Galois-Stability for Tame Abstract Elementary Classes

Grossberg, Rami; VanDieren, Monica

doi:10.1142/s0219061306000487

Cited by 74 publications

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“…We have broken this notion into three precise concepts. Following [GV06a], we have chosen tame as the name of one of these. We call the others locality and compactness.…”

Section: Definitionmentioning

confidence: 99%

“…First, the increased emphasis, signaled in [She99,SV99] and emphasized in [GV06a], on hypotheses such as amalgamation or tameness as fruitful conditions to create a workable theory of AEC, has led to a number of new results. The need for studying AEC became more clear for two reasons.…”

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confidence: 99%

“…In order to understand further progress on the categoricity transfer problem, we introduce an important notion (first named in [GV06a]; the cardinal parameters were added in [Bal05]). …”

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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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“…We have broken this notion into three precise concepts. Following [GV06a], we have chosen tame as the name of one of these. We call the others locality and compactness.…”

Section: Definitionmentioning

confidence: 99%

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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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“…[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.…”

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“…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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2011

Mathematical Logic Quarterly

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Galois-Stability for Tame Abstract Elementary Classes

Cited by 74 publications

References 14 publications

Abstract elementary classes: some answers, more questions

Abstract elementary classes: some answers, more questions

Categoricity, amalgamation, and tameness

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