saharon shelah- strong covering without squares

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Strong covering without squares

Saharon Shelah

Institute of Mathematics

The Hebrew University of Jerusalem

91904 Jerusalem, Israel

and

Department of Mathematics

Rutgers UniversityNew Brunswick, NJ 08903, USA

October 6, 2003

1 Introduction

The study of covering lemmas started with Jensen [DeJe] who proved in19745 that in the absence of 0 there is a certain degree of resemblancebetween V and L. More precisely, if 0 does not exist then for every setof ordinals X there exists a set of ordinals Y L such that X Y andV |Y| = max{|X|, 1}. There is no hope of covering countable sets bycountable ones in general, because doing Namba forcing over L will changethe cofinality of L2 to while preserving 1.

This form of covering has strong implications for the structure of V.For example Jensens theorem implies that in the absence of 0 the SingularCardinals Hypothesis holds, and that there is a special +-Aronszajn tree forevery singular . So we can conclude that the negations of these statementshave substantial consistency strength.

One subsequent line of development has involved proving covering lem-

mas over larger and larger core models, on the assumption of the non-existence of stronger and stronger large cardinals. Inevitably these coveringlemmas have much more complex statements than Jensens original theo-rem, the reason being that once the core model contains even one measurablecardinal we can start to do Prikry forcing.

The research was partially supported by Basic Research Foundation of the Israel

Academy of Sciences and Humanities. Publication 580

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This line of research has provided much information about the consis-tency strengths of combinatorial hypotheses. For example work of Gitikand Mitchell has determined the exact strength of the failure of the Singu-lar Cardinals Hypothesis, while work of Schimmerling, Mitchell and Steelhas provided a very strong lower bound (a Woodin cardinal) for the strengthof is singular and there is no special +-Aronszajn tree.

Another line of development involved getting more information about thenature of the covering set Y. For example given an ordinal and some first-order structure M V with underlying set , we can take a set X andask whether it can be covered by some Y W with |Y| max{|X|, 1} andY M. This kind of phenomenon is called strong covering (see Definition1.1).

One approach to proving strong covering theorems is to go back toJensens proof and to prove directly that there exists an appropriate Y L.This approach was taken by Carlson [Ca]. Another approach (due to theauthor) is more axiomatic; given W V two transitive class models of ZFCwhere W is sufficiently L-like and for every X V there is Y X with|Y| = max{|X|, 1}, it is proved in [Sh:b, XIII] that a certain form of strongcovering holds between V and W.

The work in this paper continues that in [Sh:b, XIII] and [Sh410] (notethat a slightly improved version of [Sh:b, XIII] has appeared as [Sh:g, VII]).The idea here is to eliminate as far as possible the structural assumptionson W. We start by outlining the structure of the paper.

1.1 Definition. Let W be an inner model of ZFC. Let be a cardinal inV.

1. -covering holds between V and W iff for all X V with X ONand V |X| < , there exists Y W such that X Y ON andV |Y| < .

2. Strong -covering holds between V and W iff for every structure M V for some countable first-order language whose underlying set is someordinal , and every X V with X and V |X| < , there isY W such that X Y M and V |Y| < .

In the first section it is proved that if is V-regular, +V = +W, andwe have both -covering and +-covering between W and V, then strong -covering holds. In fact something rather stronger is proved. The assumptionthat -covering holds is reasonable enough, but we can hope to weaken theother assumptions.

In the remainder of the paper we will prove a series of facts about cov-ering culminating in two main results; one result says that we can drop the

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assumption of

+

-covering at the expense of assuming some more absolute-ness of cardinals and cofinalities between W and V, and the other says thatwe can drop the assumption that +W =

+V and weaken the

+-coveringassumption at the expense of assuming some structural facts about W (theexistence of certain square sequences). Both these results are contained inTheorem 7.1.

The paper was written up by Uri Abraham and James Cummings, andI am grateful for their excellent work. I am also grateful to Moti Gitik forasking me about the possibility of a theorem like Theorem 7.1 after reading[Sh420].

The material in this paper represents part of some lectures given by theauthor in Jerusalem in the period MayAugust 1995. The rest of those

lectures will appear in [Sh598] and so we have retained here to some extentthe notation and terminology used in the lectures.

In particular the Jerusalem lectures introduced names for some of theimportant hypotheses. For the record, here is a complete list of those names.In the body of the paper we will recall these definitions as and when we needthem. W will always be some inner model of ZFC.

(A),: 0 < = cfV () < CARDW.

(B),: []


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(H)D,: For all (, +

), if =

i


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2 The painless strong covering theoremIn this section we will give a simple proof of a form of strong covering fromrather strong hypotheses. We begin by discussing the well-known conceptof a filtration which will be useful at several points in what follows.

2.1 Definition. Let be a regular uncountable cardinal and let X be aset of cardinality . Then a filtration of X is a sequence X = X : < such that

1. |X| < for all < .

2. < = X X.

3. For limit , X =


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Proof: We build an increasing and continuous sequence x : < ofsubsets of , where each x has size less than .

In case = 0, x0 = z.

In case = 2+ 1, x is some set such that x2 x , x W,|x| < . Such a set exists because we are assuming (B).

In case = 2+ 2, x is some set such that x2+1 x , x M,and |x| < . Such a set exists because M has a countable set ofSkolem functions and is uncountable.

In case is limit, x =


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2. Strong -covering for sequences holds between V and W for subsets of iff whenever ai : i < is an increasing (but not necessarily contin-uous) sequence of subsets of with ai W and |ai| < , then

i ai is

W-filtered.

3. Strong -covering for sequences holds between V and W iff for all strong -covering for sequences holds between V and W for subsets of.

The proof of Theorem 2.3 can be broken into two lemmas.

2.6 Lemma. Let (B) hold, let M be a structure for a countable first-order language with underlying set , and let z with |z| < . Thenz

i .The next lemma represents a variation on the main idea in Theorem

2.3. Here we are covering by a set of size which is W-filtered, rather thanactually lying in W.

3.1 Lemma. Suppose that b =

i


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Proof: Suppose that b a where a is W-filtered, as evidenced by an in-creasing and continuous sequence ai : i < such that a =

i


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3.3 Theorem. (F)P, implies (G)P,.

Proof: Let b : < be an increasing sequence of members of P, and setb =


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Recall that a function g : A On from a set of ordinals A of size intothe ordinals is W-filtered iff there exists a a filtration of A such that for alli both ai and g ai are in W.

(J), : For any a RegW ( \ +) of cardinality , ifa is W-filtered, then

for every f a there exists g a such that f g and g is W-filtered.

Here f a means that f() for every a, and f g meansthat a (f() g()). J is a weaker version of the property J definedin the introduction.

(B)+,: For every ordinal , if cf() < , then cfW() < +V .

4.2 Remark. The hypotheses (B)+, and (J), are both consequences of(B)+,.

4.3 Theorem. If (B)+,, (J), and (H) then (F),.

Proof: Let x [] be given; we shall find a W-filtered set A such that| x A |= . For this we define by induction W-filtered sets Nn for eachn < , such that x

n


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Proof: Easy, using the fact thatB

W.

In the second step, M0 is defined to be any elementary submodel ofBof size which contains x and N0. For example, M0 = Sk(N0 x) will do.

Suppose now that Nn and Mn have been defined. We first define Nn+1 asfollows. Set an = Reg

W(\+)Nn. For every an, as (B)+, holds and

as + is regular in W, cfV() +. Hence fn() = sup(Mn ) < , andthe function fn thus defined is in an. So (J)

can be applied to an, fn and

there exists gn an which is W-filtered and such that fn gn. It followsthat the set {gn() : an} is also W-filtered. Let n = sup(Mn

+),then n is W-filtered, because we are assuming (H) . Let n = sup Mn if

sup Mn < , and n = 0 otherwise. Define

Nn+1 = Sk(Nn {gn() : an} n {n})

Nn+1 is W-filtered. Finally, define

Mn+1 = Sk(Nn+1 Mn).

This ends the inductive definition of the models, and we prove now thatn


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Now < implies , and yet = is not possible since N;hence < < contradicts the minimality of .

Case: is singular in W. Let = cfW(). Then N as long asthe W-cofinality function is in B. We therefore demand that cfW isamong the functions of W.

Now as < ( being singular) < follows. We will demand thatB contains a function COF which assigns to every W-singular cardinal a cofinal sequence COF() W of length cfW(). COF() : is cofinal in and so, as M, there is < in M such that = COF()() . Since N = M by the minimality of , N. Hence N is an ordinal in [, ), in contradiction to the

minimality of .

Case: is regular in W. This is the last case. We know by (4) thatsup(N ) = sup(M ). Hence sup(N ) which is again acontradiction to = min(N \ ).

This concludes the proof of the Theorem.

We end this section by considering a variation (J)

which seems weakerthan (J) but suffices for Theorem 4.3

(J)

: If A RegW

( \ +

) of cardinality is W-filtered and f A,then there exists i < and a collection {fi : i < i

}, fi A, such thatf sup{fi : i < i

} and each fi is W-filtered.

4.5 Theorem. Theorem 4.3 still holds if (J)

replaces (J).

Proof:

Given i < , let us say that A is (i, W)-filtered iff A is a union of i

sets each of which is W-filtered.We will define Nn, Mn as before, but require now that Nn is (i

n, W)-

filtered for some in < . The construction is very similar, but in defining

Nn+1 it is not a single function that is added to Nn, but rather in func-tions, each W-filtered. Finally N = M as before, and for some n we have|Nn x| = .

As Nn is (in, W)-filtered, we may find a W-filtered set X Nn such

that |X x| = as required.

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5 The pcf induction lemmaThe results in the last section indicate the usefulness of the hypothesis (J) .In this section we prove a crucial lemma, which we will exploit in the finalsection of the paper to prove that two rather different sets of assumptionswill each imply (J).

5.1 Lemma. Let W V be two transitive class models of set theory. InV let and be regular, with < . Assume that

1. N (H, , |a| wemay find a fixed j such that g() < sup(Nj ). Since Nj N, the functiong : sup(Nj ) N.

Now we apply covering in a routine way to find a set X a a with

|X|V < , X W and X g. Since g N and N M (a structure into

which we built information about W) we may assume that X N W. Wedefine h() = sup { | (, ) X }, and then h N W and also, sincecfV() > , h a. Clearly g() < g() h() for all , and we are done.

So the missing ingredient for applying Lemma 5.1 is the existence of aset C with C W and C unbounded in N . In the next section we seetwo ways of guaranteeing the existence of such a C.

Notice that the covering assumption in Lemma 5.2 can be weakened. Allwe need is that a set of size less than can be covered by a set of size lessthan +.

6 Applying the pcf induction lemma

We begin by showing that we can use Lemma 5.1 to prove instances of theprinciple (J) which we now recall.

(J), : For any a RegW ( \ +) of cardinality , ifa is W-filtered, then

for every f a there exists g a such that f g and g is W-filtered.

As usual, W and V are transitive class models of ZFC with W V, and is a regular cardinal in V.

6.1 Lemma. Let a REGW \ ( + 1) be a W-filtered set of size , aswitnessed by a filtration a such that ai W for all i < . Let f a.Suppose that f, a,a N where N is a structure obeying the conclusionof Lemma 5.1, and such that max(pcfW(ai)) N for all i. Then ch

Na

isW-filtered and f chN

a.

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Proof: Sincea

N (because |a

| = N) it is easy to see that f chN

a .Since we built the filtration a into N we see that for all i < we haveai N W, and we may apply Lemma 5.1 to conclude that ch

Nai

W.

Therefore chNai : i < gives a filtration of chNa

and we are done.

6.2 Remark. The same idea could be used to derive the stronger property(J) defined in the introduction. That is, the assumption that a is increasingis never used.

Now we describe, in Lemmas 6.4 and 6.6, two ways of building structures

N that satisfy the hypotheses of Lemma 5.1.6.3 Definition. Let and be regular cardinals with < . Then a squaresequence on for cofinalities less than is a sequence C : < , cf() < such that C is club in , o.t.(C) < and lim(C) = C = C .

6.4 Lemma. Let W and V be two transitive class models of ZFC withW V. Let be regular in V. Suppose that

W there is a square on for cofinalities less than +V

for every W-regular cardinal . Suppose also cfW() +V = cfV()

+V

for all .

Let Ni : i < be a sequence of substructures of (H, , and < +.

2. N0 and |Ni| = .

3. Ni : i j Nj+1 for all j < .

Let N =

i .

C is defined because cfW() < +V .

If Ni and we define j = sup(Nj ) then we may argue as inLemma 5.1 that j : i j < is increasing, continuous and cofinal in .Now since cf() > and C is club in there is a club D of j < suchthat j lim(C). For each such j, Cj = C j ; since j N, |Cj | and N we see that Cj N, so C =

jD Cj N and we are done.

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6.5 Remark. By Lemma 5.2, the structure N defined in Lemma 6.4 obeysall the hypotheses of Lemma 5.1.

We now describe another way of getting the hypotheses of Lemma 5.1 tohold. Here we drop the assumption of squares but pay for this by needingto assume more resemblance between V and W.

6.6 Lemma. Let W, V be transitive class models of ZFC with W V.Let , be regular in V with < and let M = (H, ,


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desired. Now for each i let D

i =

Ni D, and define a diagonal intersectionD = { j < + | i < j = j Di }. Then if j D with cf(j) = , and

Nj , we see immediately that Ni for i < j and thus j D.

We sum up the results of this section in a corollary.

6.7 Corollary. Let W V be two inner models of ZFC. Suppose that , are regular cardinals in V with < . Suppose that cfW()

+V =

cfV() +V for all < , and that either one of the following two assump-

tions holds:

1. W there is a square on for cofinalities less than +V

for everyW-regular such that +V < .

2. +V = +W,

++V =

++W , and cfW()

++ = cfV() ++ for all

< .

Then (J), holds.

7 Conclusion

We can finally state the main theorem.

7.1 Theorem. Let W be an inner model of ZFC. Suppose that < where is regular and is a cardinal in W. Suppose that

1. (H) holds. That is, every ordinal in (, +) is W-filtered.

2. There exists P W such that W P []


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Proof: The structure of this proof can be seen by looking at the picture inthe introduction. Notice that assumption 2 implies that (B), holds. ByLemmas 6.1 and 6.4 (if we are assuming squares as in 4a) or 6.1 and 6.6(if we are assuming correctness as in 4b) we have that (J) , holds. ByLemma 4.3, (F), holds. By Theorem 3.3, (G)P, holds. By Lemma 3.2,strong -covering for sequences of subsets of holds. As in the proof ofTheorem 2.3, this implies that strong -covering holds for structures on .

8 Appendix on pcf

In this appendix we will prove the elementary facts about pcf theory usedin the paper. For more information see the book [Sh:g] or the survey paper[BuMa].

8.1 Definition. Let a be a set of regular cardinals such that |a|+ < min(a).

1. IfI is an ideal on a then

(a) Ifb, c a, b I c iffb \ c I.

(b) Iff, g a then f


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8.3 Definition. Leta

be a set of regular cardinals with |a

|

+

< min(a

).Then

1. pcfa (Potential CoFinalities ofa) is the set of regular cardinals suchthat cf(a/D) = for some ultrafilter D.

2. If is a cardinal (not necessarily regular) then J


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such that g+1 > g and g+1 is an upper bound forf in

a

/D. Thekey point here is that since D J = and f is


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3.f is bounded modulo the ideal generated by I and the

bs.

Proof: As in the proof of Lemma 8.4 we will build a sequence g of functionswhich are increasing on each coordinate, where g0 = 0 and we take thepointwise supremum at each limit stage < |a|+. Suppose we have defined

g. Then since f is unbounded, the set b = { | g() < f() } is in I

+

for all sufficiently large (say ()). Consider b : () as a

candidate for the desired sequence b; clearly it is positive and increasingmodulo I, and what is more g will be a bound for f modulo the idealgenerated by I and b : ().

So the construction is finished unless there is an () () such

that f b

() : < fails to be cofinal. In this case we will chooseg+1 > g to be a witness to this failure of cofinalness, which is to say that

{ b() | g+1() > f() } I

+ for all . The key point is that (since

the b are increasing modulo I with ) for all () and all we have

{ b | g+1() > f() } I+.

Now suppose that the construction runs for |a|+ many steps. Choose sup f() } I+, so in particular there is a such

that g() < f() < g+1(). This leads to a contradiction exactly as inthe proof of Lemma 8.4.

This shows that at some stage < |a|+ the construction terminates,

giving a sequence b as desired.

Using the facts above we can derive the key fact about pcf which is beingused in this paper.

8.7 Theorem. Let = max pcf(a). Then tcf(a/J


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other hand there is a withb

D, then f b

: < witnesses thattcf(b/I) = J, so b D J contradicting the choice of D disjoint fromJ.

This contradiction shows that a J, hence tcf(a/J

saharon shelah- strong covering without squares

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