Abstract elementary classes stable in ${\aleph }_0$

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). Assuming a locality property for 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 ${\aleph }_1$. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.

27 pages

References in corpus (8)

• Building independence relations in abstract elementary classes
• Infinitary stability theory
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Downward categoricity from a successor inside a good frame
• Chains of saturated models in AECs
• Symmetry in abstract elementary classes with amalgamation
• Quasiminimal abstract elementary classes
• Good Frames in the Hart-Shelah Example

Cited by in corpus (2)

• Universal classes near ${\aleph }_1$
• Building models in small cardinals in local abstract elementary classes