## Categoricity transfer in simple finitary abstract elementary classes.(English)Zbl 1250.03055

Summary: We continue our study of finitary abstract elementary classes, defined in [the authors, Ann. Pure Appl. Logic 143, No. 1–3, 103–138 (2006; Zbl 1112.03026)]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak $$\kappa$$-categoricity and $$\mathrm{f}$$-primary models to the framework of $$\aleph_{0}$$-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let ($$\mathbb K, \curlyeqprec_{\mathbb {K}}$$) be a simple finitary AEC, weakly categorical in some uncountable $$\kappa$$. Then ($$\mathbb K, \curlyeqprec_{\mathbb {K}}$$) is weakly categorical in each $$\lambda \geq \min\{\kappa ,\beth_{(2^{\aleph_{0}})^{+}\}}$$. If the class ($$\mathbb K, \curlyeqprec_{\mathbb {K}}$$) is also LS($$\mathbb K$$)-tame, weak $$\kappa$$-categoricity is equivalent with $$\kappa$$-categoricity in the usual sense. We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.

### MSC:

 03C48 Abstract elementary classes and related topics 03C35 Categoricity and completeness of theories

Zbl 1112.03026
Full Text:

### References:

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