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Categoricity transfer in simple finitary abstract elementary classes

Published online by Cambridge University Press:  12 March 2014

Tapani Hyttinen  and
Meeri Kesälä
Tapani Hyttinen
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin Katu 2B), FI-00014 University of Helsinki, Finland, E-mail: tapani.hyttinen@helsinki.fi, E-mail: meeri.kesala@helsinki.fi
Meeri Kesälä
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin Katu 2B), FI-00014 University of Helsinki, Finland, E-mail: tapani.hyttinen@helsinki.fi, E-mail: meeri.kesala@helsinki.fi
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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. If the class () is also -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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 759 - 806
Copyright
Copyright © Association for Symbolic Logic 2011

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