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Every Nonrecursive Many-One Degree Contains Either One or Infinitely Many Finite-One Degrees
Abstract
This paper proves that every nonrecursive many-one degree contains either exactly one or infinitely many finite-one degrees. This gives a negative answer to Open Question 2 of Richter, Stephan, and Zhang~\cite[p.~17]{RSZ}. Earlier work established that, for almost every A⊆N with respect to the standard product measure on 2N, the degree [A]m contains an infinite antichain of finite-one degrees~\cite{Cintioli}. Thus the question had already received an almost-sure negative answer; the present theorem settles it for every nonrecursive many-one degree.
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BibTeX
@misc{cintioli2026every,
title = {{Every Nonrecursive Many-One Degree Contains Either One or Infinitely Many Finite-One Degrees}},
author = {Patrizio Cintioli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00419},
eprint = {2609.00419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00419},
}
RIS
TY - UNPB
TI - Every Nonrecursive Many-One Degree Contains Either One or Infinitely Many Finite-One Degrees
AU - Patrizio Cintioli
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00419
UR - https://arxiv.org/abs/2609.00419
AB - This paper proves that every nonrecursive many-one degree contains either exactly one or infinitely many finite-one degrees. This gives a negative answer to Open Question 2 of Richter, Stephan, and Zhang~\cite[p.~17]{RSZ}. Earlier work established that, for almost every A⊆N with respect to the standard product measure on 2^N, the degree [A]_m contains an infinite antichain of finite-one degrees~\cite{Cintioli}. Thus the question had already received an almost-sure negative answer; the present theorem settles it for every nonrecursive many-one degree.
ER -