The paper demonstrates multiple forms of recursion theorem applicable in countable sets, suggesting important implications.
The paper studies the relationships between various forms of the Recursion Theorem in arbitrary numbered sets. We prove that any at most countable set [Formula: see text] with [Formula: see text] has a non-precomplete numbering that satisfies Kleene’s recursion theorem with parameters and Arslanov’s recursion theorem, as well as a numbering that satisfies Kleene’s recursion theorem but does not satisfy either Kleene’s recursion theorem with parameters, nor Arslanov’s recursion theorem.
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M. Kh. Faĭzrahmanov (2025) studied this question.
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