A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913088376143872 |
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| author | Cintioli, Patrizio |
| author_facet | Cintioli, Patrizio |
| contents | 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.) $tt$-degree contains a c.e.\ \textit{irreducible} $m$-degree (i.e., an $m$-degree consisting of only one $1$-degree). We answer this question in the negative by proving the existence of a c.e.\ $tt$-degree that does not contain any c.e.\ irreducible $m$-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.\ $m$-degree contained within the $tt$-degree of such a set consists of simple sets, which cannot be cylinders, and therefore necessarily splits into multiple $1$-degrees. Furthermore, our result demonstrates that a classical 1969 theorem by C.~G.~Jockusch Jr. -- which guarantees the existence of an irreducible $m$-degree within every c.e.\ $tt$-degree -- is strictly optimal and cannot be generally strengthened to require such an $m$-degree to be computably enumerable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_03066 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees Cintioli, Patrizio Logic 03D25, 03D30 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.) $tt$-degree contains a c.e.\ \textit{irreducible} $m$-degree (i.e., an $m$-degree consisting of only one $1$-degree). We answer this question in the negative by proving the existence of a c.e.\ $tt$-degree that does not contain any c.e.\ irreducible $m$-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.\ $m$-degree contained within the $tt$-degree of such a set consists of simple sets, which cannot be cylinders, and therefore necessarily splits into multiple $1$-degrees. Furthermore, our result demonstrates that a classical 1969 theorem by C.~G.~Jockusch Jr. -- which guarantees the existence of an irreducible $m$-degree within every c.e.\ $tt$-degree -- is strictly optimal and cannot be generally strengthened to require such an $m$-degree to be computably enumerable. |
| title | A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees |
| topic | Logic 03D25, 03D30 |
| url | https://arxiv.org/abs/2605.03066 |