The Consistency Strength of Long Projective Determinacy

Abstract

We determine the consistency strength of determinacy for projective games of length ω². Our main theorem is that Π¹n₊₁-determinacy for games of length ω² implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies A=R and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω² with payoff in RΠ¹₁ or with σ-projective payoff.