Submission date: 7 July 2015.

Abstract:

In this paper, we apply results of [Va3] and use towers to transfer symmetry from μ^+ down to μ in superstable abstract elementary classes without using extra set-theoretic assumptions or tameness.

Theorem. Suppose K is an abstract elementary class satisfying the amalgamation and joint embedding properties and that K is both μ- and μ^+-superstable. If K has symmetry for non-μ^+-splitting, then K has symmetry for non-μ-splitting.
This is a new application of towers which were introduced by Shelah and Villaveces [ShVi] and later used by VanDieren [Va1], [Va2] and Grossberg, VanDieren, and Villaveces [GVV] to prove the uniqueness of limit models.

Mathematics Subject Classification:

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Full text arXiv 1507.01991: pdf, ps.