Categoricity transfer in simple finitary abstract elementary classes

Journal of Symbolic Logic 76 (3):759 - 806 (2011)
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Abstract

We continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak κ-categoricity and f-primary models to the framework of א₀-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let (������, ≼ ������ ) be a simple finitary AEC, weakly categorical in some uncountable κ. Then (������, ≼ ������ ) is weakly categorical in each λ ≥ min { \group{ \{\kappa,\beth_{ \group{ (2^{ \aleph_{ 0 _} ^});^{ + ^} \group} _}\}; \group} . If the class (������, ≼ ������ ) is also LS(������)-tame, weak κ-categoricity is equivalent with κ-categoricity in the usual sense. We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples

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Citations of this work

Shelah's eventual categoricity conjecture in universal classes: Part I.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (9):1609-1642.
Forking in short and tame abstract elementary classes.Will Boney & Rami Grossberg - 2017 - Annals of Pure and Applied Logic 168 (8):1517-1551.
Quasiminimal structures, groups and Zariski-like geometries.Tapani Hyttinen & Kaisa Kangas - 2016 - Annals of Pure and Applied Logic 167 (6):457-505.
Building prime models in fully good abstract elementary classes.Sebastien Vasey - 2017 - Mathematical Logic Quarterly 63 (3-4):193-201.
Finding a field in a Zariski-like structure.Kaisa Kangas - 2017 - Annals of Pure and Applied Logic 168 (10):1837-1865.

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References found in this work

Model theory for infinitary logic.H. Jerome Keisler - 1971 - Amsterdam,: North-Holland Pub. Co..
Galois-stability for Tame abstract elementary classes.Rami Grossberg & Monica Vandieren - 2006 - Journal of Mathematical Logic 6 (01):25-48.
Categoricity for abstract classes with amalgamation.Saharon Shelah - 1999 - Annals of Pure and Applied Logic 98 (1-3):261-294.
Independence in finitary abstract elementary classes.Tapani Hyttinen & Meeri Kesälä - 2006 - Annals of Pure and Applied Logic 143 (1-3):103-138.

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