Abstract
We show that Martin’s conjecture on Π1-functions uniformly ≤T -order preserving on a cone implies Π1 Turing Determinacy over ZF+DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant Π2n+1functions is equivalent over ZFC to Σ2n+2-Axiom of Determinacy. As a corollary, the consistency of the conjecture for Π1-uniformly degree invariant functions implies the consistency of the existence of a Woodin cardinal.
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