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The Henkin Sentence

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The Life and Work of Leon Henkin

Part of the book series: Studies in Universal Logic ((SUL))

Abstract

In this paper we discuss Henkin’s question concerning a formula that has been described as expressing its own provability. We analyze Henkin’s formulation of the question and the early responses by Kreisel and Löb and sketch how this discussion led to the development of provability logic. We argue that, in addition to that, the question has philosophical aspects that are still interesting.

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Notes

  1. 1.

    One may compare this with the truth-teller sentence that states its own Σ 1-truth. The answer to the question whether this sentence is provable, refutable, or independent depends on assumptions on the coding, the diagonalization method, and so on [27]. So Henkin’s question for Σ 1-truth instead of provability only admits an answer that is far less robust than Löb’s answer to Henkin’s original question, which is extremely robust. Among the “Henkin-like” problems, the robustness of the answer to Henkin’s original problem may be more the exception than the rule.

  2. 2.

    In what follows, we somewhat neglect the problems involving the choice of the formal system Σ and a coding of syntax. See [27] for some additional remarks.

  3. 3.

    Feferman [17] introduced and used the term “numerate” for “weakly represent.”

  4. 4.

    See, for instance, [51] for a discussion.

  5. 5.

    Kreisel asked that the theory be Σ 1-sound, but that demand is superfluous.

  6. 6.

    See, for instance, [51] for a discussion.

  7. 7.

    Note also that the Kreisel–Henkin construction works in some very weak cases where it is not clear that we have Löb’s theorem.

  8. 8.

    It was first noted by Craig Smoryński in [42].

  9. 9.

    Actually, Löb mentions more conditions in his paper. However, upon analysis, we only need the ones given here.

  10. 10.

    It would be more appropriate to call this logic simply L. Unfortunately, L also suggests language, so the designation GL was preferred.

  11. 11.

    The substitution principle used here can be proved by induction of φ.

  12. 12.

    In her paper [36], Larisa Maksimova shows that, conversely, Beth’s theorem follows from the existence of explicit fixed points. See also [26].

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Acknowledgements

We thank Volodya Shavrukov for his comments on the penultimate version.

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Authors and Affiliations

  1. New College, OX1 3BN, Oxford, England, UK

    Volker Halbach

  2. Philosophy, Faculty of Humanities, Utrecht University, Janskerkhof 13, 3512BL, Utrecht, The Netherlands

    Albert Visser

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  1. Volker Halbach
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Correspondence to Albert Visser .

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Editors and Affiliations

  1. Department of Philosophy, Logic and Aesthetics, University of Salamanca, Salamanca, Spain

    María Manzano

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    Ildikó Sain

  3. Department of Logic and Philosophy of Science, Autonomous University of Madrid, Madrid, Spain

    Enrique Alonso

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Halbach, V., Visser, A. (2014). The Henkin Sentence. In: Manzano, M., Sain, I., Alonso, E. (eds) The Life and Work of Leon Henkin. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-09719-0_17

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