The strength of the projective Martin conjecture

Abstract

We show that Martin's conjecture on Π11 functions uniformly ≤τ-order preserving on a cone implies Π11 Turing Determinacy over ZF + DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant Π2n+11 functions is equivalent over ZFC to Σ2n+21-Axiom of Determinacy. As a corollary, the consistency of the conjecture for uniformly degree invariant Π11 functions implies the consistency of the existence of a Woodin cardinal. © Instytut Matematyczny PAN, 2010.