PreprintCC0arXiv
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 AD+DCR and inside the choiceless Solovay model. We then determine the exact consistency strength of the unrestricted order-preserving dichotomy as an axiom over ZF+ACω(R): it is equiconsistent with ZFC plus a strongly inaccessible cardinal.
Related
- An order-reversing embedding of Turing degrees into Arthur-Nimue-Merlin degrees (short version)
- Pseudo-Hyperjump Inversion Fails for Turing Degrees
- The pattern of consistency strength for membership in a computably enumerable set
- A Reflection Calculus for Bar Induction and the Linearity of Consistency Strength Ordering
- Interpretability for Turing Machines
BibTeX
@misc{yu2026orderpreserving,
title = {{Order-preserving maps of the Turing degrees: regularity and consistency strength}},
author = {Liang Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14548},
eprint = {2610.14548},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14548},
note = {CC0},
}
RIS
TY - UNPB
TI - Order-preserving maps of the Turing degrees: regularity and consistency strength
AU - Liang Yu
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14548
UR - https://arxiv.org/abs/2610.14548
AB - 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 AD+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 ZF+AC_ω(R): it is equiconsistent with ZFC plus a strongly inaccessible cardinal.
ER -