Toward a stability theory of tame abstract elementary classes

Sebastien Vasey

Published 2016-09-12Version 1

We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. We define a cardinal parameter $\chi (K)$ which is the analog of $\kappa_r (T)$ from the first-order setup. In particular, a full characterization of the (high-enough) stability cardinals is possible assuming the singular cardinal hypothesis (SCH). Using tools of Boney and VanDieren, we show that limit models of length at least $\chi (K)$ are unique and deduce results on the saturation spectrum, including a full (eventual) characterization assuming SCH. We also show (in ZFC) that if a class is stable on a tail of cardinals, then $\chi (K) = \aleph_0$ (the converse is known). This indicates that there is a clear notion of superstability in this framework.