Archive for Mathematical Logic

, Volume 55, Issue 3–4, pp 567–592 | Cite as

Infinitary stability theory

Article

Abstract

We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois (orbital) type of length less than a fixed cardinal \(\kappa \). We show:

Theorem 0.1 (The semantic–syntactic correspondence) An AEC K is fully \(({<}\kappa )\)-tame and type short if and only if Galois types are syntactic in the Galois Morleyization.

   This exhibits a correspondence between AECs and the syntactic framework of stability theory inside a model. We use the correspondence to make progress on the stability theory of tame and type short AECs. The main theorems are:

Theorem 0.2 Let K be a \(\text {LS}(K)\)-tame AEC with amalgamation. The following are equivalent:

  1. (1)

    K is Galois stable in some \(\lambda \ge \text {LS}(K)\).

     
  1. (2)

    K does not have the order property (defined in terms of Galois types).

     
  1. (3)

    There exist cardinals \(\mu \) and \(\lambda _0\) with \(\mu \le \lambda _0 < \beth _{(2^{\text {LS}(K)})^+}\) such that K is Galois stable in any \(\lambda \ge \lambda _0\) with \(\lambda = \lambda ^{<\mu }\).

     

Theorem 0.3 Let K be a fully \(({<}\kappa )\)-tame and type short AEC with amalgamation, \(\kappa = \beth _{\kappa } > \text {LS}(K)\). If K is Galois stable, then the class of \(\kappa \)-Galois saturated models of K admits an independence notion (\(({<}\kappa )\)-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.

Keywords

Abstract elementary classes Stability inside a model  Stability spectrum Forking 

Mathematics Subject Classification

Primary 03C48 Secondary 03C45 03C52 03C55 03C75 03E55 

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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