Classification theory for abstract elementary classes

Grossberg, Rami

doi:10.1090/conm/302/05080

Cited by 94 publications

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“…The definitions and concepts in the section are all part of the literature; in particular, see the books by Baldwin [Bal09] and Shelah [Sh:h], the survey article by Grossberg [Gro02], or the forthcoming book by Grossberg [Gro1X] for general information…”

Section: Preliminariesmentioning

confidence: 99%

Tameness From Large Cardinal Axioms

Boney¹

2014

J. symb. log.

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Abstract. We show that Shelah's Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ-tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called type shortness, and show that it follows similarly from large cardinals.

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Section: Preliminariesmentioning

confidence: 99%

Tameness From Large Cardinal Axioms

Boney¹

2014

J. symb. log.

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“…a kind of Galois type) but considers types over arbitrary subsets. Key to this is the observation that the proof of transitivity in establishing an equivalence relation by tp A (a) = tp B (b) if there is an automorphism of the universe taking a to b uses amalgamation not of the domain models (in the sense of Shelah's definition [She83a,She99,Gro02]) but rather amalgamation of the models in which the types are realized. They work in the general context of AEC with arbitrarily large models and amalgamation and make the additional assumptions that K has a prime model and extend amalgamation to disjoint amalgamation.…”

Section: Definition 14mentioning

confidence: 99%

“…But, in [She87] and as expounded in [Gro02,Bal] the result can be given an extremely beautiful proof in ZFC; it is not quite possible to describe the result just in the language of AEC's [She, Bal]. In contrast two other results:…”

Section: Definition 14mentioning

confidence: 99%

“…This necessity is outlined in [She87], and more clearly in the revised version of that paper [She]; see [Gro02] for a good account of the positive Theorem 25 2). Complete proofs of both results including finding counterexamples in L ω1,ω showing the necessity of weak CH appear in [Bal].…”

Section: Definition 14mentioning

confidence: 99%

“…Grossberg (e.g [Gro02]) has posed this question for AEC and for L ω1,ω . For sentences of L ω1,ω , Shelah's result reported above gives a partial answer modulo weak gch.…”

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confidence: 99%

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Abstract elementary classes: some answers, more questions

Baldwin¹

2007

Logic Colloquium 2004

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We survey some of the recent work in the study of Abstract Elementary Classes focusing on the categoricity spectrum and the introduction of certain conditions (amalgamation, tameness, arbitrarily large models) which allow one to develop a workable theory. We repeat or raise for the first time a number of questions; many now seem to be accessible.Much late 19th and early 20th century work in logic was in a 2nd order framework; infinitary logics in the modern sense were foreshadowed by Schroeder and Pierce before being formalized in modern terms in Poland during the late 20's. First order logic was only singled out as the 'natural' language to formalize mathematics as such authors as Tarski, Robinson, and Malcev developed the fundamental tools and applied model theory in the study of algebra. Serious work extending the model theory of the 50's to various infinitary logics blossomed during the 1960's and 70's with substantial work on logics such as L ω1,ω and L ω1,ω (Q). At the same time Shelah's work on stable theories completed the switch in focus in first order model theory from study of the logic to the study of complete first order theories As Shelah in [She75, She83a] sought to bring this same classification theory standpoint to infinitary logic, he introduced a total switch to a semantic standpoint. Instead of studying theories in a logic, one studies the class of models defined by a theory. He abstracted (pardon the pun) the essential features of the class of models of a first order theory partially ordered by the elementary submodel relation. An abstract elementary class AEC (K, ≺ K ) is a class of models closed under isomorphism and partially ordered under ≺ K , where ≺ K is required to refine the substructure relation, that is closed under unions and satisfies two additional conditions: if each element M i of a chain satisfiesFurther there is a Löwnenheim-Skolem number κ associated with K so that if A ⊆ M ∈ K, there is an M 1 with A ⊂ M 1 ≺ K M and |M 1 | ≤ |A| + κ.

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A topology for galois types in abstract elementary classes

Lieberman

2011

Mathematical Logic Quarterly

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We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle.We begin with a very brief introduction to abstract elementary classes (AECs), Galois types, and a few relevant properties thereof. Naturally this exposition will not be exhaustive; readers interested in further details may wish to consult [1], and, for a presentation emphasizing the context in which to situate these details, [9]. As suggested in [16] and elsewhere, AECs can be seen as a fundamentally category-theoretic generalization of elementary classes, where we excise syntactic considerations, and retain the purely diagrammatic, category-theoretic properties of the elementary submodel relation. In particular:Definition 1.1 Let L be a fixed finitary signature (one-sorted, for simplicity). A class of L-structures equipped with a strong submodel relation, (K, ≺ K ), is an abstract elementary class (AEC) if it satisfies the following axioms: (A0) The relation ≺ K is a partial order. (A1) For all M , N in K, if M ≺ K N , then M ⊆ L N . (A2) (Closure Under Isomorphism)(1) If M ∈ K, N is an L-structure, and M ∼ =L N , then N ∈ K.(2) If M i , N i ∈ K for i = 1, 2, and there are L-structure isomorphisms(A3) (Unions of Chains) Let (M α |α < δ) be an ≺ K -increasing sequence.(1) α< δ M α ∈ K.(2) For all α < δ,(A5) (Downward Löwenheim-Skolem) There exists an infinite cardinal LS(K) which is the smallest cardinal κ with the property that for any M ∈ K and subset A of M , there exists M 0 ∈ K with A ⊆ M 0 ≺ K M and |M 0 | ≤ |A| + κ.The prototypical example, of course, is the case in which K is an elementary class-the class of models of a particular first order theory T -and ≺ K is the elementary submodel relation. Naturally, the notion of AEC is far more general. For example, given a sentence φ ∈ L ∞,ω , K = Mod(φ), and ≺ K the relation of elementarity *

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Classification theory for abstract elementary classes

Cited by 94 publications

References 0 publications

Tameness From Large Cardinal Axioms

Tameness From Large Cardinal Axioms

Abstract elementary classes: some answers, more questions

A topology for galois types in abstract elementary classes

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