J. Baldwin and S. Shelah- Abstract Classes with few models have `homogeneous-universal' models

8/3/2019 J. Baldwin and S. Shelah- Abstract Classes with few models have `homogeneous-universal' models

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Abstract Classes with few models have

homogeneous-universal models

J. Baldwin

Department of Mathematics

University of Illinois, Chicago

S. Shelah Department of Mathematics

Hebrew University of Jerusalem

October 6, 2003

This paper is concerned with a class K of models and an abstract notionof submodel . Experience in first order model theory has shown the de-sirability of finding a monster model to serve as a universal domain for K.

In the original constructions of Jonsson and Fraisse, K was a universal classand ordinary substructure played the role of . Working with a cardinal satisfying


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homogenous-universal model of power

. We show (5.25) in a class with amal-gamation this dichotomy holds for the notion of K-homogeneous-universalmodel in the more normal sense.

The methods here allow us to improve our earlier results [5] in two otherways: certain requirements on all chains of a given length are replaced byrequiring winning strategies in certain games; the notion of a canonicallyprime model is avoided. A full understanding of these extensions requiresconsideration of the earlier papers but we summarise them quickly here.

Shelah emphasized in [13] that Tarskis union theorem has two compo-nents: closure under unions; each union is an amalgamation base (smooth-ness). The first is used to show the existence of a homogeneous universal

model; the second is needed for uniqueness. In this paper we show that clo-sure can be replaced by the existence of a bound for each chain and evenstronger that we need the boundedness only for a dense (in a sense madeprecise by a game defined below) set of chains.

In [5] we established a dichotomy between the smoothness of a class anda nonstructure theorem. There was a weakness in our result; although thedefinition of smooth (there is a unique compatibility class over each chain)does not involve the concept of a canonically prime model we only estab-lished the theorem for classes equipped with a notion of a canonically primemodel. We remedy that difficulty in this paper at some cost. First we require

some additional set theoretic hypotheses (all provable if V=L). Second wemust weaken the conclusion. Instead of coding stationary sets we can onlyguarantee that there are 2 models of power .

The results here generalize an earlier result proved in [12]. That paperdealt with a class K satisfying the axioms discussed here but also closedunder unions of K-chains and that was smooth. Theorem 3.5 and Claim 3.4of [12] imply that if K is categorical in and has few models of power +

then the unique model of power is an amalgamation base.We rely on many notations and definitions from [4] and [5] but only on

rudimentary results from those papers.

Section 1 contains the background notation. In Section 2 of this pa-per we introduce several games; we are able to express questions about thesmoothness or boundedness of a class K in terms of winning strategies forthese games. Section 3 describes the set theoretic hypotheses necessary forour construction. We show in Section 4 that a winning strategy for PlayerNAM in Game 2 (, ) implies the existence of many models. Section 5

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translates the existence of a winning strategy for a player (B) trying to showchains are bounded and the failure of the player trying to force nonamalga-mation (NAM) to have a winning strategy into the existence and, if there arefew models, the uniqueness of chain homogeneous-universal models. For aclass with amalgamation this yields uniqueness and existence of the (K, )homogeneous-universal models. In Section 6 we summarise our results andsuggest some open problems. Section 7 contains proofs of the combinatorialresults summarised in Section 3.

We thank Chris Laskowski and Bradd Hart for their valuable advice inpreparing this paper.

1 Setting the Scene

Most of the notions used in this paper are defined in [4] or [5]. They or minorvariants occur in earlier papers of Shelah, specifically [13].

(K, ) is an abstract class satisfying Axiom group A of [4]:

A0 If M K then M M.

A1 If M N then M is a substructure of N

A2 is transitive.

A3 If M0 M1 N, M0 N and M1 N then M0 M1.

We review here some of the less common concepts. All notions defined withcardinal parameters have the obvious variants obtained by, e.g., replacing by < . K is the class of members of K with cardinality . A (< ,)chain is a K-increasing chain of cofinality members of K (i < j impliesMi Mj), each of cardinality < . A chain M is K-bounded if there isan M K and a compatible family of maps fi mapping Mi into M. K is(< , )-bounded if each (< , )-chain is bounded. K is (< , < )-closed ifthe union of each such chain is in K. Sections 2 and 4 of [5] contain a numberof examples that illustrate these concepts.

1.1 Assumptions. We fix for this paper a cardinal with the followingproperties.

i) is a regular cardinal greater than the size of the vocabulary of K.

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ii) There are no maximal models in K


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Either player loses the game if at some stage he does not have a legalmove. Player B wins ifP is bounded; otherwise Player NB wins. PlayerB has the advantage of playing first at limit ordinals; the price is thathe must guarantee the existence of a bound at each limit stage.

ii) We say Player B has a winning strategy for Game 1 (, < R) if he hasa uniform strategy to win all plays of Game 1 (, ) for each ordinal < R.

2.2 Proposition. If K is (< , )-bounded then Player B has a winningstrategy for Game 1 (, ).

We continue to use the notations for properties of embeddings of chainsestablished in [5] and briefly reviewed in Section 1.

2.3 Definition. i) The K-increasing chain N of members ofK


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NB). A player who has no legal move loses; however, Player NAM winsinstantly if for some limit ordinal , P| is bounded but is not a-amalgamation base.

ii) Game 2 (, < ) is defined similarly, but Player NAM wins only if forsome < , P| is bounded but is not a -amalgamation base.

Note that if player NAM wins instantly at some stage then the length ofthe game is less than .

In both games the decision about a completed game is based only on thesecond players moves (NB or AM); apparently to win the first player mustforce the second to make mistakes. The following lemma shows this intuition

is misleading.

2.5 Lemma. Player B (NAM) wins Game 1 (Game 2) (, ) if and only ifthe sequence L constructed during the play is bounded (and is not an amal-gamation base).

Proof. A chain is bounded (an amalgamation base) if and only if eachcofinal subsequence is.

The preceding remark is quite straightforward; contrast it with the dif-ficulties involved in considering canonically prime models over subsequences[5].

Note that if player NAM plays a winning strategy for Game 2, he alsoplaying a winning strategy for Game 1.

3 23

We discuss in this section the principle 23 from [3]. It is a combination ofJensens combinatorial principles 2 and 3 that will be used in our mainconstruction. Some justification is necessary for the use of such strong settheory. On the one hand, arguments with a strong set theoretic hypothesis

that conclude the existence of many nonisomorphic models show it is impos-sible to prove in ZFC that K has few models. Thus any structure theorythat could be established for K in ZFC would have to allow the maximalnumber of models in a class with structure. On the other, while for ease ofstatement we assert that these combinatorial principles follows from V = L,in fact they only depend on the structure of the subsets of . Thus 2 can

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be as large as desired while keeping23

,,R for each <

and someR

.Moreover, the consistency of23 can be obtained by a forcing extension aswell as by an inner model. Some instances of23 depend only on appropriateinstances of GCH. We begin by establishing some notation.

3.1 Notation. i) For any set of ordinals C, acc[C] denotes the set ofaccumulation points of C, i.e., the C with = sup C . nacc[C]denotes C acc[C].

ii) Fix regular cardinals > ; let C(S) denote the set of S whichhave cofinality . In the following always denotes a limit ordinal.

iii) Suppose there is S and a collection C : S. Then for anyS1 S, S1 denotes S1 {C : S1}.

3.2 Definition. [23,,R(S)] We say that C = C : S and the sequenceA = A : < witness that ,,R satisfy [23,,R(S)] if for some subsetS1 of S the following conditions hold.

i) is a regular cardinal < , R is an ordinal .

ii) S is stationary in and contains all limit with cf() < R; Scontains only even ordinals.

iii) S1 C(S).

iv) Each C S.

v) If S

(a) C is a closed subset of ,

(b) if C then C = C ,

(c) otp(C) max(R, ), more precisely,

1. otp(C) = if S1 and2. C S1 = and otp(C) < R if S S1 is a limit ordinal,

(d) all nonaccumulation points ofC are even successor ordinals.

vi) If S is a limit ordinal then C is a club in .

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vii) EachA

and(a) if S1 and acc[C] then A = A ,

(b) for A and any closed unbounded subset E of , XE = { S1 : E& acc[C] E & A = A } is stationary in .

3.3 Definition. 23,,R holds if for some subset S , 23,,R(S) holds.We discuss the truth of this proposition for various choices of and .

We begin with a case proved in ZFC; some cases of the GCH are needed tofind cardinals satisfying the hypotheses. Combining the methods of [3], [11],and [8] yields the following result; we include a full proof in an appendix at

the suggestion of the referee.

3.4 Lemma. Suppose = and2 = + = . If S S() is stationarythen there is an S such that23,, holds and S C() S.

This combinatorial result is sufficient for our model theoretic construc-tions if the problematical model theoretic situation concerns chains of length ( = . To deal with chains of longer length (this is essential; see [5]), thefollowing stronger combinatorial principal is needed.

Lemma 3.5 (V=L). If = +

and , then for some stationary S,23,,(S).

4 Players B and NAM construct many mod-

els

We show in this section that if for some < and an ordinal R Player B has a winning strategy for Game 1 (, ) and Game 1 (,< R),K is (, R)-bounded, and Player NAM has a winning strategy for Game2 (< , ) then K has the maximal number of models in power . Thisis of course a technically weaker conclusion than in [5] where we showedthat if K is not smooth then K codes stationary sets. But we have weakerhypotheses here and this conclusion expresses a somewhat weaker intuition ofnonstructure. This weakening of the result is reflected in a complication of themain argument. In the earlier paper we constructed for each stationary set W

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a modelMW

codingW

. Here, we construct 2

models simultaneously anddestroy putative isomorphisms between them enroute. The second authorhas in mind a more elaborate version of our construction, which we dontexpand on here, that recovers the coding of stationary sets in this context.The stronger conclusion in [5] assumed the class K was equipped with notionsof free amalgamation and canonically prime models; here we have no suchassumption.

The many-models proof given here illustrates the role of canonically primemodel notion. When cpr is in the formal metalanguage a particular choice oflimit model is specified so one can construct a sequence of models and code astationary set by asking the question, Is the limit model at the canonically

prime model? When we remove this notion from the formal language wehave to destroy isomorphisms between the possible choices of a limit model.Diamond allows us to do this.

We thank Bradd Hart for suggesting a simplification in the proof of themain result.

4.1 Theorem. Fix regular cardinals < and an ordinal R satisfying23,,R. Suppose

i) K is (< , R)-bounded; Player B has a winning strategy for Game 1

(, < R).ii) Player NAM has a winning strategy for Game 2 (, ).

iii) Player B has a winning strategy for Game 1 (, ).

Then there are 2 members of K with cardinality . Moreover, these modelsare mutually K-non-embeddible.

4.2 Remark. The connection between hypotheses i) and iii) deserves somecomment. If > R then iii) implies the second clause of i). It is temptingto think that i) implies iii). However, if > R a game of length > might

still have cofinality < R. Neither clause of i) guarantees a winning stategyfor that game.Proof. Fix S, Ci : i S and A : < and S1 to witness 23,,R.

Using a pairing function and condition vii) of Definition 3.2 we can find, , f : < with , in 2

and f a function from to such that

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for any,

, any functionf

:

, and any closed unbounded subsetE of , for some S1 E, acc[C] E, and for every {} acc[C]we have = , = , f restricted to is f.

By induction, for each < and each 2 we define a structure Nwhose universe is an ordinal < so that if < , 2, 2 N is aproper K-submodel of N. Then we finish the construction by defining foreach 2, N as


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4.5 Construction. We split the construction into several cases. LetS1 = S1 {C : S1}.

Most cases in the construction are defined by playing the winning strategyof Player B or Player NAM on a closed unbounded subset of . Conditionv) of23,,R(S) guarantees that the various cases cohere. Certain inductiveproperties of the construction are incorporated in the description of the cases.

At stage , we have fixed , 2 and a map f from to . For each 2 we construct a model N. In many cases N is chosen by playingthe winning strategy for Player B or player NAM on N|C. To see that

these strategies do not conflict note first that if every play on N|C hasbeen played by the winning strategy in either game then it has been playedaccording to a winning strategy for Game 1 (as the winning strategy forGame 2 also wins game 1) and so inductively N can be chosen by PlayerBs winning strategy in Game 1. Moreover, since Player B has a uniformwinning strategy for all games of length less than his play does not dependon the particular game of length less than that is being considered. Theinductive hypothesis in the cases where Player NAMs winning strategy inGame 2 is used are verified below. The key step in the proof is subcase b) ofCase IV. The other stages are preserving the induction hypothesis.

Case I. is a successor ordinal + 1. We will choose N as a proper K-extension of N|. This is possible since we have assumed that thereare no maximal models in K


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Case III. is a limit ordinal and (S S1). Since cf(C) < R, we canchoose N to bound N|, by playing Player Bs winning strategy for

Game 1 on N|C. (Note C is unbounded in by Definition 3.2 vi).

Case IV. S1; a limit ordinal. The situation is interesting only if = N = N, f|N| is a K-embedding for each < and = .Unless all of these conditions hold, choose N as in Case Ib) by playingNAMs strategy on N|C or Bs winning stategy on N|C,. If theydo hold there are two subcases.

Subcase a. N is an amalgamation base. If C then by Sub-

case Ib, or Subcase IV b) at stage , N| was chosen by PlayerNAMs winning strategy on N|C. So, we are able to applyPlayer NAMs winning strategy for Game 2 (, ) on N|C tochoose N.

Subcase b. N is not an amalgamation base. N has been defined.Thus there are incompatible bounds A1, A2 K


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guarantees there is no extension of

f|

mappingN

| into any extension ofN| and thus no extension of f| mapping N| into any extension of N|.Thus there is no K-embedding of N into N and we finish.

5 Classes with few models have homogeneous-

universal models

We consider here several variants on the notion of homogeneous-universalmodel and establish the existence and uniqueness of models satisfying one ofthese notions for a class that has few models.

5.1 Assumption. In this section we assume that the class K has a notion ofstrong submodel satisfying the axioms of group A in [4] (listed in Section 1),the properties of enumerated in Section 1 and

i) K has fewer than 2 models of power .

ii) K is (< , < )-bounded.

All the results of this section go through under these assumptions. Since someof them require slightly less, many of the statements repeat these overriding

hypotheses or stipulate more technical conditions that suffice.The following obvious consequence of Theorem 4.1 is a key to this section.

5.2 Lemma. For a regular cardinal < such that some regular R , satisfies 23,,R, Assumption 5.1 implies that Player NAM does not have awinning strategy for Game 2 (, ).

5.3 Refinement. Checking the hypotheses for Theorem 4.1, we see thatinstead of assuming K is (< , < )-bounded it suffices to assume K is ( 0 in divisible by and let E denote the accumulations points of E.

We define by induction on E a sequence, increasing with , C1|( +1) = C1 : .

i) = min E: Thus, = . If is even C1 is the set of even ordinalsless than ; if is odd, C1 is empty.

ii) is a successor in E: Thus = + where E and C1|( + 1)has been defined. Fix a map h from onto

.

(a) < and 4|: C1 consists of those ordinals in (, ) thatare divisible by 4.

(b) < and is odd: C1 is empty.

(c) < and 4 | but 2|: Thus has the form + 4i + 2for some i.

otp C1h(i) < and h(i) is even: C1 = C

1h1(i)

{h(i)}

otherwise: C1 is empty.

iii) = E, i.e a limit in E: we only have to define C1 .

(a) acc[C] E

= : Necessarily cf() = 0. Choose an increasingsequence {n : n < } from E with limit . Then choose byinduction on n successor even ordinals n with n < n < n + such that C1m = {n : n < m}. (The third clause of the previouscase is the key to this induction.) Finally, let C1 be the set ofn.

(b) acc[C] E = but acc[C] E is bounded in by some .Choose n with limit

and n as before but so that C1n

= C {} {m : m < n}. Then let C

1 = C {

} {m : m < }.

(c) = sup(acc[C] E): Let C1 = CEC1.

Now one shows by induction on

that if

C1

, thenC1

=C1

.(The other requirements on the C1 are easily verified.)If < we modify the C1 defined in the first stage of the proof as follows.

If otp C1 then C2 = C

1. If otp C

1 > then C

2 = { C

1 : otp C

1 > }.

Let S1 = { < : cf() = & otp C1 = }. And S1 = { < : otp C

1

}.

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7.3 Concluding Remarks.i) Examination of our proof of Lemma 3.4 shows that in fact given a

square sequence, it is possible to add on a diamond sequence to satisfy23,,R(S). We state this explicitly in Lemma 7.4 below.

ii) The proof of Lemma 3.5 can be extended to allow for R < .

iii) In fact, the derivation of23,,R for R from the assumption that = , = 2 and there is a square on { < : cf() < R} but withoutassuming V=L will be published elsewhere by the second author.

iv) Similar results hold assuming V=L for inaccessible and > withR should follow by the methods of Beller and Litman [6] but wehave not checked this in detail.

7.4 Lemma. Suppose 2 = + = , = , and S, R, C : S havebeen chosen to satisfy all conditions of23,,R(S) except ii) and vii). Supposefurther that for each cub E of , { : acc[C] E}is stationary in . Then23,,R(S) holds.

References

[1] M. Karlowicz A. Engelking. Some theorems of set theory and theirtopological consequences. Fundamenta Mathematica, 57:275285, 1965.

[2] M. Albert and R. Grossberg. Rich models. Journal of Symbolic Logic,55:12921298, 1990.

[3] U. Avraham, S. Shelah, and R. Solovay. Squares with diamonds andSouslin trees with special squares. Fundamenta Mathematica, 127:133162, 1987.

[4] J.T. Baldwin and S. Shelah. The primal framework: I. Annals of Pureand Applied Logic, 46:235264, 1990.

[5] J.T. Baldwin and S. Shelah. The primal framework II: Smoothness.Annals of Pure and Applied Logic, 55:134, 1991.

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[6] A. Beller and A. Litman. A strengthening of Jensens2

principles.Journal of Symbolic Logic, 45:251264, 1980.

[7] J. A. Makowsky. Abstract embedding relations. In J. Barwise andS. Feferman, editors, Model-Theoretic Logics, pages 747792. Springer-Verlag, 1985.

[8] S. Shelah. Reflection of stationary sets and successor of singulars.preprint 351: to appear in Archive fur Mat. Log.

[9] S. Shelah. On the number of nonisomorphic models of cardinality L-equivalent to a fixed model. Notre Dame Journal of Formal Logic,22:510, 1981.

[10] S. Shelah. Models with second order properties IV, A general methodand eliminating diamonds. Annals of Mathematical Logic, 38:183212,1983.

[11] S. Shelah. Remarks on squares. In Around Classification Theory ofModels. Springer-Verlag, 1986. Springer Lecture Notes 1182.

[12] S. Shelah. Nonelementary classes II. In J. Baldwin, editor, ClassificationTheory, Chicago 1985. Springer-Verlag, 1987. Springer Lecture Notes

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[13] S. Shelah. Universal classes: Part 1. In J. Baldwin, editor, ClassificationTheory, Chicago 1985, pages 264419. Springer-Verlag, 1987. SpringerLecture Notes 1292.

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J. Baldwin and S. Shelah- Abstract Classes with few models have `homogeneous-universal' models

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