J. Baldwin and S. Shelah- The Primal Framework II: Smoothness

8/3/2019 J. Baldwin and S. Shelah- The Primal Framework II: Smoothness

1/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

The Primal Framework II: Smoothness

J. Baldwin

Department of MathematicsUniversity of Illinois, Chicago

S. Shelah Department of MathematicsHebrew University of Jerusalem

October 6, 2003

This is the second in a series of articles developing abstract classicationtheory for classes that have a notion of prime models over independent pairsand over chains. It deals with the problem of smoothness and establishingthe existence and uniqueness of a monster model. We work here with apredicate for a canonically prime model. In a forthcoming paper, entitled,Abstract classes with few models have homogeneous-universal models, weshow how to drop this predicate from the set of basic notions and still obtainresults analogous to those here.

Experience with both rst order logic and more general cases has shownthe advantages of working within a monster model that is both homogeneous-universal and saturated. Fra isse [6] for the countable case and J onsson [9]for arbitrary cardinalities gave algebraic conditions on a class K of modelsthat guaranteed the existence of a model that is homogeneous and universalfor K . Morley and Vaught [11] showed that if K is the class of models of a rst order theory then the algebraic conditions of homogeneity and uni-

versality are equivalent to model theoretic conditions of saturation. Firstorder stability theory works within the ction of a monster model M . Such

Partially supported by N.S.F. grant 8602558Both authors thank Rutgers University and the U.S. Israel Binational Science foun-

dation for their support of this project. This is item 360 in Shelahs bibliography.

1


2/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

a ction can be justied as a saturated model in an inaccessible cardinal,by speaking of class models, or by asserting the existence of a function f from cardinals to cardinals such that any set of data (collection of models,sets, types etc.) of cardinality can be taken to exist in a sufficiently satu-rated model of cardinality f (). This paper culminates by establishing theexistence and uniqueness of such a monster model for classes K of the sortdiscussed in [3] that do not have the maximal number of models. We avoidsome of the cardinality complications of [11] by specifying closure propertieson the class of models.

One obstacle to the construction that motivates an important closurecondition is the failure of smoothness. Is there a unique compatibility classof models embedding a given increasing chain? It is easy for a model to becompatibility prime (i.e. prime among all models in a joint embeddabilityclass over a chain) without being absolutely prime over the chain. This fail-ure of smoothness is a major obstacle to the uniqueness of a monster model.Our principal result here shows that this situation implies the existence of many models in certain cardinalities. We can improve the certain by theaddition of appropriate set theoretic hypotheses.

The smoothness problem also arose in [18]. Even when the union of achain is in K it does not follow that it can be K -embedded in every memberof K which contains the chain. The argument showing this situation implies

many models is generalized here. However, we have further difficulties. Inthe context of [18] once full smoothness (unions of chains are in the classand are absolutely prime) is established in each cardinality, one can provea representation theorem as in [17] to recover a syntactic (omitting typesin an innitary language) denition for the abstractly given class. Fromthis one obtains full information about the Lowenheim-Skolem number of K and in particular that there are models in all sufficiently large powers.The examples exhibited in Section 2 show this is too much to hope for inour current situation. Even the simplest case we have in mind, 1-saturatedmodels of strictly stable theories, gives trouble in if > . This illustrates

one of the added complexities of the more general situation. Many propertiesthat in the rst order case hold on a nal segment of the cardinals hold onlyintermittently in the general case. This greatly complicates arguments byinduction and presents the problem of analyzing the spectrum where a givenproperty holds.

This paper depends heavily on the notations established in Chapters I

2


3/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

and II of [3]; we do not use the results of Chapter III. Reference to [18] ishelpful since we are generalizing the context of that paper but we do notexpressly rely on any of the results there. Some arguments are referred toanalogous proofs in [18] and [10].

Section 1 of this paper recapitulates the properties of canonical primemodels over chains and contains some examples illustrating the efficacy of the notion. Sections 2 and 4 x some basic notations and assumptions. Sec-tion 2 deals with downward L owenheim-Skolem phenomena; the upwardsLowenheim-Skolem theorem is considered in Section 3. In Section 5 we de-scribe the combinatorial principles used in the paper. Section 7 introduces auseful game and some axioms on double chains that are used to show PlayerI has a winning strategy if K is not smooth.

In Section 6 we consider two important ideas. First we discuss the notionK codes stationary sets a particularly strong form of a nonstructure resultfor K . Then we consider two different ways that a model might code a sta-tionary set. Dealing with canonically prime models over chains rather than just unions of chains introduces subtleties into the decomposition of modelsthat make the process of taking points of continuity more complicated thanin earlier studies.

Section 8 contains the main technical results of this paper, showing thatwith appropriate set theory, if K is not smooth then it codes stationary sets.

In Section 9 we assume that K is smooth. We are then able to i) constructand prove the categoricity of a monster model, ii) introduce the notion of a type, and iii) recover the Morley-Vaught equivalence of saturation withhomogeneous-universality. We conclude in Section 10 with a discussion of further problems.

1 Prime models over chains

We begin by reviewing the discussion in [3] of prime models over chains.Let M = M , f , : < < be an increasing chain of members of K .

An embedding f of M into a structure M is a family of maps f i : M i M that commute with the f i,j . As for any diagram, there is an equivalencerelation of compatibility over M . Two triples ( M, f ,M ) and (M,g ,N ),where f (g) is a K -embedding of M into M (N ), are compatible if thereexists an M and f 1 (g1) mapping M (N ) into M such that f 1 f and g1 g

3


4/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

agree on M (i.e. on each M i ). This relation is transitive since K has theamalgamation property.Now M is compatibility prime over (M, f ) if it can be embedded over f

into every model compatible with it. In Section II.3 of [3] we introducedthe relation cpr for canonically prime, characterized it by an axiom Ch1,and then asserted the existence of canonically prime models by axiom Ch2.Before stating the basic characterization and existence axioms used in thispaper we need some further notation.

1.1 Denition.

i) The chain M i , f i,j : i < j < is essentially K -continuous at < if there is a model M that is canonically prime over M and compatiblewith M over M i , f i,j : i < j < .

ii) The chain M is essentially K -continuous if for each limit ordinal < ,M is essentially K -continuous at .

We will use the following slight variants on the existence and character-ization axiom in Section II.3 of [3]. Note that by Axiom Ch1, the M inDenition 1.1 can be K -embedded in M over M |.

1.2 Axioms for Canonically Prime Models.Axiom Ch1 cpr(M,M, f ) implies

i) M is an essentially K -continuous chain,ii) M is compatibility prime over M via f .

Axiom Ch2 If M is essentially K -continuous there is a canonically primemodel over M .

Clearly, if K satises Ch1 and Ch2 each essentially K -continuous chain

can be rened to a K -continuous chain. The following example and Exam-ple 1.5 show the necessity of introducing the predicate cpr rather than justworking with models that satisfy the denition of compatibility prime.

1.3 Example. Fix a language with 1 unary predicates L i (for level) anda binary relation . Let K be the collection of structures isomorphic to

4


5/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

structures of the form A, L i , where A is a subset of


6/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

plicated matters. In particular, in this framework we expect to discuss theclass of 1-saturated models of a strictly stable theory. We can not expectto reduce the structure of models of such theories to anything simpler thana tree with 1-levels. The following example provides another reason forintroducing the predicate cpr.

1.5 Example. Let T be the theory REI of countably many rening equiv-alence relations with innite splitting [2, page 81]. Let K be the class of 1-saturated models of T and dene M N if no E -class of M is extendedin N . (E denotes the intersection of the E i for nite i.) Now there are manychoices for the interpretation of the predicate cpr, namely the -saturated

prime model for each uncountable . (Note that if M = M i : i < themodels prime among the and saturated models respectively containingM are incompatible over M if = .)

Thus, the canonically prime model becomes canonical only with the ad-dition of t he predicate cpr. There are a number of reasonable candidatesin the basic lan guage and we have to add a predicate to distinguish one of them. The last example shows that we should demand that cpr models arecompatible. This property was not needed in [3] but we need it here to provesmoothness. Its signicance is explained in Paragraph 4.4.

1.6 Axiom Ch4. Let M be a K -continuous chain and suppose both cpr( M, M )and cpr( M, N ) hold. Then M and N are compatible over M .

2 Adequate Classes

This paper can be considered as a reection on the construction of a homoge-neous universal model as in Fraisse [6], J onsson [9], and Morley and Vaught[11]. These constructions begin with a class K that satises the amalgama-tion and joint embedding properties. They have assumptions of two furthersorts: Lowenheim-Skolem properties and closure under unions of chains.

We deal with these assumptions in two ways. Some are properties of thekinds of classes we intend to study; we just posit them. For others we are ableto establish within our context a dichotomy between the property holdingand a nonstructure result for the class. Most of this paper is dedicated to thesecond half of the dichotomy; in this section we sum up the basic propertieswe are willing to assume.

6


7/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

We begin by xing the language.2.1 Vocabulary. Recall that each class K is a collection of structures of xed vocabulary (i.e.similarity type) K . We dene a number of invariantsbelow. We will require that the cardinality of K is less than or equal to anyof our invariants. If we did not make this simplifying assumption we wouldhave to modify each invariant to the maximum of the current denition and| K | . This would complicate the notation but not affect the arguments inany essential way.

As usual we denote by K (K 1(K ) has power . Since we nowdeal with essentially innitary functions, we cannot make this demand forall . If there are -ary functions it is likely to fail in cardinals of conality. We assume a downwards Lowenheim-Skolem property in many but notall cardinals. We justify this assumption in two ways. First the conditionholds for the classes (most importantly, 1-saturated models of strictly stabletheories) that we intend to consider. Secondly, the assumption holds for anyclass where the models can be generated by -ary Skolem functions for some that depends only on K and the similarity type.

2.3 Denition. K has the -Lowenheim-Skolem property if for each M Kand A M with |A| there exists an N with A N M and |N | .

Replacing the two occurences of in the denition of the -Lowenheim-Skolem property by < we obtain the ( < )-Lowenheim-Skolem property. If

7


8/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

= +

then the -Lowenheim-Skolem property and the ( < )-Lowenheim-Skolem property are equivalent.Note that K may have the -Lowenheim-Skolem property and fail to have

the -Lowenheim-Skolem property for some > .LS(K ) denotes the least such that K has the -Lowenheim-Skolem

property.

2.4 Downward L owenheim-Skolem property.

Axiom S1 There exists a such that for every , if = then K has the-Lowenheim Skolem property.

Notation 1(K ) denotes the least such . K = (sup( 1(K ), LS (K ))+ .

2.5 Example. Examination of Example 1.5 shows that as stated it does notsatisfy the -Lowenheim-Skolem property for any . An appropriate modi-cation is to consider the class K of models of T that are 1-saturated buteach E class has less than -elements. Then K satises the -Lowenheim-Skolem property for any and we are able to apply our main results.

We easily deduce from the < -Lowenheim-Skolem property the followingdecomposition of members of K . Note that no continuity requirement isimposed on the chain.

2.6 Proposition. If K satises the < -L owenheim-Skolem property, isregular, and M K has cardinality then M can be written as i


9/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

ii) K is ( , )-bounded if every ( , )-chain is bounded.To assert K is ( , )-bounded imposes a nontrivial condition even in

the presence of axiom Ch2 because there is no continuity assumption on thechain. Moreover, a demand of boundedness is not comparable to a demandfor the Lowenheim-Skolem property; it is a demand that a certain abstractdiagram have a concrete realization. It is easy to construct examples of abstract classes where boundedness fails if there is a bound on the size of the models in the class. We describe several more interesting examples inParagraph 4.3.

2.9 Alternatives. Here is a natural further notion. K is K -weakly bounded (with appropriate parameters) if every K -continuous chain is bounded. Wewill not actually have to consider this notion because the existence of canon-ically prime models implies that K is K -weakly bounded.

2.10 Denition. The cardinal is K -inaccessible if it satises the followingtwo conditions.

i) Any free amalagam M 0, M 1, M 2 with |M 1| , |M 2| < can be extendedto M 0, M 1, M 2, M 3 with |M 3| < .

ii) Any (< , < ) chain, which is bounded, is bounded by a model withcardinality less than .

Since + is K -inaccessible if K satises the -Lowenheim-Skolem propertywe deduce from S1 the following proposition. The rst clause shows thereare an abundance of K -inaccessible cardinals. For many of the results of this paper it suffices for to be K -inaccessible rather than requiring the< -Lowenheim-Skolem property.

2.11 Lemma. Suppose is greater than K and K satises S1.

i) If 1 (K ) = then + is K -inaccessible.

ii) If is a strongly inaccessible cardinal then is K -inaccessible.

The following examples show that some of the classes we want to inves-tigate have models only in intermittent cardinalities.

9


10/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

2.12 Examples. Let K be the class of 1-saturated models of a countablestrictly stable theory T .

i) If > then there are sets with power which are contained in nomember of K with power .

ii) If T is the model completion of the theory of countably many unaryfunctions there is no member of K with power if > .

We modify our notion of adequate class from Chapter III of [3] to incor-porate these ideas.

2.13 Adequate Class. We assume in this paper axiom groups A andC, Axioms D1 and D2 from group D, (all from [3]), Axioms Ch1,Ch2 and Ch4 from Section 1, and Axioms S0 and S1 from thissection. A class K satisfying these conditions is called adequate.

One of our major uses of the Lowenheim-Skolem property is to guaranteethe existence of K -inaccessible cardinals as in Lemma 2.11. We now notethat this conclusion can be deduced from very weak model theory and anot terribly strong set theoretic hypothesis. We begin by describing the settheoretic hypothesis.

2.14 Denition. We say is Mahlo if for any class C of cardinals that isclosed and unbounded in the class of all cardinals there is a weakly inacces-sible cardinal such that C is an unbounded subset of .

In fact, the of the denition could be taken as strongly inaccessiblesince the strong limit cardinals form a closed unbounded class.

2.15 Theorem. Suppose is Mahlo and that K is a class of -stucturesthat is closed under isomorphism, satises axiom C1 (existence of free amal-gamations of pairs) and is (< , < )-bounded. Then the class of K -inaccessible cardinals is unbounded. In fact, it has nonempty intersection with any closed unbounded class of cardinals.

Proof. For any cardinal , let J 1() be the least cardinal such that forany M 0, M 1, M 2 with the universe of each M i a subset of and each pairf 1, f 2 where f i is a K -embedding of M 0 in M i (for i = 1 , 2) there is an M 3

and a pair of maps g1, g2 that complete the amalgamation with |M 3| < J 1().

10


11/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

Similarly, let J 2() be the least cardinal such that any ( , )-chain isbounded by a model of cardinality less than J 2(). Finally, let J () be themaximum of J 1(), J 2(). Now the set C = { : < implies J () < } isclosed and unbounded. Since is Mahlo, there is an inaccessible cardinal with C unbounded in . But then is K -inaccessible. It is easy to varythis argument to show there are actually a proper class of K -inaccessiblesand indeed that that class is stationary.

3 Upwards L owenheim-Skolem phenomena

As the examples in the Section 2.12 show, it is impossible to get the fullLowenheim-Skolem-Tarski phenomenon models in all sufficiently largecardinals in the most general situation we are studying. Neverthelesswe can establish an upwards L owenheim-Skolem theorem. We show that Kis a Hanf number for models of K .

These results generalize (with little change in the proof) and imply FactV.1.2 of [19].

3.1 Remark. The real signicance of the following theorem is that it doesnot rely on axiom C7 (disjointness). With that axiom the second part of the following theorem is trivial. We asserted in [3] that the use of C7 wasprimarily to ease notation; this argument keeps us true to that assertion.

Recall that we have assumed for simplicity that | K | K .

3.2 Theorem. Suppose K has the -L owenheim-Skolem property and thereis a member M of K with cardinality greater than 2 . Then

i) There exist M 0, M 1, M 2, M 3 such that NF (M 0, M 1, M 2, M 3) and thereis a nontrivial (i.e. not the identity on M 1) isomorphism of M 1 ontoM 2 over M 0.

ii) There exist arbitrarily large members of K .

Proof . The proof of conclusion i) is exactly as in [19]. That is, since |M | > 2 ,by the -Lowenheim-Skolem property, we can x M 0 M with |M 0| and choose for each c M M 0 an N c with M 0 N c M , c N c,and |N c | . Expand the language L of K to L by adding names for

11


12/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

{d : d M 0} and let L contain one more constant symbol. There are atmost 2 isomorphism types of L -structures N c, c satisfying the diagramof M 0 so there are c1 = c2 M with N c1 , c1 N c2 , c2 . Thus there isan isomorphism f from N c1 onto N c2 over M 0 with f (c1) = c2. ApplyingAxiom C2 (existence of free amalgams), we can choose M 3 and g : N c1 M 1 over M 0 with NF (M 0, M 1,M,M 3). Now by monotonicity we have bothNF (M 0, M 1, N c1 , M 3) and NF (M 0, M 1, N c2 , M 3). Let c denote g(c1). Nownot both g 1(c) = c1 and f g 1(c) = c2 equal c. So one of N c1 and N c2 canserve as the required M 2.

Our proof of the existence of arbitrarily large models actually only relieson conclusion i). Let c M 1 be such that the isomorphism f of M 1 andM 2 moves c. For any , we dene by induction on a K -continuoussequence of models M such that |M | as required. As an auxiliary in theconstruction we dene f and N such that f is a nontrivial isomorphismbetween M 3 and N . We demand NF (M 0, N , M , M +1 ).

For = 0, let M 0 = M 3. At stage + 1 we dene f , N and M +1 byinvoking the existence axiom to obtain: NF (M 0, N , M , M +1 ) and f :M 3 N . For limit , choose M canonically prime over its predecessors.

To obtain the cardinality requirement it suffices to show that if < then f (c) M . Fix and let A1 denote f (M 1) and A2 denote f (M 2).We have NF (M 0, A1, M , M +1 ) and NF (M 0, A2, M , M +1 ) by the con-

struction. Again from the construction g = f |M 2 f (f |M 1) 1 is anisomorphism between A1 and A2 over M 0. By the weak uniqueness axiom(C5) (see Lemma I.1.7 of [3]), g extends to an isomorphism g betweenA1 and A2 which xes M pointwise. Now, if f (c) M , g (f (c)) =g (f (c)) = ( f (c)). But g (f (c)) = f (f (c)) (by the denition of g ) so f xes c. This contradiction yields conclusion ii).

Noticing that the existence of a nontrivial map implies the existence of a nontrivial amalgamation and that only conclusion i) was used in the proof of conclusion ii) we can reformulate the theorem as follows.

3.3 Corollary. If K does not have arbitrarily large models then all membersM of K have cardinality less than K . Moreover, if N M K , there isno nontrivial automorphism of M xing N .

Proof. Note the denition of K (2.4) and apply Lemma 3.2 with asLS(K). Thus the models of a class with a bound on the size of its models areall almost rigid. These arguments give some more local information.

12


13/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

3.4 Denition. The structure M is a maximal model in K if there is noproper K -extension of M .

3.5 Corollary. If |M | > 2 and K has the -L owenheim-Skolem property then M is not a maximal model in K . Thus if |M | K , M is not a maximal model.

4 Tops for chains

We discuss in this section several requirements on a model that bounds a

chain. Shelah has emphasized (e.g. [17, 18]) that the Tarski union theoremhas two aspects. One is the assertion that the union of an elementary chainis an elementary extension of each member of the chain and thus a memberof any elementary class containing the chain; the second is the assertion thatthe union is an elementary submodel of any elementary extension of eachmember of the chain. First we consider the second aspect.

4.1 Denition. i) The class K is (< , )-smooth if there is a uniquecompatibility class over every ( < , )-chain.

ii) K is smooth if it (< , < )-smooth.Note that if the class K is (< , )-smooth then the canonically prime

model over any essentially K -continuous ( < , ) chain is absolutely prime.Moreover, if K is smooth every K -increasing chain is essentially K -continuous.

The next two notions represent the rst aspect of Tarskis theorem; thethird unites both aspects.

4.2 Closure under unions of chains.

i) K is (< , )-closed if for any (< , )-chain M inside N the union of M is in K and for each i, M i M .

ii) K is (< , )-weakly closed if for any (< , ) K -continuous chain M inside N the union of M is in K and for each i, M i M .

iii) K is fully (< , )-smooth if the union of every K -continuous ( < , )-chain inside M is in K and is absolutely prime over the chain.

13


14/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

The inside N in i) is perhaps misleading. We have not asserted N K ,so this is not an a priori assumption of boundedness. In fact N must exist asthe union of M . If K is (, )-closed it is (, )-bounded as the union servesas the bound.

If K is (< , )-smooth and (< , )-weakly closed then for any ( < , )K -continuous chain M inside N the union of M is the canonically primemodel over M and K is fully (< , )-smooth .

If K is the class of 1-saturated models of a strictly stable countabletheory, K is not closed under unions of countable conality but is closed underunions of larger conality. This property of a class being closed under unionsof chains with sufficiently long conality is rather common. For example anyclass denable by Skolem functions with innite but bounded arity will havethis property. We can rephrase several properties of some of the examples inSection 1 in these terms.

4.3 Examples. i) The class K of Example 1.3 is ( , 2)-bounded,(< , 2)-closed, and even fully (< , 2)-smooth. But K is not(< 1,1) bounded and not ( < 1,0) or (< 1,1)-smooth.

ii) Example 1.5 shows that for K the class of 1-saturated models of REI and a particular choice of , the class K is not smooth. For, e.g., theprime 1-saturated model over a chain and the prime 2-saturatedmodel over the same chain may be incompatible.Note that for any strictly stable countable theory and any uncountable, if K is the class of -saturated models of a countable strictly stabletheory, denotes elementary submodel, and cpr means prime amongthe -saturated models then K is smooth. In this case 1 is 0.

iii) Consider again the class K discussed in Example 2.5. The union of a countable K -chain may determine an E class that is not realized.So K is not (< , )-closed. But it is ( < , [1, < ])-closed and(< , )-bounded.

Our basic argument will establish a dichotomy between the followingweakening of smoothness and a nonstructure theorem.

4.4 Semi-smoothness. The class K is (< , )-semismooth if for each(< , )-K -continuous chain M each compatibility class over M containsa canonically prime model over M .

14


15/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

The distinction between smooth and semi-smooth quickly disappears inthe presence of Axiom Ch4.

4.5 Lemma. If K is semismooth and satises Ch4 then K is smooth.

Proof . Ch4 asserts that all M satisfying cpr( M, M ) are compatible. Sinceeach compatibility class contains such an M there is only one compatibilityclass.

4.6 Remark. By an argument similar to the main results of this paper (butmuch simpler) we can show for a proper class of that a class K that has

prime models over independent pairs and is closed under unions of chains(of any length) is fully (< ,< 1(K ))-smooth unless K codes stationarysubsets of (See Section 6.) To establish this result we need the axiomsabout independence of pairs enumerated in [3] and that there is a properclass of K -inaccessible cardinals. The last condition can be guaranteed byassuming a Lowenheim-Skolem property like Axiom S1 or by assuming isMahlo as in Theorem 2.15.

This situation is half way between the situation in [18] and that consid-ered here. We replace closed under substructure by the existence of primemodels over independent pairs but retain taking limits by unions.

5 Some variants on 2

We discuss in this section some variants on Jensens combinatorial principle 2which will be useful in model theoretic applications. We begin by establishingsome notation.

5.1 Notation. i) For any set of ordinals C , acc[C ] denotes the set of accumulation points of C the C with = sup C . The nonac-cumulation points of C , C acc[C ], are denoted nacc[C ].

ii) For any set of ordinals S , C (S ) denotes the elements of S with co-nality .

iii) For any cardinal , sing(), the set of singular ordinals less than isthe collection of limit ordinals less than that are not regular cardinals.

15


16/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

iv) In the following always denotes a limit ordinal.The following denition is a version to allow singular cardinals of the 2principle for + in [5]. We refer to it as full 2 . This principle has beendeduced only from strong extensions of ZFC such as V = L. [5]. We willalso use here weaker versions, obtained by relativizing to a stationary set,that are provable in ZFC.

When = + Jensen called the condition here a 2 on . The versionhere also applies to limit ordinals and since we will deal with inaccessiblesseems preferable.

5.2 Denition. The sequence C : sing() witnesses that satises2 if satises the following conditions.

Each C is a closed unbounded subset of .

otp( C ) < .

If acc[C ] then C = C .

Now we proceed to the relativized versions of 2 . The relativization iswith respect to two subsets, S and S + . It is in allowing the C to be indexedby S + rather than all of that this principal weakens those of Jensen and

can be established in ZFC.We will consider two relativizations. In Section 8 we will see that the twoset theoretic principles will allow us two different model theoretic hypothesesfor the main result. They, in fact correspond to two different ways of assigninginvariants to models.

5.3 Denition. We say that S + and C : S + witness that thesubset S of C () satises 2 a, (S ) if S S + and

i) S is stationary in .

ii) For each S +

, C S +

S .iii) If S + is not a limit ordinal, C is a closed subset of .

iv) If S + is a limit ordinal then

(a) C is a club in .

16


17/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

(b) otp( C ) .(c) otp( C ) = if and only if S .(d) All nonaccumulation points of C are successor ordinals.

v) For all S + , if C then C = C .

5.4 Denition. 2 a, holds if for some subset S , 2 a, (S ) holds.

5.5 Fact. Suppose > are regular cardinals. If is a successor of aregular cardinal greater than or = + and = then for any stationaryS C () there is a stationary S S such that 2 a

,(S ) holds.

The proof with = is on page 276 of [16] (see also the appendix to[10]); for regular see Theorem 4.1 in [12].

Fact 5.5 is proved in ZFC; if we want to make stronger demands on thestationary set S we must extend the set theory.

5.6 Denition. The subset S of a cardinal is said to reect in S if S is stationary in . We say S reects if S reects in some S .

Thus a stationary set that does not reect is extremely sparse in that itsvarious initial segments are not stationary.

5.7 Fact. If > are regular, is not weakly compact, and V = L then forany stationary S 0 C () there is a stationary S S 0 that does not reectsuch that 2 a, (S ) holds.

This is a technical variant on the result of [4]. Although this result followsfrom V=L it is also consistent with various large cardinal hypotheses.

We now consider the other relativization of 2 . For it we need a new lteron the subsets of .

5.8 Denition. Let the stationary subset S of the regular cardinal indexthe family of sets C = {C : S }. Then ID( C ) denotes the collection of subsets B of such that there is a cub C of satisfying: for every B,C is not contained in C . We denote the dual lter to ID( C ) by FIL( C ).

It is easy to verify that ID( C ) is an ideal. Note that B ID(C ) if andonly if for every club C , there is a B with C C .

17


18/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

5.9 Denition. [2 b

,,,R (S, S 1, S 2)]: Suppose = , and R are four reg-ular cardinals with < , + < , R and S is a subset of containingall limit ordinals of conality < R . We say 2 b,,,R (S, S 1, S 2) holds if thefollowing conditions are satised for some C = C : S .

i) C = C : S is a sequence of subsets of satisfying

(a) C is a closed subset of .

(b) C S .

(c) If is a limit ordinal then C is unbounded in .

(d) If is an accumulation point of C then C = C .(e) If < 1, 2 and C 1 C 2 then C 1 = C 2 .

ii) S 1 and S 2 are disjoint subsets of C (S ) with union S 0 S .

(a) If S 0, otp( C ) = .

(b) If nacc[C ] and S 0 then cf( ) = .

iii) The ideal ID(C ) is nontrivial; S 1 and S 2 are not in ID( C ).

5.10 Remark. Note that if S 0 and C then cf( ) < so C S 0 = .

5.11 Denition. [2 b,,,R ]: We say 2 b,,,R holds if there exist subsets S and C S for S that satisfy the following conditions.

i) S contains each of a family T i : i < of sets; each T i C () andthe T i are distinct modulo ID( C ).

ii) For each A there exist S 1, S 2 S such that

(a) S 1 = iA T i and S 2 = iA T i and

(b) 2 b,,,R (S, S 1, S 2) holds with C = C : S .

Now the set theoretic strength required for these combinatorial principlescan be summarised as follows.

5.12 Theorem.

18


19/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

i) If is a successor of a regular cardinal, +

< , and +

< then 2 b,,, 0 is provable in ZFC.

ii) If is a successor cardinal, < , and < then 2 b,,,R is provablein ZFC + V=L for any R .

Proof. Case 1 is proved in III.6.4, III.7.8 F (3) of [19] and in [12]. ForCase 2 consult III.7.8 G of [19].

5.13 Alternative set theoretic hypotheses. There are a number of re-nements on conditions sufficient to establish 2 b,,,R .

i) If is a successor of a regular cardinal or even just not Mahlo, The-orem 5.12 ii) can be strengthened by replacing V= L, by there is asquare on . See [19, III.7.8 H].

ii) In fact the conclusion of Theorem 5.12 ii) holds for any that is notweakly compact. (Similar to [4].)

iii) The existence of a function F such that 2 b,,,R holds for any regular > F ( + + R) and < is consistent with ZFC + there is a classof supercompact cardinals.

6 Invariants

As a rst approximation we say a class K has a nonstructure theorem if formany , K has 2 models of power . But this notion can be rened. Forsome classes K it is possible to code stationary subsets of regular by modelsof K in a uniform and absolute way while other classes have many modelsfor less uniform reasons. The distinction between these cases is discussed in[15, 20]. In the stronger situation we say, informally, that K codes stationary sets. We dont give a formal general denition of this notion, but the two

coding functions we describe below sm, sm1 epitomize the idea.The basic intention is to assign to each model a stationary set so that atleast modulo some lter on subsets of nonisomorphic models yield distinctsets. Historically (e.g. [18]) to assign such an invariant one writes the modelM as an ascending chain of submodels and asks for which limit ordinals isthe chain continuous. The replacement of continuity by K -continuity in this

19


20/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

paper makes this procedure more difficult. We can succeed in two ways.Either we add an additional axiom about canonically prime models and pro-ceed roughly as before or we work modulo a different lter. Both of thesesolutions are expounded here.

We deal only with classes K that are reasonably absolute. That is, theproperty that a structure M K should be preserved between V and L andbetween V and reasonable forcing extensions of V . Of course a rst orderclass or a class in a L , meets this condition (a reasonable forcing in thiscontext would preserve the family of sequences of length < of ordinals ). But somewhat less syntactic criteria are also included. For example, if K is the class of 1-saturated models of a strictly stable theory membershipin K is preserved if we do not add countable sets of ordinals.

Clearly, K codes stationary sets implies K has 2 models of power .But it is a stronger evidence of nonstructure in several respects. First, theexistence of many models is preserved under any forcing extension that doesnot add bounded subsets of and does not destroy the stationarity of subsetsof . Secondly, the existence of many models on a proper class of cardinals isnot such a strong requirement; for example, a multidimensional (unboundedin the nomenclature of [2]) theory has 2 models of power whenever = . However, the class of models of a rst order theory codes stationarysets only if T is not superstable or has the dimensional order property or has

the omitting types order property.6.1 Denition.

i) A representation of a model M with power (with regular) is anincreasing chain M = M i : i < of K -substructures of M such thateach M i has cardinality less than and M = M .

ii) The representation is proper if M | M implies M = M |.

We showed in Proposition 2.6 that if is a regular cardinal greater than 1(K ) and K satises the -Lowenheim-Skolem property then each model

of power has a representation. We will not however have to invoke theLowenheim-Skolem property in our main argument because we analyze mod-els that are constructed with a representation. Using Axiom A3, it is easyto perturb any given representation into a proper representation.

We now show how to dene invariant functions in our context. For therst version we need an additional axiom.

20


21/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

6.2 Axiom Ch5. For every K -continuous chain M = M i , f i,j : i, j < and every unbounded X , a K -structure M is canonically prime over(M, f ) if and only if M is canonically prime over (M |X, f |X ).

We refer to this axiom by saying that canonically prime models behaveon subsequences.

6.3 Denition. Let M be a representation of M . Let sm(M, M ) denote theset of limit ordinals < such that for some X unbounded in a canonicallyprime model N over M |X is a K -submodel of M .

6.4 Lemma. If M and N are representations of M and axiom Ch5 holds

then sm(M, M ) = sm( N, M ) modulo the closed unbounded lter on .

Proof. Since |M | is regular, there is a cub C on such that every C is a limit ordinal and M | = N | for C . Now we claim that for C , sm(M, M ) if and only if sm(N, M ). To see this choose an increasingsequence Li alternately from M and N . Since cpr behaves on subsequencesthe canonically prime model over the common subsequence of L and M is aK -submodel of M if and only if the canonically prime model over L is andsimilarly for the common subsequence of L and N . Thus, sm(M, M ) if and only if sm(N, M ).

This lemma justies the following denition.6.5 Denition. Denote the equivalence class modulo cub( ) of sm(M, M )for some (any) representation M of M by sm(M ). We call sm(M ) the smooth-ness set of M .

We now will describe a second way to assign invariants to models. Thisapproach avoids the reliance on Axiom Ch5 at the cost of complicating (butnot increasing the strength of) the set theory. Recall from Section 5 the idealID(C ) assigned to a family of sets C = {C : S }. Fix for the followingdenition and arguments subsets S, S 1, S 2 satisfying 2 b,,,R (S, S 1, S 2). Wedene a second invariant function with C as a parameter. It distinguishesmodels modulo ID(C ).

6.6 Denition. Fix a subset S of and C = {C : S }. Let N be arepresentation of M . sm1(N, C , M ) is the set of S such that

i) for every nacc[C ], N =


22/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

ii) N |C is K -continuous,iii) there is an N canonically prime over N | nacc[C ] that can be K -

embedded into M over N | nacc[C ].

We are entitled to choose an N canonically prime over N | nacc[C ] in con-dition iii) because condition i) guarantees that N | nacc[C ] is K -continuous.

6.7 Lemma. If M and N are proper representations of M then

sm1(N, C , M ) = sm 1(M, C , M ) modulo FIL( C ).

Proof. Let X 1 denote sm 1(N, C , M ) and X 2 denote sm 1(M, C , M ).Without loss of generality we can assume the universe of M is . There isa cub C containing only limit ordinals such that for C , =


23/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

i) Player I chooses a model L in K of power less than that is a properK -extension of all the structures P for < . If is a limit ordinalless than , L must be chosen canonically prime over P : < .

ii) Player II chooses a model P in K of power less than that is a K -extension of L .

Any player who is unable to make a legal move loses. Player I wins thegame if there is a model P K that extends each P for < but thesequence P i : i is not essentially K -continuous.

In order to establish that nonsmoothness implies a winning strategy for

Player I we need to consider certain properties of double chains. We introducehere some notation and axioms concerning this kind of diagram.

7.2 Denition. i) M = M 0, M 1 = { M 0i , M 1i : i < } is a doublechain if each M 0i M 1i and M 0, M 1 are K -increasing chains. Wesay M is (separately) ( K )-continuous if each of M 0 and M 1 is (K )-continuous.

ii) M is a free double chain if for each i < j < , M 0 j M 0i M 1i inside M 1 j +1 .

iii) M = M 0, M 1, = { M 0i , M 1 j : i + 1 , j < } is a K -continuous

augmented double chain inside N if i < implies M 0i M

1i , andM 0, M 1 are increasing K -continuous chains inside N .

iv) An augmented double chain is free inside N if for each i < ,

M 1i M 0i

M 0+1 inside N.

We extend the existence axiom Ch2 for a prime model over a chain toassert the compatibility of the prime models guaranteed for each sequence ina double chain.

7.3 Axioms concerning double chains.

DC1 If M is an essentially K -continuous free double chain and M 1 is canon-ically prime over M 1 then there is an M 0 that is canonically prime overM 0 such that M 0 and M 1 are compatible over M 0.

23


24/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

DC2 If M is an essentially K -continuous free augmented double chain of length in M then there is an N with M N and an M 1 N suchthat

M 1 M 0

M 0+1 inside N

and the chain M 1 {M 1 } is essentially K -continuous.

We will refer to versions of these axioms for chains of restricted length;we may denote the variant of the axiom for chains of length less than asDCi(< ).

Note that it would be strictly stronger in DC2 to assert that M 1 is canon-ically prime over M 1 since under DC2 as stated the canonically prime modelover M 1 inside M 1 need not contain M 0 .

Since we are going to use these axioms to establish smoothness we indicatesome relationships between the properties. K is (< , ) smooth means thatevery K -continuous chain of conality has a single compatibility class overit necessarily there will be a canonically prime model in that class. DC1would hold if there were many compatibility classes over a chain but each hada canonically prime model (i.e. K is semismooth). In particular it holds at if K is (< , )-smooth (sometimes read smooth at ). Thus, the followinglemma is easy.

7.4 Lemma. If K is an adequate class that is (< , )-smooth then Ksatises DC1 for chains of conality .

Now we come to the main result of this section.

7.5 Lemma. Let K be an adequate class that is (< , )-bounded and sup-pose K is (< , < )-smooth but not (< , )-smooth. Suppose further that K satises DC1 and DC2 and > K is K -inaccessible. Then Player I hasa winning strategy for Game 1 (, ).

Proof. Since K is (< , < ) smooth, we can choose a counterexampleN = N i : i < that is essentially K -continuous. Then N is boundedby two models N and N that are incompatible over N . If there exist M and M canonically prime over N and embeddible in N and N respectively,Axiom Ch4 requires that M and M are compatible. But then so are N andN . From this contradiction we conclude without loss of generality that each

24


25/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

N i N but that no canonically prime model over N can be K -embeddedinto N . That is, K is not semismooth (4.4). Now Players I and II willchoose models Li : 1 i < and P i : i < for a play of Game 1.

We describe a winning strategy for Player I. The construction requiresauxiliary models P i , N i , and Li and isomorphisms i : Li Li . They willsatisfy the following conditions.

i) P and L are essentially K -continuous sequences and the i are anincreasing sequence of maps.

ii) P i N i N inside N i+1 .

iii) L i+1 is prime over P i N i+1 inside N i+1 .

iv) i is an isomorphism between L i and Li mapping P j onto P j for j < i .

v) The N i form an essentially K -continuous sequence with N i N i .

Let L0 = N 0. Each successor stage is easy. Player II has chosen P i K


26/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

7.6 Remarks. i) Instead of assuming Axiom Ch4 (part of the deni-tion of adequate) we could have assumed that K was not (< , )-semismooth.

ii) It is tempting to think that by choosing the minimal length of a se-quence witnessing nonsmoothness, we could apply Lemma 7.4 andavoid assuming DC1. However, DC1 is applied for chains of length so this ploy is ineffective.

iii) Why is Li+1 a proper extension of P i? Since Li+1 was chosen as anamalgam of P i and N , this is immediate if we assume the disjointness

axiom (C7). To avoid this hypothesis we can demand that each modelin the construction have cardinality > K and so not be maximal (byLemma 3.5). That is why we assumed > K .

iv) Note that DC1 is used to derive the contradiction at the end of theproof; DC2 is used to pass through limit stages of the construction.Thus in the important case when = we have

7.7 Lemma. Let K be an adequate class that is not (< , ) smooth. Sup-pose that K satises DC1. Then Player I has a winning strategy for Game1 (, ).

The choice of L i according to the winning strategy of Player I dependsonly on the sequence L j , P j , for j < i (not, for example on some guessabout the future of the game).

In the remainder of this section section we consider a third axiom DC3on double chains. The following axiom bears the same relation to DC2 thatC5 bears to C2.

7.8 Weak uniqueness for prime models over double chains.

DC3 Suppose that M and N are essentially K -continuous augmented doublechains that are free in M and N respectively and f is an isomorphismfrom M onto N . Suppose also that M 1 M 0 M

0+1 inside M and N 1 N 0

N 0+1 inside N . Then there is an M K and K -embeddings h0 of M and h1 of N into M with h1 f = h0|M .

26


27/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

Just as Lemma I.1.8 of [3] rephrased the weak uniqueness axiom for amal-gamation over vees we can reformulate DC3 as follows.

7.9 Lemma. Assume DC2 and DC3. Suppose that M and N are essentially K -continuous augmented double chains that are free in M and N respectively and f is an isomorphism from M onto N . Suppose also that for some M 1 N , M 1 M 0 M

0+1 inside M .

Then there exist a model N and an isomorphism h : M N such that h f and h(M 1 ) N 0 N

0+1 inside N .

1 Question. If DC2 and DC3 hold and K is not smooth does Player I havea winning strategy for Game 1?

8 Nonsmoothness implies many models

We show in this section that if the class K is not smooth then K codesstationary sets. These results involve several tradeoffs between set theory andmodel theory. The main result is proved in ZFC. Here there are two versions;one uses 2 a and requires the hypothesis that cpr behaves on subsequences.The second uses 2 b and replaces cpr behaves on subsequences with strongerhypotheses concerning the closure of K under unions of chains. By workingin L we can reduce our assumptions on which chains are bounded in bothcases.

8.1 Invariants modulo the CUB lter

In this subsection we show if K is not smooth then for many we can codestationary subsets of by assigning invariants in the cub lter by the functionsm. Our general strategy for constructing many models is this. We builda model M W for each of a family of 2 stationary subsets W of S that arepairwise distinct modulo the cub lter. The key point of the construction is

that, modulo cub( ), we can recover W from M W as sm(M W ).We need one more piece of notation.

8.1 Notation. Fix a square sequence C : S . Suppose Player I hasa winning strategy for Game 1( , ). In the proof of Theorem 8.2 and somesimilar later results we dene a K -increasing sequence M . We describe here

27


28/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

what is meant by saying a certain M is chosen by playing Player Is strategyon M |C .Let c : < 0 enumerate C . We regard M |C as two sequences

L, P by setting for any ordinal + n with a limit ordinal and n < :

L+ n is M c +2 n .

P + n is M c +2 n +1We say M for = or C is chosen by Player Is winning strategy onM |C if the sequence L, P associated with C { } is

i) an initial segment of a play of Game 1 (, ) andii) Player Is moves in this game follow his winning strategy.

Here is the technical version of the main result with the parameters andreliance on the axioms enunciated in Section 2 and 4 stated explicitly.

Although the assumption that is K -inaccessible is weaker than theassumption that K satises the -Lowenheim Skolem property it plays therole of the Lowenheim-Skolem property in the following construction. Weassume > K and apply Corollary 3.5 to avoid the appearance of maximalmodels in the construction.

8.2 Theorem. Fix regular cardinals < . Suppose the following condi-tions hold.

i) K is an adequate class.

ii) Player I has a winning strategy for Game 1 (, ).

iii) is a K -inaccessible cardinal, for some stationary S , 2 a, (S )holds and > K ,

iv) K is (< , < )-bounded.

v) cpr behaves on subsequences (Ch5).

vi) K is (< , )-closed.

Then for any stationary W S there is a model M W and a representation M W with W sm(M W , M W ) and S + W sm(M W , M W ).

28


29/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

Proof . Fix S +

and C

= C i : i S +

to witness2 a

, (S ). Without loss of generality, 0 C . Fix also a stationary subset W of S . For each < wedene a model M W . The model M W =


30/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

a winning strategy on M |C . So, since cpr behaves on subsequences, neitheris the canonically prime model on M |A for any A unbounded in . Thus sm(M, M W ). All other limit S + are in sm(M, M W ) and we nish.

The next theorem rephrases Theorem 8.2 to avoid technicalities. It showsthat reasonable K that are not smooth have many models in all sufficientlylarge successor cardinals. In fact we have the stronger result that K codesstationary subsets of such cardinals.

Theorem 8.3 (ZFC). Let K be an adequate class and suppose that K sat-ises DC1, DC2 and cpr behaves on subsequences (Ch5). Suppose there exist , 1 with < 1 such that K is not (1, )-smooth. Then for every K -inaccessible > sup( K , 1) such that

i) is a successor of a regular cardinal,

ii) K is (< , < )-bounded,

iii) K is (< , )-closed,

K has 2 models in power .

Proof . K is not (< 1, )-smooth trivially implies K is not (< , )-smooth.We assumed DC1 and DC2 so by Lemma 7.5, Player I has a winning strategyin Game 1 (, ). By Fact 5.5, there is a stationary S C () such that2 a

, holds. The result now follows from the previous theorem, choosing 2 stationary sets W S that are distinct modulo the cub ideal. (In moredetail, let V and W be two of these stationary sets. Then sm( M W , M W )sm(M V , M V ) W V . Thus by Lemma 6.4, ( M W ) (M V ).)

8.4 Remark. If we add the requirement 1 (K ) = we can deduce that isK -inaccessible from Lemma 2.11. Applying Lemma 7.7 we could omit DC2from the hypothesis list if = .

The assumption in Theorem 8.2 that K is (< , < )-bounded is usedonly for the construction of the M for S + . We can weaken this modeltheoretic hypothesis at the cost of strengthening the set theoretic hypothesis.We noted in Fact 5.7 that the set theoretic hypotheses of the next theoremfollow from V=L.

30


31/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

8.5 Theorem. Suppose2 a

, (S ) holds for an S that does not reect. Then the hypothesis that K is (< , < )-bounded can be deleted from Theorem 8.2.

Proof . The only use of this hypothesis is the construction of M for S + .In this case we make our construction more uniform by demanding for S +that M is canonically prime over M : < . If S does not reect in then there is a club C with C S = . By induction, for each C , wehave M canonically prime over M : < . Thus the chain M : C is K -continuous and we can choose M canonically prime over it. By AxiomCh5, M is canonically prime over M : < as required.

Recall that K is (< , [, ])-bounded if every chain with conality be-tween and inclusive of models that each have cardinality < is bounded.In a number of the examples we have adduced (4.3), K is (< , [, < ])bounded for appropriate . Thus the model theoretic hypothesis of the fol-lowing theorem is reasonable. The existence of stationary sets that do notreect in of small conality is provable if V = L and is consistent withlarge cardinal hypotheses.

8.6 Theorem. Fix , < . Suppose 2 a, (S ) holds for some stationary subset S of that satises:

if S reects in then cf() .

Then the hypothesis that K is (< , < )-bounded can be replaced in Theo-rem 8.2 by assuming that K is (< , [, ])-bounded.

Proof . Again we must construct M for S + . If S reects in , cf() so M | is bounded. Since is K -inaccessible (Denition 2.10) we can chooseM to bound M | and with |M | < . If S does not reect in , write asa limit of ordinals i of smaller conality. By induction M i is canonicallyprime over M | i and taking M canonically prime over the M i (using AxiomCh5) we nish.

8.2 Invariants modulo ID(C )In this subsection we show if K is not smooth then for many we cancode stationary subsets of by assigning invariants modulo the ideal ID( C )

31


32/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

by the function sm 1. We now replace the hypothesis that cpr behaves onsubseqences by assuming K is weakly (< , )-closed for certain ; we use 2 brather than 2 a but these have the same set theoretic strength.

Again we rst give the technical version of the main result with the pa-rameters and reliance on the axioms enunciated in Section 2 and Section 4stated explicitly. We need to vary the meaning of the phrase, a winningstrategy against M |C by changing the game played on C .

8.7 Notation. Fix a square sequence C : S . Suppose Player I hasa winning strategy for Game 1( , ). We modify our earlier notion of whatis meant by saying a certain M is chosen by playing Player Is strategy on

M |C to a notion that is appropriate for the proof of the next theorem.Let c : < 0 enumerate C . Denote C { +1 : nacc[C ]} by C .

We attach to M |C two sequences Li : i < otp( C ) and P i : i < otp( C )by setting

P = M cL = M c +1 if nacc[C ]

L = M c if acc[C ]

We say M for = or C is chosen by Player Is winning strategy onM |C if the sequence L, P associated with C { } is

i) an initial segment of a play of Game 1 (, ) and

ii) Player Is moves in this game follow his winning strategy.

In dening this play of the game we have restrained Player IIs movessomewhat (as P = L if acc[C ]. But this just makes it even easier forPlayer I to play his winning strategy.

8.8 Theorem. Fix regular cardinals ,,R, with = , < , R ,and K , + < . Suppose the following.

i) K is an adequate class.

ii) Player I has a winning strategy for Game 1 (, ).

iii) is a K -inaccessible cardinal and for some S, S 1, S 2 and C =C : S , 2 b,,,R (S, S 1, S 2) is witnessed by C . Let S 0 = S 1 S 2

and C = C |S 0.

32


33/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

iv) K is (< , [R, ])-bounded.v) K is (< , )-closed.

vi) K is (< , )-closed.

Then there is a model M with sm1(M, C ) = ( S 2/ FIL( C )) .

Proof . For each < we dene M as follows. We will guarantee

sm1(M,M,C ) = S 2 modulo FIL( C ).

The construction proceeds by induction. There are a number of cases de-pending on whether S 1, S 2 etc. Choose M by the rst of the followingconditions that applies to .

Case I. (S 1 S 1 acc[C ] S 1 { + 1 : nacc[C ]}: Chooseany S with C (if is in S let = ). Apply PlayerIs winning strategy for Game 1 to M |C to choose M . Again thecoherence conditions in the denition of the square sequence guaranteethat the particular choice of is immaterial. Note that for not inS 1, the denition of Player I winning the game guarantees that M iscanonically prime over C .

Case II. For any successor ordinals not yet covered, say = + 1, chooseM as a K -extension of M . Thus in the rest of the cases we mayassume is a limit ordinal.

Case III. S 0 nacc[C ]: Then cf() = so by the fth hypothesiswe may choose M =


34/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

Case VI. is a limit ordinal and S : Then Denition 5.9 guaranteescf() R. Choose M as a K -extension of M for each < by(< , R)-boundedness.

Now we show that sm1(M, C , M ) intersects the stationary set S 0 in S 2.If S 1 the play of Game 1 guarantees that there is no M such thatM | nacc[C ] {M } is essentially K continuous. Thus by condition iii) of Denition 6.6 sm1(M, C , M ) But if S 2, is in neither S 1 nor S 0 nacc[C ] (since all elements of the second set have conality andcf() = ). Thus, by Case IV of the construction M is canonically primeover M | nacc[C ] and since K is (< , )-closed M M . Condition ii) in the

denition of sm1 is guaranteed since M |C is K -continuous (as C S 1 = and all points of non K -continuity are in S 1). Condition i), M = K and such that

i) is K -inaccessible,ii) is a succcessor of a regular cardinal and + is less than ,

iii) and K is

(a) (< , < )-bounded,(b) weakly (< , )-closed,(c) (< , )-closed.

If K has fewer than 2 models with cardinality and + < then K is(< , < )-smooth.Proof . Fix with + < such that K is not (< , )-smooth. By Lemma 7.5Player I has a winning strategy in Game 1 ( , ). By Theorem 5.12i), 2 ,,, 0holds. Now a very slight variant of the proof of 8.8 shows there exist 2models M i with the sm 1(M i ) distinct modulo FIL( C ). (Namely, since K is

34


35/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

(< , < )-bounded we do not need to worry about the conality of inCase VI.)

This shows that if K is not smooth at some then there will be manymodels in power for many > satisfying certain model theoretic hy-potheses.

If V=L we can waive the boundedness hypothesis.

Theorem 8.11 (V=L). Let > K and not weakly compact beK -inaccessible.Suppose the adequate class K satises DC1, DC2 and is

i) weakly (< , )-closed, for some ,

ii) (< , )-closed.

If K has fewer than 2 models in power and + < then K is (< , )smooth.

Proof . Since V=L, Lemma 5.12 implies 2 b,,, holds. The result now followsfrom Theorem 8.8 taking as and as R. We observed after Denition 4.2that ( < , )-closed implies (< , )-bounded.

We have shown in this section that each of the variants of 2 discussed in

Section 5 suffice to show that a nonsmooth K codes stationary subsets of for many .

9 The Monster Model

We begin this section by recapitulating the assumptions that we will make indeveloping the structure theory. Then we show that under these assumptionswe can prove the existence of a monster model and prove the equivalencebetween homogenous-universal and saturated.

9.1 Convention. = K is a cardinal satisfying Axioms S0 and S1 statedin Section 2. We assume throughout this section.

9.2 Notation. D denotes a compatibility class of K .

35


36/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

9.3 Denition. We say K satises the joint embedding property if for anyA, B K there is a C K and K -embeddings of A and B into C .At this point we extend the axioms from Section 2 by adding the smooth-

ness hypothesis we have justied in the last few sections. We have shownthat under reasonable set theoretic hypotheses the failure of these smooth-ness conditions allows us to code stationary subsets of for a proper class of .

9.4 Assumptions. We assume in this section axiom groups A and C,Axioms D1 and D2 from group D, (all from [3]), axioms Ch1, Ch2,Ch4, L1, S0 and S1 from Section 2 and the following smoothnessconditions. K is (< , 1(K )) smooth and (< , > 1(K )) fullysmooth. Thus, we assume K is (< , )-smooth. Finally we as-sume that K satises the joint embedding property. We call sucha class fully adequate.

The assumption of the joint embedding property is purely a notationalconvenience. We have just restricted from K to a single compatibility classin K . Thus, the notions that in [18] are written, e.g., ( D, )-homogeneoushere become (K , )-homogeneous with no loss in generality. We could in factdrop the K altogether.

This class is called fully adequate because (modulo V = L, see Con-

clusion 9.8 ii)) any fully adequate class either has a unique homogeneous-universal model or many models. We did not include the axiom Ch5, cpr be-haves on subsequences, since we can rely on Theorem 8.8 to obtain smooth-ness without that hypothesis. We did assume that K is (< , ) closed forlarge so the other hypotheses of that theorem are fullled.

The following observation allows us to perform the required constructions.

9.5 Lemma. If K satises the -L owenheim-Skolem property, 1(K ),M i : i < is a chain of models inside M with each |M i | < and cf() <

then there is a canonically prime model M over M i : i < with |M | < .

Proof . If cf() K , M = i


37/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

i) for every N 0 M and every N 1 with N 0 N 1 K and |N 1 | thereis a K -embedding of N 1 into M over N 0;

ii) every N K with cardinality less than can be K -embedded in M .

There is a certain entymological sense in labeling this notion a kind of saturation. The argument for homogeneity is that naming K describes thelevel of universality and we need only indicate the homogeneity again. In anycase Shelah established this convention almost twenty years ago in [13]. Weidentify this algebraic notion with a type realization notion in Theorem 9.12.

9.7 Lemma. Suppose K is (< , < )-bounded and (< , )-closed.i) If K , is regular, K -inaccessible and satises


38/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

stage , one takes the canonically prime model M over an initial segment of the sequence of submodels of M and embeds it as a submodel N of N . Inorder to continue the induction we must know M is a strong submodel of M and this is guaranteed by smoothness.

9.8 Conclusion. i) For any fully adequate K that is ( < , < )-bounded there is (in some cardinal ) a unique (K , )-homogeneousmodel.

ii) If V = L, we can omit the boundedness hypothesis (by Theorem 8.11.

iii) We will call unique (K , )-homogeneous model, M , the monster model.From now on all sets and models are contained in M .

9.9 Remark. This formalism encompasses the constructions by Hrushovski[7] of 0-categorical stable pseudoplanes. An underlying (but unexpressed)theme of his constructions is to generalize the Fraisse-J onsson constructionby a weakening of homogeneity. He does not demand that any isomorphismof nite substructures extend to an automorphism but only an isomorphismof submodels that are strong substructures (where strong varies slightlywith the construction). This is exactly encapsuled in the formalism here.This viewpoint is pursued in [1]. Of course in Hrushovskis case the real

point is the delicate proof of amalgamation and is trivially K -inaccessible.We assume amalgamation and worry about inacessibility and smoothness inlarger cardinals.

9.10 Denition. i) The type of a over A (for a, A M ) is the orbitof a under the automorphisms of M that x A pointwise. We write p = tp(a; A) for this orbit.

ii) p is a k-type if lg(a) = k.

iii) The type of B over A (for B, A M ) is the type of some (xed)enumeration of B .

iv) S k (A) denotes the collection of all k-types over A.We will often write p, q, etc. for types. This notion is really of interest

only when lg(a) ; despite the suggestive notation, k-type, we may dealwith types of innite length. We will write S (A) to mean S k (A) for somek < whose exact identity is not important at the moment.

38


39/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

9.11 Denition. i) The type p S (A) is realized by c N with A N M if c is member of the orbit p.

ii) N M is (K , )-saturated if for every M N with |M | < , every1-type over M is realized in N .

9.12 Theorem. Let K be K -inaccessible. Then M is (K , )-saturated if and only if M is (K , )-homogeneous.

The proof follows that of Proposition 2.4 of [10] line for line with oneexception. If we consider those stages in the construction where cf( ) < 1(K ), we cannot form M just by taking unions. However, any canoni-

cally prime model over the initial segment of the construction will work bysmoothness and Lemma 9.5.

10 Problems

2 Question. Can one give more precise information on the class of cardinalsin which an adequate class K has a model.

In 1.5 we gave a denition of on the class of 1-saturated models of the theory T = REI under which this class is not ( < 1, )-smooth. Thus,

by our main result T has the maximum number of 1-saturated models inpower (if e.g. = + ).

3 Question. Dene on the class of 1-saturated models of an strictly stable with didop [14] (or perhaps if K is not nitely controlled in the senseof [8]) so the class is not smooth.

There are strictly stable theories with fewer than the maximal number of 1-saturated models in most . See: Example 8 page 8 of [2].

4 Question. Formalize the notion of coding a stationary set to encompassthe examples we have described and clarify the distinctions described at thebeginning of Section 6.

We have developed this paper entirely in the context of cpr models. In aforthcoming work we replace this fundamental concept by axioms for winninggames similar to Game 1 ( , ) and establish smoothness in that context. Thecost is stronger set theory (but V=L suffices).

39


40/41

360

r e v

i s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

References[1] J.T. Baldwin. New families of stable structures. technical report: Notes

on Hrushovskis construction; intended for lectures in Beijing.

[2] J.T. Baldwin. Fundamentals of Stability Theory . Springer-Verlag, 1988.

[3] J.T. Baldwin and S. Shelah. The primal framework: I. Annals of Pureand Applied Logic, 46:235264, 1990.

[4] A. Beller and A. Litman. A strengthening of Jensens 2 principles.Journal of Symbolic Logic, 45:251264, 1980.

[5] Keith Devlin. Constructibility . Springer-Verlag, 1984.

[6] R. Fraisse. Sur quelques classications des systemes de relations. Publ.Sci. Univ. Algeria Ser. A , 1:35182, 1954.

[7] E. Hrushovski. A stable 0-categorical pseudoplane. preprint, 1988.

[8] E. Hrushovski. Finitely based theories. The Journal of Symbolic Logic,54:221225, 1989.

[9] B. Jonsson. Homogeneous universal relational systems. Mathematica Scandinavica , 8:137142, 1960.

[10] Michael Makkai and Saharon Shelah. Categoricity of theories in l ,with a compact cardinal. Annals of Pure and Applied Logic, 47:4197,1990.

[11] M. Morley and R.L. Vaught. Homogeneous universal models. Mathe-matica Scandinavica , 11:3757, 1962.

[12] S. Shelah. Reection of stationary sets and successor of singulars.preprint 351: to appear in Archive fur Mat. Log.

[13] S. Shelah. Finite diagrams stable in power. Annals of Mathematical Logic, 2:69118, 1970.

[14] S. Shelah. The spectrum problem I, -saturated models the main gap.Israel Journal of Mathematics , 43:324356, 1982.

40


41/41

vi s

i o n :

1989 -

01 -

14

m o

d i f i

e d

: 1 9 9 5

- 0 9

- 0 4

[15] S. Shelah. Classication of rst order theories which have a structuretheory. Bulletin of A.M.S., 12:227232, 1985.

[16] S. Shelah. Remarks on squares. In Around Classication Theory of Models. Springer-Verlag, 1986. Springer Lecture Notes 1182.

[17] S. Shelah. Nonelementary classes II. In J. Baldwin, editor, Classication Theory, Chicago 1985 . Springer-Verlag, 1987. Springer Lecture Notes1292.

[18] S. Shelah. Universal classes: Part 1. In J. Baldwin, editor, Classication Theory, Chicago 1985 , pages 264419. Springer-Verlag, 1987. SpringerLecture Notes 1292.

[19] S. Shelah. Universal classes: Part 2. preprint, 1989.

[20] S. Shelah. Classication Theory and the Number of NonisomorphicModels. North-Holland, 1991. second edition.

41

J. Baldwin and S. Shelah- The Primal Framework II: Smoothness

Documents

Transcript of J. Baldwin and S. Shelah- The Primal Framework II: Smoothness