Categoricity From One Successor Cardinal in Tame Abstract Elementary Classes

Grossberg, Rami; VanDieren, Monica

doi:10.1142/s0219061306000554

Cited by 56 publications

(106 citation statements)

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“…Grossberg and VanDieren originated this trend in their analysis of the stability spectrum for tame AEC in [GV06a]; it continued in further work on the stability spectrum [BKV00] and the analysis of categoricity in [GV06b,GV,BL00] and under even stronger hypotheses in [HV, Hyt]. This kind of work suggests several directions of inquiry.…”

Section: Definition 14mentioning

confidence: 97%

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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Section: Definition 14mentioning

confidence: 97%

Abstract elementary classes: some answers, more questions

Baldwin¹

2007

Logic Colloquium 2004

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“…This analysis shows the exact point that tameness fails. Grossberg pointed out that after establishing amalgamation in Section 3, non-tameness at some (µ, κ) could have been deduced from eventual failure of categoricity of the example and the known upward categoricity results [6,13]. However, one could not actually compute the value of κ without the same technical work we used to show tameness directly.…”

Section: Proofmentioning

confidence: 99%

“…Roughly speaking, K is (µ, κ)-tame if distinct Galois types over models of size κ have distinct restrictions to some submodel of size µ. For classes with arbitrarily large models, that satisfy amalgamation and tameness, strong categoricity transfer theorems have been proved [7,8,6,13,4,10]. In particular these results yield categoricity in every uncountable power for a tame AEC in a countable language (with arbitrarily large models satisfying amalgamation and the joint embedding property) that is categorical in any single cardinal above ℵ 2 ([6]) or even above ℵ 1 ([13]).…”

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Categoricity, amalgamation, and tameness

Baldwin

Kolesnikov

2009

Isr. J. Math.

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ABSTRACT. Theorem. For each 2 ≤ k < ω there is an Lω 1 ,ω -sentence φ k such that:(1) φ k is categorical in µ if µ ≤ ℵ k−2 ; (2) φ k is not ℵ k−2 -Galois stable; (3) φ k is not categorical in any µ with µ > ℵ k−2 ; (4) φ k has the disjoint amalgamation property;indeed, syntactic first-order types determine Galois types over models of cardinality at mostWe adapt an example of [9]. The amalgamation, tameness, stability results, and the contrast between syntactic and Galois types are new; the categoricity results refine the earlier work of Hart and Shelah and answer a question posed by Shelah in [17].Considerable work (e.g. [14,15,16,7,8,6,18,12,11]) has explored the extension of Morley's categoricity theorem to infinitary contexts. While the analysis in [14,15] applies only to L ω 1 ,ω , it can be generalized and in some ways strengthened in the context of abstract elementary classes.Various locality properties of syntactic types do not generalize in general to Galois types (defined as orbits under an automorphism group) in an AEC [5]; much of the difficulty of the work stems from this difference. One such locality properties is called tameness. Roughly speaking, K is (µ, κ)-tame if distinct Galois types over models of size κ have distinct restrictions to some submodel of size µ. For classes with arbitrarily large models, that satisfy amalgamation and tameness, strong categoricity transfer theorems have been proved [7,8,6,13,4,10]. In particular these results yield categoricity in every uncountable power for a tame AEC in a countable language (with arbitrarily large models satisfying amalgamation and the joint embedding property) that is categorical in any single cardinal above ℵ 2 ([6]) or even above ℵ 1 ([13]).In contrast, Shelah's original work [14,15] showed (under weak GCH) that categoricity up to ℵ ω of a sentence in L ω 1 ,ω implies categoricity in all uncountable cardinalities. Hart and Shelah [9] showed the necessity of the assumption by constructing sentences φ k which were categorical up to some ℵ n but not eventually

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“…Grossberg and VanDieren [GV06a] have recently approached this problem from an exciting new approach. They isolated a model theoretic property called tameness.…”

Section: Introductionmentioning

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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Categoricity From One Successor Cardinal in Tame Abstract Elementary Classes

Cited by 56 publications

References 14 publications

Abstract elementary classes: some answers, more questions

Abstract elementary classes: some answers, more questions

Categoricity, amalgamation, and tameness

Tameness From Large Cardinal Axioms

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