A review of the Gödel fixed-point lemma, with generalizations and applications

This bried unpublished note (11 pages) contains an overview of the Gödel fixed-point lemma, along with several generalizations and applications, written for use in the Week 3 lecture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

Computational self-reference and the universal algorithm, Queen Mary University of London, June 2019

This will be a talk for the Theory Seminar for the theory research group in Theoretical Computer Science at Queen Mary University of London. The talk will be held 4 June 2019 1:00 pm, ITL first floor.

Abstract. Curious, often paradoxical instances of self-reference inhabit deep parts of computability theory, from the intriguing Quine programs and Ouroboros programs to more profound features of the Gödel phenomenon. In this talk, I shall give an elementary account of the universal algorithm, showing how the capacity for self-reference in arithmetic gives rise to a Turing machine program $e$, which provably enumerates a finite set of numbers, but which can in principle enumerate any finite set of numbers, when it is run in a suitable model of arithmetic. In this sense, every function becomes computable, computed all by the same universal program, if only it is run in the right world. Furthermore, the universal algorithm can successively enumerate any desired extension of the sequence, when run in a suitable top-extension of the universe. An analogous result holds in set theory, where Woodin and I have provided a universal locally definable finite set, which can in principle be any finite set, in the right universe, and which can furthermore be successively extended to become any desired finite superset of that set in a suitable top-extension of that universe.

Self reference in computability theory and the universal algorithm, Ouroboros: Formal Criteria of Self-Reference in Mathematics and Philosophy, Bonn, February 2018

This will be a talk for the conference: Ouroboros: Formal Criteria of Self-Reference in Mathematics and Philosophy, held in Bonn, February 16-18, 2018.

Abstract. I shall give an elementary account of the universal algorithm, due to Woodin, showing how the capacity for self-reference in arithmetic gives rise to a Turing machine program $e$, which provably enumerates a finite set of numbers, but which can in principle enumerate any finite set of numbers, when it is run in a suitable model of arithmetic. Furthermore, the algorithm can successively enumerate any desired extension of the sequence, when run in a suitable top-extension of the universe. Thus, the algorithm sheds some light on the debate between free will and determinism, if one should imagine extending the universe into a nonstandard time scale. An analogous result holds in set theory, where Woodin and I have provided a universal locally definable finite set, which can in principle be any finite set, in the right universe, and which can furthermore be successively extended to become any desired finite superset of that set in a suitable top-extension of that universe.

Ouroboros Bonn 2018 Conference Poster | Slides