Kleene's recursion theorem

Kleene's recursion theorem, named after mathematician Stephen Cole Kleene, is a fundamental result in the field of computability theory. It addresses the existence of computable functions that can be defined recursively. The theorem states that for any total computable function \( f \), there exists a program (or particular index in the sense of the arithmetical hierarchy) that produces itself as an output when given its own index (or code) as input.