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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

Citations:

Zbl 1112.03026
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References:

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