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Henkin on Completeness

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The Life and Work of Leon Henkin

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

Abstract

The Completeness of Formal Systems is the title of the thesis that Henkin presented at Princeton in 1947 under the supervision of Alonzo Church. A few years after the defense of his thesis, Henkin published two papers in the Journal of Symbolic Logic: the first, on completeness for first-order logic (Henkin in J. Symb. Log. 14(3):159–166, 1949), and the second one, devoted to completeness in type theory (Henkin in J. Symb. Log. 15(2):81–91, 1950). In 1963, Henkin published a completeness proof for propositional type theory (Henkin in J. Symb. Log. 28(3):201–216, 1963), where he devised yet another method not directly based on his completeness proof for the whole theory of types.

In this paper, these tree proofs are analyzed, trying to understand not just the result itself but also the process of discovery, using the information provided by Henkin in Bull. Symb. Log. 2(2):127–158, 1996.

In the third section, we present two completeness proofs that Henkin used to teach us in class. It is surprising that the first-order proof of completeness that Henkin explained in class was not his own but was developed by using Herbrand’s theorem and the completeness of propositional logic. In 1963, Henkin published An extension of the Craig–Lyndon interpolation theorem, where one can find a different completeness proof for first-order logic; this is the other completeness proof Henkin told us about.

We conclude this paper, by introducing two expository papers on this subject. Henkin was an extraordinary insightful professor, and in 1967, he published two works that are very relevant for the subject addressed here: Truth and provability (Henkin in Philosophy of Science Today, pp. 14–22, 1967) and Completeness (Henkin in Philosophy of Science Today, pp. 23–35, 1967).

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Notes

  1. 1.

    The paper was dedicated to his maestro Alonzo Church on the occasion of his 91 birthday; it was to be a book chapter, but the book was never published.

  2. 2.

    See [5, p. 556].

  3. 3.

    See [5, p. 557].

  4. 4.

    See [5, p. 557].

  5. 5.

    This idea is developed with some details in [19] and [21].

  6. 6.

    As presented in [2].

  7. 7.

    As you can see in [2, p. 61].

  8. 8.

    See [16, p. 146].

  9. 9.

    These pages refer to the reprint [28].

  10. 10.

    See [7, p. 85].

  11. 11.

    See [6, p. 162].

  12. 12.

    See [16, p. 146].

  13. 13.

    See [16, p. 149].

  14. 14.

    See [16, p. 151]. In this paper, the type of individuals is 1, that is why he writes (01=01)′ instead of (0 ι =0 ι )′.

  15. 15.

    The quotes in the following paragraph all belong to: Henkin [7, p. 85].

  16. 16.

    Such a definition can be given; for type theory I gave one in [17] and it is also included in the book [18].

  17. 17.

    See [9, p. 22].

  18. 18.

    See [10, p. 138].

  19. 19.

    See [10, p. 139].

  20. 20.

    See [10, p. 140].

  21. 21.

    Recall that at that time logicians include in the logic a bunch of axioms that allow the formulation of natural numbers and even real numbers.

  22. 22.

    See [7, p. 89].

  23. 23.

    In The little mermaid [20], we ended the paper, devoted to second order logic, saying:

    It is clear that you can have both: expressive power plus good logical properties. You cannot be a mermaid and have an immortal soul.

    […]

    And the little mermaid got two beautiful legs (with a lot of pain, as you might know). But even in stories everything has a price; you know, the poor little mermaid lost her voice.

  24. 24.

    In [6, p. 159].

  25. 25.

    In [6, p. 163].

  26. 26.

    Gödel papers are [3] and [4].

  27. 27.

    In [12, p. 325].

  28. 28.

    Here \(\Im _{X_{\beta }}^{\mathbf{x}}= \langle \mathfrak{PT},g_{X_{\beta }}^{\mathbf{x}} \rangle \) where \(g_{X_{\beta }}^{\mathbf{x}}\) is an X β -variant of g.

  29. 29.

    In [15, p. 31].

  30. 30.

    We would use the symbol ≡ instead of \(\mathcal{Q}_{(0\alpha )\alpha } \) for any α.

  31. 31.

    See [12, p. 326].

  32. 32.

    This calculus was improved by Andrews [1]. Please read the beautiful paper in this book where Andrews himself tell us the whole personal business involved.

  33. 33.

    See [12, p. 341].

  34. 34.

    See [13, p. 14].

  35. 35.

    See [13, p. 15].

  36. 36.

    See [13, p. 18].

  37. 37.

    See [13, p. 19].

  38. 38.

    See [13, p. 19].

  39. 39.

    See [13, p. 19].

  40. 40.

    See [13, p. 19].

  41. 41.

    See [13, p. 19].

  42. 42.

    See [13, p. 19].

  43. 43.

    See [13, p. 19].

  44. 44.

    See [13, p. 21].

  45. 45.

    See [13, p. 21].

  46. 46.

    In [13, p. 22].

  47. 47.

    See [13, p. 22].

  48. 48.

    In [22], we analyze the evolution of the completeness theorem from Gödel to Henkin in some detail.

  49. 49.

    The previously mentioned anathema is even stronger when Gödel’s incompleteness result is mentioned.

  50. 50.

    In [14, p. 25].

  51. 51.

    See [14, p. 25].

  52. 52.

    See [14, p. 25].

  53. 53.

    See [14, p. 26].

  54. 54.

    See [14, p. 26].

  55. 55.

    See [14, p. 26].

  56. 56.

    In [14, p. 27].

  57. 57.

    See [14, p. 28].

  58. 58.

    See [14, p. 29].

  59. 59.

    See [14, p. 31].

  60. 60.

    See [14, pp. 31–32].

  61. 61.

    In [14, p. 33].

  62. 62.

    See [14, p. 34].

  63. 63.

    See [14, p. 34].

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Authors and Affiliations

  1. Universidad de Salamanca, Salamanca, Spain

    María Manzano

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  1. María Manzano
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Correspondence to María Manzano .

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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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Manzano, M. (2014). Henkin on Completeness. 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_12

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