Determinacy and \Delta^1_3-degrees

Abstract

Let D = { d_n } be a countable collection of Delta^1_3 degrees. Assuming that all co-analytic games on integers are determined (or equivalently that all reals have ``sharps''), we prove that either D has a Delta^1_3-minimal upper bound, or that for any n, and for every real r recursive in d_n, games in the pointclasses Delta^1_2(r) are determined. This is proven using Core Model theory.