Classification Theory for Abstract Elementary Classes [2] 9781904987727

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Studies in Logic: Mathematical Logic and Foundations, Vol 20

Classification Theory for Abstract Elementary Classes Volume 2

Saharon Shelah

Contents

A Annotated Contents

1

V.A Universal Classes: Stability theory for a model – Sh300A

38

V.B Universal Classes: Axiomatic Framework – Sh300B

97

V.C Universal Classes: A frame is not smooth or not χ-based – Sh300C

140

V.D Universal Classes: Non-Forking and Prime Models – Sh300-D

189

V.E Universal Classes: Types of finite sequences – Sh300E

234

V.F Universal Classes: The Heart of the Matter – Sh300-F

295

V.G Universal Classes: Changing the framework – Sh300g

356

VI Categoricity of an a.e.c. in two successive cardinals revisited – E46

VII Non-structure in λ++ using instances of WGCH – Sh838

Bibliography

365

482

683

Paper Sh:E53, Annotated Content

ANNOTATED CONTENTS

Annotated Content for Ch.N (E53): Introduction (This chapter appeared in book 1.) Abstract §1 Introduction for model theorists, (A) Why to be interested in dividing lines, (B) Historical comments on non-elementary classes, §2 Introduction for the logically challenged, (A) What are we after? [We first explain by examples and then give a full definition of an a.e.c. (abstract elementary class), central in our context, K = (K, ≤K ), with K a class of models (= structures), ≤K a special notion of being a submodel, it means having only the quite few of the properties of an elementary class (like closure under direct limit). Such a class is (ModT , ≺) with M ≺ N meaning “being an elementary submodel”; but also the class of locally finite groups with ⊆ is O.K. Second, we explain what is a superlimit model (meaning mainly that a ≤K -increasing chain of models isomorphic to it has a union isomorphic to it (if not of larger cardinality). We can define “an a.e.c. is superstable” if it has a superlimit model in every large enough cardinality. For first order class this is an equivalent definition. A stronger condition (still equivalent Typeset by AMS-TEX

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for elementary classes) is being solvable: there is a PCλ,λ class, i.e the class of reducts of some ψ ∈ Lλ+ ,ω which, in large enough cardinality, is the class of superlimit models; similarly we define being (µ, λ)-solvable. Of course we investigate the one cardinal version (hoping for equivalent behaviour) in all large enough cardinals, etc. We state the problem of the categoricity spectrum and the solvability spectrum. We finish explaining the parallel situation for first order classes and explain “dividing lines”.] (B) The structure/non-structure dichotomy, ˙ K) counting the number of non[We define the function I(λ, isomorphic models from K of cardinality λ, define the main gap conjecture, phrase and discuss some thesis explaining an outlook and intention. We then explain the main gap conjecture and the case it was proved and list the possible reasons for having many models. We then discuss dividing lines and their relevance to our problems.] (C) Abstract elementary classes, [We shall deal with a.e.c., good λ-frames and beautiful λframes. The first is very wide so we have to justify it by showing that we can say something about them, that there is a theory; the last has excellent theory and we have to justify it by showing that it arises from assumptions like few nonisomorphic models (and help prove theorems not mentioning it); the middle one needs justifications of both kinds. In this part, we concentrate on the first, a.e.c., explain the meaning of the definition, discuss examples, phrase our opinion on its place as a thesis, and present two theorems showing the ˙ κ) is not “arbitrary” under mild set theoretic function I(λ, conditions.] (D) Toward good λ-frames, [We explain how we arrive to “good λ-frame s” mentioned above, which is our central notion; it may be considered a “bare bone case of superstable class in one cardinal”. We

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choose to concentrate on one cardinal λ, so Ks = Kλ . Also we may assume Kλ has a superlimit model, and that it has amalgamation and the joint embedding property, so only in λ! Amalgamation is an “expensive” assumption, but amalgamation in one cardinal is much less so. This crucial difference holds because it is much easier to prove amalgamation in one cardinality (e.g. follows from having one model in λ (or a superlimit one) and few models in λ+ up to isomorphism and mild set theoretic assumptions). We are interested in something like “M1 , M2 are in non-forking (= free) amalgamation over M0 inside M3 ”. But in the axioms we only have “an element a and model M1 are in non-forking amalgamation over M0 inside M3 , equivalently tps (a, M1 , M3 ) does not fork over M0 ”, however the type is orbital, i.e. defined by the existence mapping and not by formulas. There are some further demands saying non-forking behave reasonably (mainly: existence/uniqueness of extensions, transitivity and a kind of symmetry). So far we have described a good λframe. Now we consider a dividing line - density of the class of appropriate triples (M, N, a) with unique amalgamation. ˙ ++ , K s ) is large if 2λ < 2λ+ < 2λ++ , Failure of this gives I(λ from success (i.e. density) we derive the existence of nonforking amalgamation of models in Ks . After considering a further dividing line we get s+ , a good λ+ -frame such that + Kµs ⊆ Kµs for µ ≥ λ+ . All this (in Chapter II) gives the + +n theorem: if 2λ < 2λ < . . . < 2λ , LS(K) ≤ λ, K categorical in λ, λ+ , has a model in λ+2 and has not too many models in λ+2 , . . . , λ+n then K has a model in λ+n+1 . If this holds for every n, we get categoricity in all cardinals µ ≥ λ. For the first result (from Chapter II) we just need to go from s to s+ , for the second (from Chapter III) need considerably more.] §3 Good λ-frames, (A) getting a good λ-frame, [We deal more elaborately on how to get a good λ-frame

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starting with few non-isomorphic models in some cardinals. + If K is categorical in λ, λ+ and 2λ < 2λ we know that K has amalgamation in λ. Now we define the (orbital) type tpKλ (a, M, N ) for M ≤K N, a ∈ N . Instead of dealing with SKλ (M ), the set of such types, we deal with Kλ3,na = {(M, N, a) : M ≤Kλ N and a ∈ N \M }, ordered naturally (fixing a!) The point is of dealing with triples, not just types, is the closureness under increasing unions, so existence of limit. Now we ask: are there enough minimal triples? (which means with no two contradictory extensions). If no, we have a non-structure result. If yes, we can deduce more and eventually get a good λ-frame. Here we consider Ks3,bs = {(M, N, a) : M ≤Ks N, tp(a, M, N ) ∈ Ssbs (M ), i.e. is a basic type} (this is part of the basic notions of a good λ-frame s).] (B) the successor of a good λ-frame, [We elaborate the use of successive good frames in Chapter II. If s is a good λ-frame, we investigate “N is a brimmed extension of M in Ks = Ksλ ”, it is used here instead of satus rated models, noting that as K µ ≥ cf(µ) > LS(K) and K N , the ∆type which it realizes over M inside N is the average of some (∆, χ+ )-convergent set of cardinality µ+ inside M . We give an alternative definition of being a submodel (in V.A.4.4) when M has an appropriate non-order property, prove their equivalence and note some basic properties supporting the thesis that this is a reasonable notion of being a submodel. We then define “stable amalgamation of M1 , M2 over M0 inside M3 ” and investigate it to some extent.] V.A.§5 On the non-order implying the existence of indiscernibility [We give a sufficient condition for the existence of “large” indiscernible set J ⊆ I, in which |J| < |I|, but the demand on the non-order property is weaker than in V.A.§2 speaking only on non-order among singletons. Even for some first order T which are unstable, this gives new cases e.g. for ∆ = the set of quantifier free formulas.] Annotated Content for Ch.V.B (300b): Axiomatic framework V.B.§0 Introduction [Rather than continuing to deal with universal classes per se, we introduce some frameworks, deal with them a little and show that universal classes with the (χ, < ℵ0 )-non-order property fit some of them (for suitable choices of the extra relations). In the rest of Chapter II almost always we deal with AxFr1 only.]

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V.B.§1 The Framework [We suggest several axiomatizations of being “a class of models K with partial order ≤K with non-forking and possibly the submodel generated by a subset” (so being a submodel, non-forking and hAign M for A ⊆ M are abstract notions). The main one here, AxFr1 is satisfied by any universal class with (χ, < ℵ0 )-non-order; (see §2). For AxFr1 if M1 , M2 are in non-forking amalgamation over M0 inside M3 then the union M1 ∪ M2 generate a ≤K -submodel of M3 . In such contexts we define a type as an orbit, i.e. by arrows (without formulas or logic); to distinguish we write tp (rather than tp∆ ) for such types. Also “Tarski-Vaught theorem” is divided to components. On the one hand we consider union existence Ax(A4) which says that: the union of an ≤K -increasing chain belongs to the class and is ≤K -above each member. On the other hand we consider smoothness which says that any ≤K upper bound is ≤K -above the union.] V.B.§2 The Main Example [We consider a universal class K with no “long” linear orders, e.g. by quantifier free formulas (on χ-tuples), we investigate the class K with a submodel notion introduced in V.A§4, and a notion of non-forking, and prove that it falls under the main case of the previous section. We also show how the first order case fits in and how (D, λ)-homogeneous models does.] V.B.§3 Existence/Uniqueness of Homogeneous quite Universal Models [We investigate a model homogeneity, toward this we define Dχ (M ), Dχ (K), D′K,χ and define “M is (D, λ)-model homogeneous”. We show that being λ+ -homogeneous λ-universal model in K can be characterized by the realization of types of singletons over models (as in the first order case) so having “the best of both worlds”.]

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Annotated Content for Ch.V.C (300c): A frame is not smooth or not χ-based V.C.§0 Introduction [The two dividing lines dealt with here have no parallel in the first order case, or you may say they are further parallels to stable/unstable, i.e. stability “suffer from schizophrenia”, there are distinctions between versions which disappear in the first order case, but still are interesting dividing lines.] V.C.§1 Non-smooth stability [This section deals with proving basic facts inside AxFr1 . On the one hand we assume we are hampered by the possible lack of smoothness, on the other hand the properties of h−ign M are helpful. These claims usually say that specific cases of smoothness, continuity and non-forking hold. So it deals with the (meagre) positive theory in this restrictive context.] V.C.§2 Non-smoothness implies non-structure [We start with a case of failure of κ-smoothness, copy it many times on a tree T ⊆ κ≥ λ; for each i < κ for every η ∈ T ∩ i λ we copy the same things while for η ∈ T ∩ κ λ we have a free choice. This is the cause of non-structure, but to prove this we have to rely heavily on §1. If we assume the existence of unions, for any χ++ of cofinality χ+ , we build a model in Ksλ which codes any subset S of S ∗ (modulo the club filter) hence get a non-structure theorem. Naturally we use the stable constructions from the previous section, §4 and have some relatives.]

Annotated Content for Ch.V.D (300d): Non-forking and prime models V.D.§0 Introduction [Here we deal with types of models (rather than types of single elements). This is O.K. for parallel to some properties of stable first order theories T , mainly dealing with |T |+ saturated models.] V.D.§1 Being smooth and based propagate up [By Chapter V.C we know that failure of smoothness and failure of being χ-based are non-structure properties, but they may fail only for some large cardinal. We certainly prefer

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to be able to prove that faillure, if it happens at all, happens for some quite small cardinal; we do not know how to do it for each property separately. But we show that if s is (≤ χ, ≤ χ+ )-smooth and (χ+ , χ)-based and LSP(χ) then for every µ ≥ χ, s is (≤ µ, ≤ µ)-based, and (≤ µ, ≤ µ)-smoothed and has the LSP(µ). So it is enough to look at what occurs in cardinality LS(Ks ) for the non-structure possibility (rather than “for some χ”). We then by Chapter V.C get a nonstructure result from the failure of the assumption above. We also investigate when Ks has arbitrarily large models. So being “(≤ χ, ≤ χ+ )-smooth, (χ+ , χ)-based, LSP(χ)” is a good dividing line.] V.D.§2 Primeness [We define prime models (over A), isolation (for types of the form N/M + c) and primary models. We prove the existence of enough isolated types; the difference with the first order case is that we need to deal with M