Odifreddi's Problem 3 on Irreducible m-Degrees
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Statement
Odifreddi asked, as Problem 3 in his surveys "Strong Reducibilities" (1981) and "Reducibilities" (1999), whether every computably enumerable -degree contains a c.e. irreducible -degree, meaning an -degree consisting of a single -degree. Answered negatively: there is a c.e. -degree containing no c.e. irreducible -degree. This also shows Jockusch's 1969 theorem, which produces an irreducible -degree inside every c.e. -degree, is strictly optimal and cannot be strengthened to make that degree c.e.
Context
A numbered problem from Odifreddi's standard survey lists on strong reducibilities, open since 1981 and restated in 1999.
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Credited at the level of the paper rather than the lemma. The author describes the work as the result of an extended human-AI interaction in which several structural ideas and technical arguments emerged from exploratory sessions with Gemini Deep Think, after which he fully reworked and verified all arguments and takes sole responsibility for their correctness. Nothing is attributed step by step, so the contribution is real but unitemised.
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