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Abstract elementary classes stable in $\aleph_0$

Research paper by Saharon Shelah, Sebastien Vasey

Indexed on: 27 Feb '17Published on: 27 Feb '17Published in: arXiv - Mathematics - Logic



Abstract

We study abstract elementary classes (AECs) that, in $\aleph_0$, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). We prove that such classes exhibit superstable-like behavior at $\aleph_0$. More precisely, there is a superlimit model of cardinality $\aleph_0$ and the class generated by this superlimit has a type-full good $\aleph_0$-frame (a local notion of nonforking independence) and a superlimit model of cardinality $\aleph_1$. This extends the first author's earlier study of $\operatorname{PC}_{\aleph_0}$-representable AECs and also improves results of Hyttinen-Kes\"al\"a and Baldwin-Kueker-VanDieren.