This brief unpublished note (11 pages) contains an overview of the Gödel fixed-point lemma, along with several generalizations and applications, written for use in the Week 3 lecture of the Graduate Philosophy of Logic seminar that I co-taught with Volker Halbach at Oxford in Hilary term 2021. The theme of the seminar was self-reference, truth, and consistency strengths, and in this lecture we discussed the nature of Gödel’s fixed-point lemma and generalizations, with various applications in logic.

Contents

1. Introduction
2. Gödel’s fixed-point lemma
   An application to the Gödel incompleteness theorem
3. Finite self-referential schemes
   An application to nonindependent disjunctions of independent sentences
4. Gödel-Carnap fixed point lemma
   Deriving the double fixed-point lemma as a consequence
   An application to the provability version of Yablo’s paradox
5. Kleene recursion theorem
   An application involving computable numbers
   An application involving the universal algorithm
   An application to Quine programs and Ouroborous chains
6. References