James Cummings and Saharon Shelah- Cardinal invariants above the continuum

8/3/2019 James Cummings and Saharon Shelah- Cardinal invariants above the continuum

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Cardinal invariants above the continuum

James Cummings

Hebrew University of Jerusalem

[email protected]

Saharon Shelah

Hebrew University of [email protected]

October 6, 2003

Abstract

We prove some consistency results about b() and d(), which arenatural generalisations of the cardinal invariants of the continuum b

and d. We also define invariants bcl() and dcl(), and prove thatalmost always b() = bcl() and d() = dcl()

1 Introduction

The cardinal invariants of the continuum have been extensively studied. They

are cardinals, typically between 1 and 2, whose values give structural in-

formation about . The survey paper [2] contains a wealth of information

about these cardinals.

In this paper we study some natural generalisations to higher cardinals.Specifically, for regular, we define cardinals b() and d() which generalise

the well-known invariants of the continuum b and d.

Supported by a Postdoctoral Fellowship at the Hebrew University.Partially supported by the Basic Research Fund of the Israel Academy of Science.

Paper number 541.

1


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For a fixed value of, we will prove that there are some simple constraintson the triple of cardinals (b(), d(), 2). We will also prove that any triple

of cardinals obeying these constraints can be realised.

We will then prove that there is essentially no correlation between the

values of the triple (b(), d(), 2) for different values of, except the obvious

one that 2 is non-decreasing. This generalises Eastons celebrated

theorem (see [3]) on the possible behaviours of 2; since his model was

built using Cohen forcing, one can show that in that model b() = + and

d() = 2 for every .

b() and d() are defined using the co-bounded filter on , and for >

we can replace the co-bounded filter by the club filter to get invariants bcl()

and dcl(). We finish the paper by proving that these invariants are essen-

tially the same as those defined using the co-bounded filter.

Some investigations have been made into generalising the other cardinal

invariants of the continuum, for example in [7] Zapletal considers s() which

is a generalised version of the splitting number s. His work has a different

flavour to ours, since getting s() > + needs large cardinals.

We are indebted to the referee and Jindrich Zapletal for pointing out a

serious problem with the first version of this paper.

2 Definitions and elementary facts

It will be convenient to define the notions of bounding number and domi-

nating number in quite a general setting. To avoid some trivialities, all par-

tial orderings P mentioned in this paper (with the exception of the notionsof forcing) will be assumed to have the property that p P q P p


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d

(P) is the least cardinality of a dominating subset ofP.The next lemma collects a few elementary facts about the cardinals b(P)

and d(P).

Lemma 1: Let P be a partial ordering, and suppose that = b(P) and = d(P) are infinite. Then

= cf() cf() |P|.

Proof: To show that is regular, suppose for a contradiction that cf () < .

Let B be an unbounded family of cardinality , and write B =


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Let D P be dominating. If |D| < then

{ y | (, y) D } = ,and this is impossible, so that d(P) . On the other hand GCH holds andcf() , so that |P| =


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Let V[G] be a generic extension ofV such that every set of ordinals ofsize less than in V[G] is covered by a set of size less than in V.

Then V[G] b(P) = .

Let V[G] be a generic extension of V such that every set of ordinals

of size less than in V[G] is covered by a set of size less than in V.

Then V[G] d(P) = .

Proof: We do the first part, the second is very similar. The hypothesis

implies that is a cardinal in V[G], and since B is unbounded is upwards

absolute from V to V[G] it is clear that V[G] b

(P) . Suppose for acontradiction that we have C in V[G] unbounded with V[G] |C| < . Byour hypothesis there is D V such that C D and V |D| < , but now

D is unbounded contradicting the definition of .

With these preliminaries out of the way, we can define the cardinals which

will concern us in this paper.

Definition 3: Let be a regular cardinal.

1. Iff, g then f f() < g().

2. b() =def b((,


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d

() 2

.

cf(2) > .

Proof: The first claim follows from the following trivial fact.

Fact 1: Let { f | < } . Then there is a function f such that

< f < f.

Proof: Define f() = sup { f() + 1 | < }. Then f() > f() for

< < .

The next three claims follow easily from our general results on b(P) andd(P), and the last is just Konigs well-known theorem on cardinal exponen-tiation.

We will prove that these are essentially the only restrictions provable in

ZFC. One could view this as a refinement of Eastons classical result (see [3])

on 2.

3 Hechler forcing

In this section we show how to force that certain posets can be cofinally

embedded in (,


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Definition 4: Let be regular. D() is the notion of forcing whose con-ditions are pairs (s, F) with s


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3. Let f

V, and let (t, F) be an arbitrary condition. Let us defineF : (F() f()) + 1, then (t, F) refines (t, F) and forces that

f() < fG() for all dom(t).

If = cf() > =


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(b) For all b dom(p), p(b) = (t,

F) where t I().

iii. p

(Q/b)Pb

H()

I().

We define D(,Q) = Ptop, and verify that this forcing does what weclaimed. The verification is broken up into a series of claims.

Claim 1: D(,Q) is -closed.

Proof: Let < and let p : < be a descending -sequence of con-

ditions. We will define a new condition p with dom(p) = dom(p). For

each b dom(p) let p(b) = (t(b), F(b)).

Let p(b) = (t,

F) where t =

{ t(b) | b dom(p) } and

F(b) is a Pb-name for the pointwise supremum of { F(b) | b dom(p) }. Then it it iseasy to check that p is a condition and is a lower bound for p : <

Claim 2: D(,Q) is +-c.c.

Proof: Let p : < + be a family of conditions. Since =


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Claim 3: If c < a then Pc is a complete subordering of Pa, and the mapp p Q/c is a projection from Pa to Pc.

Proof: This is routine.

IfG is D(,Q)-generic, then for each a Q we can define faG V[G]

by faG =

{ t(a)0 | t G }. It is these functions that will give us a cofinal

embedding ofQ into ( V[G],


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Now it is easy to extend p2 to a condition which forces fa

G() > fb

G().

Claim 6: The map a faG embeds Q cofinally into (,


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4 Controlling the invariants at a fixed cardi-nal

In this section we show how to force that the triple (b(), d(), 2) can be

anything reasonable for a fixed value of .

Theorem 2: Let = .

Then there is a forcing M(,,,) such that in the generic extensionb() = , d() = and 2 = .

Proof: In V define Q = P(, ), as in lemma 2. We know that V b(Q) = and V d(Q) = . Fix Q a cofinal wellfounded subset ofQ, and then definea new well-founded poset R as follows.

Definition 5: The elements of R are pairs (p,i) where either i = 0 andp or i = 1 and p Q. (p,i) (q, j) iff i = j = 0 and p q in ,i = j = 1 and p q in Q, or i = 0 and j = 1.

Now we set M(,,,) = D(,R). It is routine to use the closure andchain condition to argue that M makes 2 = . Since R contains a cofinalcopy ofQ, it is also easy to see that M forces b() = and d() = .

5 A first attempt at the main theorem

We now aim to put together the basic modules as described in the previous

section, so as to control the function (b(), d(), 2) for all regular .A naive first attempt would be to imitate Eastons construction from [3]; this

almost works, and will lead us towards the right construction.

Let us briefly recall the statement and proof of Eastons theorem on the

behaviour of 2.

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Lemma 8 (Eastons lemma): IfP is -c.c. and Q is -closed then P is -c.c. in VQ and Q is -dense in VP. In particular


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absurd as P is +

-c.c. and +

< < . A very similar argument will workin case is a non-Mahlo inaccessible.

Suppose that we replace Add(, F()) by P() = M(, (), (), ()),where ((), (), ()) is a function obeying the constraints given by

Lemma 6. Let P be the Easton product of the P(). Then exactly as inthe proof of Eastons theorem it will follow that P preserves cardinals andcofinalities, and that 2 = () in VP.

Lemma 9: If is inaccessible or the successor of a regular cardinal then

b() = (), d() = () and 2 = () in VP.

Proof: For any , VP = VP. b() and d() have the right values

in VP() by design, and these values are not changed by -c.c. forcing. So

assuming P


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right not to take all P-names for members ofQ (as long as we take enoughnames that the set of their denotations is forced to be dense). For exampleifQ V we will only take names q for q Q, so P Q will just be P Q.

We will describe a kind of iteration which we call Easton tail iteration

in which at successor stages we choose iterands from V, but at limit stage

we choose Q in a different way; possibly Q / V, but we will arrange thingsso that the generic G factors at many places below and any final segment

ofG essentially determines Q. This idea comes from Magidor and Shelahspaper [5].

We assume for simplicity that in the ground model all limit cardinals

are singular or inaccessible. In the application that we intend this is no

restriction, as the ground model will obey GCH.

Definition 6: A forcing iteration P with iterands Q : + 1 < is anEaston tail iteration iff

1. The iteration has Easton support, that is to say a direct limit is taken

at inaccessible limit stages and an inverse limit elsewhere.

2. Q = 0 unless is a regular cardinal.

3. If is the successor of a regular cardinal then Q V.

4. For all regular , P+1 is +-c.c.

5. For a limit cardinal P is ++-c.c. if is singular, and +-c.c. if isinaccessible.

6. For all regular with + 1 < there exists an iteration P dense inP such that P+1 = P+1, and for with + 1 <

(a) P factors as P+1 P ( + 1, ).(b) If is inaccessible or the successor of a singular, and p P, then

p() is a name depending only on P ( + 1, ).

(c) P ( + 1, ) is +-closed.

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Clause 6 is of course the interesting one. It holds in a trivial way ifPis just a product with Easton supports. Clauses 6a and 6b should really beread together, as the factorisation in 6a only makes sense because 6b already

applies to < , and conversely 6b only makes sense once we have the

factorisation from 6a.

The following result shows that Easton tail iterations do not disturb the

universe too much.

Lemma 10: Let P be an Easton tail iteration. Then

1. For all regular < ,

ON VP

=

ON VP+1

.

2. P preserves all cardinals and cofinalities.

Proof: Exactly like Theorem 3.

In the next section we will see how to define a non-trivial Easton tail iter-

ation. If+ is the successor of a regular then it will suffice to choose Q+ Vas any +-closed and ++-c.c. forcing. The interesting (difficult) stages are

the ones where we have to cope with the other sorts of regular cardinal, herewe will have to maintain the hypotheses on the chain condition and factorisa-

tion properties of the iteration. It turns out that slightly different strategies

are appropriate for inaccessibles and successors of singulars.

7 The main theorem

Theorem 4: Let GCH hold. Let ((), (), ()) be a class function

from REG to CARD3, with + () = cf(()) cf(()) () ()

and cf(()) > for all .Then there exists a class forcing P, preserving all cardinals and cofi-nalities, such that in the generic extension b() = (), d() = () and

2 = () for all .

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Proof: We will define by induction on a sequence of Easton tail iterationsP, and then let take a direct limit to get a class forcing P. The proof thatP has the desired properties is exactly as in [3], so we will concentrate ondefining the P. As we define the P we will also define dense subsets Pintended to witness clause 6 in the definition of an Easton tail iteration.

Much of the combinatorics in this section is very similar to that in Section

3. Accordingly we have only sketched the proofs of some of the technical

assertions about closure and chain conditions.

The easiest case to cope with is that where we are looking at the successor

of a regular cardinal. So let be regular and assume that we have defined P+(which is equivalent to P+1 since we do trivial forcing at all points between and +), and P+ (which is equivalent to P

+1) for all .

Definition 7: Q+ = M(+, (+), (+), (+)) as defined in Section 4.P++1 = P+ Q+ , and P++1 = P

+ Q+ for .

It is now easy to check that this definition maintains the conditions

for being an Easton tail iteration. Since P+ is +-c.c. and we know that++ VP =

++ VP++1, we will get the desired behaviour at + in

VP

.Next we consider the case of a cardinal +, where is singular. Sup-

pose we have defined P+ (that is P) and P+ appropriately. Let R be awell-founded poset of cardinality () with b(R) = () and d(R) = (),as defined in Section 4. Let R+ be R with the addition of a maximal ele-ment top. We will define P+ Qa by induction on a R+, and then setP++1 = P+ Qtop. In the induction we will maintain the hypothesis that,for each < , P+ Qa can be factored as P+1 (P

( + 1, +) Qa).Let us now fix a, and suppose that we have defined P+ Qb for all b belowa in R+.

Definition 8: Let b < a, and let be a for a function from + to +. Then

is symmetric iff for all < , whenever G0 G1 and G0 G1 are two

generics for P+1 (P ( + 1, +) Qb), then G0G1 = G0G1.

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Of course the (technically illegal) quantification over generic objects inthis definition can be removed using the truth lemma, to see that the collec-

tion of symmetric names really is a set in V.

Definition 9: (p,q) is a condition in P+ Qa iff

1. p P+.

2. q is a function, dom(q) R+/a and |dom(q)| .

3. For each b dom(q), q(b) is a pair (s, F) where s


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Lemma 11: The function added by P+ Qtop at b R

+

eventually domi-nates all functions in

++ VP+Qb .

Proof: It suffices to show that if F is a P+ Qb-name for a function from+ to + then there is a symmetric name F such that F() F().

For each regular < we factor P+ Qb as P+1 (P (+1, +)Qb).In VP

(+1,+)Qb we may treat F as a P+1-name and define

G() = sup({ | p P+1 p F() = }).

Notice that we may also treat G as a P+ Qb-name, and that if < then G G. Now let F be the canonical name for a function such that

F() = sup


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the hypothesis that for each < there is a dense subset ofP

Qa whichfactorises as P+1 (P ( + 1, ) Qa ). Let us fix a, and suppose that we

have defined everything for all b below a.

Definition 12: (p, (, q)) is a condition in P Qa iff

1. p P.

2. < , is regular.

3. q is a function, dom(q) R+/a and |dom(q)| < .

4. For each b dom(q), q(b) is a pair (s, F) where s


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8 VariationsIn this appendix we discuss the invariants that arise if we work with the club

filter in place of the co-bounded filter. It turns out that this does not make

too much difference. All the results here are due to Shelah.

Definition 16: Let be regular.

1. Let f, g . f


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Now let = min(n .

Fix some > . For each n we know that Cn = , so that if we define

n = sup(Cn (+ 1)) then n is the largest point of Cn less than or equal

to . Notice that min(Cn (+ 1)) = min(Cn (n + 1)).

Since Cn+1 Cn, n+1 n, so that for all sufficiently large n (say

n N) we have n = for some fixed . We claim that g0() fN+1(),

which we will prove by building a chain of inequalities. Let us define

= min(CN (+ 1)) = min(CN (+ 1)).

Then

g0() gN() gN() < fN() < gN+1() < fN+1() fN+1(),

where the key point is that CN+1 and hence gN+1() < fN+1().

Now it is easy to manufacture a family of size dcl() which is dominating

with respect to


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for all and gn bounds Un modulo clubs. We choose clubs { Cn

f | f Un }such that

1. Cnf = f() < gn().

2. Iff Un1, then Cnf C

n1f .

3. If n 2 and f Un2 then Cnf Cn1f[n2]

, where this makes sense

because in this case f[n2] Un1.

For each f Un we define f[n] : f(min(Cnf (+ 1))) to finish round

n of the inductive construction.Now we claim that the pointwise sup of the sequence gn : n < is an

upper bound for U0 with respect to . We define n = sup(Cnf (+ 1)) and observe that n+1 n, so we

may find N and such that n N = n = .

Let = min(CNF (+ 1)) = min(CNF (+ 1)). We now get a chain of

inequalities

f() f() = f[N]() < gN+1() gN+1().

This time the key point is that CN+2f CN+1f[N]

, so that f[N]() < gN+1().

We have proved that > = f() n gn(), so that every function

f U0 is bounded on a final segment of by n gn(). This contradicts

the choice of U0 as unbounded with respect to . Then [] = for all sufficiently large . Then d() = dcl().

Proof: Let = dcl(). Then > , so that we may apply Theorem 7 tofind a regular < such that [] = . let us fix P P() such that|P| = and A P() C P |A C| = .

Now let D

be such that |D| = and D is dominating in (

,


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6. If < , then g() > f(min(C (+ 1))) for all .This is easy, because < . By the choice of P we may find a set A P

such that |{ h | A } { f | < }| = . Enumerate the first many

such that f { h | A } as n : n < .

We may now repeat the proof of Theorem 5 with fn , gn and Cnin place of fn, gn and Cn. We find that for all sufficiently large we

have g0() g0() n fn(). By the definition of hA and the fact

that { fn | n < } { h | A },n fn() hA() for all , so that

g0 < hA. This shows { hA | A P } to be dominating, so we are done.

We do not know whether it can ever be the case that dcl() < d(). This

is connected with some open and apparently difficult questions in pcf theory.

References

[1] J. E. Baumgartner, Iterated forcing, Surveys in set theory, (A. Mathias,

editor), LMS Lecture Notes 87, Cambridge University Press, Cambridge,

1983, pp. 159.

[2] E. K. van Douwen, The integers and topology, Handbook of Set-

Theoretic Topology, (K. Kunen and J. E. Vaughan, editors), North-

Holland, Amsterdam, 1984, pp. 111167.

[3] W. Easton, Powers of regular cardinals, Annals of Mathematical Logic,

vol. 1 (1964), pp. 139178.

[4] S. Hechler, On the existence of certain cofinal subsets of , Axiomatic

Set Theory, Proceedings of Symposia in Pure Mathematics, vol. 13 partII, American Mathematical Society, Providence, Rhode Island, 1974,

pp. 155173.

[5] M. Magidor and S. Shelah, When does almost free imply free?, Journal

of the American Mathematical Society (to appear).

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[6] S. Shelah, The Generalised Continuum Hypothesis revisited, Israel Jour-nal of Mathematics (to appear).

[7] J. Zapletal, Splitting number and the core model, (to appear).

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James Cummings and Saharon Shelah- Cardinal invariants above the continuum

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