Categoricity transfer in simple finitary abstract elementary classes
Tapani Hyttinen, Meeri Kesälä · Journal of Symbolic Logic · 2011
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 weakκ-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 uncountableκ. Then ( ) is weakly categorical in eachλ≥ min . If the class ( ) is also -tame, weakκ-categoricity is equivalent withκ-categoricity in the usual sense. We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.