Determinacy in the difference hierarchy of co-analytic sets Original Research Article

Equivalents have been established between the determinacy of games played at various intervals of the difference hierarchy of co-analytical sets, with embeddings of inner models, by the work of Martin. Taken together with a theorem of Harrington, these yield a strictly level-by-level description for most levels. We complete this analysis by establishing suitable equivalences for the remaining cases. Namely, we show that ω2α-Π11 Determinacy, for α < ω1, is equivalent to the existence of a generalised “sharp” (or “mouse”) generating embeddings of particular inner models. For α = δ + 1, such determinacy follows from (but is strictly weaker than) the existence of δ measurable cardinals, with an “almost” View the MathML source cardinal above their supremum. The results are provable uniformly in any parameters arising, and we give proofs of the corresponding “lightface” versions.