The strength of the projective Martin conjecture
Volume 207 / 2010
Fundamenta Mathematicae 207 (2010), 21-27
MSC: 03D28, 03E35, 28A20.
DOI: 10.4064/fm207-1-2
Abstract
We show that Martin's conjecture on $\Pi^1_1$ functions uniformly $\leq_T$-order preserving on a cone implies $\Pi^1_1$ Turing Determinacy over $\hbox{ZF}+{\hbox{DC}}$. In addition, it is also proved that for $n\ge 0$, this conjecture for uniformly degree invariant $\mathbf{\Pi}^1_{2n+1}$ functions is equivalent over ZFC to $\mathbf{\Sigma}^1_{2n+2}$-Axiom of Determinacy. As a corollary, the consistency of the conjecture for uniformly degree invariant $\Pi^1_1$ functions implies the consistency of the existence of a Woodin cardinal.
