic for PI (respectively forPI0) induced by G.

184

V [H] Ult(V [H], I)

Ult(V [H], I0)

jK

jK0

iVν,Vλ V [K] V [H][G] V [H][K] V [H][K0]

⊆

⊆= = =

Proof. The proofs of items 1, 2 are standard and follow the same lines of the proof ofitem 2 in 5.4.2; we leave the details to the reader. 3a follows from 11.2.3. 3c follows fronForeman’s duality theorem. 3d is immediate if one unveils the definitions of the naturalisomorphisms of PI and PI0 with Q/i[H] given by Foreman’s duality theorem (using thesame patterns of the proof of the last item in Foreman’s duality theorem).

We are left with 3b: the fact that PI0 induces a < ν-closed generic ultrapower followsfrom 11.2.2; the unique part which requires a new separate argument is the proof that PI0is self-generic. This can be proved as follows:

Let k0 : PI0 → Q/i[H] and k : PI → Q/i[H] denote the natural isomorphisms of theseforcing notions given by Foreman’s duality theorem.

Let G be V [H]-generic for Q/i[H] and K0 = i−10 [G],K = k−1[G] be the corresponding

V [H]-generic ultrafilters on PI0 ,PI .Now

SGI0 =M ≺ HV [H]

ν+ : GM ∩ P (P (Vν [H]))V [H] is M -generic for PI0.

Let N = jK [Vλ[H]], and observe that jK [K0] = jK0 [K0] is jK [Vλ[H]]-generic for jK(PI0)since K0 is V [H]-generic for PI0 . By definition:

jK [K0] =jK(S) : S ∈ P (P (Vν [H]))V [H] , jK [Vν [H]] ∈ jK(S)

=

=jK(S) : S ∈ P (P (Vν [H]))V [H] , (N ∩ jK(Vν [H])) ∈ jK(S)

=

=T ∈ P (P (j(Vν [H])))Ult(V [H],K) ∩N, (N ∩ jK(Vν [H])) ∈ T

=

= GN ∩ P (P (j(Vν [H])))Ult(V [H],K) .

Hence GN ∩ P (P (j(Vν [H])))Ult(V [H],K) is N -generic for j(PI0). This gives that N ∈jK(SGI0 ↑ Vλ[H]), and therefore that SGI0 ↑ Vλ[H] ∈ K is stationary in V [H]. Weconclude that SGI0 is stationary in V [H]. Similarly one can argue that T ∧ SGI0 is

stationary in V [H] for all stationary T ∈ P (P (Vν [H]))V [H] \ I0.

11.3.1 Forcing the existence of presaturated towers: a guiding example

We can now produce the simplest example of a presaturated ideal forcing inducing analmost huge generic embedding with critical point a successor cardinal. More sophisticatedapplications of the same type of ideas forcing the existence of presaturated towers orideals with certain given properties are ubiquitous in the literature, Foreman’s handbookchapter [18] is one of the most up-to-date source of references; for our part we will repeatedlyapply these results in chapter 15.

Recall that Coll(ω,< κ) is the partial order given by the finite functions f : κ× ω → κsuch that f(α, n) ∈ α for all (α, n) ∈ dom(f). The order on Coll(ω,< κ) is reverseinclusion.

185

Example 11.3.6. Let δ be huge and G ∈ V be a normal ultrafilter on Vλ for someλ > δ witnessing this. Let also H be V -generic for Coll(ω,< δ). Then Coll(ω,<λ)/H ∼=Coll(ω,<λ) is a presaturated ideal forcing in V [H] induced by a normal ideal J on Vλdefined by the following property:

Let for p ∈ Coll(ω,<λ)

Tp =X ∈ P (Vλ)V : (X,∈) ∼= (Vδ,∈) and πX(p) ∈ H

.

Then[Tp]J : p ∈ Coll(ω,<λ)

defines a dense subset of P (P (Vλ))V [H] /J .

Aggiungerereferenza – M

Proof. Let j = jG be the ultrapower embedding induced by G. Then j : V → Ult(V,G) =M has critical point δ with j(δ) = λ. Observe that Coll(ω,< λ)V = Coll(ω,< λ)V [H].Moreover let i be the inclusion map:

i : Coll(ω,<δ) −→ Coll(ω,<λ)

f 7−→ f.

Then i extends to a regular embedding on the respective boolean completions.Let K be V [H]-generic for Coll(ω,<λ)/i[H]. It is an instructive exercise to check that:

• Coll(ω,<λ) ∼= Coll(ω,<δ) × Coll(ω, [δ,< λ) (where f ∈ Coll(ω, [δ,<λ)) if f ∈Coll(ω,<λ) and dom(f) ∩ δ × ω = ∅);

• H = K ∩ Vδ = i−1[H] is V -generic for Coll(ω,<δ);

• K ∩ Coll(ω, [δ,<λ)) is V [H]-generic for Coll(ω, [δ,<λ)).

– M Hence the map j Coll(ω,<δ)= i extends to a regular embedding of Coll(ω,< δ) intoColl(ω,< λ). By Theorem 11.3.1 and Lemma 11.3.2 we conclude that Coll(ω,< λ)/H isan ideal forcing PJ which induces a generic ultrapower which is < λ-closed. Now observethat Foreman’s duality theorem gives that

K =S ∈ P (P (Vλ))V [H] : iG,H,Coll(ω,<δ)([S]J) ∈ K

induces the V [H]-generic filter for PJ . Since K ⊇ G by Foreman’s duality theorem, we getthat (by 5.4.2)

X ∈ P (Vλ)V : πX [X] = Vκ and X ∩ κ ∈ κ = otp(X)∈ K.

Since this occurs for all K V [H]-generic for Coll(ω,<λ)/i[H], Claim 10 applied in thepresent context shows that

iG,H,Coll(ω,<δ)([Tp]J) = p

for all p ∈ Coll(ω,< δ).

The reader can play checking what happens if in the variety of Lemmas proved so farP is replaced by Coll(ω,< κ).

186

11.4 Self-genericity and presaturation of the stationary towerforcings

A remarkable result of Woodin shows that there are provably in ZFC+large cardinalsnatural examples of self-generic presaturated towers, and moreover that these towers play akey role in order to establish generic absoluteness results for second order number theory. Inthe remainder of this chapter we give a proof of Woodin’s result stating that the stationarytowers of height a supercompact cardinal are presaturated and self-generic.

Woodin’s is able to show that one can reduce significantly the large cardinal assumptionsreplacing supercompacts by Woodin cardinals (see [30, Theorem 2.5.8]). However we decideto present Woodin’s result using the supercompactness assumption, since on the one handthe proof of the presaturation of the relevant towers is greatly simplified, and on the otherhand it brings forward many ideas which rely on supercompact and huge embeddings whichwill be developed in the final part of this book, where we show in a variety of ways how onecan employ presaturated towers and ideal forcings to derive generic absoluteness results.

11.4.1 Basic properties of tower forcings concentrating on sets of ordertype at most λ

Notation 11.4.1. Let λ be a a regular cardinal.

Pλ = Z : Z ∩ λ ∈ λ > |Z|

andPλ(X) = Pλ ∩ P (X) .

IλX = NS Pλ(X) and Iλ is the tower of normal ideals IλX .We also use the following notational conventions:

T Iλδ = T λδ .

SGIλ∩Vδ := SGλδ .

Lemma 11.4.2. Let λ be a a regular cardinal and I be a tower concentrating on sets Mof size at most λ such that M ∩ λ ∈ λ. Let

SIδ =M ∈ SGI : otp(M ∩ δ) = λ

.

Assume SIδ ∧ T is stationary for all T ∈ Vδ. Then

SGI ∧ Pλ(Vδ+1) ∧ T

is also stationary for all T ∈ Vδ and T Iδ is < δ-presaturated.

To prove the Lemma we need the following piece of notation:

Definition 11.4.3. Let N ⊇ M and X be a transitive set. We say that N is an X-endextension of M if N ∩X = M ∩X.

Notation 11.4.4. Given an ordinal δ and a set M , δM = sup(M ∩ Vδ).

Remark 11.4.5. Observe that if N is a VδM -end extension of M and α ≤ δM , GN ∩ Vα =GM ∩ Vα, since:

S ∈ GN ∩ Vα ⇐⇒ N ∩⋃S ∈ S ⇐⇒ (N ∩ Vα) ∩

⋃S ∈ S

⇐⇒ (M ∩ Vα) ∩⋃S ∈ S ⇐⇒ M ∩

⋃S ∈ S ⇐⇒ S ∈ GM .

187

Proof. Let S = SIδ .

SGI ∧ Pλ(Vδ+1) ∧ T is stationary for all stationary T ∈ Vδ such that TI ∈ T Iδ .

Assume M ∈ S ↑ Hθ ∧ T for some large θ. Let jM : πM [M ]→M be the inverse ofthe transitive collapse of M .

Since δ is inaccessible, fix in M g : δ → Vδ bijection. We have that M ∩Vδ = g[M ∩ δ]has size λ. Fix in M f : H<ω

δ+ → Hδ+ Skolem function such that any N ∈ Cf iselementary in Vδ+1 and such that f [Z<ω] ∈ Cf for any infinite set Z (f exists by 2.2.3).Let Nα = f [g[α ∩M ]<ω]. Let N =

⋃α∈M∩δNα. Now N ⊆ M , N ∩ Vδ = M ∩ Vδ,

N = f [M ∩ V <ωδ ]. Hence GM ∩ Vδ = GN ∩ Vδ, therefore N ∈ SGI ∧ T . Moreover

Nβ =⋃α∈M∩β Nα for all β ∈M ∩ Vδ+1, Therefore we can find a club subset C of λ

such that for any α ∈ C

NjM (α) ∩ Vδ = g[jM (α) ∩M ],

NjM (α) = f [NjM (α) ∩ V <ωδ ],

and GM ∩ A ∈ NjM (α) for any A ∈ NjM (α), i.e. GNjM (α)∩ Vδ is NjM (α)-generic for

T Iδ .

Let for each α ∈ C δα = supNjM (α) ∩ δ.Observe that NjM (β) is a Vδα-end extension of NjM (α) for any α < β both in C since:

NjM (β) ∩ Vδ = g[δβ ∩M ],

NjM (α) ∩ Vδ = g[δα ∩M ].

andg[δβ ∩M ] ∩ Vδα = g[δ ∩M ] ∩ Vδα = g[δα ∩M ]

since α ∈ C.

Pick α ∈ C large enough so that T ∈ NjM (α). Then

NjM (α) ∈ SGI ∧ Pλ(Vδ+1) ∧ T.

Since this can be repeated for any M ∈ S ↑ Hθ ∧ T , the stationarity of SGI ∧Pλ(Vδ+1) ∧ T is established.

T Iδ is < δ-presaturated.

By Propositions 4.2.3, 11.2.1, it is enough to show the following:

Claim 11. Given any sequence of γ-many maximal antichains Aξ : ξ < γ for someγ < δ,

A =R ≤I T ∈ T Iδ : ∀i < γ| U ∈ Ai : U ∧R is stationary | < δ

is dense.

Proof. Let T0 ∈ T Iδ . We look for T ∈ A such that T ≤I T0.

Given M ≺ Hθ with M ∈ S ∧ T0 ↑ Hθ, Ai : i < γ ∈ M and S, T0 ∈ M , we haveM ∩

⋃T0 ∈ T0 and for any i ∈ γ ∩M , GM ∩ Ai ∩M 6= ∅, since GM is M -generic

for T Iδ . For any i ∈ M ∩ γ, let Si ∈ GM ∩ Ai, and find αi ∈ M ∩ δ be such that

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Si ∈ Vαi . Since otp(M ∩ δ) = λ and γ ∈M ∩ δ, otp(M ∩ γ) < otp(M ∩ δ) = λ, thusα = sup αi : i ∈M ∩ γ < sup(M ∩ δ). Therefore there exists β ∈M ∩ δ such thatαi < β for any i ∈M ∩ γ. Put

T ∗ = N ∩ Vβ : ∀i ∈ N ∩ γ(Ai ∩GN ∈ Vβ) ∧ T0.

Then M ∩ Vβ ∈ T ∗, and T ∗ ∈ GM . This gives that T ∗ ∈ T Iδ refines T0.

Subclaim 1. For any i ∈ γ:

U ∈ Ai : U ∧ T ∗ is stationary ⊆ Ai ∩ Vβ.

By the subclaim we get that T ∗ ∈ A and we are done. So we prove it.

Proof. Assume that U ∈ Ai and U /∈ Vβ for some i < γ, we will show that U ∧T ∗∧Sis non-stationary: Assume by contradiction it is stationary, let N ∈ U ∧ S ∧ T ∗ withi ∈ N , then U ∈ GN ∩ (Ai \ Vβ), on the other hand GN ∩Ai ∩ Vβ is non-empty, sinceT ∗ ∈ GN as well. Thus |GN ∩ Ai| ≥ 2. Since Ai is an antichain and GN is a filter,their intersection can have at most one element, a contradiction. We conclude thatU ∧ T ∗ ∈ ∪I, and we are done.


The Lemma is proved.

11.4.2 The stationary towers are presaturated and self-generic

We prove the following theorem:

Theorem 11.4.6. Let δ be supercompact. Then T γ+

δ is < δ-presaturated and self-genericfor any γ < δ.

We prove the following stronger result:

Theorem 11.4.7. Assume δ is supercompact and λ = γ+ < δ. Let

Sδ = SGλδ ∩ M ≺ Hδ+ : otp(M ∩ δ) = λ .

Then Sδ ∧ T is stationary for any stationary set T ∈ T λδ .

By Lemma 11.4.2, we get that T λδ is < δ-presaturated; clearly it is also self-generic,since Sδ ∧ T ≤NS SGλ

δ ∧ T for all T ∈ Vδ.We proceed with the proof of Theorem 11.4.7.

Proof. We start our proof introducing the following definitions and sets:

Notation 11.4.8. Let A ⊆ Vδ be a family of stationary sets, and M ≺ Hδ+ .M captures A ∈M if A ∩GM is non-empty, i.e.:

M ∈hA =

N ≺ H+

δ : ∃T ∈ A ∩N,N ∩⋃T ∈ T

.

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Definition 11.4.9. Let A ⊆ Vδ be a family of stationary sets, then

S(A) = M ≺ Hδ+ : A ∩GM 6= ∅ =hA,

C(A) = M ≺ Hδ+ : ∃N ∈ S(A) VδM -end extension of M ,

where δM = sup(M ∩ δ).

Lemma 11.4.10. Let δ be inaccessible, θ δ be a strong limit of cofinality larger thanδ, λ < δ be regular. Let A ⊆ Vδ be a family of stationary sets with S ⊆ Pλ for all S ∈ A.The following are equivalent:

1. Given any M ≺ Hθ in Pλ(Hθ) with A ∈M , there exists N ≺ Hθ which is a VδM -endextension of M such that S(A) ∈ GN .

2. C(A) is a club subset of Pλ(Hδ+).

Proof. We prove this equivalence as follows:

(1)⇒ (2) Assume by contradiction that C(A) is not a club subset of Pλ(Hδ+). Let Sbe the complement in Pλ(Hδ+), and let M be an elementary substructure of Hθ

such that M ∈ S ↑ Hθ. By (1) there exists N VδM -end extension of M such thatN ≺ Hθ and N ∩Hδ+ ∈ S(A). Then M ∩Hδ+ ∈ C(A) and this is a contradictionsince M ∈ S ↑ Hθ.

(2)⇒ (1) Assume that (2) holds, and assume M ⊆ Hθ with A ∈ M . Then C(A) ∈ Mand so M |= C(A) is a club. Therefore there exists

f : H<ωδ+ → Hδ+

such that f ∈M and C(A) ⊇ Cf . Since f ∈M , for any s ∈M∩H<ωδ+ f(s) ∈M∩Hδ+ .

This implies that M ∩ Hδ+ ∈ Cf ⊆ C(A). By definition of C(A), there existsN ′ ∈ S(A) such that

N ′ ∩ VδM = M ∩ Vδ,

N ′ ≺ Hδ+ , and N ′ ⊇M ∩Hδ+ . Put

N =g(x) : x ∈ N ′ ∩ Vδ, g ∈M, g : Vδ → V

.

We want to show that N witnesses (1) for M . We have the following:

• N ∩ Vδ = N ′ ∩ Vδ. Clearly N ∩ Vδ ⊇ N ′ ∩ Vδ. For the other inclusion letx ∈ Vδ ∩ N . Then x = g(y) for some g : Vδ → Vδ, g ∈ M ∩ H+

δ ⊆ N ′ andy ∈ N ′ ∩ Vδ. Then g(y) ∈ N ′ ∩ Vδ, since g, y ∈ N ′.• N ≺ Hθ: We prove this using the Tarski-Vaught criterium. Assume

Hθ |= ∃yφ(y, f1(x1), . . . , fn(xn)),

where f1, . . . , fn ∈M , and x1, . . . , xn ∈ N ′ ∩ Vδ. W.l.o.g. we can assume thatx1 = · · · = xn = x, letting x = 〈x1, . . . , xn〉 and eventually replacing fj(xj) withfj πj(x) where πj is the projection on the j-th coordinate.

For each x ∈ Vδ, let αx < θ be the least β such that

Hθ |= ∃y ∈ Vβ ∧ φ(y, f1(x), . . . , fn(x))

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if such a y and β can be found, αx = 0 otherwise. The map x 7→ αx is definablein Hθ and maps Vδ into θ. Recall that θ is strong limit and cof(θ) > δ = |Vδ|by our assumptions. Hence the range of this map is bounded below θ by someβ. Therefore Hθ models that there exists β such that for all x ∈ Vδ there existsyx ∈ Vβ such that

Hθ |= φ(yx, f1(x), . . . , fn(x))

(if there is any any y such that Hθ |= φ(y, f1(x), . . . , fn(x))). Since M ≺ Hθ, wecan find such a β ∈M . Let g : Vδ → Vβ be in M be defined for z ∈ Vδ by

g(z) = 0 if Hθ |= ∀y¬φ(y, f1(z), . . . , fn(z));

g(z) = y such that Hθ |= φ(y, f1(z), . . . , fn(z)) otherwise.

Since g ∈M and x ∈ N ′ ∩ Vδ, g(x) ∈ N , moreover since

Hθ |= ∃yφ(y, f1(x), . . . , fn(x)),

we get thatHθ |= φ(g(x), f1(x), . . . , fn(x)).

This concludes the proof.

The following is the key-step in the proof of 11.4.7. We feel free to confuse an elementof Vδ with its equivalence class in T λδ to simplify notation.

Lemma 11.4.11. Let δ be supercompact and λ = γ+ < δ. Then for any A ⊆ Vδ predensesubset of T λδ , C(A) is a club of models in Pλ (Vδ+1).

Proof. Let

A =C(A)c ∩ Pλ (Hδ+) =

=N ≺ Hδ+ : N ∈ Pλ (Hδ+) , and ∀N ′ ≺ Hδ+ VδN end-extension of N,N ′ /∈ S(A)

Assume by contradiction that C(A) is not a club. Then A is stationary.

By the supercompactness of δ, we can find j : V → M with M<η ⊆ M , crit j = δ,η δ. Since P

(P(H+δ

))⊆ M , U ⊆ P (Hδ+) is stationary in V if and only if U is

stationary in M . Therefore if A ∈ V is stationary, then A ∈M and it is also stationary 2

in M . Hence A ∈ (T λj(δ))M . Since M |= j(A) is a maximal antichain of T λj(δ), there exists

T1 in M such thatM |= T1 ∧A is stationary and T1 ∈ j(A).

Let θ ∈M be large enough (i.e. above j(δ)) and let3 in M

N ∈ T1 ∧A ↑ (Hθ)M ,

with j Hδ+∈ N . Take4 N0 = N ∩Hδ+ . Then:

• j[N0] = j(N0) ∈ M , since j[N0] has size less than λ < δ and M is closed underη-sized sequences with η > δ.

2This conclusion can be obtained just assuming δ measurable so that Hδ+ ⊆M , since A ⊆ [Hδ+ ]ℵ0 ⊆ Hδ+is definable with parameters in Hδ+ and M ⊆ V grants that A remains stationary in M .

3To find such an N we need j to be a (2δ)+- supercompact embedding to grant that j Hδ+∈M .

4Note that Mδ ⊆M grants that (Hδ+)M = Hδ+ .

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• N ∩ (Hj(δ)+)M ⊇ j[N0], since j Hδ+ ∈ N (hence j(δ+) ∈ N and j(δ+) > δ+.

• j[N0]∩ Vj(δ) = N0 ∩ Vδ = N ∩ Vδ, since δ is the critical point of j (hence all elementsof N0 ∩ Vδ are not moved by j).

• j(δ)j[N0] = sup(j[N0] ∩ j(δ)) = sup(N0 ∩ δ) < δ, since N0 ∩ Vδ = j[N0] ∩ Vj(δ) hassize less than λ.

• N∩(Hj(δ)+)M ∈ S(j(A)), since N∩∪T1 ∈ T1 ∈ j(A) and N∩(Hj(δ)+)M ≺ (Hj(δ)+)M .

Hence N ∩ (Hj(δ)+)M is in M a Vj(δ)-end extension of j[N0] witnessing that

M |= j[N0] ∈ C(j(A)).

On the other hand j[N0] = j(N0) ∈ j(A), since N0 ∈ A.We conclude that j[N0] witnesses that j(A) and C(j(A)) have non-empty intersection.

But, by elementarity of j, j(A) is disjoint from C(j(A)) = j(C(A)), since A is disjointfrom C(A). This is a contradiction.

The proof of the Lemma is completed.

To conclude our proof it suffices to show the following:

Claim 12. LetS = SGλ

δ ∩ M ≺ Hδ+ : otp(M ∩ δ) = λ .

Then S ∧ T is stationary for any T ∈ T λδ .

We prove the lemma in case λ = ω1 = ω+ to simplify slightly the book-keeping devices.The reader can supply the proof of this Claim for arbitrary λ = γ+ with minor modifications.With these assumption on λ, Pλ consists of the countable sets whose intersection with ω1

is an ordinal.

Proof. Let M ≺ Hθ be countable and such that M ∩⋃T ∈ T (i.e. T ∈ GM ). Put M0 = M .

Let

ϕ : N −→ N2

n 7−→ 〈m, j〉,

be a surjection such that the preimage of every ordered pair is infinite and m ≤ n for alln in the preimage of 〈m, j〉. Assume that

A0i : i ∈ ω

is an enumeration of the maximal

antichains of M0. Let ϕ(0) = 〈0, j〉 and α = sup(M0 ∩ δ), then by Lemmas 11.4.11, 11.4.10,we can find M1 which is a Vα end-extension of M = M0 such that A0

j ∩ GM1 6= ∅ andM1 ≺ Hθ. Observe that GM1 ∩ Vδ ⊃ GM0 ∩ Vδ.

We proceed by induction on γ ≤ ω1 as follows:

• Assume γ = β + n with β limit and that we have defined Mγ . LetAβ+ni : i ∈ ω

be an enumeration of the maximal antichains of Mβ+n. Let ϕ(n) = 〈m, j〉 andα = sup(Mβ+n ∩ δ), then by Lemmas 11.4.11, 11.4.10, we can find a Vα end-

extension of Mγ , Mβ+n+1, such that Aβ+mj ∩GMβ+n+1

6= ∅ and Mβ+n+1 ≺ Hθ. ThenGMβ+n+1

⊃ GMn .

• At a limit stage γ ≤ ω1 let Mγ =⋃β<γMβ.

Let N =⋃Mγ : γ ∈ ω1. Then N ∈ S ∧ T .

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The proof of Theorem 11.4.7 is completed.

Remark 11.4.12. Regarding the proof of the last Claim, following the same notation, weobserve that in ω-many steps we consider all the possible maximal antichains of Mω (andfor all β in ω-many steps we consider all the possible maximal antichains of Mβ+ω, inparticular along the way Mγ ∈ SGω1

δ for all limit γ). This shows that an induction oflength ω suffices to prove that T ω1

δ is self-generic, but it is not yet sufficient to infer thatT ω1δ is presaturated. Hence we continued the induction for ω1 + 1-many steps in order

to have otp(N ∩ δ) = ω1, so that we can use Lemma 11.4.2 to get the presaturation ofT ω1δ . To prove the Claim for arbitrary λ = γ+ one has to organize the above inductive

construction using λ+ 1-many steps and use a bookeeping surjection from γ to γ2 suchthat the preimage of each ordered pairs has size γ. We leave the details to the reader.

11.4.3 Other properties of stationary towers

We outline a few other properties of the stationary towers we will need in the sequel:

Lemma 11.4.13. Assume δ is supercompact and γ+ = λ < δ. Then for all X ∈ V

DX =

SGλα : ∃j : Vη → Vλ elementary with crit(j) = α, j(α) = δ and X ∈ j[Vη]

is predense in T λδ .

Proof. Fix T stationary in T λδ . Given X ∈ V , find j : Vη → Vλ elementary with crit(j) =α > λ, j(α) = δ, X ∈ j[Vλ], T ∈ Vα. This is possible by the supercompactness of δ. Thenwe have that

Sδ = SGλδ ∩ M ≺ Hδ+ : otp(M ∩ δ) < λ

is equal to j(Sα), where

Sα = SGλα ∩ M ≺ Hδ+ : otp(M ∩ α) < λ .

By Theorem 11.4.7 and Lemma 11.4.2, Sδ ∧ T is stationary, and by elementarity of j weget that Sα ∧ T is stationary. Hence SGλ

α ∧ T is stationary and SGλα belongs to DX . Since

T was chosen arbitrarily in T λδ , the thesis follows.

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Chapter 12

Generic absoluteness for L([Ord]ω)

Woodin’s generic absoluteness for second order number theory is in our eyes one of themajor achievements of set theory of the last fourty years. The aim of this chapter is toprovide a detailed proof of this result. More precisely we will prove the following theorem:

Theorem 12.0.1 (Woodin). Assume V is a (transitive1) model of ZFC which also modelsthat there are class many supercompact cardinals2. Let φ(x1, . . . , xn) be a formula in thefree variables x1, . . . , xn. The following are equivalent for any r1, . . . , rn ∈ 2ω:

• L([Ord]ω)V models φ(r1, . . . , rn).

• For some boolean algebra B ∈ V

JL([Ord]ω) |= φ(r1, . . . , rn)KB = 1

holds in V .

The theorem will be an easy corollary of the following:

Theorem 12.0.2. Assume V is a (transitive) model of ZFC and δ is supercompact in V .Then there is a complete boolean algebra B ∈ V and injective complete homomorphisms

i0 : T ω1δ → B

i1 : RO(Coll(ω,< δ))→ B

in V such that for all H V -generic for B

L([Ord]ω)V [i−10 [H]] = L([Ord]ω)V [H] = L([Ord]ω)V [i−1

1 [H]].

We prove Theorem 12.0.1 assuming Theorem 12.0.2. First we need to recall someproperties of the forcing Coll(ω,< δ):

Remark 12.0.3. Coll(ω,< δ) has the following properties whenever δ is inaccessible (mostof them hold just assuming δ regular and uncountable):

• It is < δ-cc and collapses δ to become ω1 (see [28, Theorem 15.7(iii), Thm 15.22]).

1We carry the proof of this and of the subsequent theorem assuming that V is transitive, but thisassumption is redundant.

2The large cardinal assumption can be reduced to ask just for the existence of class many Woodincardinals which are a limit of Woodin cardinals (see [30, Theorem 3.1.2, Corollary 3.1.7]).

194

• The inclusion map of Coll(ω,< γ) into Coll(ω,< δ) extends to an injective completehomomorphism of the respective boolean completions for all cardinals γ < δ (see theproof of [28, Theorem 15.22]).

• It is homogeneous, i.e. for any p, q ∈ RO(Coll(ω,< δ))+ there is an automor-phism of RO(Coll(ω,< δ)) mapping p to q (see [28, Thm 26.12, Cor. 26.13], Thm.26.12 actually proves that any π : B → C isomorphism of complete subalgebras ofRO(Coll(ω,< δ)) can be extended to an isomorphism of RO(Coll(ω,< δ))).

• It absorbs any forcing in Vδ if δ is inaccessible, i.e.: for any B ∈ Vδ there is a completeinjective homomorphism of B into RO(Coll(ω,< δ)) (Kripke [28, Corollary 26.8]).

• For homogenous forcings B, Jφ(a1, . . . , an)KB is either 1B or 0B (see [28, Lemma14.37]).

• Whenever G is V -generic for some B ∈ Vδ, Coll(ω,< δ)V [G] = Coll(ω,< δ)V andRO(Coll(ω,< δ))V [G] ∼= RO(Coll(ω,< δ))V /i[G], where i : B→ RO(Coll(ω,< δ))V isany complete injective homomorphism in V (see the Factor Lemma [28, Cor. 26.11]).

Proof. Let W be a transitive model of ZFC, H be W -generic for B (where B is the cbagiven in Theorem 12.0.2 for W and δ. Let Gj = i−1

j [H], by 11.4.6 there is in W [G] anelementary j : W → Ult(W,G0) such that Ult(W,G0)ω ⊆ Ult(W,G0). In particular weobtain that

L([Ord]ω)Ult(W,G0) = L([Ord]ω)W [G0] = L([Ord]ω)W [H] = L([Ord]ω)W [G1].

Moreover k = j L([Ord]ω)W is still an elementary embedding

k : L([Ord]ω)W → L([Ord]ω)W [H].

This gives that the structures

〈L([Ord]ω)W ,∈, (2ω)W 〉

〈L([Ord]ω)W [G1],∈, (2ω)W 〉

are elementarily equivalent in W [H]. Thus for any first order formula φ(x1, . . . , xn) andr1, . . . , rn ∈ (2ω)W , we get that

〈L([Ord]ω)W ,∈, (2ω)W 〉 |= φ(r1, . . . , rn)

if and only if〈L([Ord]ω)W [G1],∈, (2ω)W 〉 |= φ(r1, . . . , rn).

Since Coll(ω,< δ) is homogeneous, we get that

W |= φ(r1, . . . , rn)L([Ord]ω)W

if and only if (since

sφ(r1, . . . , rn)L([Ord]ω)

W [GColl(ω,<δ)]∈ G1)

W |=sφ(r1, . . . , rn)L([Ord]ω)

W [GColl(ω,<δ)]> 0RO(Coll(ω,<δ))

195

if and only if (by the homogeneity of Coll(ω,< δ))

W |=sφ(r1, . . . , rn)L([Ord]ω)

W [GColl(ω,<δ)]

= 1RO(Coll(ω,<δ)).

We conclude that for all transitive models W , and for all supercompact cardinal δ in W ,and all formulae φ(x1, . . . , xn) in the free variables x1, . . . , xn, and all r1, . . . , rn ∈ (2ω)W :

L([Ord]ω)W |= φ if and only ifrφ(r)L([Ord]ω)

z

RO(Coll(ω,<δ)= 1.

Now let V be a fixed transitive model of ZFC, C be any complete boolean algebra in V ,and δ be a supercompact cardinal in V such that C ∈ Vδ. Let G be V -generic for C. Thenin V [G] δ remains supercompact (by Lemma 11.3.3) and

Coll(ω,< δ)V = Coll(ω,< δ)V [G].

Moreover any V [G]-generic filter H for Coll(ω,< δ) is also V -generic for the same forcingnotion and V [G][H] = V [H]. By the above observations, we get that for any formulaφ(x1, . . . , xn) in the free variables x1, . . . , xn, and all r1, . . . , rn ∈ (2ω)V , letting H beV [G]-generic for Coll(ω,< δ):

L([Ord]ω)V |= φ(r1, . . . , rn)

if and only if

L([Ord]ω)V [H] |= φ(r1, . . . , rn)

if and only if

L([Ord]ω)V [G] |= φ(r1, . . . , rn).

Since this holds for any G V -generic for B and H V [G]-generic for Coll(ω,< δ), we getthat

L([Ord]ω)V |= φ(r1, . . . , rn)

if and only ifrL([Ord]ω)V [G] |= φ(r1, . . . , rn)

z= 1B.


We now prove Theorem 12.0.2.

Proof. Let G be V -generic for T ω1δ with δ supercompact in V .

In V [G] consider the forcing notion P given by functions

g : ω × γg → γg

such that γg < δ is a cardinal in V and

H(g) = s ∈ Coll(ω,< γg) : s ⊆ g

is V -generic for Coll(ω,< α) for all α ≤ γg which are regular cardinals in V . The order onP is reverse inclusion. We will be done once we prove the following:

Claim 13. The following holds for any V [G]-generic filter K for P :

196

1. H = H(∪K) is V -generic for Coll(ω,< δ),

2. For any α we can find g ∈ K such that dom(g) ⊇ ω × γ and G ∩ Vα ∈ V [g].

Consequently HV [H]ω1 ⊇ HV [G]

ω1 , since

HV [G]ω1

=τG : τ ∈ Vδ ∩ V T

ω1δ

=⋃α<δ

τG∩Vα : τ ∈ Vα ∩ V T

ω1δ

by the presaturation of T ω1

δ and 4.2.4.

3. V [G] models that P is < ω1-distributive. Consequently HV [G]ω1 = H

V [G][K]ω1 ⊇ HV [H]

ω1 .

Therefore HV [G]ω1 = H

V [G][K]ω1 = H

V [H]ω1 .

4. All countable sequences of ordinals in V [G][K] are in V [H]. Hence V [G][K] is aV [H]-generic extension by a < δ-distributive forcing in V [H].

It is clear that the proof of this Claim suffices to prove the Theorem.

Proof. We prove each item of the Claim as follows:

1. Any maximal antichain A of Coll(ω,< δ) is a maximal antichain of Coll(ω,< γ)for all cardinals γ such that A ∈ Vγ , due to the < δ-CC of Coll(ω,< δ) and to thefact that the inclusion map of Coll(ω,< γ) into Coll(ω,< δ) extends to a injectivecomplete homomorphism of the respective boolean completions for all cardinals γ < δ.It is also immediate to check that

Dα = h ∈ P : ω × α ⊆ dom(h)

is an open dense subset of P in V [G] for all α < δ. This gives that if A ∈ Vα is amaximal antichain of Coll(ω,< δ) and g ∈ Dα, H(g) ∩ A is non-empty. An easydensity argument on P shows that H(∪K) is V -generic for Coll(ω,< δ).

2. It suffices to show that (2ω)V [G] = (2ω)V [H(∪K)] for any K V [G]-generic for P , since

G ∩ Vα is a countable set in HV [G]ω1 for any α < δ. To this aim given h ∈ P let

(γ+n : n ≤ ω) list the first ω + 1-many cardinals of V greater or equal than γh. Tosimplify notation let γω = γ. Notice that (γn : n < ω) ∈ V and that Coll(ω,< γ)is not < γ-CC in V since γ is a strong limit cardinal of countable cofinality in V .Nonetheless this will not harm our construction below.

Fix r ∈ (2ω)V [G] and build a sequence (hn : n ∈ ω) in V [G] such that:

• h0 = h,

• hn+1 ⊇ hn,

• hn+1(0, γn) = r(n),

• H(hn) is V -generic for Coll(ω,< γn) for all 0 < n < ω.

To build hn+1 given hn, find in V [G] a V [hn]-generic filter U for the poset given byfinite partial functions s : ω×γn+1 → (γn+1 \γn) such that s(0, γn) = r(n) ordered byreverse inclusion — this forcing is easily seen in V to be isomorphic to Coll(ω,< γn+1)

— let hn+1 = hn ∪⋃U .

Let g = ∪n∈ωhn. Then H(g ω×γn) is V -generic for Coll(ω,< γn) for all 0 < n < ω,giving that g ∈ P refines h.

197

Thus in V [G] for any given h ∈ P and r ∈ (2ω)V [G], we can find g ⊇ h in P such thatr ∈ V [H(g)], since r is computable in the parameters (γ+n

h : n < ω) ∈ V and g.

A standard density argument will now give that HV [H(∪K)]ω1 = H

V [G]ω1 for any K

V [G]-generic for P . The desired conclusion follows.

3. We have to prove that V [G] models that P is < ω1-distributive for any G V -genericfor T ω1

δ .

Let S ∈ T ω1δ and fix G V -generic for T ω1

δ such that S ∈ G. Let An : n ∈ ω be acountable family of maximal antichains of P in V [G] and h ∈ V [G] ∩ P . Notice thateach An is a subset of Vδ[G]. By the < δ-presaturation of T ω1

δ in V , we can find in

V a familyAn : n ∈ ω

∈ V and h ∈ V such that

• valG(h) = h,

• valG(An) = An for all n ∈ ω.

• An ⊆ Vδ, h ∈ Vδ.

By Lemmma 11.4.13 the setD〈An:n∈ω,H,P 〉

is dense in T λδ . Hence we can find j : Vγ → Vη in V elementary and such that

• j(crit(j)) = δ, α = crit(j) > ξ, h ∈ Vα,

•An : n ∈ ω

, P ∈ j[Vγ ],

• SGIα ∈ G, so that G ∩ Vα is V -generic for T ω1α .

Notice that j(An ∩ Vα) = An for all n since An ⊆ Vδ.Now in V [G] let (αn : n ∈ ω) be an increasing sequence converging to α of countableordinals of V [G] which are inaccessible cardinals in V (this sequence exists in V [G]since α is in V an inaccessible limit of inaccessible cardinals by elementarity of j,while it is a countable ordinal in V [G]).

In V [G ∩ Vα] build an increasing sequence hm : m ∈ ω ⊆ P ∩ Vα such that

• h0 = h

• h2m+1 extends some h∗ ∈ Am ∩ Vα = (Am ∩ Vα)G∩Vα ,

• H(h2m+2) is V -generic for Coll(ω,< αi) with αi chosen so that h2m+1 ⊆ Vν forsome ν < αi.

Then ∪m∈ωhm = g ∈ P : Coll(ω,< α) is < α-CC in V , hence for any maximalantichain B ∈ V on Coll(ω,< α), there is i < ω such that B ⊆ Coll(ω,< αk) isa maximal antichain of Coll(ω,< αk) for all k ≥ i. therefore B has been met bythe filter H(hk) for eventually all k ∈ ω. Thus H(g) is V -generic for Coll(ω,< α).Clearly g meets all the An since it extends h2m+1 for all m.

4. Let us now argue that f ∈ V [H] for any function f : ω → Ord in V [G][K]. So letus fix f : ω → Ord in V [G][K]. We aim to show that f ∈ V [H]: first observe thatf ∈ V [G] by the < δ-distributivity of P over V [G].

Next, since δ is supercompact, we have that

D =

SGIα : α < δ T ω1α is self-generic and presaturated

198

is predense in T ω1δ by Lemma 11.4.13, in particular it is immediate to check that

β < δ : SGIβ ∈ G ∩D

is unbounded in δ in V [G].

Now f = fG for some f ∈ V Tω1δ name for a function with domain ω and range into

the ordinals. Since δ = ωV [G]1 , the countably many maximal antichains contained

in Vδ needed to decide the values of f in V [G] are met by G all in some Vα forsome α < δ. We can suppose that α is such that SGIα ∈ G. Hence Gα = G ∩ Vα isV -generic for T ω1

α by Lemma 11.1.8.

This gives that f ∈ V [Gα], since fG = hG∩Vα where h ∈ V Tω1α is the T ω1

α -namesatisfying the property

rh(n) = α

z≥ S iff (S ∈ T ω1

α andrf(n) = α

z≥ S ∧ SGIα in T ω1

δ ).

We conclude that f ∈ V [g] ⊆ V [H] for any g ∈ K such that Gα ∈ V [g].

Now we can show that V [G][K] is a generic extension of V [H] by a < δ-distributiveforcing in V [H]: V [G][K] is a forcing extension of V [H] by [28, ?, Theorem XXX]for some forcing Q ∈ V [H]. Assume L ∈ V [G][K] is V [H]-generic for Q; since allcountable sequences of ordinals in V [G][K] are in V [H] and δ = (ω1)V [G][K], we getthat Q q is < δ-distributive over V [H] for some q ∈ L. W.l.o.g. we can assume thatQ = Q q.

The proof of the Claim is completed.

The proof of the Theorem is completed.

199

Chapter 13

Forcing axioms II

This chapter dwelves further in the analysis of forcing axioms. The first aim is to provethe consistency of MM++ relative to a supercompact cardinal. The remaining part of thechapter outlines that many forcing axioms can be formulated as density properties of thecategory forcings (Γ,≤Γ) for suitably chosen Γ: for example in the last part of the chapterwe show that MM++ can be formulated as a density property of the class partial order(SSP,≤SSP). Towards this aim we first show that assuming the existence of class manysupercompact cardinals the natural embedding of the category forcing (Ωκ,≤Ω) of forcingsP satisfying FAκ(P ) into the class partial order given by stationary sets concentrating onPκ+ (V ) ordered by the ≤NS-relation is order and incompatibility preserving and preservesset sized suprema.

Recall that (see Def. 8.6.2 for details):

For M ≺ Hθ of size ω1 and containing ω1 and B ∈ M a cba, G ultrafilteron B ∩M is (M,SSP)-correct if G is M -generic and interprets correctly theB-names for stationary subsets of ω1 in M .



13.1 Consistency of MM++

Theorem 13.1.1 (Foreman, Magidor, Shelah). Assume there exists a supercompact cardi-nal. Then NBG + MM++ is consistent.

Proof. We use the following property of supercompact cardinals1 [28, Laver, Theorem20.21]:

Assume δ is supercompact. Then there exists g : δ → δ such that for allα > δ there exists j : V → M ⊆ V elementary with δ = crit(j), j(g)(δ) > α,Mα ⊆M .

Now consider the lottery preparation forcing PSP,gδ (see Def. 7.5.4).

Lemma 13.1.2. Assume G is V -generic for PSP,gδ . Then V [G] |= MM++ and δ = ω

V [G]2 .

1Literally the cited theorem proves something different for a certain function f : δ → Vδ, nonethelessletting g(α) =

∣∣Vrank(f(α)+1)

∣∣, we have that g satisfies the quoted sentence.

200

Proof. We first prove the following:

Claim 14. Assume that for all semiproper forcings B FA++(B) holds. Then MM++ holds.

Proof. Notice that every countably closed forcing is semiproper. Moreover FA++(B) clearlyentails FA+(B). By Theorem 9.4.2, we conclude that every stationary set preserving forcingis semiproper. Hence the thesis.

Now assume G is V -generic for PSP,gδ . Then ωV1 = ω

V [G]1 since PSP,g

δ is semiproper in

V by Theorems 10.2.7, 10.2.8 and Proposition 7.5.5. Moreover PSP,gδ is < δ-CC, again by

Proposition 7.5.5. Hence δ is regular in V [G] and is the successor of ω1, since for all even

successor α < δ, PSP,gα adds a bijection of α with ω1.

To complete the proof of the Lemma we must show that MM++ holds in V [G].By the first Claim it suffices to prove the following:

Claim 15. Let B ∈ V [G] be a semiproper forcing. Then FA++(B) holds in V [G].

Proof. Find B ∈ Hθ with θ > 2|B| regular and such thatrB ∈ SP

z= 1

PSP,gδ

and BG = B.

Find j : V →M ⊆ V such that M2θ ⊆M and j(g)(δ) > θ. Observe that j Hθ ∈Mand that HM

θ = Hθ. Hence HMθ models that

rB ∈ SP

z= 1

PSP,gδ

, which gives that M as

well models thatrB ∈ SP

z= 1

PSP,gδ

. Hence B is one of the factors used to define in M

(PSP,j(g)δ+1 )M = PSP,g

δ ∗∏

C ∈ VMj(g)(δ) : M |=

rC ∈ SP

z= 1

PSP,gδ

.

Consider in V the forcing (PSP,j(g)j(δ) )M . Now (PSP,g

δ )M = PSP,gδ is a complete sub-forcing

of (PSP,j(g)j(δ) )M and is < δ-presaturated in V . Moreover letting q = iδ+1,j(δ)(〈1PSP,g

δ ,B, B〉),

we have that the map p 7→ j(p) ∧ q defines a complete homomorphism of PSP,gδ into

(PSP,j(g)j(δ) )M q in V .

We let H be V -generic for (PSP,j(g)j(δ) )M with G ⊆ H and q ∈ H. By Theorem 11.3.1 j

lifts to an elementary j : V [G]→M [H] with j(σG) = j(σ)H .Now observe that q in H grants that

K =bG ∈ B : iδ+1,j(δ)(〈1PSP,g

δ, b〉) ∈ H

is V [G]-generic for B. Let KG = K. Since M [H] is a generic extension of M [G][K] by the

semiproper (in M [G][K]) forcing (PSP,j(g)j(δ) )M/G∗K (with G ∗ K =

(p, b) : p ∈ G, bG ∈ K

V -generic for (P

SP,j(g)δ+1 )M ), we get that all B-names S in M [G] for stationary subsets of ω1

are such that SK remains stationary in M [H]. Moreover each such B-name S can be found

in HVθ [G] = HM

θ [G] = HM [G]θ , since HV

θ = HMθ and B ∈ Hθ. Now j[HV

θ [G]] ∈M [H], sinceG ∈M [H] and j[Hθ] ∈M , and

M [H] |= j[HVθ [G]] ≺ HM [H]

j(θ) .

Moreover j[HVθ [G]] has size ω

M [H]1 = ωV1 in M [H] (again because (P

SP,j(g)j(δ) )M is semiproper

in M) and contains ωM [H]1 , since ω

M [H]2 = j(δ) >

∣∣HVθ [G]

∣∣M [H]and crit j = ω

V [G]2 > ωV1 .

We get that HVθ [G] is the transitive collapse of j[HV

θ [G]] and that K is HVθ [G]-generic

for B and evaluates correctly in M [H] all B-names for stationary subsets of ω1.

201

We conclude that M [H] models that j[K] is (SSP, j[HVθ [G]])-correct for j(B). By

Theorem 8.6.4 applied in M [H], FA++(j(B)) holds in M [H].By elementarity of j, we get that FA++(B) holds in V [G].The proof of the Claim is concluded.

The proof of the Lemma is completed.

The Theorem is proved.

13.2 Selfgeneric towers, forcing axioms, and category forc-ings

Recall that for a self-generic ideal forcing induced a by a tower of normal ideals I of heightδ SGI is the set of M ≺ Hδ+ such that

GM ∩ Vδ = S ∈M ∩ Vδ : M ∩ ∪S ∈ S

is M -generic for T I . Recall also that a presaturated tower of normal ideals which concen-trates on

Pκ (V ) = X : X ∩ κ ∈ κ > |X| .

is such that its generic ultrapower embedding has critical point κ and maps the criticalpoint to δ.

We start remarking the following fundamental property of the class SSP:

Theorem 13.2.1 (Woodin, Theorem 2.53 [55]). Assume δ is inaccessible and I is aself-generic and presaturated tower of normal ideals of height δ concentrating on Pλ+ (V ).For each cba B ∈ SSP ∩ Vδ the following are equivalent:

1. FAλ(B)

2. There is some S ∈ T I and a complete homomorphism k : B→ T I [S]I .

Proof. By Theorem 8.2.2 for any B ∈ SSP, FAλ(B) holds iff

SB,λ =M ∈ Pκ

(H|B|+

): there exists H M -generic filter for B

is stationary.

First assume SB,λ is stationary and for each M ∈ SB,λ fix HM M -generic for B. Define

k :B→ T I [SB,λ]I

b 7→ M ∈ SB,λ : b ∈ HM

We claim that k is a complete homomorphism. It is an homomorphism, since

k(b ∧ c) =NS M ∈ SB,λ : b ∈ HM ∩ M ∈ SB,λ : c ∈ HM

andk(¬b) =NS M ∈ SB,λ : b /∈ HM .

Now we show that it is complete: PickG V -generic filter for T I [SB,λ]I and letH = k−1[G].Then H is a filter on B. Now fix A maximal antichain of B. Then M ∈ SB,λ : A ∈Mis a club subset of SB,λ, hence it is in G as well. Remark that HM ∩A = bM for some

202

unique bM ∈ HM , since HM is M -generic for B. By the V -normality of G there exists aunique b ∈ B such that Sb = M ∈ SB,λ : bM = b ∈ G. Then k(b) = Sb ∈ G, hence b ∈ H.Therefore H is V -generic for B. Clearly H ∈ V [G]. Since this occurs for all V -genericfilters for T I [SB,λ]I , we conclude that k is a complete homomorphism by Lemma 6.1.2.

Conversely assume there is [T ]I ∈ T I and a complete homomorphism k : B→ T I [T ]I .W.l.o.g. we can assume H|B|+ = ∪SB,λ ⊆ ∪T = Hθ.

For any M ∈ T ∧ SGI , let HM = b ∈M ∩ B : k(b) ∈ GM.We claim that HM is M -generic for B for any M ∈ T ∧SGI with k ∈M : Let A ∈M be

a maximal antichain of B. Then k[A] ∈M is a maximal antichain of T I [T ]I , thereforeGM ∩ k[A] 6= ∅, giving that HM ∩A 6= ∅.

We conclude that for some M ∈ T ∧ SGI there exists an M -generic filter for B. HenceSB,λ is stationary by Theorem 8.2.2.

We can prove the analogue result for presaturated self-generic ideal forcings:

Theorem 13.2.2. Assume δ is inaccessible and I is a normal ideal concentrating onPλ+ (δ) and such that PI is presaturated and self-generic. For each cba B ∈ Ωλ ∩ Vδ thefollowing are equivalent:

1. FAλ(B)

2. There is some S ∈ PI and a complete homomorphism k : B→ PP [S]I .

Proof. Identical to the previous one and left to the reader.

In case we focus on the case λ+ = ω2 and we replace FAω1(B) with FA++(B) we cangather much more informations on the corresponding category forcing.

Lemma 13.2.3. Let I be a self-generic and presaturated tower of normal ideals of heightδ which concentrates on

Pω2 (V ) = X : X ∩ ω2 ∈ ω2 > |X| .

For each M ∈ SGI , GM ∩ Vδ is (M,SSP)-correct for T I .

Proof. Since I is a self-generic tower, SGI ∧ T is stationary for all T ∈ T ω2δ . In particular

FAω1(T ω2δ ) holds by Theorem 8.2.2.

Moreover whenever G is V -generic for T ω2δ , we have that

HUlt(V,G)ω2

= HV [G]ω2

= Vδ[G],

where the first equality holds since Ult(V,G)<δ ⊆ Ult(V,G) and δ = ωV [G]2 holds in V [G],

and the second equality holds since T I is < δ-presaturated.To simplify the argument (even if the assumption that θ is inaccessible is not necessary)

assume that θ > δ is inaccessible. We can replicate the above argument for each M ∈SGI ↑ Hθ as follows: Let πM : M → ZM be the transitive collapse of M , a = πM (a) forall a ∈M . Then ZM is a transitive model of ZFC in which δ is supercompact. MoreoverHM = πM [GM ] is ZM -generic for T I . Hence

ZM [HM ] |= HUlt(ZM ,HM )ω2

= HZM [HM ]ω2

= (Vδ)ZM [HM ].

Now fix τ ∈ V T ω2δ -name for a stationary subset of ω1.

203

Then for each M as above with τ ∈ M τHM is a stationary subset of ω1 in ZM [HM ].

Hence it belongs to HUlt(ZM ,HM )ω2 ; therefore there exists some SM ∈ HUlt(ZM ,HM )

ω2 such thatτHM = SM with SM = [fMτ ]HM for some fMτ : P

(P(XM

))→ P (ω1) in M ∩ Vδ.

By pressing down we can find a fixed f and a fixedX ∈ Vδ such that P (P (X)) = dom(f)and the set T of M with fMτ = f is stationary. Let S = T ↓ Vβ for some β < δ withX ∈ Vβ . Then S ∈ T I . Now assume G is V -generic for T I with S ∈ G. Then V [G] modelsthat τG is a stationary subset of ω1 and also that τG = [f ]G. Since

HUlt(V,G)ω2

= HV [G]ω2

= Vδ[G],

we get that Ult(V,G) models that [f ]G is a stationary subset of ω1. By Los theorem weconclude that

M ∈ S : fτ (M ∩X) is a stationary subset of ω1 ∈ G.

Since this occurs for all V -generic filter G for T I with S ∈ G, we conclude that

M ∈ S : fτ (M ∩X) is a stationary subset of ω1 =I S.

Hence fτ (M ∩X) = SM is a stationary subset of ω1 for all M ∈ T .Now remark that

SGI =NS

∨Tf : dom(f) ∈ Vδ

where Tf is the set of M ∈ SGI such that fMτ = f for some fixed f and for all M ∈ Tf .Repeating the above argument for all Tf ⊆ SGI we obtain that SM = fMτ is a stationary

subset of ω1 for all M ∈ SGI .We conclude that for each M ∈ SGI , for each T I-name τ for a stationary subset of ω1

in M , letting HM = πM [GM ∩ Vδ], SM = τHM , we have that SM is a stationary subset ofω1.

By definition GM ∩ Vδ is an M -generic filter for T I . Hence GM is (M,SSP)-correct forT I . The proof is concluded.

Similarly one can prove:

Lemma 13.2.4. Let I be a self-generic normal ideal which concentrates on

Pω2 (δ) = X ⊆ δ : X ∩ ω2 ∈ ω2 > |X|

and such that PI is presaturated.For each M ∈ SGI , GM ∩ Vδ is (M,SSP)-correct for PI .

We can now prove the following characterization of MM++:

Theorem 13.2.5. Assume δ is a supercompact cardinal. For each cba B ∈ SSP ∩ Vδ thefollowing are equivalent:

1. FA++ω1

(B)

2. There is some S ∈ T ω2δ and a complete and SSP-correct homomorphism i : B →

T ω2δ S.

204

Proof. By Lemma 13.2.3 SGω2δ ∧ T is stationary for all T ∈ (T ω2

δ )+. In particularFA++

ω1(T ω2δ T ) holds for all supercompact cardinals δ and all T ∈ (T ω2

δ )+.Notice also that for any B ∈ SSP, FA++

ω1(B) holds iff TB is stationary (where the latter

is the set of M ≺ H|B|+ admitting an (SSP,M)-generic filter for B).Now assume 2 holds. Given some B ∈ SSP ∩ Vδ and some SSP-correct i : B→ T ω2

δ T ,Fix M ∈ SGω2

δ ∧ T with k ∈M and let HM = b ∈ B : i(b) ∈ GM. Let τ ∈ V B ∩M be a

B-name for a stationary subset of ω1. Then σ = k(τ) is a T ω2δ -name for a a stationary subset

of ω1 since i is SSP-correct. By Lemma 13.2.3, πM (σ)πM [GM∩Vδ] is a stationary subsetof ω1. Clearly πM (σ)πM [GM∩Vδ] is equal to τπM [HM ], hence the latter is also a stationary

subset of ω1. Since this occurs for all τ ∈ V B ∩M B-names for a stationary subset of ω1,we conclude that HM is an (SSP,M)-correct filter for B.

Conversely assume 1 holds for B ∈ Vδ. Fix θ inaccessible with δ > θ > |B|+ + 2ω1 .Then S = TB ↑ Hθ is stationary.

For each M ∈ S fix HM (M,SSP)-correct filter for M . Let k : B→ T ω2δ S be defined

by b 7→ M ∈ S : b ∈ HM. We check that k is SSP-correct: First of all we remark thatall B-names for stationary subsets of ω1 are in H|B|++2ω1 ∈ Vδ (since they are given by

ω1-many maximal antichains of size 2 of B).Fix S ∈ Hθ B-name for a stationary subset of ω1. Let in V ,

fS :S → P (ω1)

M 7→ (πM (S))πM [HM ]

Since HM is (M, SSP)-correct for all M ∈ S, we get that fS(M) is stationary for all M ∈ S.Fix G V -generic for T ω2

δ with S ∈ G and let H = k−1[G]. We claim that:

SH = [fS ]G.

Proof. Observe that

α ∈ SHif and only if

rα ∈ S

z

B∈ H

if and only if

k(rα ∈ S

z

B) =

M ∈ S :

rα ∈ S

z

B∈ HM

∈ G

if and only if M ∈ S : α ∈ fS(M)

∈ G

if and only if

α ∈ [fS ]G.

Remark thatHUlt(V,G)ω2

= HV [G]ω2

= Vδ[G],

where the first equality holds since Ult(V,G)<δ ⊆ Ult(V,G) and δ = ωV [G]2 holds in V [G],

and the second equality holds since T ω2δ is < δ-presaturated.

Now for all S ∈ Hθ B-name for a stationary subset of ω1

Ult(V,G) |= [fS ]G is stationary

205

by Los Theorem, henceV [G] |= SH is stationary

for all S ∈ Hθ B-name for a stationary subset of ω1. Since this holds for all G V -genericfor T ω2

δ with S ∈ G, we conclude that k is SSP-correct. The proof is completed.

We have the following immediate corollary:

Corollary 13.2.6. Assume there are class many supercompact cardinals. Then the follow-ing are equivalent:

1. MM++;

2. the class of presaturated normal towers is dense in (SSP,≤SSP).

Duality between SSP-forcings and stationary sets

We can now prove the following two theorems:

Theorem 13.2.7. Assume there are class many supercompact cardinals. Let for each

cardinal λ and B ∈ Ωλ SB,λ be the set of M ∈ Pλ+

(H|B|+

)admitting an M -generic filter

for B.The map

Iκ :Ωκ → T κ+

B 7→ SB,κ

is order and incompatibility preserving and maps set sized suprema to set sized suprema.

Theorem 13.2.8. Assume there are class many supercompact cardinals and MM++ holds.The map

ISSP :SSP→ T ω2

B 7→ TB

is order and incompatibility preserving (with respect to the ≤SSP-order on the class SSP)and maps set sized suprema to set sized suprema.

Both theorems are immediate corollaries of Lemma 8.7.3 (respectively Lemma 8.7.4 forthe case of MM++) and Lemma 13.2.9 (respectively Lemma 13.2.10 for the case of MM++)to follow.

Lemma 13.2.9. Assume there are class many supercompact cardinals. Then B0,B1 arecompatible conditions in (Ωκ,≤Ω) if and only if SB0,κ ∧ SB1,κ is stationary.

Lemma 13.2.10. Assume MM++ and there are class many supercompact cardinals. ThenB0,B1 are compatible conditions in (SSP,≤SSP) if and only if TB0 ∧ TB1 is stationary.

We prove just Lemma 13.2.10 and leave the other proof to the reader.

Proof. First assume that C ≤ B0,B1. We show that TB0 ∧TB1 is stationary. Let ij : Bj → Cbe SSP-correct homomorphisms Fix γ > |C|+|B0|+|B1| inaccessible, and for all M ∈ TC ↑ γsuch that ij ∈M for j = 0, 1 pick GM (SSP,M)-correct filter for C. Let Hj

M = i−1j [GM ],

then both HjM are (SSP,M)-correct filter for Bj for both j = 0, 1, hence

TB0 ∧ TB1 ≥SSP TC

206

is stationary.Conversely assume that S = TB0 ∧ TB1 is stationary. For each M ∈ S pick Hj

M

(M,SSP)-correct filter for Bj for j = 0, 1. Fix a supercompact cardinal δ > |S|. Letij : Bj → T ω2

δ S map

b→ M ∈ S : b ∈ HjM.

Then (by Theorem 8.6.4) each ij is an SSP-correct homomorphism, therefore B0 and B1

are compatible conditions with respect to ≤SSP.

All in all assuming MM++ and class many supercompact cardinals we are in thefollowing situation:

1. MM++ can be defined as the statement that the class PT of presaturated towers ofnormal filters is dense in the category forcing (SSP,≤SSP).

2. We also have in the presence of MM++ a functorial map

ISSP : SSP→ T ω2

defined by B 7→ TB which

• is order and incompatibility preserving,

• maps set sized suprema to set sized suprema in the respective class partialorders.

It is now tempting to conjecture that it is possible to reflect this to some Vδ and obtain thatthe map F Vδ defines a complete embedding of USSP

δ = (SSP ∩ Vδ,≤SSP ∩Vδ) into T ω2δ .

However we just have that F preserves suprema of set sized subsets of USSP which wouldreflect to the fact F Vδ defines a < δ-complete embedding of USSP

δ into T ω2δ . However

we have no reason to expect that F Vδ extends to a complete homomorphism of therespective boolean completions because neither of the above posets is < δ-CC.

Nonetheless we have now many reasons to investigate to a full extent this correspondenceand this is what we will do in the last part of the book.

207

Part VI

Forcing Axioms and CategoryForcings

208

The aim of this last part of the book is to study certain categories of forcings Γ as forcingnotions. We divide our analysis in two chapters: “Category forcings” (chapter 14), and“Category forcing axioms” (chapter 15). Chapter 14 shows that any category Γ of forcingnotions satisfying certain natural properties defines a well behaving class forcing. Thistype of class forcings will be used in chapter 15 to prove the consistency of a forcing axiomwhich makes the theory of L([Ord]κ) invariant with respect to forcings in Γ preservingthe axiom for a cardinal κ which depends on Γ. We will also argue that these genericabsoluteness results are close to optimal extensions of Woodin’s generic absoluteness resultsfor L([Ord]ω). The chapters are organized as follows:

Category forcings (chapter 14): We introduce and analyze category forcings: specif-ically we study subcategories of the category of complete boolean algebras withcomplete homomorphisms. Given a category (Γ,→Θ) (where Γ is the class of objectsand →Θ the class of arrows) we associate to it the partial order (Γ,≤Θ) whoseelements are the objects in Γ ordered by B ≥Θ C iff there is an i : B → C in →Θ.We feel free to confuse a set sized partial order with its uniquely defined booleancompletion.

Depending on the choice of Γ and →Θ these partial orders can be trivial or not,for example, by Remark 6.3.1 and Fact 8.7.1, the category (Ω,→Ω) of all completeboolean algebras and complete homomorpshims between them gives raise to a trivialclass forcing (Ω,≤Ω), while the category (SSP,→Ω) whose objects are the stationaryset preserving complete boolean algebras, and whose arrows are the complete homo-morphisms between them gives raise to a non trivial class partial order (SSP,≤Ω).

We focus on the analysis of category forcings of type (Γ,→Γ) with Γ a definable class

of posets and→Γ the complete homomorphisms i : B→ C such thatrC/i[GB] ∈ Γ

z

B=

1B. Among the classes (Γ,→Γ) we analyze we find: proper, semiproper, stationaryset preserving, and many more.

The reasons guiding our selection of classes of forcings are twofolds:

• We aimed firstly (see [3, 52, 51]) at a generic absoluteness result for a strength-ening of Martin’s maximum or of the proper forcing axiom. This naturally ledus to an analysis of the category of forcings which are relevant for these forcingaxioms, i.e. the SSP-forcings, the semiproper forcings, the proper forcings.

• Along the way, and in joint work with Aspero [2], we realized that the machinerywe set forth works smoothly for a variety of other category forcings which sharea certain amount of similarity with the three classes mentioned above. Inparticular our machinery gives the means to provide modular proofs of a genericabsoluteness result which applies to a variety of forcing classes Γ.

The following list sums up the main concepts and results on the combinatorialproperties of these category forcings we isolate in this chapter:

• We introduce the key concept (at least for our aims) of Θ-rigid element of acategory (Γ,→Θ).

B ∈ Γ is Θ-rigid if it is fixed by any automorphism of some complete booleanalgebra in Γ which absorbs B using an arrow in →Θ. We can formulate thisproperty in purely categorical terms as follows:

B object of Γ is Θ-rigid if for all Q ∈ Γ there is at most one arrowi : B→ Q in →Θ.

209

We show that for suitably defined classes of forcings Γ, in the presence of classmany large cardinals, the class of Γ-rigid partial orders is dense in (Γ,≤Γ).

• We show that for these classes Γ the cut-off UΓδ = Γ ∩ Vδ of the category forcing

(Γ,≤Γ) is itself an element in Γ, is Γ-rigid, and absorbs all forcings in Γ ∩ Vδ(provided that δ is inaccessible and Vδ+1 satisfies some extra axioms other thanMK).

• We show that for the relevant classes Γ, the quotient of the category forcing(Γ,≤Γ)V with respect to a V -generic filter G for some B ∈ ΓV is the categoryforcing (Γ,≤Γ)V [G] as computed in the generic extension V [G].

Category forcing axioms (chapter 15): We introduce and analyze the forcing axiomCFA(Γ). To do so we proceed as follows:

• We have observed (13.2.6) that in the presence of class many supercompactcardinals the forcing axiom MM++ can be formulated as the assertion thatthe class of self-generic presaturated towers is dense in the category forcing(SSP,≤SSP). Moreover assuming MM++ we have shown that there is a naturalembedding of the class forcing (SSP,≤SSP) into the class tower T ω2 concentratingon stationary sets whose elements have size ω1 and contain ω1 (13.2.8). Thisembedding is order and incompatibility preserving (with respect to the ≤NS

order on T ω2), but we cannot prove by means of MM++ that the embeddinghas a dense image.

• These observations raise a number of questions:

– What can be said in general about the intersection of the class of Γ-rigidposets and the class of self-generic presaturated towers? In case Γ = SSPand assuming MM+++class many supercompacts we can prove that both ofthese classes are dense in (Γ,≤Γ), but we do not see how to prove rightawayin MK + MM+++large cardinals that their intersection is non-empty.

– Can we prove that the natural embedding of (SSP,≤SSP) into the classforcing (T ω2 ,≤NS) has a dense image? More generally can we classify whichclass forcings Γ admit a class sized tower of normal ideals which we cannaturally associate to Γ and in which Γ can be densely embedded?

– Can UΓδ be forcing equivalent to a self-generic presaturated tower or ideal

forcing inducing an almost huge generic embedding?

The forcing axiom CFA(Γ) arises as a positive answer to these questions andis a slight strengthening of the assertion that the class of presaturated towerforcings which are also Γ-rigid is dense in the category forcing (Γ,≤Γ).

• We prove (mimicking Woodin’s arguments of Chapter 12) that over any modelof CFA(Γ)+large cardinals any forcing in Γ which preserves CFA(Γ) does notchange the theory of L([Ord]≤κΓ) with parameters in P (κΓ), where κΓ is acardinal associated to Γ (κΓ = ω1 for Γ the class of proper and/or semiproperforcings). We can also argue that our generic result is optimal outlining that noforcing out of Γ can even preserve the Σ1-theory of L([Ord]≤κΓ) with parametersin P (κΓ) (at least for the case Γ = SSP).

• We finally prove the consistency of CFA(Γ) for many classes Γ: this is doneshowing that any of the standard forcing methods to produce a model of FAκΓ(Γ)collapsing an inaccessible δ to become κ+

Γ actually produces a model of CFA(Γ)provided that δ is a large enough cardinal (2-super huge suffices).

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Further notational conventions

From now on we will always assume that V (the universe of sets) is the “standard” modelof set theory and we focus on first order theories T ⊇ ZFC which we believe to hold inV . We need to handle proper classes, therefore it will be convenient to assume that Vis a model of the Morse-Kelley theory of sets with classes, theory which we denote byMK, i.e. we consider the first order theory of the hyperuniverse (V, C,∈,=) where C is thefamily of classes contained in V . Strictly speaking in what follows, we will just work withclasses definable by formulae with parameters which are sets, and quantifiers which rangejust over sets; hence the use of MK is somewhat an overshot and has the drawback thatit creates some ambiguity on the precise scope of range of quantifiers (i.e. do quantifiersrange over all sets or over all sets and proper classes?), at points where this ambiguitymay generate confusion we will be explicit on how to solve it. We will denote set sizedtransitive models (M,X ,∈,=) ∈ V of MK by their family of classes X since their familyof sets M can be recovered inside X as those classes which belong to some element. Forexample Vδ+1 for δ inaccessible will be a standard example of a transitive set-sized modelof MK. For the purpose of these notes we will also be focusing on theories T holding in Vwhich are extensions of MK by a finite explicit number of axioms. For example T can beMK+ the statement that there a exists a proper class of large cardinals of a certain kind(supercompact, Woodin, huge, etc....).

Finally it is important to note that MK is preserved in set sized forcing extensions of amodel of MK (by the results of Section 4.3). When we focus on a class of forcings Γ, wewill be interested in theories T which are preserved by all forcings in Γ. For example if Γ isthe class of semiproper posets the theory T = MK+ ω1 is a regular cardinal is preservedby all forcings in Γ. We will soon give precise definitions encompassing the theories thatare of interest for us.

To fix ambiguities we may encounter in the semantic interpretation of logical formulae,we adopt the following conventions (most of which have already been introduced in theprelimiaries of this book):

• We use a two sorted language with variables for sets x, y, z..., constants for setsa, b, c..., variables for classes X,Y, Z, ...., and constants for classes A,B,C, ..... Aquantifier of type Qx ranges only over sets, a quantifier of type QX ranges over setsand classes.

• (V, C,∈) is the universe of sets and classes which is the standard model of MK.

• Vδ ≺Σn V asserts that all Σn-formulae with set parameters in Vδ and quantifierswhich range only over sets are absolute between Vδ and V .

• Vδ+1 ≺Σn V asserts that all Σn-formulae with set parameters in Vδ and quantifierswhich range over sets and classes are absolute between (Vδ+1,∈) and (V, C,∈).

Notation 13.2.11. Let Qn-denote the complexity class Σn, Πn or ∆n of formulae of thelanguage of MK with quantifiers ranging only over sets and with just set parameters.

• ∃Qn is the complexity class Qn if Qn ∈ Σn or Σn+1 if Qn ∈ Πn;

• ∀Qn is the complexity class Qn if Qn ∈ Πn or Πn+1 if Qn ∈ Σn;

• ¬Σn is Πn and ¬Πn is Σn.

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Chapter 14

Category forcings

14.1 Definitions and main results

All the classes Γ we consider in these notes are defined as the extension of a formulaφΓ(x, aΓ) in the language of set theory enriched with a constant symbol for a set aΓ; aΓ is aset parameter, all the quantifiers in φΓ range over sets, the free variable x ranges over sets.

Notation 14.1.1. Given a class Γ defined by a formula with quantifiers and parametersranging over sets, φΓ(x, aΓ) and aΓ will always denote the formula and the parameter usedto define it.

Remark 14.1.2. Our official definition of a class forcing Γ assumes that Γ consists ofcomplete boolean algebras. This is the case since most of our definitions and calculationson such class forcings Γ are much easier to state and compute if Γ consists solely of completeboolean algebras. On the other hand in some cases there are posets Q, which are not evenseparative, and whose boolean completion RO(Q) is in Γ. As it is often the case in forcingarguments, we have a clear grasp of what Q is and how its combinatorial properties work,while this is much less transparent when we pass to its boolean completion RO(Q). It willbe convenient in these situations to assume Q ∈ Γ even if this actually holds just for itsboolean completion RO(Q). So we feel free in many cases to assume that the extension ofa class forcing Γ consists of all the posets Q whose boolean completions satisfy the definingproperty of Γ. If we feel that this can generate misunderstandings we shall be explicitlymore careful in these situations.

The following definitions introduce the key properties of a class of forcings Γ we areinterested in.

Definition 14.1.3. Let Γ be a definable class of forcing notions.Let B,C be complete boolean algebras.

• A complete homomorphism i : B→ C is Γ-correct if1

rC/i[GB] ∈ Γ

z

B=

rφΓ(C/i[GB], aΓ)

z

B≥B coker(i).

• C ≤Γ B if there is a Γ-correcti : B→ C.

1Notice that we do not require a priori neither B nor C to be in Γ, even if in what follows we shall mostlybe interested in the case in which this occurs for both of them.

212

• C ≤∗Γ B if there is an injective Γ-correct complete homomorphism

i : B→ C.

• Assume further that k : B→ C is Γ-correct and B,C ∈ Γ.

k Γ-freezes B if for all Γ-correct i0, i1 : C→ D we have that i0 k = i1 k.

• B is Γ-rigid if the identity map Id : B→ B Γ-freezes B.

Notation 14.1.4. Given a category forcing (Γ,≤Γ) with Γ a definable classes of completeboolean algebras and ≤Γ the order induced on Γ by the Γ-correct homomorphisms betweenelements of Γ, we denote the incompatibility relation with respect to ≤Γ by ⊥Γ, and thesubclass of Γ given by its Γ-rigid elements by RigΓ.

Remark the following:

Remark 14.1.5. Assume Γ ⊆ ∆ are definable classes of forcings. Then ≤Γ⊆≤∆ and⊥∆ ⊆ ⊥Γ. Hence if i : B→ C is Γ-correct and ∆-freezes B, we also have that i Γ-freezes B.Aspero proved that the classes Γ of proper (from now on denoted as Proper), semiproper(denoted by SP), and stationary set preserving forcings (denoted by SSP) all have theΓ-freezeability property establishing that for all B ∈ SSP, there is i : B → C which isProper-correct and SSP-freezes B (Theorem 14.2.3).

A rough comparison of ≤∗Γ and ≤Γ

Given a definable class Γ as above, we can define two natural order relations ≤Γ and ≤∗Γon Γ. The first one is given by Γ-correct homomorphisms i : B→ Q (which is the one wedescribed before) and the other given by injective Γ-correct homomorphisms i : B → Q.Both notion of orders are interesting and as set theorists we are used to focus on thesecond stricter notion of order since it is the one suitable to develop a theory of iteratedforcing. However the ≤Γ notion of ordering induced by complete (but possibly non-injective)Γ-correct homomorphisms grants that whenever we add a V -generic filter for a C ≤Γ B, wewill also be adding a V -generic filter for B by 6.1.1 applied to the i : B → C witnessingC ≤Γ B. ≤Γ will grant us that whenever B is put into a V -generic filter for Γ, then thisV -generic filter for Γ will also add a V -generic filter for B. We can replicate this feature of≤Γ at a suitable inaccessible cardinal δ and (repeating the above reasoning in Vδ+1) getthat the partial order (Γ∩Vδ,≤Γ ∩Vδ) absorbs all its elements as complete suborders, muchin the same way as Coll(ω,< δ) absorbs all complete boolean algebras in Vδ as completesuborders.

On the other hand if Γ is a weakly iterable class (see Def. 7.5.3), such as the classof semiproper forcings, we get quite easily that the class forcing (Γ,≤∗Γ) is (at leaststrategically) closed under set sized descending sequences. This observation is useful toprove that (Γ,≤Γ) has nice weak distributivity properties; however as a forcing notionby itself (Γ,≤∗Γ) is not very interesting: its closure under set-sized descending sequencesentails that any V -generic filter for Γ adds just new proper classes but no new sets, inparticular it cannot add V -generic filters for any of its elements.

We will show that the merging of the nice closure properties of (Γ,≤∗Γ) with thefreezeability property for (Γ,≤Γ) gives a powerful tool to analyze the forcing (Γ,≤Γ).

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Syntactic analysis of definable class forcings

Definition 14.1.6. Let Γ be a definable class of forcing notions. An inaccessible cardinalδ is Γ-correct2 if

• Γ ∩ Vδ = ΓVδ+1 ,

• ≤Γ ∩Vδ =≤Vδ+1

Γ .

Definition 14.1.7. We say that P (x) is absolutely Σ2 for T ⊇ MK if there is a Σ0 formulaφ(x, y, z) such that for all inaccessible δ and A ∈ Vδ in any model V of T

Vδ |= ∃y∀zφ(A, y, z) if and only if P (A) holds in V .

Fact 14.1.8. The statements “B is a complete boolean algebra” and “i : B → C is acomplete homomorphism” are absolutely Σ2 for MK in the relevant parameters (they arealso ∆2).

Assume T ⊇ MK proves that φΓ(x, aΓ) is an absolutely Σ2-property in the parameteraΓ. Then T proves that all inaccessible cardinals δ are weakly Γ-correct.

Proof. The second assertion is self-evident. Regarding the first observe that 3

i : B → C is a complete homomorphism can be expressed either by the absolutelyΣ2-property

∃θ [θ = max|trcl(B)|+ , |trcl(C)|+

∧ (Hθ |= i : B→ C is a complete homomorphism)]

or by the Π2-property

∀Hθ[(i,B,C ∈ Hθ)→ (Hθ |= i : B→ C is a complete homomorphism)]

in the relevant parameters. Similarly for the statement “B is a complete boolean algebra”.

Proposition 14.1.9. Assume Γ is a definable class of forcing with φΓ(x, aΓ) a Qn-propertywith Qn either Σn, Πn or ∆n (with n > 1)

Then:

• “i : B→ C is Γ-correct” is a Qn-statement in the set parameters i,B,C, aΓ. Moreoverif φΓ(x, aΓ) is an absolutely Σ2-property so is “i : B→ C is Γ-correct”.

• B ≤Γ C is a ∃Qn-statement in the set parameters B,C, aΓ. Moreover if φΓ(x, aΓ) isan absolutely Σ2-property so is B ≤Γ C.

• “B,C are incompatible in Γ” is a ∀¬Qn-statement in the set parameters B,C, aΓ (withquantifiers ranging only over sets),

• “i : B→ C is Γ-freezing B” is a ∀¬Qn-statement in the set parameters i,B,C, aΓ, κ(with quantifiers ranging only over sets).

• “B is Γ-rigid” is a ∀¬Qn-statement in the set parameters B, aΓ, κ (with quantifiersranging only over sets).

2Notice that we do not require the absoluteness of the incompatibility relation ⊥Γ, which is (as we willsee below) in general harder to obtain between V and Vδ for the classes Γ we are interested in.

3The formula X = Hθ for some regular cardinal θ can be expressed by: “∀a(|trcl(a)| < θ ↔ a ∈ X)”which is a Π1-formula in parameter θ and variable X.

214

Hence for all inaccessible δ such that Vδ ≺Σn+1 V , we have that

Γ ∩ Vδ = ΓVδ+1

≤Γ ∩Vδ =≤Vδ+1

Γ

⊥Γ ∩ Vδ = ⊥Vδ+1

Γ

RigΓ ∩ Vδ = (RigΓ)Vδ+1

Moreover if φΓ(x, aΓ) is provably ∆n in some theory T holding in V , we have that theabove holds already if Vδ ≺Σn V .

Proof.

• “i : B→ C is Γ-correct” is the conjuctions of the statement “i : B→ C is a completehomomorphism” and the statement

rφΓ(C/i[GB], aΓ)

z

B= 1B.

The latter has the same complexity of φΓ(x, aΓ) i.e. it is Qn in the relevant parameters,moreover in case φΓ(x, aΓ) is absolutely Σ2, it remains absolutely Σ2.

On the other hand we have already seen that the former can be expressed either byan absolutely Σ2-property or by a Π2-property.

Hence the the required statement is the conjuction of a Qn-statement with a ∆2-statement, therefore it is Qn and is absolutely Σ2 if so is φ(x, aΓ).

• B ≤Γ C is the statement “∃i : B → C Γ-correct” which is ∃Qn in the relevantparameters and it remains absolutely Σ2 if so is φΓ(x, aΓ) in view of the previousitem.

• “B,C are incompatible in Γ” is the assertion

∀R∀i0∀i1¬[(i0 : C→ R is Γ-correct) ∧ (i1 : C→ R is Γ-correct)].

Since (i0 is Γ-correct) ∧ (i1 is Γ-correct) is a Qn-property in the relevant parameters,the above sentence is ∀¬Qn in the relevant parameters.

• “i : B→ C is Γ-freezing B” is the assertion

∀R∀i0∀i1¬[(i0 : C→ R is Γ-correct) ∧ (i1 : C→ R is Γ-correct)∧∧ (i0 i : B→ R is Γ-correct) ∧ (i1 i : B→ R is Γ-correct)∧∧ i0 i 6= i1 i].

Observe that

(i0 : C→ R is Γ-correct) ∧ (i1 : C→ R is Γ-correct)∧∧(i0 i is Γ-correct) ∧ (i1 i is Γ-correct)∧

∧(i0 i 6= i1 i)

is a Qn-property in the parameters B,C,R, i0, i1, i. The desired property hence is∀¬Qn in the relevant parameters.

215

• “B is Γ-rigid” is immediately seen to be ∀¬Qn since it is obtained substituting theidentity map on B in the place of i in the assertion that i is Γ-freezing B.

The last part of the Lemma is immediate since all the relevant notions are either Σn+1 orΠn+1, hence are absolute between Vδ and V . In case φΓ(x, aΓ) is ∆n, the relevant notionsbecome either Σn or Πn. Hence in this case we can decrease by 1 the degree of elementarityrequired between Vδ and V .

κ-suitable class forcings

Definition 14.1.10. Let Γ ⊆ V be a definable class of posets.

• Γ is closed under complete subalgebras if, given any B ∈ Γ, and a complete injectivehomomorphism i : C→ B, we also have that C ∈ Γ.

• Γ is closed under two step iterations if for all B ∈ Γ and all C ∈ V B such thatrC ∈ Γ

z

B= 1B we have that B ∗ C ∈ Γ.

• Γ is closed under lottery sums if every set A ⊂ Γ has an exact upper bound∨

ΓAin ≤Γ (

∨ΓA is the lottery sum of the posets in A, equivalently - if A consists of

complete boolean algebras - the product of the boolean algebras in A).

• Γ has the Γ-freezeability property if for every B ∈ Γ there is k : B→ C Γ-freezing B.

• Γ is weakly κ-iterable if for each ordinal α Player II has a winning strategy in thegames Gα(Γ) defined as follows:

– at successor stages α players I and II play Γ-correct injective embeddingsiα,α+1 : Bα → Bα+1;

– Player I plays at odd stages, player I at even stages (0 and all limit ordinalsare even);

– at stage 0, II plays a Γ-correct embedding i0,1 : 2→ B1 (i.e. a B1 ∈ Γ);

– at limit stages λ, II must play lim−→Bα : α < λ if cof(λ) = κ or λ is regularand all boolean algebras in F have size less than λ;

– II wins Gα(Γ) if she can play at all stages up and including α.

There is a tight interaction between the properties of a class of forcings Γ and thetheory T ⊇ MK in which we analyze this class. For example in our analysis of Γ, we arenaturally led to work with theories T which extend MK but which are not preserved by allset sized forcings. For example this occurs for T = MK + ω1 is a regular cardinal whichis not preserved by Coll(ω, ω1), but is preserved by all stationary set preserving forcings.

The following definition outlines the key correlations between a theory T ⊇ MK anda class of forcings Γ we want to bring forward, and allows us to prove that Γ is a wellbehaved class forcing. The reader may keep in mind, while reading the definition below,that semiproperness and properness will be the simplest examples of ω1-suitable classes Γ,and that for these classes Γ a useful Γ-canonical theory is any enlargment of MK + ω1 isa regular cardinal by large cardinal axioms.

Definition 14.1.11. Let T ⊇ MK be a theory and Γ be a definable class of forcings bymeans of the formula φΓ(x, aΓ) in parameter aΓ ⊆ κ.

216

• T ⊇ MK is κ-canonical if it extends MK by a finite list of axioms expressible withoutquantifiers ranging over classes (but with no bound on the number of quantifiersranging over sets) among which the axiom “κ is a regular cardinal”.

• Γ is (Qn, κ)-suitable for a κ-canonical theory T if:

1. T proves that φΓ(x, aΓ) is a equivalent to a Qn-property in the parameter aΓ ⊆ κ.Hence, if Γ is provably in T an absolutely Σ2-property in the parameter aΓ ⊆ κ,T proves that δ is Γ-correct for all inaccessible cardinals δ > κ.

2. T proves that all forcing notions in Γ preserve T .

3. T proves that Γ is closed under two step iterations, lottery sums and completesubalgebras.

4. T proves that Γ contains all < κ-closed posets.

5. T proves that Γ is weakly κ-iterable.

6. T proves that Γ has the Γ-freezeability property,

T is (Qn,Γ)-canonical if it is κ-canonical and Γ is (Qn, κ)-suitable for T .If Γ is provably in T an absolutely Σ2-property in the parameter aΓ ⊆ κ, we just say

that T is Γ-canonical and Γ is κ-suitable for T .

Remark 14.1.12. We remark the following:

• Theories T of the form MK+ the statement that there a exists a proper class of largecardinals of a certain kind (supercompact, Woodin, huge, etc....) are Ω-canonical,where Ω is the class of all set sized forcings.

• A key feature of a canonical theory T we will exploit is that once it holds in Vδ+1 forsome inaccessible δ it holds also in Wδ+1 for any W ⊇ V such that:

– δ remains inaccessible in W ,

– Wδ+1 is a model of MK,

– Wδ = Vδ.

This is the case since the extra axioms in T \MK are defined by properties which donot take into consideration (in order to evalute their truth) the new proper classesappearing in Wδ+1 \ Vδ+1.

The main theorem of this chapter is:

Theorem 14.1.13. Assume Γ is κ-suitable for a κ-canonical T . Let δ be inaccessible andsuch that Vδ+1 |= T .

Let UΓδ = Γ ∩ Vδ with the inherited order ≤Γ ∩Vδ. Then:

• UΓδ is a forcing notion in Γ,

• UΓδ preserves the regularity of δ and makes it the successor of κ,

• B ≥Γ UΓδ B for all B ∈ UΓ

δ .

We are also going to prove that for a κ-suitable Γ the quotient of Γ by a V -genericfilter G for a forcing in Γ is forcing equivalent to the class forcing ΓV [G] in V [G]. Since theprecise formulation of this theorem is rather delicate, we defer its statement to section 14.4(Theorem 14.4.3).

217

14.2 κ-suitable category forcings

14.2.1 Proper, semiproper and stationary set preserving forcings areω1-suitable

We show that the classes of proper, semiproper and stationary set preserving forcings areω1-suitable. The first two are ω1-suitable for the ω1-canonical theory

T0 = MK+“ω1 is regular”

The latter is ω1-suitable with respect to the ω1-canonical theory

T1 = T0+ there exists class many supercompact cardinals”

Lemma 14.2.1. Let Γ be the class of proper or semiproper forcings and T0 be the theoryMK+“ω1 is regular”. Then T0 proves the following:

• Γ is the extension of a provably in T0 absolutely Σ2-property;

• all forcing notions in Γ preserve T0;

• Γ is closed under two step iterations, lottery sums, and preimages of completehomomorphisms;

• Γ contains all < ω1-closed posets;

• Γ is ω1-iterable.

Proof.

Γ is the extension of a T0-provably absolutely Σ2-property: The following two state-ments give different equivalent definitions of Γ:

• The absolutely Σ2-statement in the parameters B, ω:

There is Hθ with |B|+ = θ, and C ∈ Hθ club such that P (B) ,B ∈ ∪Cand for all countable M ∈ C with B ∈M , there exists r ∈ B which isM -(semi)generic and is compatible with all q ∈ B ∩M .

• The Π2-statement in the parameters B, ω:

For all Hθ with P (B) ∈ Hθ, there is C club subset of H|B|+ such thatfor all countable M ∈ C with B ∈M and M ≺ H|B|+ there exists r ∈ Bwhich is M -(semi)generic and is compatible with all q ∈ B ∩M .

Hence Γ is ∆2 and provably absolutely Σ2 with respect to T0.

All forcing notions in Γ preserve T0: Proper and semiproper forcings preserve ω1 andMK.

Γ is closed under two step iterations, lottery sums, preimages of complete homomorphisms:

clear from the results of Chapter 10.

Γ contains all < ω1-closed posets: every < ω1-closed partial order is proper and there-fore also semiproper.

Γ is weakly ω1-iterable: clear from the results of Chapter 10.

218

Lemma 14.2.2. Let Γ be the class of stationary set preserving forcings and T1 be thetheory

MK+“ω1 is regular and there exists class many supercompact cardinals”

Then T1 proves the following:

• Γ is the extension of a T1-provably ∆2-property which is also absolutely Σ2;

• all forcing notions in Γ preserve T1;

• Γ is closed under two step iterations, lottery sums, and preimages of completehomomorphisms;

• Γ contains all < ω1-closed posets;

• Γ is ω1-iterable.

Proof. All the above items except the last one go through with minor modifications withrespect to the proof of the previous Lemma. For the last item we proceed as follows:

Let SP denote the class of semiproper forcings and SSP denote the class of stationaryset preserving forcings. Then SP ⊆ SSP.

Recall that MM++ proves that stationary set preserving forcings are semiproper(by 13.1.1) and that for every stationary set preserving forcing B ∈ Vδ with δ-supercompactthere is an iB,δ : B→ Cδ,B SP-correct homomorphism such that

qMM++

yCδ,B

= 1Cδ,B and

iB,δ in Vδ+1 (by 13.1.2).Now let player I play at odd stages α+ 1 whichever SSP-correcti : Bα → Bα+1 he prefers and player II answer at all even non limit stages with iBα+1,δ

where δ is the least supercompact cardinal such that Bα+1 ∈ Vδ. At limit stages player IIalways takes the RCS-limit. We leave to the reader to check that the above is a winningstrategy for player II in the game Gα(SSP) for all ordinals α.

Hence the above lemmas show that the classe of proper semiproper and stationary setpreserving forcings lack just the freezeability property to be ω1-suitable with respect tothe appropriate ω1-canonical T .

The following two results in combination with the above remark yield the freezeabilityproperty for the relevant classes Γ.

Theorem 14.2.3 (Aspero). Let T0 = MK+“ω1 is regular”. Then T0 proves that for allB ∈ SSP there exists i : B→ C such that

qC/i[GB] is proper

yB

= 1B

and i is SSP-freezing for B.

The above grants that the classes of proper and semiproper forcing are ω1-suitablewith respect to T0 = MK+“ω1 is regular” and the class of SSP-forcings is ω1-suitable withrespect to

T1 = MK+“ω1 is regular and there are class many supercompact cardinals”

We will limit ourselves to prove the following weaker result which grants that the class ofsemiproper and stationary set preserving forcings are ω1-suitable with respect to T1.

219

Theorem 14.2.4. Let T0 = MK+“ω1 is regular”. Then T0 proves that for all B ∈ SSPthere exists i : B→ C such that

qC/i[GB] ∈ SSP

yB

= 1B

and i is SSP-freezing B.

The proof the above theorem is deferred to a later stage: first we need to give therelevant facts about the freezeability property.

14.2.2 Γ-Freezeability and Γ-rigidity

The aim of this section is the proof of the following:

Theorem 14.2.5. Assume Γ is κ-suitable for the κ-canonical theory T . Then T provesthat the class of Γ-rigid partial orders is dense in (Γ,≤∗Γ).

We need some preliminary Lemmas. All over this section we assume that Γ is κ-suitablefor the κ-canonical theory T .

Lemma 14.2.6. The following are equivalent characterizations of Γ-rigidity for a B ∈ Γ:

1. for all b0, b1 ∈ B such that b0 ∧B b1 = 0B we have that B b0 is incompatible withB b1 in (Γ,≤Γ).

2. For all C ≤Γ B and all H, V -generic filter for C, there is just one G ∈ V [H] Γ-correctV -generic filter for B.

3. For all C ≤Γ B in Γ there is only one Γ-correct homomorphism i : B→ C.

Remark 14.2.7. Γ-rigidity entails rigidity by its very definition. Nonetheless it is conceivablethat even if B is Γ-rigid, there could be k : B → B b complete (and non-surjective)homomorphism which is not Γ-correct. If H is V -generic for B, k−1[H] = G ∈ V [H] is alsoV -generic for B. Hence in V [H] there could be distinct V -generic filters for B even if Bis Γ-rigid. This is not in conflict with 14.2.6(2), since G ∈ V [H] is not Γ-correct for B inV [H].

Proof. We prove these equivalences by contraposition as follows:

2 implies 1: Assume 1 fails as witnessed by ij : B bj → Q for j = 0, 1 with b0incompatible with b1 in B. Pick H V -generic for Q. Then Gj = i−1

j [H] ∈ V [H] aredistinct and Γ-correct V -generic filters for B in V [H], since bj ∈ Gj \G1−j .

1 implies 3: Assume 3 fails for B as witnessed by i0 6= i1 : B → C. Let b be suchthat i0(b) 6= i1(b). W.l.o.g. we can suppose that r = i0(b) ∧ i1(¬b) > 0C. Thenj0 : B b→ C r and j1 : B ¬b→ C r given by jk(a) = ik(a) ∧ r for k = 0, 1 anda in the appropriate domain witness that B ¬b and B b are compatible in (Γ,≤Γ)i.e. 1 fails.

3 implies 2: Assume 2 fails for B as witnessed by some C ≤Γ B, H V -generic filter for C,and G1 6= G2 ∈ V [H] Γ-correct V -generic filters for B in V [H]. Let G1, G2 ∈ V C besuch that

(G1)H = G1 6= (G2)H = G2 are Γ-correct V -generic filters for B in V [H] for both j = 1, 2.(14.1)

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Find q ∈ G forcing that b ∈ G1\G2 for some fixed b ∈ B. Then, by Lemma 6.2.2(2 =⇒3), for some r ∈ H refining q both homomorphisms ij = iGj ,r : B → C defined by

a 7→ra ∈ Gj

z

C∧ r are Γ-correct. However i1(b) = r = i2(¬b), hence i1 6= i2 witness

that 3 fails for B.

We also need the following characterizations of Γ-freezeability:

Lemma 14.2.8. Let k : B→ Q be a Γ-correct homomorphism. The following are equivalent:

1. For all b0, b1 ∈ B such that b0 ∧B b1 = 0B we have that Q k(b0) is incompatible withQ k(b1) in (Γ,≤Γ).

2. For all R ≤Γ Q, all H V -generic filter for R, there is just one G ∈ V [H] Γ-correctV -generic filter for B such that G = k−1[K] for all K ∈ V [H] Γ-correct V -genericfilters for Q.

3. For all R ≤Γ Q in Γ and i0, i1 : Q → R witnessing that R ≤Γ Q we have thati0 k = i1 k.

Proof. Left to the reader, along the same lines of the proof of the previous Lemma.

A Γ-freezeable B ∈ Γ can be embedded in Γ C for some k : B → C Γ-freezing B asfollows:

Lemma 14.2.9. Assume Γ is a class of posets having the Γ-freezebility property. Letk : B→ C be a Γ-correct freezing homomorphism of B into C. Then the map kB : B→ Γ Cwhich maps b 7→ C k(b) defines a complete embedding4 of the partial order (B+,≤B) into(Γ C,≤Γ).

Proof. Left to the reader. It is immediate to check that kB preserve predense sets and the≤B-order relation. The Γ-freezeability property of k is designed exactly in order to getthat kB preserve also the incompatibility relation on B.

Proof of Theorem 14.2.4

The results of this section assume the reader is familiar with the standard facts regardingstationary set preserving forcings.

Definition 14.2.10. For any regular cardinal κ ≥ ω2 fix

Siα : α < κ, i < 2

a partition of Eωκ (the set of points in κ of countable cofinality) in pairwise disjointstationary sets. Fix

Aα : α < ω1

partition of ω1 in ω1-many pairwise disjoint stationary sets such that min(Aα) > α andsuch that there is a club subset of ω1 contained in⋃

Aα : α < ω1.

4We do not (as yet) assert that kB is Γ-correct. We will be able to prove that the embedding is Γ-correctup to δ under some further assumptions on Γ and δ.

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Given P a stationary set preserving poset, we fix in V a surjection f of the least regularκ > |P | with P . Let gP : κ→ 2 be the P -name for a function which codes GP using f , i.e.for all α < |P |

p P (gP (α) = 1↔ f(α) ∈ GP ).

Now let QP be the complete boolean algebra RO(P ∗ RP ) where RP is defined as followsin V P :

Let G be V -generic for P . Let g = valG(gP ). R = valG(RP ) in V [G] is the poset givenby pairs (cp, fp) such that for some countable ordinal αp

• fp : αp + 1→ κ,

• cp ⊆ αp + 1 is closed,

• for all ξ ∈ cp

ξ ∈ Aβ and g fp(β) = i if and only if sup(fp[ξ]) ∈ Sifp(β).

The order on R is given by p ≤ q if fp ⊇ fq and cp end extends cq. Let

• fQP : ω1 → κ be the P ∗ RP -name for the function given by⋃fp : p ∈ GP∗RP ,

• CQP ⊂ ω1 be the P ∗ RP -name for the club given by⋃cp : p ∈ GP∗RP ,

• gQP ⊂ ω1 be the P ∗ RP -name for the function gP coding a V -generic filter for Pusing f .

We are ready to show that all stationary set preserving posets are freezeable.

Theorem 14.2.11. Assume P is stationary set preserving. Then P forces that RPis stationary set preserving and QP = RO(P ∗ RP ) freezes P as witnessed by the mapk : RO(P )→ QP which maps p ∈ P to 〈p, 1RP 〉.

Proof. It is rather standard to show that RP is forced by P to be stationary set preserving.We briefly give the argument for R = valG(RP ) working in V [G] where G is V -generic forP . First of all we observe that Siα : α < κ, i < 2 is still in V [G] a partition of (Eωκ )V inpairwise disjoint stationary sets (since P is < κ-CC), and also that Aα : α < ω1 is still amaximal antichain on P (ω1)/NSω1 in V [G] (since P ∈ SSP and Aα : α < ω1 contains aclub subset of ω1).

Claim 16. R is stationary set preserving.

Proof. Let E be an R-name for a club subset of ω1 and S be a stationary subset of ω1.Then we can find α such that S ∩ Aα is stationary. Pick p ∈ R such that α ∈ dom(fp),Let β = fp(α) and i = g(β) where g : κ→ 2 is the function coding G by means of f . By

standard arguments find M ≺ HV [G]θ countable such that

• p ∈M ,

• M ∩ ω1 ∈ S ∩Aα,

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• sup(M ∩ κ) ∈ Siβ.

Working inside M build a decreasing chain of conditions pn ∈ R ∩M which seals all densesets of R in M and such that p0 = p. By density we get that

fω =⋃n<ω

fpn : ξ = M ∩ ω1 →M ∩ κ

is surjective and that ξ is a limit point of

cω =⋃n<ω

cpn

which is a club subset of ξ. Set

q = (fω ∪ 〈ξ, 0〉, cω ∪ ξ).

Now observe that q ∈ R since ξ ∈ Aα and sup(fq[ξ]) ∈ Sg(fp(α))fp(α) and cq is a closed subset

of ξ + 1. Now by density q forces that ξ ∈ E ∩ S and we are done.

Claim 17. QP SSP-freezes P .

Proof. We prove this by means of Lemma 14.2.8(2).Assume that R ≤SSP QP , let H be V -generic for R and pick G0, G1 ∈ V [H] distinct

correct V -generic fiters for QP . It is enough to show that

G0 = G1

whereGj = p ∈ P : ∃q ∈ V P such that 〈p, q〉 ∈ Gj

Let gj : κ→ 2 be the evaluation by Gj of the function gP which is used to code Gj asa subset of κ by letting gj(α) = 1 iff f−1(α) ∈ Gj . Let

hj =⋃fp : p ∈ Gj,

Cj =⋃cp : p ∈ Gj,

In particular we get that C0 and C1 are club subsets of ω1 in V [H], h0, h1 are bijectionsof ω1 with κ.

Now observe that κ has size and cofinality ω1 in V [Gj ] and thus (since V [H] is a genericextension of V [Gj ] with the same ω1) κ has size and cofinality ω1 in V [H]. Observe alsothat in V [H]

S is a stationary subset of κ iff S ∩ suph[ξ] : ξ < ω1 is non empty for anybijection h : ω1 → κ.

Now the very definition of the hj gives that for all α ∈ Cj :

hj(α) = η and gj(η) = i if and only if suphj [ξ] ∈ Siη for all ξ ∈ Aα ∩ Cj .

Now the setE = ξ ∈ C0 ∩ C1 : h0[ξ] = h1[ξ]

is a club subset of ω1, and the above observations show that

Sgj(η)η ⊇ suphj [ξ] : ξ ∈ E ∩Aα 6= ∅

for both j. In particular g0(η) = g1(η) for all η < κ, else S0η ∩ S1

η is non-empty for some ηcontradicting the very definition of the family of sets Siη. Thus G0 = G1.

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This completes the proof of Theorem 14.2.4.

14.2.3 From freezeability to total rigidity

Lemma 14.2.12. Assume

iαβ : Bα → Bβ : α < β ≤ δ

is a complete iteration system such that for each α there is β > α such that

• iα,β Γ-freezes Bα.

• Bδ is the direct limit of the iteration system and is in Γ.

Then Bδ is Γ-rigid.

Proof. Assume the Lemma fails. Then there are f0, f1 incompatible threads in Bδ suchthat Bδ f0 is compatible with Bδ f1 in (Γ,≤Γ). Now Bδ ∈ Γ is a direct limit, hence f0, f1

have support in some α < δ. Thus f0(β), f1(β) are incompatible in Bβ for all α < β < δ.Now for eventually all β > α Bβ Γ-freezes Bα as witnessed by iα,β. In particular, sincefi = iα,δ fi(α) for i = 0, 1 we get that Bδ f0 cannot be compatible with Bδ f1 in(Γ,≤Γ), contradicting our assumption.

We can now prove Theorem 14.2.5:

Proof. Given B ∈ Γ let A ⊆ B be a maximal antichain such that for all b ∈ A there iskb : B→ Cb Γ-freezing B with coker(kb) = b. Let

k =∨A

kb :B→ C =∨Γ

Cb : b ∈ A

a 7→ (kb(a) : b ∈ A)

Then k : B→ C Γ-freezes B and is injective.Now given B0 let

F = kij : Bi → Bj : i ≤ j < κ

be a decreasing sequence in ≤∗Γ such that kii+1 Γ-freezes Bi. Then lim−→F ∈ Γ is Γ-rigid andrefines B0 in ≤∗Γ.

14.2.4 Key properties of Γ

The density of the class of Γ-rigid forcings gives us the means to unfold the key propertiesof the forcing (Γ,≤Γ) for a given κ-suitable Γ and a Γ-canonical theory T .

The results of the previous section show that RigΓ (the class of Γ-rigid elements of Γ)is dense in (Γ,≤∗Γ). It is immediate to check the following facts:

Fact 14.2.13. The following holds:

1. RigΓ is closed under lottery sums,

2. RigΓ is κ-iterable.


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Fact 14.2.14. For any C ∈ RigΓ the map kC : C→ UΓ C which maps c ∈ C to C c is acomplete injective homomorphism.

Proof. We get that Id : C → C is a Γ-freezing homomorphism for C. Now we can useLemma 14.2.9.

The following Lemmas condense the main features of the forcing (Γ,≤Γ).

Lemma 14.2.15. Let D be a dense open subset of (RigΓ,≤Γ). Then for every B ∈ Γ,there is C ∈ RigΓ, an injective Γ-correct complete homomorphism i : B→ C, and A ⊂ Cmaximal antichain of C such that kC[A] ⊆ D and kC[A] is a maximal antichain of Γ C.

Proof. Given B ∈ Γ find C0 ≤Γ B in D (which is possible since RigΓ is a dense subclassof Γ). Let i0 : B → C0 be a complete and Γ-correct homomorphism of B into C0. Letb0 = coker(i0) ∈ B, so that i0 b0 : B b0 → C0 is an injective Γ-correct homomorphism.Proceed in this way to define Cl and bl such that:

• il bl : B bl → Cl is an injective Γ-correct homomorphism,

• Cl ∈ D ⊆ Γ,

• bl ∧B bi = 0B for all i < l.

This procedure must terminate in η < |B|+ steps producing a maximal antichainbl : l < η of B and injective Γ-correct homomorphisms il : B bl → Cl such thatCl ∈ D ⊆ RigΓ refines B in the ≤Γ order. Then we get that

• C =∨l<η Cl ∈ RigΓ.

• i is an injective Γ-correct homomorphism, where

i =∨k<η

ik : B→∨k<η

Ck

c 7→ 〈ik(c ∧B bk) : k < η〉

is such thatJC/i[GB] ∈ ΓKB = 1B,

since i is the lottery sum of the injective Γ-correct homomorphisms il such that

JCl/il[GB] ∈ ΓKCl = 1Cl .

• C i(bk) ∈ D for all k < η.

In particular we get that A = i[bk : k < η] is a maximal antichain of C ∈ RigΓ such thatC c ∈ D for all c ∈ A. Moreover since kC : C→ Γ C is a complete embedding, kC[A] is amaximal antichain in Γ C, as was to be shown.

Lemma 14.2.16. Γ is a class forcing notion preserving the regularity of Ord.

Proof. Let f be a Γ-name for an increasing function from η into Ord for some ordinal η.Given B ∈ Γ let Ai ⊂ RigΓ be a dense subclass of RigΓ contained in the open dense class ofpartial orders in Γ B which decide that f(i) = α for some α. Then using the previouslemma we can build a downward directed system with respect to ≤∗Γ

iα,β : Cα → Cβ : α ≤ β ≤ η

225

such that for all i < η Ci+1 ∈ RigΓ and there is Bi maximal antichain of Ci+1 ≤∗Γ Ci suchthat kCi+1

[Bi] ⊂ Ai and kCi+1[Bi] is a maximal antichain of Γ Ci+1. Then Cη ∈ Γ forces

that f has values bounded by

supα : ∃i ∈ η, ∃c ∈ Bi ⊆ Ci+1 such that Ci+1 c forces that f(i) = α ∈ Ord.

Lemma 14.2.17. Assume f ∈ V Γ is a name for a function with domain α and rangecontained in V for some ordinal α. Then there is a dense set of C ∈ RigΓ with an fC ∈ V C

such thatC Γ kC(fC) = f .

Proof. Given f as above, let for all ξ < α

Dξ = C ∈ RigΓ : ∃a ∈ V C Γ f(ξ) = a.

Let B ∈ RigΓ be arbitrary. By the previous lemma we can find C ∈ RigΓ below B such thatfor all ξ < α there is a maximal antichain Aξ ⊂ C such that kC[Aξ] ⊂ Dξ and kC[Aξ] is amaximal antichain in Γ C. Now let fC be the C-name

〈(ξ, a), c〉 : c ∈ Aξ and C c Γ f(ξ) = a.

It is immediate to check that for all ξ < α and c ∈ Aξ

c C fC(ξ) = a iff C c Γ f(ξ) = a.

The Lemma is proved.

Lemma 14.2.17 can be refined to the following more useful versions:

Lemma 14.2.18. For every τ ∈ V Γ such that

Γ τ is a set

we get that

Dτ =C ∈ RigΓ : ∃σ ∈ V C C Γ τ = kC(σ)

is dense open in RigΓ.

Proof. Left to the reader: observe that any Γ-name τ as in the assumptions can be codedin a provably ∆1-way by a Γ-name for a well-founded relation on some ordinal α, which inturn can be coded in a provably ∆1-way by a Γ-name for a function in 2(α2). Now applyLemma 14.2.17.

14.3 When is (Γ ∩ Vδ,≤Γ) a partial order in Γ?

Notation 14.3.1. For any class Γ, UΓδ denotes the partial order Γ ∩ Vδ with the order

relation ≤Γ ∩Vδ.

This is the main result of the section.

Theorem 14.3.2. Assume Γ is a κ-suitable property for the Γ-canonical theory T . ThenUΓδ ∈ Γ for any inaccessible cardinal δ which models T .

To prove the theorem we must relativize the results regarding Γ to the forcing Γ ∩ Vδ.

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Transferring to Vδ+1 the properties of Γ

We need the corresponding results for UΓδ of all the results for the class forcing Γ we

established in subsection 14.2.4. This is rather straightforward once we assume that δ isΓ-correct for T . Hence we state the results we need and we leave the proofs to the reader.

Lemma 14.3.3. Assume T is Γ-canonical and Γ is κ-suitable for T .Assume δ is inaccessible and Γ-correct with Vδ+1 a model of T . The following holds:

1. Let RigΓδ = (RigΓ)Vδ+1. Then RigΓ

δ is dense in5 UΓδ .

2. Assume f ∈ V UΓδ is a UΓ

δ -name for a function with domain α < δ and range containedin V . Then there is a dense set of C ∈ RigΓ

δ with an fC ∈ V C such that

C Γ kC(fC) = f .

3. UΓδ forces that δ is the least regular cardinal larger than κ.

4. For every τ ∈ V UΓδ such that

UΓδ τ is a set whose transitive closure has size less than δ

we get that

Dτ =C ∈ RigΓ ∩ Vδ : ∃σ ∈ V C ∩ Vδ C UΓ

δτ = kC(σ)

is dense in UΓ

δ .

Proof. Most proofs are obtained by straightforward modifications of the correspondingresults for the class forcing.

1: We can repeat verbatim the relevant proofs in Vδ+1 (which models T ).

2: Left to the reader.

3: The previous item shows that the regularity of δ is preserved by Uδ since Uδ forces allfunctions with domain α < δ and range contained in δ to be bounded below δ.

On the other hand for all κ ≤ α < δ,

Dα =B ∗ ˙Coll(κ, α) : B ∈ Γ ∩ Vδ

is dense in UΓ

δ (where Coll(κ, α) is the forcing ordered by reverse inclusion whoseconditions are partial functions from κ → α with domain less than κ): this is thecase since Γ is closed under two step iterations and contains all < κ-closed posets(and Coll(κ, α) is < κ-closed for all α ≥ κ).

4: Left to the reader.

5We cannot exclude that RigΓ ∩ Vδ 6= RigΓδ for some Γ. This does not occur however for the classes of

proper, semiproper and SSP-forcings.

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14.3.1 Proof of Theorem 14.3.2

We can now prove Theorem 14.3.2.

Lemma 14.3.4. Assume T is Γ-canonical and Γ is κ-suitable for T . Assume δ is inacces-sible and Γ-correct with Vδ+1 a model of T . Let Q ⊆ Vδ be the forcing whose conditions areiteration systems

F = Bα : α < γ ∈ Vδsuch that:

• Bβ ≤∗Γ Bα for all α < β < γ,

• Bβ ∈ RigΓδ for all β < γ,

• F = Bα : α < γ ∈ Q if F has been built according to the winning strategy for II inGδ(Γ) (hence Bα = lim−→(F α) for all α < γ of cofinality κ or for all regular cardinalsα such that |Bβ| < α for all β < α).

Let Q be ordered by end extension.Then Q is < δ-closed. Moreover for all D dense subclass of Γ the set X(D) of

F = Bα : α < γ ∈ Q

such that for some (eventually all) α < γ there is A maximal antichain of Bα such thatkBα [A] ⊆ D is open dense.

Proof. The first assertion is immediate since all conditions in Q are iteration system obeyingthe winning strategy for II in Gδ(Γ), hence lower bounds for any sequence of conditions inQ can always be found. To prove the second, let D be a dense subset of Γ and

F = Bα : α < γ ∈ Q

be a condition in Q. Find B ≤∗Γ Bα for all α < γ with B ∈ RigΓδ (which is possibe since

RigΓδ is weakly κ-iterable in Vδ+1, so it has lower bounds for all iteration systems in RigΓ

δ

which obey the winning strategy for II in Gδ(Γ)) and find C ≤∗Γ B also in RigΓδ such that

for some A maximal antichain of C kC[A] ⊆ D. Then

F ∪ 〈γ,B〉, 〈γ + 1,C〉 ∈ X(D)

as well as any of its refinements in Q.

Lemma 14.3.5. Assume T is Γ-canonical and Γ is κ-suitable for T . Assume δ is inacces-sible and Γ-correct with Vδ+1 a model of T . Let G be V -generic for Q and

F = Bα : α < δ = ∪G.

Then in V [G], F is an iteration system whose direct limit lim−→(F) is such that whenever His a V [G]-generic filter for lim−→(F)

Bα b : ∃f ∈ H f(α) = b

generates a V -generic filter for UΓδ .

228

Proof. Notice that Q is < δ-closed. Let G and H be as in the assumptions of the Lemma.We get that (Vδ)

V [G] = Vδ. Clearly

Bα b : ∃f ∈ H f(α) = b

is a family of compatible conditions in (UΓδ )V . By the previous Lemma we can also easily

check thatBα b : ∃f ∈ H f(α) = b ∩D

is non-empty for all D ∈ V dense subset of UΓδ . The Lemma is proved.

We have proved the following:

Corollary 14.3.6. Assume P is κ-suitable for the Γ-canonical theory T . Let δ > κ beinaccessible and Γ-correct for T . Then UΓ

δ ∈ Γ.

Proof. With the notation of the previous Lemma, Q ∈ Γ since it is < δ-closed in V .Let G be V -generic for Q. Then:

• V [G] |= T since Q ∈ Γ and T is Γ-canonical.

• δ is inaccessible in V [G] and Vδ[G] = (Vδ)V [G], since no new sets of size less than δ

are added by Q (the latter forcing being < δ-closed).

• V V [G]δ+1 |= T , since T holds in Vδ+1 and we can apply Remark 14.1.12 to the Γ-canonical

theory T , given that Vδ[G] = (Vδ)V [G].

• δ is Γ-correct in V [G] as well, since it is inaccessible in V [G] and Γ is absolutelyΣ2-definable in T .

By the previous Lemma, we also get that UΓδ is a regular subforcing of Q ∗ F .

Since T is Γ-canonical, δ is Γ-correct in V , V [G], and VV [G]δ = V V

δ , we also have that

RigΓδ ⊆ Γ ∩ Vδ = ΓVδ+1 = ΓVδ+1[G] = V

V [G]δ ∩ ΓV [G]. (14.2)

The first and third equalities come from the observation that Γ is absolutely Σ2-definablein models of T (among which V [G] and V ), and δ is inaccessible both in V and V [G].

The second equality follows form the fact that in order to check whether B ∈ Γ holdsin Vδ+1[G] or in Vδ+1, we must check a formula which do not quantify over classes and thetwo structures have the same sets.

By the same arguments we can also infer that

≤Γ ∩Vδ =≤Vδ+1

Γ =≤Vδ+1[G]Γ =≤V [G]

Γ ∩V V [G]δ . (14.3)

Hence we get that F = ∪G ∈ V [G] is an iteration system in V [G] of length δ ofΓ-correct complete homomorphism (in V [G]) of boolean algebras in Vδ ∩ Γ = ΓVδ+1[G].

By assumption Γ is provably weakly κ-iterable for T . This gives that V [G] models alsothat lim−→F ∈ ΓV [G], since lim−→F is the direct limit of inaccessible length δ > κ of posets in

ΓV [G] ∩ Vδ.Finally remark that ΓV is closed under two-step iterations and preimages of complete

homomorphisms. We conclude that Q ∗ lim−→F ∈ ΓV , and therefore also that UΓδ ∈ ΓV .

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14.4 The quotient of (Γ B)V by a V -generic G for a B ∈ ΓV

is ΓV [G]

All over this section we assume Γ is a (Qn, κ)-suitable class of posets for a κ-canonicaltheory T .

Notation 14.4.1. For each R ∈ RigΓ let

kR : R→ Γ R

be given by r 7→ R r. Then kR is an order and incompatibility preserving embedding of Rin the class forcing Γ R which maps maximal antichains to maximal antichains. Moreoverfor every B ≥Γ C with B ∈ RigΓ, let

iB,C : B→ C

denote the unique Γ-correct homomorphism between B and C.

By the results of the previous sections RigΓ is a dense subclass of Γ and is a separativepartial order. Hence to simplify slightly calculations we focus on RigΓ rather than on Γwhen analyzing this class forcing.

Definition 14.4.2. Given B0 ∈ Γ, fix k0 : B0 → B Γ-freezing B0 and such that B ∈ RigΓ.Let iC = iB,C k0 and

k = kB k0 :B0 → Γ B

b 7→ B k0(b)

Given G a V -generic filter for B0, define in V [G] the class quotient forcing

PB0 = ((RigΓ B)V /k[G],≤Γ /k[G])

as follows:C ∈ PB0

if and only if C ∈ (RigΓ B)V and letting J be the dual ideal of G we have that 1C 6∈ iC[J ](or equivalently if and only if coker(iC) ∈ G).

We letC ≤Γ /k[G]R

if C ≤Γ R holds in V .

Theorem 14.4.3. Assume Γ is (Qn, κ)-suitable for a κ-canonical T , B0 ∈ Γ, and letk0 : B0 → B be a Γ-freezing homomorphism for B0 with B ∈ RigΓ. Set k = kB k0 andiC = iB,C k0 for all C ≤Γ B in Γ.

V

B0 B Γ Bk0 kB

k

Let G be V -generic for B0. Then:

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1. The class forcingPB0 = ((RigΓ B)V /k[G],≤Γ /k[G])

is in V [G] forcing equivalent to the class forcing

QB = (Γ (B/k0[G]))V [G]

via the map

i∗ :PB → QB

C 7→ C/iC[G].

V [G]

B0/G B/k[G] Γ B/k[G]

2 ΓV [G]

[C]k[G]

C/iC[G]

k0

∼=

kB

∼=

2. Moreover let δ > |B| be inaccessible and such that Vδ+1 models T and Vδ ≺n V . Then:

(a) (UΓδ (B/k0[G]))

V [G] is forcing equivalent in V [G] to (UΓδ B)V /k[G] via the same

map.

(b) V models that kB : B→ UΓδ B is Γ-correct.

Proof. We prove the theorem for the case of Γ being κ-suitable. We leave to the readerto check that this same proof works for the general case of a (Qn, κ)-suitable Γ, since thesyntactic complexity of Γ plays no role in any of the argument to follow.

Part 2a of the Theorem follows immediately from its part 1 relativizing every assumptionin part 1 to Vδ+1. To prove part 2b, first observe that if B = B0, k0 is necessarily theidentity map, G is V -generic for B, and this gives that (B/k0[G]) is the trivial completeboolean algebra 2 = 0, 1, i.e.:

(UΓδ (B/k0[G]))

V [G] = (UΓδ )V [G].

Now (UΓδ )V [G] ∈ (Γ)V [G], since V [G] and (Vδ+1)V [G] are both models of T and δ is inaccessible

in V [G], so we can apply Theorem 14.3.2 in V [G]. By part 2a for the case B0 = B (sothat k = kB), we get that (UΓ

δ B)V /kB[G]∼= (UΓ

δ )V [G] holds in V [G] for all G V -genericfor B. This concludes the proof of 2b in case B = B0. The desired conclusion 2b for anarbitrary B0 ∈ UΓ

δ follows using the fact that the set of B ≤Γ B0 in RigΓ is dense in UΓδ and

applying 2b to all such B.We are left to prove part 1: Following the notation introduced in 14.4.2, we let iR

denote the Γ-correct homomorphism iB,R k0 for any R ≤Γ B, and we let k denote the mapkB k0 : B0 → Γ B given by b 7→ B k0(b).

Let G be V -generic for B0 and J denote its dual prime ideal. We first observe that inV [G]

↓ k[J ] = R ∈ (Γ)V : ∃q ∈ J R ≤VΓ B k0(q).

We show that in V [G] the map i∗ is total, order and incompatibility preserving, andwith a dense target. This suffices to prove this part of the Theorem.

231

i∗ is total and with a dense target: By Theorem 6.1.11, any Q ∈ QB is isomorphicto C/iC[G] for some C ∈ (Γ B)V such that 1C /∈↓ iC[J ], since Q is a non-trivial

complete boolean algebra in V [G]. Let in V R ∈ RigΓ refine C in the ≤∗Γ-order. IfR/iR[G]

6∈ QB, we would get that 1R ∈ iR[J ]. Therefore for any Γ-correct injective

u : C→ R witnessing that R ≤∗Γ C, we would have that iC[J ] = u−1[iR[J ]], giving that1C ∈ iC[J ], and contradicting our assumption that 1C /∈↓ iC[J ]. Therefore 1R /∈ iR[J ],and

u/J : C/iC[G] → R/iR[G]

witnesses that i∗(R) refines Q in QB. Hence i∗ has a dense image.

Moreover if R ∈ PB0 , 1R 6∈ iR[J ], hence R/iR[G] is a non-trivial complete boolean

algebra in (Γ)V [G]. Thus i∗ is well defined on all of (RigΓ B)V /k[G].

i∗ is order and compatibility preserving: Let iQ0Q : Q0 → Q be a Γ-correct completehomomorphism in V with Q0,Q ∈ PB0 witnessing that Q ≤Γ /k[G]Q0. This occursonly if 1Q /∈ iQ[J ]. Hence iQ0Q/J : Q0/iQ0

[J ] → Q/iQ[J ] is well defined and witnesses

that Q0/iQ0[J ] ≥Γ Q/iQ[J ] in V [G]. This shows that i∗ is order preserving and maps

non trivial conditions to non trivial conditions. In particular we can also concludethat i∗ maps compatible conditions to compatible conditions.

i∗ preserves the incompatibility relation: We prove it by contraposition. Assumejh : Qh/iQh [G]

∼= Rh → Q for h = 0, 1 witness that Q0/iQ0[G] and Q1/iQ1

[G] are

compatible in (Γ)V [G]. We can assume that Q ∼= C/iC[G].

By Proposition 6.2.5 applied for both h = 0, 1 to B, iQh, jh we have that jh = lh/G

for some Γ-correct homomorphism lh : Qh → Ch in V such that:

• lh iQh = iCh for both h = 0, 1,

• C1/iC1[G]∼= Q ∼= C0/iC0[G]

in V [G],

• 0Ch 6∈ iCh [G] for both h = 0, 1.

By Proposition 6.1.14, we can find sj /∈ iCj [J ] such that C1 s1 and C0 s0 areisomorphic. Without loss of generality we can suppose that Ch sh = C ∈ Γ. Thisgives that (modulo the refinement via sh) lh iQh = iC for both h = 0, 1, since bothlh iQh factor through k0 which is Γ-freezing B0.

In particular each lh witnesses in V that Qh ≥Γ C and are both such that 1C 6∈ iC[J ].Find in V R ≤∗Γ C with R ∈ RigΓ. Then iR[J ] = u iC[J ] for some (any) Γ-correctinjective u : C→ R. Hence 1R 6∈ iR[J ], else 1C ∈ u−1[iR[J ]] = iC[J ].

This grants that R is a non trivial condition in (RigΓ B)V /k[G] refining Qh for bothh = 0, 1.


14.5 Other properties of the class forcing Γ and of UΓδ

Lemma 14.5.1. Assume Γ is κ-suitable, T is Γ-canonical, δ is inaccessible and Vδ+1 |= T ,G is V -generic for UΓ

δ . Then for all B ∈ UΓδ , B ∈ G if and only if there is H ∈ V [G]

Γ-correct V -generic filter for B.Moreover G is the unique V -generic filter on UΓ

δ in V [G].

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Remark 14.5.2. The latter part of the proposition does not follow from Lemma 14.2.6applied to UΓ

δ (see its following remark). UΓδ has an even stronger property than rigidity:

not only there are no non-trivial automorphisms of UΓδ , it is also the case that there are no

complete homomorphismsk : RO(UΓ

δ )→ RO(UΓδ Q)

other than the ones given by the maps kQ : B 7→ B ∧RO(UΓδ ) Q. Otherwise if

k : RO(UΓδ )→ RO(UΓ

δ Q)

is a complete homomorphism different from kQ and G is V -generic for UΓδ with Q ∈ G,

k−1[G] = H ∈ V [G] is also V -generic for UΓδ and is different from G. The Proposition rules

out this possibility.

Proof. It suffices to prove the Lemma for the B ∈ RigΓ ∩ Vδ, since this is a dense subset ofUΓδ .

For one direction observe that if B ∈ G we get that kB : B→ UΓδ B is Γ-correct and

k−1[G] is a Γ-correct V -generic filter for B in V [G]. Conversely assume H ∈ V [G] is a

Γ-correct V -generic filter for B. Let H ∈ V UΓδ be such that HG = H. Let also C ∈ G refine

rH is a Γ-correct V -generic filter for B

z

UΓδ

.

SinceC UΓ

δH ⊆ B,

by Lemma 14.3.3(4) applied in Vδ+1, we can further refine C to some E ∈ G and find someK ∈ V E ∩ Vδ such that r

kE(K) = Hz

UΓδ

≥Γ E.

We leave to the reader to check that i : B → E defined by b 7→rb ∈ K

z

Eis a Γ-correct

homomorphism, giving that B ≥Γ E ∈ G, as was to be shown.

We are left with the proof of the uniqueness of G: Assume H ∈ V [G] is V -generic forUΓδ . By what we have shown in V [H] it holds that

B ∈ H if and only if there exists k : B→ UΓδ and K V -generic for B such that

in V [H] it holds thatV [K] |= B/k[K] ∈ ΓV [K]

Now V [H] ⊆ V [G] since H ∈ V [G], hence K ∈ V [G] as well. The statement

V [K] |= B/k[K] ∈ ΓV [K]

is absolute between V [G] and V [H], hence it holds also in V [G]. Therefore H ⊆ G.Since G,H are both V -generic filters for UΓ

δ , the inclusion entails equality. The proof iscompleted.

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14.6 MM++ and the relation between the stationary towersand the category forcing (SSP,≤SSP)

We now bring forward the simplest case of a surprising duality linking category forcingsto towers of normal ideals. The intuition bringing us to the results of the next chapter islargely driven by this duality. All over this section we let T be the ω1-canonical theory

MK + ω1 is a regular cardinal + there are class many supercompact cardinals.

Recall that (see Def. 8.6.2 for details):

For M ≺ Hθ of size ω1 and containing ω1 and B ∈ M a cba, G ultrafilteron B ∩M is (M,SSP)-correct if G is M -generic and interprets correctly theB-names for stationary subsets of ω1 in M .



14.6.1 A rough analysis of the forcing axiom MM++

Theorem 14.6.1. Let T = MK+there exist class many supercompact cardinals. AssumeV models that δ is supercompact and is such that Vδ+1 is a model of T . Then USSP

δ forcesMM++.

Proof. USSPδ ∈ SSP since Vδ+1 being a model of T models that SSP is an ω1-weakly iterable

class.Let R ∈ V USSP

δ be a name for a complete boolean algebra in SSP. Given B in USSPδ find

k : Vγ+1 → Vλ+1 in V such that λ is inaccessible and Vλ+1 models T , crit(k) = α, B ∈ Vα,k(crit(k)) = δ, and R ∈ k[Vγ ].

Let Q ∈ V USSPα be such that k(Q) = R. Since Vδ + 1 models T and δ is inaccessible, we

get that α is inaccessible and Vα+1 models T . This gives that USSPα ∈ SSP. Let

Q = (USSPα B) ∗ Q ≤SSP USSP

α ,B.

We get that for all G V -generic for USSPδ with Q ∈ G, USSP

α ∈ G as well, hence we get thatG∩ Vα = G0 is an SSP-correct V -generic filter for USSP

α (since the unique SSP-correct mapkUSSP

α: USSP

α → USSPδ USSP

α is given by B 7→ USSPα B ≤SSP B). Thus k lifts to

k :Vγ [G0]→ Vλ[G]

τG0 7→ k(τ)G.

Since Q ∈ G, in V [G] there is an SSP-correct V -generic filter H for Q.Now Q = USSP

α ∗ Q, and USSPα forces Q is isomorphic to Q/k[G0] ∈ SSP where k : USSP

α →USSPα ∗ Q given by B 7→ (B, 1Q) is the natural SSP-correct embedding. Since H is an

SSP-correct V -generic filter for Q, we conclude that

K0 = [q]k[G0] : q ∈ H ∈ V [G]

is an SSP-correct V [G0]-generic filter for Q/k[G0].

We also have that in V [G0] (and thus as well in V [G]) Q/k[G0]∼= QG0 . Therefore in

V [G] there is K0 SSP-correct V [G0]-generic filter for QG0 .

234

Finally we get that in V [G], j[K0] is an SSP-correct j[Vγ [G0]] ≺ Vλ[G]-generic filter forRG = k(QG0) showing that T SSP

R,λ is non-empty in V [G]. Since this holds for all V generic

filter G to which Q ≤SSP B belongs, we have shown that for any R USSPδ -name for a cba

in SSP, for all inaccessible λ > δ, |R|, below any condition B ∈ USSPδ , there is a Q in USSP

δ

which forces that T SSPR,λ is non-empty in V [GUSSP

δ].

The thesis follows.

In the presence of MM++ and class many supercompact cardinals we have a furthercharacterization of SSP-rigidity:

Proposition 14.6.2. Assume MM++ and there are class many Woodin cardinals . Thenthe following are equivalent for a B ∈ SSP:

1. B is SSP-rigid.

2. G(M,B) = b ∈M : M ∈ TBb is the unique correct (M,SSP)-correct filter for B fora club of M ∈ TBb.

Proof. We first show that G(M,B) is an (M,SSP)-correct filter for B iff there is a uniquesuch (M, SSP)-correct filter.

So assume there are two distinct (M,SSP)-correct filters for B H0, H1. Let b ∈ H0 \H1.Then M ∈ TBb ∧ TB¬b as witnessed by H0, H1, thus b,¬b ∈ G(M,B) and G(M,B) is noteven a filter.

Conversely assume H is the unique (M, SSP)-correct filter for B. Then b ∈ H givesthat M ∈ TBb. Thus H ⊆ G(M,B). Now if c ∈ G(M,B) \H there is a correct M -genericfilter H∗ for M with c ∈ H∗ \H. This contradicts the uniqueness assumption on H. ThusH = G(M,B) as was to be shown.

Now we prove the equivalence of SSP-rigidity with 2.Assume first that 2 fails. Let S ⊂ TB be a stationary set such that for all M ∈ S there

are at least two distinct correct M -generic filters HM0 , HM

1 . For each such M we can findbM ∈M ∩ (HM

0 \HM1 ). By pressing down on S and refining S if necessary, we can assume

that bM = b∗ for all M ∈ S. Let δ > |B| be a supercompact cardinal. For j = 0, 1 defineij : B→ T ω2

δ S byb 7→ M ∈ S : b ∈ HM

j .

Then i0, i1 are complete homomorphisms such that

B T ω2δ S/ij [GB] is stationary set preserving.

and i0(b∗) = S = i1(¬b∗). In particular we get that i0 witnesses that B b∗ ≥SSP T ω2δ S

and i1 witnesses that B ¬b∗ ≥ T ω2δ S. All in all we have that B b∗ and B ¬b∗ are

compatible conditions in (SSP ≤SSP), i.e. B is not SSP-rigid.Now assume that B is not SSP-rigid. Let i0 : B b→ C and i1 : B ¬b→ C be distinct

SSP-correct homomorphisms of B into C.Then for some inaccessible γ > |C| + |B| and all M ∈ T SSP

C,γ such that i0, i1 ∈ M we

can pick HM (M,SSP)-correct filters for C. Thus Gj = i−1j [HM ] for j = 0, 1 are both

(M,SSP)-correct filters and such that b ∈ G0 and ¬b ∈ G1. In particular we get that fora club of M ∈ T SSP

C,γ ≤SSP TB there are at least two M -generic filter for B, i.e. 2 fails forTB.

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14.6.2 Duality between SSP-forcings and stationary sets

MM++ permits a simple representation of SSP-rigid boolean algebras and a characterizationof the SSP-correct homomorphisms between SSP-rigid complete boolean algebras.

Fact 14.6.3. Assume B is SSP-rigid and TBb is stationary for all b ∈ B+. Then B isisomorphic to the complete boolean subalgebra [TBb]NS : b ∈ B of the boolean algebraP (TB) /NS.

Notice that in the above setting, P (TB) /NS may not be a complete boolean algebraand may not be stationary set preserving, while [TBb]NS : b ∈ B is a subalgebra which isSSP and complete.

Fact 14.6.4. Assume MM++ holds. Assume B ≥SSP Q are SSP-rigid and complete booleanalgebras. Let i : B→ Q be the unique SSP-correct homomorphism between B and Q. Thenfor all b ∈ B and q ∈ Q and all inaccessible γ > |B|+ |Q|, T SSP

Bb,γ ∧ T SSPQq,γ is stationary if

and only if i(b) ∧Q q > 0Q.


All in all assuming MM++ and class many supercompact cardinals we are in thefollowing situation:

1. MM++ can be defined as the statement that the class PT of presaturated towers ofnormal filters is dense in the category forcing (SSP,≤SSP).

2. We also have in the presence of MM++ a functorial map

ISSP : SSP→ S ∈ V : S is stationary and concentrates on Pω2(V )

defined by B 7→ TB which

• is order and incompatibility preserving,

• maps set sized suprema to set sized suprema in the respective class partialorders,

• gives a neat representation of the separative quotients of SSP-rigid partial ordersand of the SSP-correct embeddings between them.

We have shown in the previous sections that in the presence of class many supercompactcardinals we have that the class RigSSP of SSP-rigid posets is dense in the category forcing(SSP,≤SSP). What about the intersection of the classes RigSSP and PT? Can there bedensely many presaturated towers which are also SSP-rigid in (SSP,≤SSP)? Let us nowexamine in more details this question.

14.6.3 SSP-superrigidity

We continue to work in the base theory T = MK + MM+++there are stationarily manysupercompact cardinals.

Definition 14.6.5. A self-generic ideal forcing PI ∈ SSP (respectively a self-generic towerforcing T Iδ ) is SSP-superrigid if TPI = SGI (respectively TT Iδ

= SGI).

Theorem 14.6.6. Assume G ∈ V is a V -normal ultrafilter on Vλ such that its inducedj = jG : M → Ult(V,G) is huge with critical point δ, j(δ) = λ, and Ult(V,G)λ ⊆ Ult(V,G).Let H be V -generic for USSP

δ . Then in V [H] (USSPλ )V [H] is SSP-superrigid.

236

Proof. Remark that j(USSPδ ) = USSP

λ since Ult(V,G)λ ⊆ Ult(V,G). Moreover the map6

B 7→ infUSSPδ ,B

= USSP

δ B implements the unique SSP-correct embedding of USSPδ into

USSPλ USSP

δ. Remark also that B ∈ Vδ grants that j(B) = B. Since USSP

λ is < λ-presaturated,

we are in the hypothesis of Foreman’s duality theorem 11.3.1, hence (USSPλ USSP

δ)/j[H]

is forcing equivalent to an ideal forcing PI in V [H]. By theorem 14.4.3, we get that(USSP

λ USSPδ

)/j[H] is forcing equivalent to (USSPλ )V [H] in V [H]. By Proposition 11.2.2,

we actually get that for all K V [H]-generic for PI the induced ultrapower embeddingjK : V [H]→ Ult(V [H],K) is < λ-closed in V [K].

It remains to argue that PI is SSP-superrigid self-generic in V [H]. For this weinvoke 14.5.1 and we work in V [H] (hence all definitions are computed in V [H] unlessotherwise specified). Let i : USSP

λ∼= PI be in V [H] the unique isomorphism between these

two forcing notions. Let K be V [H]-generic for USSPλ and L =↑ i[K] be the corresponding

V [H]-generic filter for PI . Now B ∈ K if and only if in V [H][K] there exists HB SSP-correctV [H]-generic filter for B by 14.5.1.

Since jK [Vλ[H]] ∈ Ult(V [H],K) and L = [S]I : jK [Vλ[H]] ∈ jK(S) (by 11.3.1), weget that in V [H]

[T SSPB,λ ]I ∈ L if and only if B ∈ K,

since:

jK [Vλ[H]] ∈ jK(T SSPB,λ )

if and only if

Vλ[H] = πjK [Vλ[H]][jK [Vλ[H]]] admits in V [H][K] (and thus also inUlt(V [H],K) ⊆ V [H][K]) an SSP-correct V -generic filter HB for B

if and only if

B ∈ K.

We conclude thatL =↑

[T SSP

B,λ ]I : B ∈ K

andK =

B : [T SSP

B,λ ]I ∈ L.

The above shows that B = i([T SSPB,λ ]I) for all B ∈ USSP

λ .We get that TPI =NS TUSSP

λ. Moreover for all M ∈ TPI with i ∈M

G(M,PI) =

[S]I ∈M : [S]I ≥ [T SSPB,λ ]I for some B ∈ i[G(M,PI)] =↑ i−1[G(M,USSP

λ )].

But for any M ∈ TUSSPλ

B ∈ G(M,USSPλ ) if and only if (by 14.6.2) M ∈ TUSSP

λ B=NS TB ∧ TUSSP

λif and

only if M ∩HB+ ∈ TB if and only if T SSPB,λ ∈ GM ∩ P (P (Vλ)).

This gives that

G(M,PI) = [S]I : S ∈ GM ∩ P (P (Hλ)) ,

for all M ∈ TUSSPλ

=NS TPI . Hence TPI =NS SGI , concluding the proof.

6By 13.2.10 TB ∧ TUSSPδ

= TUSSPδ

B, by 14.6.3 USSPδ B is the infimum of B,USSP

δ in USSPλ .

237

Corollary 14.6.7. Assume δ is a superhuge cardinal and there are class many super-compacts cardinals. Let H be V -generic for USSP

δ . Then in V [H] there are densely manySSP-superrigid self-generic forcings.

Proof. By the previous Theorem any G ∈ V inducing a huge embedding jG : V → Ult(V,G)with critical point δ is such that (USSP

j(δ))V [H] is Γ-superigid self-generic in V [H]. The thesis

follows.

In the next chapter we will see that the existence of class many SSP-superrigid self-geneirc forcings is intertwined with the search of generic absoluteness results for the Changmodel L([Ord]ω1) which are the correct and natural generalization of Woodin’s absolutenessresults for L([Ord]ω).

238

Chapter 15

Category forcing axioms

The aim of this chapter is to leverage on the results we got on category forcings to replicatefor the Chang model L(Ordκ) Woodin’s generic absoluteness results for L(Ordω). Let usremind the reader the salient features of Woodin’s argument:

1. If δ is inaccessible Coll(ω,< δ) absorbs all forcings in Vδ: i.e. there is a completehomomorphism i : Q→ RO(Coll(ω,< δ)) for all complete boolean algebras Q ∈ Vδ[28, ?, Thm. XXX]).

2. If δ is inaccessible, Q ∈ Vδ, i : Q→ RO(Coll(ω,< δ)) is a complete homomorphism,and L is V -generic for Q, in V [L] we get that RO(Coll(ω,< δ))/i[L] ∼= RO(Coll(ω,<δ))V [L] (again Theorem [28, ?, Thm. XXX]). Hence any G which is V [L]-generic forColl(ω,< δ) is also V -generic for Coll(ω,< δ).

3. If δ is supercompact in V and G is V -generic for Coll(ω,< δ), there exists in V [G]

j : L(Ordω)V → L(Ordω)V [G]

with crit(j) = ω1 and j(ω1) = δ (an immediate by-product of Theorem 12.0.2).

To complete Woodin’s argument, assume L is V -generic for Q ∈ Vδ, and δ is supercompactin V . We get that δ-remains supercompact in V [L] (by 11.3.3) and (by the second item) ifG is V [L]-generic for Coll(ω,< δ), it is also V -generic for Coll(ω,< δ). By the third itemapplied in V [G] or in V [L][G]:

(L(Ordω)V ,∈,P (ω)V ) ≡ (L(Ordω)V [G],∈,P (ω)V )

and(L(Ordω)V [L],∈,P (ω)V [L]) ≡ (L(Ordω)V [G],∈,P (ω)V [L]).

Therefore we conclude that

(L(Ordω)V ,∈,P (ω)V ) ≡ (L(Ordω)V [L],∈,P (ω)V ).

holds for all V -generic filters L for B, and we get Woodin’s generic absoluteness Theo-rem 12.0.1.

Observe what the results of the previous chapter have brought us:

1. Assume Γ is κ-suitable and δ is inaccessible. Then UΓδ ∈ Γ (Theorem 14.3.2) and

absorbs all B ∈ Γ ∩ Vδ (Lemma 14.2.9).

239

2. Assume Γ is κ-suitable and δ is inaccessible. Let G be V -generic for Q ∈ Γ ∩ Vδ.Then (UΓ

δ )V [G] ∼= (UΓδ )V /G (Theorem 14.4.3).

Assume we were able to grant by some type of axiom AX(Γ) that:

3 AX(Γ) entails that for all cardinals δ satisfying a certain large cardinal property (e.g.in the case of Woodin’s argument for L(Ordω), we asked δ being supercompact), UΓ

δ isforcing equivalent to a presaturated tower of height δ (hence when G is V -generic forUΓδ , there exists j : L(Ordκ)V → L(Ordκ)V [G] with crit(j) = κ+ and j(κ+) = δ).

Then we could replicate mutatis-mutandis Woodin’s argument in our setting as follows:Assume AX(Γ) holds in V , given some B ∈ Γ forcing AX(Γ), find δ > |B| such that δsatisfies the required property (e.g. supercompactness) in V and in V B (for the case ofsupercompactness this holds since this property of δ is preserved by forcings in Vδ). Nowreplace Coll(ω,< δ) with UΓ

δ . Then we would get that

(L(Ordκ)V ,∈,P (κ)V ) ≡ (L(Ordκ)V [L],∈,P (κ)V ).

holds for all V -generic filters L for B.In this chapter we propose to isolate an axiom CFA(Γ) which grants that the above

occurs for enough inaccessible cardinals δ, and for all κ-suitable Γ. Let us first addressthe question of how to render UΓ

δ a self-generic presaturated ideal forcing. We have seenthat this is possible in case Γ = SSP. It is also not difficult to get the same result for anarbitrary κ-suitable Γ, by means of Foreman’s duality theorem:

Proposition 15.0.1. Let Γ be κ-suitable forcing for the κ-canonical theory T . AssumeG ∈ V is a V -normal ultrafilter on λ such that its induced j = jG : M → Ult(V,G) ishuge with critical point δ, j(δ) = λ, and Ult(V,G)λ ⊆ Ult(V,G). Assume δ is such thatVδ+1 |= T . Let H be V -generic for USSP

δ . Then in V [H] (USSPλ )V [H] is forcing equivalent to

an ideal forcing PI .

Proof. Follow line by line the proof of the first part of 14.6.6 using Foreman’s dualitytheorem 11.3.1, Theorem 14.4.3 and Proposition 11.2.2. Else look at the proof of 15.1.9 tofollow.

We obtain as a corollary:

Corollary 15.0.2. Let Γ be κ-suitable forcing for the κ-canonical theory T . Assume δ isa superhuge cardinal such that Vδ+1 |= T . Let H be V -generic for UΓ

δ . Then in V [H] thereare densely many Γ-rigid presaturated ideal forcings.

Nonetheless this does not suffice to give the desired AX(Γ). The problem is the following:

Remark 15.0.3. The above corollary gives an axiom CFA0(Γ) stating that for unboundedlymany λ UΓ

λ is a Γ-rigid presaturated ideal forcing. This is not yet sufficient though to runWoodin’s argument:

Assume CFA0(Γ) holds in V . Let B ∈ Γ force CFA0(Γ), and H be V -generic for B.There is no reason to expect that the unbounded class of λ ∈ V such that (UΓ

λ)V is aΓ-rigid presaturated ideal forcing in V have a non-emtpy intersection with the unboundedclass of λ ∈ V [H] such that (UΓ

λ)V [H] is a Γ-rigid presaturated ideal forcing in V [H]. Ifthere is no δ in this intersection the last step in Woodin’s argument cannot be run.

To overcome this issue we develop a notion of Γ-superrigidity (which generalize thenotion of SSP-superrigidity to arbitrary κ-suitable Γ) and we prove the following key result(see 15.1.7):

240

Whenever δ is such that Vδ+1 |= T+ there are class many Γ-superrigid self-generic ideal forcings, then UΓ

δ is by itself a self-generic presaturated towerforcing.

Since the axiom CFA(Γ) stating that there are class many Γ-superrigid self-genericideal forcings reflects to a club-subset of the inaccessible cardinals in V , we now get thatthere will be plenty of δ such that UΓ

δ is a self-generic presaturated tower forcing in anymodel of CFA(Γ). This will grant that the class of λ ∈ V such that (UΓ

λ)V is a Γ-rigidpresaturated ideal forcing in V will have a non-emtpy intersection with the unbounded classof λ ∈ V [H] such that (UΓ

λ)V [H] is a Γ-rigid presaturated ideal forcing in V [H] wheneverH is V -generic for a B ∈ Γ forcing CFA(Γ).

There is one main key issue related to the notion of Γ-superrigidity: given a cba B wehave a clear idea of what is an (SSP,M)-correct filter for an M ≺ Hθ containing ω1 andof size ω1. However if Γ 6= SSP is just κ-suitable, it is not at all clear whether a sensiblenotion of what is a (Γ,M)-correct filter for B for an M ≺ Hθ containing κ and of size κcan at all be defined (even if this is still possible for the case of proper and semiproperforcings). We side-step this difficulty defining relative to a self-generic ideal forcing PI ∈ Γwhat it means that H is a (Γ,M)-correct filter for B for an M ∈ SGI . Then we can run anargument similar to what has been done for the SSP-case centering our characterization ofthe generic filter for PI using the stationary sets of M ∈ SGI admitting a (Γ,M)-correctfilter for B ∈ UΓ

δ .

15.1 Γ-superrgidity and category forcing axioms

Notation 15.1.1. We say that S is a presaturated self-generic forcing of height δ if:

• either S = PI for some self-generic ideal forcing PI with I a normal ideal concentratingon some X ⊇ Vδ, and PI preserving the regularity of δ and inducing a < δ-closedgeneric ultrapower (in which case we say that PI has height δ),

• or S = T Iδ for some presaturated self-generic tower forcing T Iδ of height an inaccessibleδ with I a tower of normal ideals of height δ.

We denote the elements of a presaturated self-generic forcing S by [S]S (rather then [S]Ifor S = PI or [S]I for S = T Iδ ).

For a presaturated self generic forcing S of height δ, an inaccessible λ > |S|, an M ≺ Vλ

GM,S = [S]S ∈ S : M ∩ ∪S ∈M ,

SGSλ = M ≺ Vλ : GM,S is M -generic for S

SGS = SGSλ

for λ = λS the least inaccessible cardinal larger than |S|.

Definition 15.1.2. Let

• Γ be a definable class of forcings,

• δ < λ be inaccessible cardinals,

• B ∈ Γ ∩ Vδ,

• S ∈ Γ a presaturated self-generic forcing of height δ,

241

• M ∈ SGSλ .

H is an (M,S)-correct filter for B if letting:

• πM : M → NM be the Mostowski collapse of M onto a transitive set NM ,

• KM = πM [GM,S ],

πM [H] ∈ NM [KM ] and

NM [KM ] |= πM [H] is a ΓNM -correct filter for πM (B).

We define

TB,S,λ = M ∈ TS,λ : ∃HM (M,S)-correct filter for B

and TB,S = TB,S,λS .

Definition 15.1.3. A self-generic S ∈ Γ of height an inaccessible δ is a Γ-superrigidself-generic forcing if for all [S]S ∈ S+ there exists B ∈ RigΓ

δ such that

TB,S ≤NS S ∧ SGS .

Definition 15.1.4. Let Γ be a definable class of forcings. CFA(Γ) holds if there is a denseclass of Γ-superrigid self generic forcings in (Γ,≤Γ).

15.1.1 Basic properties of Γ-superrigid self-generic forcings

These are the key basic properties of Γ-superrigid self-generic forcings:

Lemma 15.1.5. Assume Γ is κ-suitable and there are class many supercompact cardinals.Then:

1. For any self-generic forcing S of height δ the map

iS : [S]S 7→ S ∧ SGS

defines an isomorphism of S with its target i[S] = ES where ES ⊆ P (P (VλS )) isendowed with the order ≤NS.

2. LetDS =

B ∈ RigΓ

δ : TB,S is stationary.

Then for any B ∈ DS the map kB,S : b 7→ TBb,S defines the unique Γ-correct completehomomorphism of B into ES .

3. Assume moreover that S is Γ-superrigid. Then:

(a) S is Γ-rigid and the map

jS :(DS ,≤Γ)→ (ES ,≤NS)

B→ TB,S

is order and incompatibility preserving with a dense target. Hence for a Γ-superrigid self-generic forcing S, we can freely identify S with DS or ES accordingto our convenience, since these three forcings have isomorphic boolean completions.

242

(b) For all K V -generic for S and modulo the identification of S with DS

K =↑B ∈ DS : ∃H ∈ V [K] Γ-correct V -generic filter for B

.

(c) For all B ∈ DS , S B is also a Γ-superrigid self-generic forcing and TB,S =SGSB.

4. Assume S ≤∗Γ T are both Γ-superrigid self-generic with rank(T ) < rank(S). Thenthe inclusion of DT ⊆ DS extends to the unique injective Γ-correct homomorphismof T into S.

More generally S ≤Γ T if and only if there is an antichain A in DT such that forall B ∈ A the inclusion of DT B into DS B extends to an injective Γ-correcthomomorphism, and for all Q ∈ DT orthogonal to all elements of A, Q is alsoorthogonal to all elements of DS .

Proof.

1: Left to the reader, almost self-evident.

2: Fix a supercompact cardinal δ > |S|. Let G be V -generic for T κ+

δ with TBb,S ∈ G. LetH ∈ V [G] be V -generic for S. We get that Ult(V,G) models that in VλS [H] there isa Γ-correct V -generic filter K for B with b ∈ K. Since Γ is absolutely Σ2-definableand VλS [H] ≺1 V [H] with λS still inaccessible in V [H], we get that V [H] modelsthat there is a Γ-correct V -generic filter K for B with b ∈ K. By the forcing theoremapplied to V [H] and 6.2.2(2 =⇒ 3) we get that the map b 7→ TBb,S defines aΓ-correct embedding.

3: Almost self-evident. 3a follows from 2. 3b follows from 3a and 2. 3c follows from 3a,3b, and 2.

4: Almost self-evident.

Notation 15.1.6. Given a self-generic forcing S either of the form T Iγ or of the form PI :

• [S]S ≥S B for B ∈ DS signifies that S ∧ SGS ≥NS TB,S ,

• for B ∈ DS , S B denotes the Γ-superrigid self-generic T with DT = DS B.

15.1.2 Main results

We have the following theorems, whose proofs are deferred to later parts of this chapter:

Theorem 15.1.7. Let Γ be a κ-suitable class of forcings for a κ-canonical theory T .Assume CFA(Γ) holds and Vδ+1 |= CFA(Γ) + T for some inaccessible δ.

Let G be V -generic for UΓδ . Then in V [G]

K =S ∈ Vδ : S ≥NS SGS for some Γ-superrigid self generic S ∈ G

is a V -normal tower of ultrafilters and Ult(V,K) is < δ-closed in V [G].

Remark 15.1.8. We do not assert that the tower forcing in V induced by the normal filterK in V [G] is forcing equivalent in V to UΓ

δ . This can be proved to be the case if δ is Mahloand there is a stationary set of inaccessible γ < δ with Vγ+1 a model of T + CFA(Γ). Butwe won’t need this stronger result here and hence we omit its proof.

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We now reinforce Proposition 15.0.1 to the following theorem which takes into accountthe concept of Γ-superrigidity we introduced.

Theorem 15.1.9. Assume Γ is κ-suitable for a Γ-canonical theory T . Assume λ > ν ≥γ > δ are inaccessible and G ∈ V is a V -normal ultrafilter concentrating on

X ⊆ Vλ : (X,∈) ∼= (Vγ ,∈), X ∩ δ ∈ δ, otp(X ∩ λ) = γ, otp(X ∩ ν) = δ .

1. Let j = jG : V → Ult(V,G) be the embedding induced by G. Then crit(j) = δ,j(δ) = ν, and Ult(V,G)λ ⊆ Ult(V,G).

2. Furthermore let G0 = πVν ,Vλ [G]. Then G0 ∈ V is a V -normal ultrafilter concentratingon

X ⊆ Vν : (X,∈) ∼= (Vδ,∈), X ∩ δ ∈ δ .

and jG0 : V → Ult(V,G0) is a huge embedding such that jG = k jG0, wherek : [f ]G0 7→ [f ↑ Vλ]G is also elementary between Ult(V,G0) and Ult(V,G).

3. Assume that H is V -generic for UΓδ .

Then in V [H] (UΓj(δ))

V [G] is a Γ-superrigid self-generic ideal forcing.

Corollary 15.1.10. Let Γ be a κ-suitable class of forcings. Then CFA(Γ) is consistentrelative to the existence of a 2-superhuge cardinal.

Proof. Assume δ is 2-superhuge in V and H is V -generic for UΓδ . Then in V [H] (by repeated

applications of 15.1.9),(UΓ

j(δ))V [H] B : j : V →M is 2-huge in V and B ∈ (RigΓ

j(δ))V [H]

witnesses CFA(Γ) in V [H].

Theorem 15.1.11. Let Γ be a κ-suitable class of forcings with respect to the Γ-canonicaltheory

T = MK + κ is regular +aP ⊆ κ+ CFA(Γ).

Assume in V there exists stationarily many inaccessible cardinals δ.Then CFA(Γ) entails FAκ(Γ) and makes the theory of the Chang model L(Ordκ) invariant

with respect to forcings in Γ preserving CFA(Γ).

Proof. CFA(Γ) entails FAκ(Γ) by 13.2.1.For club many inaccessible δ, Vδ+1 |= T , since all the axioms of T not in MK holds on

Vδ+1 for a club subset of δ ∈ Ord. Now for any B ∈ Γ forcing T , any V -generic filter forUΓδ B induces both in V and V B generic ultrapower embeddings of V (respectively V B)

with critical point κ+ (respectively (κ+)VB) sent to δ and both closed under < δ-sequences

in V UΓδ B.

Given any B ∈ Γ forcing T , we can run the Woodin’s generic absoluteness templatedescribed in the introduction of the present chapter with UΓ

δ in the place of Coll(ω,< δ),where δ is inaccessible and such that Vδ+1, V

Bδ+1 are both models of T .

δ exists since in V there are stationarily many inaccessible cardinals and the classes ofγ with Vγ+1 a model of T and η with V B

η+1 a model of T are both closed unbounded.

Hence for all inaccessible δ which are in the intersection of these two clubs Vδ+1, VBδ+1 |=

CFA(Γ), therefore we can apply Theorem 15.1.7 to UΓδ both in V and in V B.

244


Proof. Notice that for all Γ-superrigid self-generic T ∈ UΓδ , T ∈ G entails that (the upward

closure of) G ∩DT is the unique Γ-correct filter for T in V [G]. Notice also that T B isstill Γ-superrigid self-generic for all B ∈ DT .

We prove both parts as follows:

K is a V -normal ultrafilter: Assume S ∈ K is a subset of P (X) and f : P (X) → Xis regressive. Pick S ∈ G Γ-superrigid self-generic tower forcing of height γ withS ∈ Vγ , with S ≥NS SGS (this is possible since for all η < δ the set of Γ-superrigidself-generic S of height at least η is dense in UΓ

δ ).

Then [S]S ∈ S+, given that S ∧ SGS = SGS .

NowDf = [T ]S : f T is constant and T ⊆ S

is a predense subset of S below [S]S . Since S = S [S]S ∈ G is Γ-superrigidself-generic, the set GS of [T ]S ∈ S such that S [T ]S ∈ G is a Γ-correct V -genericfilter for S in V [G] to which [S]S belongs. So there is [T ]S ∈ GS ∩ Df refining[S]S . W.l.o.g we can suppose [T ]S is refined by some B ∈ GS ∩DS ⊆ G, since S isΓ-superrigid selfgeneric. Then S B ∈ G is also a Γ-superrigid selfgeneric forcing and

T ≥NS T ∧ SGS ≥NS T ∧ TB,S = TB,S = SGSB

witnesses that T ∈ K.

Now S ≥NS T , and f T is constant.

Ult(V,K)<δ ⊆ Ult(V,K) holds in V [G]: It is enough to show the following:

Claim 18. For any γ-sequence (fi : i < γ) in V [G] of functions fi : P (Xi) → Vwith fi ∈ V and Xi ∈ Vδ for all i < γ, there is an f ∈ V with domain P (X) forsome X ∈ Vδ such that for all [h] ∈ Ult(V,K) [h]K ∈ [f ]K if and only if for somei < γ [h]K = [fi]K .

Proof. Let us fix a sequence (fi : i < γ) ∈ V [G] of elements of V which are functionswith domain P (Xi) for some Xi ∈ Vδ. By the presaturation of UΓ

δ (given byLemma 14.3.3(3)), the map i 7→ rank(Xi) is bounded in δ, thus

⋃i∈I Xi ⊆ Vα for

some α < δ. Now we can let gi(Z) = Z ∩Xi for each i < γ and Z ⊆ Vα. Then eachgi ∈ V and it is immediate to check that Z ∈ Vα : gi(Z) = fi(Z ∩Xi) ∈ K. Thusit suffices to prove the conclusion of the Claim for the sequence (gi : i < γ) ∈ V [G]to have it also for the sequence (fi : i < γ).

Assume now τ ∈ V UΓδ is a UΓ

δ -name for (gi : i < γ). Then w.l.o.g. we can assumeτ is a UΓ

δ -name for a function with domain γ < δ and range contained in V . ByLemma 14.3.3(2), there is some C ∈ G and a σ ∈ V C such that

C UΓδkC(σ) = τ

holds. By further refining C ∈ G if necessary, we can further assume that C = S is aΓ-superrigid self-generic forcing. Then

H = [S]S : S [S]S ∈ G

is the unique Γ-correct V -generic filter for S ∈ V [G] by Lemma 14.5.1.

245

First of all we observe that σ ∈ V S can be chosen so that

σ = (〈i, gS〉, [S]S) : [S]S ∈ Ai, i < γ ,

where, Ai is a maximal antichain in S for each i < γ, and each gS : P (Vα)→ V is inV for each [S]S ∈ Ai.Next notice that σH = kS(σ)G = τG = (gi : i < γ), since S ∈ G.

Finally observe the following:

K ⊇ S ∈ Vη : [S]S ∈ H . (15.1)

Proof. Since H is an ultrafilter, either [S]S ∈ H or [¬S]S ∈ H. Thus it suffices toshow that [S]S ∈ H gives that S ∈ K for all [S]S ∈ S. Now [S]S ∈ H if and onlyif for some B ∈ DS ∩G [S]S ≥S B. Hence S ≥NS SGSB and S B ∈ G. ThereforeS ∈ K.

The equation is proved.

For each i < γ, let Si ∈ K be the unique element of K such that [Si]S ∈ Ai ∩ H.This forces (〈i, gSi〉 : i < γ) = σH = τG, hence gi = gSi for all i < γ.

For each M ∈ SGS with Ai : i < γ ∈M , let g(M) : M ∩ γ → V be defined by

g(M) =gSiM

(M ∩ Vα) : i ∈M ∩ γ,

where for each i ∈M∩γ, SiM ∈ GM is the unique such that [SiM ]S ∈ G(M,S)∩Ai∩M .For any other M ⊆ VλS , let g(M) = ∅. Then g ∈ V and the following holds:

Ult(V,K) |= [g]K = [gi]K : i < γ (15.2)

Proof. Given h : Y → V in V , we have that [h]k ∈ [g]K iff for some X ∈ Vδ containingY and Vα and some fixed i < γ,

T =M ∈ P (X) : h(M ∩ Y ) = gSiM

(M ∩ Vα)∈ K.

Find T ∈ G refining S with [T ]T ∈ T and T ≥NS SGT . Since Si ∈ K and SiM = Sifor all M ∈ SGS ∧Si ≥NS SGT ∧Si with Ai : i < γ ∈M , we get that K 3 SGS ∧Si,and for a club of M in this set h(M ∩Y ) = gSi(M ∩Vα) = gi(M ∩Vα). Equation 15.2is proved.


The proof that Ult(V,K)<δ ⊆ Ult(V,K) is completed.


246


Proof. We start to show that we are in the assumption of 11.3.5 with UΓδ in the place of P

and UΓν U

Γδ in the place of j(P ) q:

Since j(δ) = ν and Ult(V,G)λ ⊆ Ult(V,G), by elementarity of j, the theory T holdsin (Vν+1)Ult(V,G) = Vν+1, since it holds in Vδ+1. Therefore UΓ

j(δ) ∈ Γ holds in V , and (by

the Σ2-absolute definability of Γ) it holds also in Ult(V,G) (since (Vλ)Ult(V,G) = Vλ andΣ2-properties are upward absolute between Ult(V,G) and Vλ).

In VUΓδ ≥Γ UΓ

j(δ) UΓδ

as witnessed by the map B 7→ UΓδ B. It also holds that UΓ

δ B = infB,UΓ

δ

both in (Γ,≤Γ)

and in (UΓj(δ),≤Γ) (we leave this to the diligent reader). Hence the map i : B 7→ B∧UΓ

j(δ)UΓδ

defines a complete homomorphism. Therefore the assumptions of Proposition 11.3.5 aresatisfied.

Moreover by Theorem 14.4.3

(UΓj(δ) U

Γδ )V /i[H]

∼= (UΓj(δ))

V [H].

Now we combine the results of 11.3.5 and 14.4.3 with the present context.Let H be V -generic for UΓ

δ . In V [H] we get that:

1. U = (UΓj(δ))

V [H] ∼= (UΓj(δ) U

Γδ )V /i[H].

2. I is the normal ideal induced on P (P (Vλ[H]))V [H] by Foreman’s duality theoremapplied to P,G,H.

3. I0 is the ideal induced on P (P (Vν [H]))V [H] by Foreman’s duality theorem appliedto P,G0, H.

4. PI a complete boolean algebra in V [H] inducing a λ-closed generic ultrapowerembedding.

5. PI0 a self-generic ideal forcing inducing a < ν-closed generic ultrapower embedding.

6. PI ∼= U ∼= PI0 .

7. The map iVν [H],Vλ[H] : [S]I0 7→ [S ↑ Vλ[H]]I implements in V [H] the isomorphism

between PI0 and PI obtained by the composition of the natural isomorphisms of PIand PI0 with U given by Foreman’s duality theorem.

We show that PI0 is a Γ-superrigid ideal forcing in V [H]. By 11.3.5 we already knowthat PI0 is self-generic. Remark that (λ being the least inaccessible above ν) for eachB ∈ U

TB,PI0 =M ∈ SGI0 ↑ Vλ[H] : there is a ΓNM -correct NM -generic filter H ∈ NM [KM ] for B

,

where for each M in the above set KM = πM [GM ∩ P (P (Vλ[H]))V [H]].We must show that for each S 6∈ I0 we can find B ∈ RigΓ

δ such that

TB,PI0 ≤NS S ∧ SGI0 .

Let K be V [H]-generic for PI , K0 be the induced V [H]-generic filter for PI0 , L be theinduced V [H]-generic filter for U. By 11.3.5 we know that jK [Vλ[H]] ∈ jK(SGI0 ↑ Vλ[H]).

247

Moreover by 14.5.1 Vλ[H][L] = Vλ[H][K0] models that there exists H Γ-correct Vλ[H]-generic filter for B if and only if B ∈ L.

Therefore for each B ∈ U, we have that

jK [Vλ[H]] ∈ j(TB,PI0 )

if and only if Vλ[H][L] = Vλ[H][K0] models that there exists H Γ-correct Vλ[H]-genericfilter for B if and only if B ∈ L.

This gives that the natural isomorphism of PI with U defined in V [H] by meansof 11.3.5 and 14.4.3 is such that [TB,PI0 ]I 7→ B for all B ∈ U.

This gives that [TB,PI0 ]I : B ∈ U

is a dense subset of PI .

Hence given [S]I0 ∈ (PI0)+ find B ∈ U such that [S ↑ Vλ[H]]I ≥ [TB,PI0 ]I . Then

S ∧ SGI0 =NS (S ∧ SGI0) ↑ Vλ[H] ≥NS TB,PI0 .

This proves that PI0 is Γ-superrigid self-generic in V [H], completing the proof of theTheorem.

248

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Iterated forcing, category forcings, generic ultrapowers ... · of chapter 1 gives the basic facts...

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