rami grossberg and saharon shelah- on hanf numbers of the infinitary order property

8/3/2019 Rami Grossberg and Saharon Shelah- On Hanf numbers of the infinitary order property

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On Hanf numbers of the infinitary order property

Rami Grossberg

Department of Mathematics

Carnegie Mellon University

Pittsburgh, PA 15213

Saharon Shelah

Institute of Mathematics

The Hebrew University of JerusalemJerusalem, 91094 ISRAEL

&

Department of Mathematics

Rutgers University

New Brunswick, NJ 08902

October 6, 2003

The authors thank the United States - Israel Binational Science foundation for sup-porting this research as well as the Mathematics department of Rutgers University wherepart of this research was carried out.

This paper replaces item # 259 from Shelahs list of publications.

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Abstract

We study several cardinal, and ordinalvalued functions that arerelatives of Hanf numbers. Let be an infinite cardinal, and let T L+, be a theory of cardinality , and let be an ordinal

+.For example we look at

1. T(, ) := min{ : L,, with rk() < , if T has

the (, )-order property then there exists a formula

(x; y) L+,, such that for every , T has the (

, )-order prop-erty }.

2. (, ) := sup{T(, ) | T L+,}.

We discuss several other related functions, sample results are:

It turns out that if T has the (,

(, ))-order propery for some L,, with rk() < then for every > we have thatI(, T) = 2 holds.

For every and as above there exists an ordinal (, ) suchthat (, ) = (,),

(, ) (||)+,

for with uncountable cofinality, we have that (, ) > ||

and

the ordinal (, ) is bounded below by the GalvinHajnal rankof a reduced product.

For many cardinalities we have better bounds, some of the boundsobtained using Shelahs PCF theory. The function (, ) is used tocompute bounds to the values of the function (, ) we studied in aprevious paper.

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1 Introduction

Let be an infinite cardinality, and suppose that T L, (notice thatwhen = we are dealing with first-order theories).

The fundamental meta-problem in the area of classification theory canbe stated as:

Problem 1.1 What is the structure of M od(T)?

A more precise (and concrete) test-question is:

Problem 1.2 What are the possible functions I(, T) : Card Card?(where I(, T) stands for the number of isomorphism types of models forT of cardinality )

A much more precise (and a very difficult) particular case of 1.2 is thefollowing

Conjecture 1.3 (Shelah about 1976) Let L1, be given. If thereexists a cardinality > 1 such that I(, ) = 1 then for every > 1,I(, ) = 1 holds.

A possible approach to Problem 1.1 and its relatives, is to try to imitateClassification theory for elementary classes (see [Sh c]). Namely it wouldbe desirable to find properties parallel to stability, superstability etc. Muchwork has been done in the last 25 years (see for example [Sh 48],[Sh 87a],[Sh 87b],[Sh 88],[Sh 300],[MaSh 285],[GrSh 238], or [Sh 299] for a generalsurvey). In this article we concentrate on dealing with the parallel (forinfinitary languages) to instability. The following can be viewed as a defini-tion of stability for first-order theories:

Fact 1.4 ([Sh 16]) LetT be a complete first-order theory. The following areequivalent:

1. T is unstable

2. There exist a formula (x; y) L(T), a model M for T, and a set{an : n < } M such that l(x) = l(y) = l(an) for every n < , and

for all n,k < we have n < k M |= [an; ak].

Condition 2 in Fact 1.4 is sometimes called the order-property. One ofthe most important properties of unstable theories is the following:

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Fact 1.5 [Sh 12] Let T be a complete first-order theory. If T is unstable

then for every > |T| we have that I(, T) = 2.

An inspection of the proof of 1.5 shows that the hypothesis that T isa complete first-order unstable theory could be replaced by the followingproperty:

()T

There is an expansion L ofL(T) with built-in Skolem functions andan L-structure M, a formula (x; y) , and there exists I := {ai :i < } M a sequence ofL-indiscernibles such that l(x) = l(y) =l(an) for every n < , M is the Skolem Hull of I, M L(T) |= T,and for every n,m < we haven < m M |= [an; am].

By formula we mean that is in any logic (over the vocabulary L

) suchthat is preserved by isomorphisms of L(T)-structures.

The condition in Fact 1.4 seems to be a natural candidate for a definitionof instability for infinitary logics. Since the compactness theorem fails evenfor L1, the next definition is the replacement of the (the first-order) order-property.

Definition 1.6 Let T L,, (x; y) L,, and let be a cardinality.

1. We say that M has the (, )-order property iff there exists {ai : i I(, ) = 2.

Already in [Sh 16] Shelah realized the importance of the above concept(it did not appear there explicitly. Only in [Gr] (see [GrSh]) we realized theimportance of functions of this kind). The previous definition is a gener-alization of one of our definitions from [GrSh], see Definition 1.8. Shelahsfundamental result from [Sh 16] can be restated as:

Fact 1.10 ([Sh 16]) For every , we have (, ) 0(, 1)1.

Let us mention here the following dramatic improvement (for = 0) of1.10:

Theorem 1.11 ([GrSh]) For every 0, we have (, 0) +.

It turns out that even for first-order theories the above question is inter-esting (for = |L| = 0, T is a complete first-order theory in L, we couldask what is an upper bound of T(, 0)?). Since there are cases when T isstable (i.e. there is no first-order formula defining an -sequence in a modelof T) but still T has a hidden instability (like in the case of stable theories

with the omitting-types order property).Namely there is a natural class of examples of theories that do not have a

first-order formula exemplifying the order-property but do have an infinitaryorder property. Any stable first order theory that has the omitting types

10(, ) is the usual Morley number to be introduced in Definition 2.2 below. It iseasy to see that 0(,) = 0(, 1)

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order-property has the (L1,, )-order property but not the (L, , 0)-

order property (see [Sh 200]).Already from Morleys omitting-types theorem it follows that given T

and as above there exists := (T, ) such that if T has the (, )-order property then , T has the (, )-order property. The boundobtained from repeating the argument in the proof of Morleys omittingtypes theorem (see [Sh 16]) is: (T, ) max{Hanf(T),Hanf()}. WhereHanf(T) and Hanf() are the Hanf numbers of T and the logic containing (respectively).

Let > 0 (T still may be first-order). Our object is to find upperbounds on . It turns out that for L, L, there is a cardinality := (T, )2, such that the following implication holds: If T has the

(, )-order property then there exists a formula L, (it is a collapseof ) such that T has the (, )-order property for every .

In this paper we present a systematic study of several cardinal and or-dinal valued functions related to the infinitary order property. This is acontinuation of [GrSh], we deal with similar problems and improve manyof our results. This is achieved via a generalization of the original problem(dealing with new cases) while obtaining often better estimates to our earlierbounds. The reader is not expected to be familiar with [GrSh].

Notation: Everything is standard. Often x, y, and z will denote freevariables or finite sequences of variables, when x is a sequence l(x) denote

its length. It should be clear from the context whether we deal with vari-ables or sequences of variables. L will denote a similarity type (also knownas-language or signature), will stand for a set of L formulas. M and Nwill stand for L - structures, |M| the universe of the structure M, Mthe cardinality of the universe of M. Given a fixed structure M, subsetsof its universe will be denoted by A, B, C, and D. So when we writeA M we really mean that A |M|, while N M stands for N is asubmodel of M. Let M be a structure. By a M we mean a |M|,when a is a finite sequence of elements then a M stands for all the ele-ments of the sequence a are elements of |M|. For cardinalities , letS


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EC(T, ) = P C(T, , L(T)). ,,, and will stand for infinite cardinal-

ities; ,,,,, and are ordinals. References of the form Theorem IV3.12 are to [Sh c]. For L,, let Sub() be the set of subformulas of, now let

rk() :=

0 if is atomicSup{rk() + 1 : Sub()} otherwise.

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2 Review

C.C. Chang in [Ch] made the following fundamental observation:

Fact 2.1 Let be an infinite cardinality, and let L be a similarity type ofcardinality no more than . Given L+,, there exist a similarity typeL L, a first-order theory T in L, and a set of T-types (all three ofcardinality less than or equal to ) such that M od() = P C(T, , L).

Namely instead of studying M od() directly for an infinitary theory it is enough to look at a class of reducts of models of a first-order theorythat omits a set of types.

W. Hanf and M. Morley [Mo], recognized the importance of the followingconcept:

Definition 2.2 Let T be a first-order theory, and let be a set of T-types.The Morley number3 of T and , is the following:

1. 0(T, ) := min{ : M EC(T, ) M |T| N EC(T, ) of cardinality }.

2. Let , be cardinalities.0(, ) := sup{0(T, ) : |T| , a set of T-types of cardinality }4.

Morley (among other things) have shown that 0(0, 0) = 1. Hismost general result is stated as Theorem 2.4 below. Shelah in [Sh 78] havedealt with what can be viewed to be an interpolant of 0(T, ) and 0(, ):

0(T, ) := sup{0(T, ) : a set of T-types, || }.

It is not difficult to conclude from the proof of Morleys categoricitytheorem that when T is a countable and 0-stable theory then 0(T, ) 1.Shelah in [Sh 78] studied the effect that stability of T has on the upperbounds on 0(T, ). This work was continued about ten years later byHrushovski and Shelah in [HrSh 334].

In this paper, since our main goal is the study of unstable theories (ortheories that are not stable in a weak sense) we will ignore the effect thatthe stability of T may have on the function 0(T, ).

3Some authors call this the Hanf number ofT and the 4We hope that the reader is not bothered by this abuse of notation. We are using the

same letter 0 to denote entirely different (but related) functions. They can be distin-guished by the type of the arguments they take.

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The modern era in the study of Hanf numbers began with the paper of

Barwise and Kunen [BaKu]. They studied systematically the relationshipbetween the function 0 and the first ordinal that exemplify the undefinabil-ity of well ordering in classes of models that omit a set of types. Below weintroduce an ordinal-valued function 0(, ) that turns out to be related to0(, ) in a nice way.

Definition 2.3 Let and be infinite cardinalities, T varies over consis-tent first-order theories such that L(T) {P,


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From now on we concentrate on the case that = and we will work

with the functions 1(, ) and 1(, ). The arguments for the functionswith three parameters are essentially similar (they require an additionaltechnical effort, but require no new ideas). Note that by [Ch] working withtwo parameter functions is sufficient for L+,. The new point is that we areable to show that 1(, ) 1(,). The proof of Theorem 2.6 is similarto that of Theorem VII 5.5, we skip its proof, since later we will prove arelated theorem (Th. 2.25) whose proof is similar (and little harder).

The next proposition provides us with a lower bound for 1(, ), itfollows immediately from the definitions (in Theorem 2.27 we show a betterlower bound).

Proposition 2.7 For we have that 1(, ) 1(, ) 0(, ) =0(, 1).

In the following proposition the connection between the last definitionand the order property is clarified.

Theorem 2.8 Let , be cardinalities. + (, ) 1(, ).

Proof. First we show that (, ) 1(, ). Let L+,, and (x; y) L+, be given. Suppose has the (, 1(, ))-order property we need tofind a formula L+, such that has the (

, )-order property.

By Fact 2.1 there exists a first-order theory T in a similarity type L(T)that extends L, and there is a set of T-types of cardinality suchthat P C(T, , L) = M od(). By following the inductive definition of theformula we may identify with a function f from the set P into the setL {, , (, ), =} {xi : i < }.

Let be a regular large enough such that

{L+,, L+,, T, , L, , f , P, , +, 1(, ), 1(, )} 1(, ) H().

In addition we require that the structure H(), reflects all the relevantproperties of the above sets. Let P be the rank of the formula , note thatit is an ordinal less than +. Let A H(), , . . . of cardinality 1(, )such that 1(, )A

= 1(, ) (so 1(, ) + 1 A), fix a bijection G from1(, ) onto the universe ofA

, and let A := A, G. By the definition of1(, ), for every there exists B A of cardinality such that Bomits the types from , B = , and P is an ordinal less than + (justapply the Mostowski collapse on B). Using P

B , and fB we know (in

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Definition 2.12 1. Let T1, T2 be well-founded trees, and let be an

ordinal. By induction on define when T1 T2:

(a) For = 0, always T1 T2.

(b) For = 0, if for every < and for every x1 SucT1(rt(T1))there exists x2 SucT2(rt(T2)) such that T1[x1] T2[x2], and

for everyx2 SucT2(rt(T2)) there exists x1 SucT1(rt(T1)) suchthat T2[x2] T1[x1].

2. A tree T is called simple iff there are no distinct x1, x2 SucT(rt(T))such that DpT(x1) = DpT(x2) and T[x1] DpT(x1) T[x2].

Proposition 2.13 Let T1, T2 be trees, and let be an ordinal. If T1 T2then one of the following conditions holds:

1. Dp(T1) = Dp(T2), or

2. Dp(T1) and Dp(T2) .

Proof. Easy, by induction on . 2.13

Claim 2.14 For every ordinal there exists a family of simple trees {Ti :i < }, such that for every i <

1. Ti ,

2. i = j Ti + Tj,

3. Dp(Ti) = + + 1.

Proof. By induction on :For = 1; First construct 0 simple trees {Tn

> : n < } suchthat T0 := {}, Tn+1 := {n + 1 } T n when the order is an extention ofthe order on Tn; is the root, and n + 1 is a new immediate successor ofthe root incomparable with the elements of SucTn(). Now for every A let

TA := { : Tn, n A} { Tk : k A}.The order of TA is defined as follows: is a new immediate sucessor

of the root, the elements and are pairwise incomparable when Tn, (n A) and Tk, (k A), and we require that 1 < 2 1


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that n A B. Since TA[, n] TB[] for any SucTB () (this is

because Tk and Tn are inequivalent for k = n).For = 1; By the inductive hypothesis let {{Ti : i < } : < }

be disjoint trees satisfying the statement of the Theorem. Denote by Sthe set {, i : i < , < }. Fix an injective mapping from S into

On Sup(

,i Ti ), denote by ,i the image of the pair , i. For every

< , and for every A S cardinality define

TA := {} {,i : Ti , , i A}.

The order on TA is defined in the natural way: is the root, and, i 1


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there exists y Suc(y) such that for every x Suc(x) we have that

T[x] ++1 T[y] holds.In order to complete the proof of Theorem 2.8, it suffices to prove the

following:

SubClaim 2.15 1. (x, y) L+,,

2. M has the (,)-order property,

3. There do not exist a formula (x, y) L+, such that has the((x, y), )-order property.

Proof.

1. Let T be a well founded tree, and let < + be given, it is enoughto show by induction on that there exists a formula (x, y) L+,such that for every a, b T we have that T |= [a, b] iff T[a] T[b].It is easy to check that the relation is definable in L+,.

2. Check that for every i1, i2 < we have that

i1 < i2 iff M |= [i1, i2].

3. For the sake of contradiction suppose that there exists a formula(x, y) L+, such that has the (

, )-order property. Supposethat is a limit ordinal < + such that the formula has quantifier

depth<

. Denote by

the cardinality (

+1(|L

|))

+

. LetN

|=be a model of cardinality such that there exists {ai : i < }

such that l(x) = l(y) = l(ai) = n < and for every i1, i2 < we have i1 < i2 N |=

[ai1 , ai2] holds. For every i < fixbil : l < n = ai. By the L1,-part of the definition of wehave that N |= (x)

m


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1. 2(, , ) := min{ : is a set of T-types , || , |T|

if < M EC(T, ) with otp(PM,


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Theorem 2.27 (A better lower bound): Suppose J is an1-complete ideal

on such that(*) J as an ideal is generated by sets or at least we have(**) there exists a model B (of an expansion of set theory) with universe, |L(B)| and (P) L+,, when L = L(B) {P}, P is a unarypredicate; having the following property:

J for every A , we have that A J B, A |= (P)

or at least

J for every A , we have that A J iff for some A we have thatA A J, B, A |= (P)

then for every ordinal > we have that J < 2(, ).

Remark 2.28 1. One way to see that Theorem 2.26 is a special case ofTheorem 2.27 is by using the same argument. Another formal argu-ment (using the statement of 2.27) we can take B := ,


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For l() > 0; If i ds() then f is defined, and by the inductive hy-

pothesis we have that f iJ > i (as i ds() and fJ = ), bythe definition of the GalvinHajnal rank there exists f


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3 Concluding Remarks

It is natural to ask whether the lower bound from Theorem 2.27 is equal tothe one in Fact 2.21. The following seems to be a reasonable

Conjecture 3.1 For cardinalities of uncounable cofinality and suchthat 2 < we have 1(, ) = (cov(, ) + 2

)+.

Remark 3.2 1. Notice that when is strong limit singular of cofinality0 then the conjecture holds.

2. Why 2 < See Barwise-Kunen for independence results.

3. The conjecture can to large extent be traslated to a one on pcf; it isevident that e.g. (***) below is a sufficient condition:

(***) for any set a of regular cardinals which are > 2 the setpcf(a) has cardinality at most , or at least the set pcf1complete(a)has cardinality at most .

This is because by [Sh g] II 5.4 if 2 < then = cov(, ) is thefirst such that if {i : i < } is a set of regular cardinalities in theinterval (2, ) and J is an 1-complete ideal on and cf(

i


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References

[BaKu] K. J. Barwise and K. Kunen, Hanf numbers for fragments of L,,Israel J. of Mathematics, 10, 306320, 1971

[Ch] C. C. Chang, Some remarks on the model theory of infinitary lan-guages, The Syntax and Semantics of Infinitary Languages,Springer-Verlag Lecture Notes in Mathematics 72, edited by J. Bar-wise (1968) pages 3663.

[Gr] Rami Grossberg, Classification theory for non elementaryclasses, Ph.D. Thesis (in Hebrew), The Hebrew University of Jerusalem1986.

[GrSh] Rami Grossberg and Saharon Shelah, On the number of noniso-morphic models of an infinitary theory which has the infinitary or-der property. I, The Journal of Symbolic Logic, 51:302322, 1986. MR: 87j:03037, (03C45)

[GrSh 238] Rami Grossberg and Saharon Shelah, A nonstructure theoremfor an infinitary theory which has the unsuperstability property, IllinoisJournal of Mathematics, 30:364390, 1986. Volume dedicated to thememory of W.W. Boone; ed. Appel, K., Higman, G., Robinson, D. andJockush, C. MR: 87j:03036, (03C45)

[HrSh 334] Ehud Hrushovski and Saharon Shelah, Stability and omit-ting types, Israel Journal of Mathematics, 74:289321, 1991. MR: 92m:03049, (03C45)

[Ma] Menachem Magidor, Changs conjecture and powers of singular cardi-nals, The Journal of Symbolic Logic, 42: 272276, 1977.

[MaSh 285] Michael Makkai and Saharon Shelah, Categoricity of theoriesin L, with a compact cardinal, Annals of Pure and Applied Logic,47:4197, 1990. MR: 92a:03054, (03C75)

[Mo] Michael Morley, Omitting classes of elements, The Theory of Mod-

els, edited by Addison, Henkin and Tarski, North-Holland Publ Co(1965) pages 265273.

[Sh c] Saharon Shelah. Classification theory and the number of nonisomor-phic models, volume 92 ofStudies in Logic and the Foundations of Math-ematics. North-Holland Publishing Co., Amsterdam, xxxiv+705 pp,1990.

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[Sh g] Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic

Guides. Oxford University Press, 1994.

[Sh 12] Saharon Shelah. The number of non-isomorphic models of an unsta-ble first-order theory. Israel Journal of Mathematics, 9:473487, 1971. MR: 43-4652.

[Sh 16] Saharon Shelah, A combinatorial problem; stability and order formodels and theories in infinitary languages, Pacific Journal of Mathe-matics, 41:247261, 1972. MR: 46:7018, (02H10)

[Sh 87a] Saharon Shelah. Classification theory for nonelementary classes, I.The number of uncountable models of L1,. Part A. Israel Journal

of Mathematics, 46:212240, 1983.

[Sh 87b] Saharon Shelah. Classification theory for nonelementary classes, I.The number of uncountable models of L1,. Part B. Israel Journalof Mathematics, 46:241273, 1983.

[Sh 88] Saharon Shelah. Classification of nonelementary classes. II. Abstractelementary classes. In Classification theory (Chicago, IL, 1985), volume1292 ofLecture Notes in Mathematics, pages 419497. Springer, Berlin,1987. Proceedings of the USAIsrael Conference on Classification The-ory, Chicago, December 1985; ed. Baldwin, J.T.

[Sh 300] Saharon Shelah. Universal classes. In Classification theory(Chicago, IL, 1985), volume 1292 of Lecture Notes in Mathematics,pages 264418. Springer, Berlin, 1987. Proceedings of the USAIsraelConference on Classification Theory, Chicago, December 1985; ed.Baldwin, J.T.

[Sh 299] Saharon Shelah. Taxonomy of universal and other classes. In Pro-ceedings of the International Congress of Mathematicians (Berkeley,Calif., 1986), volume 1, pages 154162. Amer. Math. Soc., Providence,RI, 1987. ed. Gleason, A.M.

[Sh 78] Saharon Shelah. Hanf number of omitting type for simple first-ordertheories. The Journal of Symbolic Logic, 44:319324, 1979.

[Sh 48] Saharon Shelah. Categoricity in 1 of sentences in L1,(Q). IsraelJournal of Mathematics, 20:127148, 1975.

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a structure theorem. American Mathematical Society. Bulletin. NewSeries, 12:227232, 1985.

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rami grossberg and saharon shelah- on hanf numbers of the infinitary order property

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