Juliette Kennedy, Saharon Shelah and Jouko Vaananen- Regular Ultrafilters and Finite Square...

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Regular Ultrafilters and Finite Square

Principles

Juliette Kennedy

Department of Philosophy

Utrecht University, Netherlands

Department of Mathematics and StatisticsUniversity of Helsinki, Finland

Saharon Shelah

Institute of Mathematics

Hebrew University

Jerusalem, Israel

Jouko Vaananen

Institute for Logic, Language and Computation

University of Amsterdam

Amsterdam, Netherlands

August 7, 2007

Abstract

We show that many singular cardinals above a strongly compactcardinal have regular ultrafilters D that violate the finite square prin-

ciple fin,D introduced in [3]. For such ultrafilters D and cardinals

Research partially supported by a grant from NWO and grant 40734 of the Academyof Finland.

The second author would like to thank the Israel Science Foundation for partial sup-port of this research (Grant no. 242/03). Publication 912.

Research partially supported by grant 40734 of the Academy of Finland.

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there are models of size for which M/D is not ++-universal and

elementarily equivalent models M and N of size for which M/Dand N/D are non-isomorphic. The question of the existence of suchultrafilters and models was raised in [1].

1 Introduction

In [3] and [4] the equivalence of a finite square principle fin,D with vari-ous model theoretic properties of structures of size and regular filters wasestablished.

The model theoretic properties were the following: Firstly, if D is an

ultrafilter, then fin,D is equivalent to M/D being ++-universal for eachmodel M in a vocabulary of size . Secondly, if 2 = +, then fin,Dis equivalent to M/D and N/D being isomorphic for any elementarilyequivalent models M and N of size in a vocabulary of size . Theexistence of such ultrafilters and models is related to Open Problems 18 and19 in [1].

The consistency of the failure offin,D at a singular strong limit cardinal was proved in [4] relative to the consistency of a supercompact cardinal. Thedrawback of the result in [4] was that only regular filter D with fin,D wasobtained, while the existence of such an ultrafilter D would be relevant from

the point of view of model theoretic consequences. When we have failure offin,D for an ultrafilter, we get the failure of

++-universality of M/D for

some M, as well as failure of isomorphism of some regular ultrapowers M/Dand N/D. In this paper we indeed construct regular ultrafilters with fin,Dfor singular above a strongly compact. And thus the present paper answersnegatively problems 18 and 19 of [1] modulo large cardinal assumptions. Theuse of large cardinals is justified by [3], [4] and [6] as the failure offin,D forsingular strong limit implies the failure of, which implies the consistencyof large cardinals.

A filter D on a set I is said to be regular if there is a family E D, such

that |E| = |I| and

F = for all infinite F E. We then say that E isa regular family in D. Regular filters have given rise to hard set theoreticalproblems but at the same time they are very useful in model theory.

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2 An equivalent condition for fin,D

The following finite square principle was introduced in [3]:

fin,D : D is a filter on a cardinal and there exist finite sets C and integers

n for each < + and < such that for each ,

(i) C + 1

(ii) If B + is a finite set of ordinals and < + is such thatB + 1, then { : B C} D

(iii) C implies C = C

(+ 1)

(iv) |C| < n

By results in [3] and [6], on the one hand,


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Proof. Let us first assume (1) and derive (2). For < , < +, let

C = { < : B,} {}. We show that (i)-(iv) offin

,D hold. Clause(i) holds by construction. To prove (ii), assume X + 1 is finite. Notethat

X

B, { : X C}.

Since D is a filter, we get { : X C} D. To prove (iii), assume C.

If C and < , then

B, B, = B, B,,

whence C. Conversely, if C and < ,

B, B, = B, B,,

whence C. Finally, (iv) is true by condition (1.3) above, since the setsC are finite for each and moreover, the number sup{|C

| : <

+} isfinite.

Let us then assume (2) and prove (1). Let B, = { < : C}. By

(ii) offin,D, B, D. To prove (1.1), let < < . If B, B,,

then C and C. By (iii), C

= C

(+ 1). Thus we may conclude

C and C, i.e. B, B,. Conversely, if B, B,, then

C

and C

. Again by (iii), C

= C

(+ 1). Thus C

and C, i.e. finally, B, B,.

To prove (1.2), assume n < n+1 < for n < , but

n Bn, = , say

n Bn,. Then each n is in C, which is impossible because the latter

is finite. 2

In Theorem 6 below we will construct a regular ultrafilter which does nothave the strong regularity property of Theorem 1.

3 A partition property

We define a particular partition property Pr2(, ) which turns out to beuseful when we show that the ultrafilter we construct in the next sectiondoes not have fin,D. A similar partition property is used in [5, Theorem 6.1].

Definition 2 Let Pr2(, ) denote the following property of and with < (See Figure 1):

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Suppose c : []2 E, where E is a filter on . Then there is an

i < such that for all < there is an increasing sequence , < , of ordinals < such that for all 1 < 2 < there is > 2 such that i c({1, }) (2, })

1 .

E

i

1

2

c

c

c({2, })

c({1, })

Figure 1: The partition property Pr2(, ).

If is weakly compact, then Pr2(, ) holds for all < . What isinteresting about Pr2(, ) is that it can hold also for successor cardinals :

Proposition 3 Suppose < and E is a +-complete uniform ultrafilteron +. Then Pr2(

+, ).

Proof. Fix < +. Then [+ 1, +) E. Obviously,

[+ 1, +) =

i 2 such that i = c({1 , }) = (2 , }).

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Then D = ED is a regular ultrafilter on with fin

,D. Moreover, (1.1)

and (1.2) of Theorem 1 cannot be satisfied.

Proof. It is easy to see (and known) that D is regular. We assume now Dsatisfies condition (2) of Theorem 1, and derive a contradiction. Let B,, < < +, be as in Theorem 1. Since B, D,

a(, ) =df { < : B, D} E.

Claim 1: If 1 < 2 < 3, then a(1, 3) a(2, 3) a(1, 2).

To prove the Claim, assume 1 < 2 < 3 and a(1, 3) a(2, 3). ThusB1,3 D and B2,3 D. Hence B1,3 B2,3 D. Nowwe use the fact that B1,3 B2,3 = B1,2 B2,3, i.e. (1.1) of Theorem 1,to conclude that B1,2 D, and thereby a(1, 2). The Claim isproved.

Let c({, }) = a(, ) E. By Pr2(+, ) there are an i < and anincreasing sequence , < , =

++i , of ordinals <

+ such that for all1 < 2 < we have i c({1, }) c({2, }), for some [2,

+),recalling that ++

i

< . Let Y = { : < }.

Claim 2: If 1 < 2 in Y, then B1,2 i Di, i.e. i a(1, 2).

To prove the Claim, assume 1 < 2 in Y. Then for some > 1, 2 wehave i a(1, ) a(2, ). By Claim 1, a(1, ) a(2, ) a(1, 2), whencei a(1, 2) i.e. B1,2 i Di. The Claim is proved.

Let Y such that |Y | > i, and for < i

Z = { Y : B, i}.

Claim 3: Y =

{Z : < i}.

To prove this, assume Y . By Claim 2, i a(, ), i.e. B, i Di,which implies that we may pick B, i. Now Z. Claim 3 isproved.

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As |Y | > i, there is such that Z is infinite. Let 0 < 1 < ... be

an infinite increasing sequence in Z. Then

n Bn,. This contradicts(1.2) of Theorem 1, as

n Bn, = . 2

Corollary 7 Suppose is strongly compact. Then every cardinal > ofcofinality < has a regular ultrafilter D such thatfin,D fails.

5 Model theory

The background offin,D is the following question, asked by Chang and Keisleras Conjecture 18 in [1]:

Let M and N be structures of cardinality in a language ofsize and let D be a regular ultrafilter over . If M N,then M/D = N/D.

The question is a natural one as most of the model theory regardingultrapowers is centered on the regular ultrafilters. It is reasonable to assumeGCH in this question, although it is not part of the question.

Another open problem that motivated the formulation offin,D is Conjec-ture 19 of [1]:

IfD is a regular ultrafilter over , then for all infinite M, M/Dis ++-universal.

The original motivation for the study of fin,D was its equivalence withthe above conjectures:

Theorem 8 ([4]) Assume D is a regular ultrafilter on. Then the followingconditions are equivalent:

(i) fin,D.

(ii) If M and N are elementarily equivalent models of a language of car-dinality , then the second player has a winning strategy in theEhrenfeucht-Frasse game of length + on M/D and Nl/D.

(v) If M is a structure in a language of cardinality , then M/D is++-universal.

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Juliette Kennedy, Saharon Shelah and Jouko Vaananen- Regular Ultrafilters and Finite Square...

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