Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 671–690 | Cite as

Saturation and solvability in abstract elementary classes with amalgamation

Article

Abstract

Theorem 0.1Let\(\mathbf {K}\)be an abstract elementary class (AEC) with amalgamation and no maximal models. Let\(\lambda > {LS}(\mathbf {K})\). If\(\mathbf {K}\)is categorical in\(\lambda \), then the model of cardinality\(\lambda \)is Galois-saturated.

This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: \(\mathbf {K}\) has a unique limit model in each cardinal below \(\lambda \), (when \(\lambda \) is big-enough) \(\mathbf {K}\) is weakly tame below \(\lambda \), and the thresholds of several existing categoricity transfers can be improved.

We also prove a downward transfer of solvability (a version of superstability introduced by Shelah):

Corollary 0.2Let\(\mathbf {K}\)be an AEC with amalgamation and no maximal models. Let\(\lambda> \mu > {LS}(\mathbf {K})\). If\(\mathbf {K}\)is solvable in\(\lambda \), then\(\mathbf {K}\)is solvable in\(\mu \).

Keywords

Abstract elementary classes Superstability Saturation Solvability Categoricity Indiscernibles Order property 

Mathematics Subject Classification

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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