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

Liang Yu

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arXiv · 2610.14548

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

Liang Yu

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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 $\mathsf{ZF}+\mathsf{AC}_ω(\R)$: it is equiconsistent with $\mathsf{ZFC}$ plus a strongly inaccessible cardinal.

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Liang Yu. 2026-10-10. Order-preserving maps of the Turing degrees: regularity and consistency strength. https://arxiv.org/abs/2610.14548

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