Submission date: 7 July 2015.

Abstract:

In this paper we extend the work of [ShVi], [Va1], [Va2], and [GVV] by weakening the structural assumptions on K_{μ^+} to something more closely resembling superstability from first order logic in order to derive the uniqueness of limit models of cardinality μ.
Moreover, we identify a necessary and sufficient condition for symmetry of non-splitting that involves reduced towers. This new condition is particularly interesting since it does not have a pre-established first-order analog. Additionally, this condition provides a mechanism for deriving symmetry in abstract elementary classes without having to assume set-theoretic assumptions or tameness:

Corollary: Suppose that K satisfies the amalgamation and joint embedding properties and μ is a cardinal ≥ ℶ_{(2^{Hanf(K)})^+}. If K is categorical in λ=μ^+, then K has symmetry for non-μ-splitting.

Mathematics Subject Classification:

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Full text arXiv 1507.01990: pdf, ps.