Fixed Point Theorems in Computability Theory

TL;DR

This survey analyzes fixed-point phenomena in computability across Kleene's recursion theorem, its effective variants, and generalizations to partial combinatory algebras and numberings. It compares conditions under which fixed points exist, including Feferman-type and Ershov-type theorems, and extends to Arslanov's completeness criterion and ADN-type results. It discusses effectiveness: the basic recursion theorem is effective, while Arslanov and ADN are not, and it highlights open questions about precomplete numberings and joint generalizations. The work emphasizes categorical and descriptive set-theoretic perspectives and connections to embeddings of pcas.

Abstract

We give a quick survey of the various fixed point theorems in computability theory, partial combinatory algebra, and the theory of numberings, as well as generalizations based on those. We also point out several open problems connected to these.

Figures (2)

• Figure 1: The relation between various forms of the recursion theorem
• Figure 2: Generalizations of the recursion theorem

Theorems & Definitions (9)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• proposition 1
• proof
• theorem 5
• theorem 6
• theorem 7