English

Abstract elementary classes stable in $\aleph_0$

Logic 2018-05-31 v5

Abstract

We study abstract elementary classes (AECs) that, in 0\aleph_0, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at 0\aleph_0. More precisely, there is a superlimit model of cardinality 0\aleph_0 and the class generated by this superlimit has a type-full good 0\aleph_0-frame (a local notion of nonforking independence) and a superlimit model of cardinality 1\aleph_1. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.

Keywords

Cite

@article{arxiv.1702.08281,
  title  = {Abstract elementary classes stable in $\aleph_0$},
  author = {Saharon Shelah and Sebastien Vasey},
  journal= {arXiv preprint arXiv:1702.08281},
  year   = {2018}
}

Comments

27 pages

R2 v1 2026-06-22T18:29:24.446Z