Recursion Theorems in Numbered Sets

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.