A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees
Abstract
In this paper, we provide a negative solution to Problem 3 formulated by P.~Odifreddi in his survey articles \textit{``Strong Reducibilities''} (1981) and \textit{``Reducibilities''} (1999). The problem asks whether every computably enumerable (c.e.) -degree contains a c.e.\ \textit{irreducible} -degree (i.e., an -degree consisting of only one -degree). We answer this question in the negative by proving the existence of a c.e.\ -degree that does not contain any c.e.\ irreducible -degree. Our proof relies on the structural properties of c.e.\ semirecursive sets with a rigid complement, originally constructed by A.~N.~Degtev. We show that the unique c.e.\ -degree contained within the -degree of such a set consists of simple sets, which cannot be cylinders, and therefore necessarily splits into multiple -degrees. Furthermore, our result demonstrates that a classical 1969 theorem by C.~G.~Jockusch Jr. -- which guarantees the existence of an irreducible -degree within every c.e.\ -degree -- is strictly optimal and cannot be generally strengthened to require such an -degree to be computably enumerable.
Cite
@article{arxiv.2605.03066,
title = {A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees},
author = {Patrizio Cintioli},
journal= {arXiv preprint arXiv:2605.03066},
year = {2026}
}
Comments
5 pages