Abstract elementary classes stable in $\aleph_0$

by Shelah and Vasey. [ShVe:1119]

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 super stable-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 PC_{aleph_0}-representable AECs and also improves results of Hyttinen-Kesala and Baldwin-Kueker-VanDieren.


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