Transferring Symmetry

الأصل · EN

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.