Order-preserving maps of the Turing degrees: regularity and consistency strength

Abstract

We formulate a proof of Part~I of Martin's conjecture for order-preserving maps directly on the quotient of the reals by Turing equivalence. The assumptions are countable choice for reals, the perfect-set property for sets of reals, and continuous uniformization on a comeager domain. The argument applies under $\mathsf{AD}+\mathsf{DC}_{\R}$ and inside the choiceless Solovay model. We then determine the exact consistency strength of the unrestricted order-preserving dichotomy as an axiom over $\mathsf{ZF}+\mathsf{AC}_{\omega}(\R)$: it is equiconsistent with $\mathsf{ZFC}$ plus a strongly inaccessible cardinal.