Kecheng Liu and Saharon Shelah- Cofinalities of Elementary Substructures of Structures on ℵω

8/3/2019 Kecheng Liu and Saharon Shelah- Cofinalities of Elementary Substructures of Structures on

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Cofinalities of Elementary Substructures of

Structures on

Kecheng LiuDepartment of Mathematics

University of CaliforniaIrvine, CA 92717

and

Saharon Shelah

Institute of MathematicsThe Hebrew University

Jerusalem, IsraelDepartment of Mathematics

Rutgers UniversityNew Brunswick, NJ

Revisions made 3/29/96

1. INTRODUCTION.

Let 0 < n < and f : X n + 1 be a function where X \(n + 1) is infinite.Lets consider the following set

Sf = {x : |x| n (n X)cf(x n) = f(n)}.

The question is whether Sf is stationary in [] n and

f1 : X n + 1 agrees with f on dom(f)\(n1 + 1), then Sf1 is stationary provided thatSf is. So we are interested in the case that f is not eventually constant. You may wonderhow strong the statement that Sf is stationary is. Magidor (unpublished, but close to[Mg3]) has shown that if f : \2 2 assumes the values 0 and 1 alternatively and Sf

Partially supported by the United States-Israel Binational Science Foundation publ484.

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is stationary (here n = 1), then theres an inner model with infinitely many measurablecardinals.

Liu has proven (in [Liu]) that under the existence of huge cardinals, it is consistentthat Sf is stationary together with GCH for many fs assuming (any) two different values.In this paper, we are going to prove a few results concerning the above question, which are

due to the second author except for Proposition 3.1 and a small remark of the first authorimproving Theorem 4.2.

Theorems here (3.1, 4.2) are the first results for function f with more than 2 valuesgotten infinitely many times. Also the present results have relatively small consistentstrength. A version of 4.1 was proved in the mid eighties and forgotten and we thank M.Gitik for reminding. The main results are as follows:

Theorem 3.1. Assume sup(pcf({n : n < })) = +n . Let 1 < m < . Let I be theideal of finite subsets of . Let Ai : 1 i n

be pairwise disjoint subsets of \(m + 1)such that

kAik/I has true cofinality +i for 1 i n

. Let i : 1 i n be a

sequence of uncountable cardinals below m+1. Then the set

S = {x : |x| m (i)[1 i n (k Ai)cf(x ) = i]}

is stationary in [] (

n )+ be a regular cardinal. LetM H, , S , , . . . ,M + 1 and |M| = , where is a well-ordering of H. Then the following areequivalent:

1. S is stationary in [] j cf(B m) = ];

(3) (m X\Xi0) [m > j cf(B

m) = cf(B m)].Proof. For each i = i0, for each m Xi, let am be a cofinal subset of B m with ordertype f(i). Now we can build a sequence B : < such that

(1) {am : m X\Xi0} (B j) B0;

(2) m Xi0 [sup(B m) < sup(B+1 m)];

(3) B B+1 B and |B| = n .

The construction is obvious. Now let B =


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We are going to prove the following theorem using pcf theory (see [Sh:g]).

Theorem 3.1. Assume max(pcf({n : n < })) = +n . Let 1 < m < . LetI be

the ideal of finite subsets of . Let Ai : 1 i n be a sequence of pairwise disjoint

subsets of \(m + 1) such thatkAi k/I has true cofinality +i for 1 i n. Let

i : 1 i n

be a sequence of uncountable cardinals below m

+1. Then the set

S = {x : |x| m (i)(1 i n (k Ai)[cf(x k) = i])}

is stationary in [] +n , no harm is done since we can use Levy

collapse to collapse pp to +n and no new subset of is added.

4. The theorem can be generalized to other singular cardinals. Also, we can use other

regular cardinals in (, pp) in the proof of the theorem.5. Consistency results giving the assumptions are well know, starting with [Mg]; see

history and references on this in [Sh:g].

The proof of Theorem 3.1 uses the following lemma:

Lemma 3.1. For each 1 i n, there is a sequence Ci =def

Ci : < +1 such that

(1) Ci and o.t.(Ci) i

(2) Ci implies Ci = C

i

(3) Si =def

{ < +i : cf() = i and = sup(Ci)} is stationary in +i.

Remarks. Note that the Cis are not necessarily closed.

For a proof of Lemma 3.1, see [Sh 351, 4.4] for successor of regular cardinals and ingeneral [Sh 420, 1.5] which rely on [Sh 351, 4.4].

For each 1 i n, let Si and Ci = Ci : < +i be as in Lemma 3.1. We nowproceed to prove Theorem 3.1.

Proof of Theorem 3.1. For 1 i n, let fi = fi : < +i kAikbe increasing and cofinal in kAik/I. Let >> pp be a regular cardinal. Letsconsider the structure A = H, , , . . . with skolem functions closed under composition,

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where is a well-ordering on H. We define X, N by induction on n as follows. Letx = { fi, Ai, Ci}1in . For each = i : 1 i n 1in+i, let

X ={ : < n} ( 1in

Cii) {N :

n

i=1

Cii}

(hence Ci X if : 1 i n Cii)

and let N = skA(X). Note that |N| = n and

ni=1 C

ii

N N and N N.

Claim. There is = i : 1 i n 1inSi such that all 1 i n

1) For all 1inCii

, we have sup(N +i) < i for all 1 i < n.

2) sup(N +i) = i.

3) For some ni < , {fi(k) : C

ii

} is cofinal in N k for all k Ai\ni.

4) For some mi ni, cf(N k) = cf(i) = i for all k Ai\mi.

Proof of Claim. We first construct as required by part 1) of the claim. The constructionis as follows: Let En = { < +n : 1in+i, if n < , then sup(N +n) < }. Then En is clearly closed unbounded in +n . Since Sn is stationary in+n we have Sn En = . Pick some n Sn En .

Suppose we have defined j for n j > i. We now define i. Let Ei = { < +i :

1n+ if i < and j = j for all i < j n then sup(N +i) < }.

Its easy to see that Ei is closed unbounded in +i. So we can find i Si Ei since Siis stationary in +i.

We now show satisfies clause (1). Let 1inCii . Let 1 i n

be fixed, andwe want to show that sup(N +i) < i. Consider = 1, . . . i, i+1, . . . n. By thechoice of i, we have that sup(N +i) < i. Since

1kn C

kk

clearly X X.

So sup(N +i) sup(N +i) < i.

Lets now prove clause (2) of the claim. Fix 1 i n. Let N +i. Then = (y) for some y [X]


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Finally, lets prove clause (4) again by contradiction. Suppose clause (4) is not true.Then there is an unbounded set b Ai\ni such that cf(sup{fi(k) : C

ii

}) < i for

some k b. So as o.t.(Cii) = i for each k b, there is k Cii

such that for all Ciiwith k we have: sup{f

i(k) : C

ii

} = sup{fi(k) : Cii

}. Let Cii be

such that > sup({k : k b}). For each k Ai, let k = sup({f

i

(k) : C

i

}). Thenk : k Ai N kAik since Ci N. So as above there is

N +i such

that k : k Ai fi . So we have sup(N k) : k b = k : k B

fi b

which contradicts to fi N kAik (the initiated could have used wlogfi obeys

Ci). This completes the proof of the claim.Now, Theorem 3.1 follows from the claim, Proposition 2.1 and Lemma 2.1.

By the remarks following Theorem 3.1, the theorem is not trivial only when pp >+1. In particular GCH does not hold at . But using the following observation, wecan make GCH hold at by collapsing 2

and still have the desired conclusion in the

forcing extension, i.e., the set S in Theorem 3.1 is still stationary in the generic extension.(So by well-known consistency results we can even have GCH.)

Proposition 3.1. Let P be an < +1-closed forcing notion. Suppose S is stationary in[] 0, the first player partitions the set


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Lemma 4.1. Let T,I be a tagged tree such that for each T, I is a proper idealsuch that SucT() I. Let be a regular, uncountable cardinal and for every T, Iis -indecomposable, i.e. if A SucT()A = mod I and f : A then for some < we have {x A : f(x) < } = mod I. (This holds if for every T, I is a

+

complete ideal or |SucT()| < ).

(*) for every function F : T , there is a subtree T1 of T such that for all T1,SucT1() I and Sup(F

T1) < .

We now proceed to prove Theorem 4.1.

Proof of Theorem 4.1. Let mi : i < be such that each mi A and foreach m A there are infinitely many i with m = mi. Let T = { T : T is a subtree of

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move of the second player following his winning strategy in the game GMn (Flh()). LetTi+1 T be such that Bi, = SucT() for each T. Let i be such that Sup(F

Ti+1) be a regular cardinal and A = H(), , , < ni : i E >, (i)i n, let hi : Wi [i]


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1) For each i E, Ai,0, Pi,, Bi, : < m is a play of the game GMm

(Fi) with thesecond player following his winning strategy;

2) Ai,0 = Wi, A0 = skA({}) and A =


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A Bn1 since Bn in1 A

. So f = f Bn1. Therefore, = f(x0, . . . , xn1) Bn1 since xk Bn1 for all k < n. We have thus proved part a) of the claim.

Clause b) follows from Lemma 2.1 and sup(Bn in) = in . By Lemma 2.1, sup(Bn in) = sup(B

n in) = in .)

We prove c) by induction on n. If n = 0, clause c) follows from Lemma 2.1. Now

suppose c) holds for n1. We want to show c) holds for n. By a) and induction hypothesis,sup(Bn j) = sup(Bn1 j) = sup(A j) if i0 < j < in and j E. For in < j E, sup(Bn j) = sup(B

n j) = sup(A

j) by Lemma 2.1. This finishes the proof ofthe claim.

We now can complete the proof of Theorem 4.2. Let B = n n and

i in, then B i = Bn i by the claim.)

Finally, let B = skA(B n). B is as required again by Lemma 2.1. So we havefinished the proof of Theorem 4.2.

Concluding Remarks. The most natural context (at least for the second author) ishaving a constant cardinal , set a of regular cardinals. Let = sup(a) and we look forstationary subsets of [] sup(a).Assume further f Fa , b1, , bn is a partition of a, and f be is constant ande = tcf(

be, |a|n

+2, then by the proof of 3.1 if 1 i n f(i) > |a| we succeed. Lastly use ()3possibly several times.]

()5 Assume (a) b a is countable, f Fa , f b is constantly 0 and f (a\b) isconstantly , is a unit ordinal of cofinably but < and is divisible by |a|.(b) for a, the second player has a winner strategy in the game GM0(F, (a\b))(see second remark to Theorem 4.1).

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Then Sf is stationary.[Why? repeat the proof of Theorem 4.1, but we let i : i < list b, each appearinginfinitely often, and T = {T : T subtree of e


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REFERENCES.

[B7] James E. Baumgartner. On the Size of Closed Unbounded Sets. preprint.

[Liu] Kecheng Liu. Stationary subsets of [] 2. Notre DameJournal of Formal Logic, 29:117, 1988.

[Sh 351] Saharon Shelah. Reflecting stationary sets and successors of singular cardinals.Archive for Mathematical Logic, 31:2553, 1991.

[Sh 420] Saharon Shelah. Advances in Cardinal Arithmetic. In Finite and InfiniteCombinatorics in Sets and Logic, pages 355383. Kluwer Academic Publishers,1993. N.W. Sauer et al (eds.). 0708.1979.

[Sh:g] Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Ox-ford University Press, 1994.

[Sh 542] Saharon Shelah. Large Normal Ideals Concentrating on a Fixed Small Cardi-nality. Archive for Mathematical Logic, 35:341347, 1996. math.LO/9406219.

[Sh:f] Saharon Shelah. Proper and improper forcing. Perspectives in MathematicalLogic. Springer, 1998.

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Kecheng Liu and Saharon Shelah- Cofinalities of Elementary Substructures of Structures on ℵω

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