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Henkin’s Theorem in Textbooks

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The Life and Work of Leon Henkin

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

Abstract

Our aim in this paper is to examine the incorporation and acceptance of Henkin’s completeness proof in some textbooks on classical logic. The first conclusion of this paper is that the inclusion of Henkin’s completeness proof into the standards of Logic was neither quick nor easy. Surprising as it may seem today, most of the textbooks published in the 1950s did not include a section for this proof, nor presented it in any way. A question we should try to answer is at what moment does Henkin’s proof of completeness for first order logic begin to be considered as a part of the standards of elementary logic. This point brings us to a discussion on the way in which the specific gains of Henkin’s proof have been assessed in literature. The possibility of using Henkin’s methods in a wide variety of formal systems made completeness a general property belonging to foundations of logic, leaving the realm of model theory for quantification languages where it was previously located.

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Notes

  1. 1.

    The list is featured prominently in Sect. 8.

  2. 2.

    For details on the dissemination of logic in Spain, see [2].

  3. 3.

    See [16, p. 132].

  4. 4.

    See [12].

  5. 5.

    The translation was elaborated on that occasion by Professor Stefan Bauer-Mengelberg (see [35, p. 593]) and was reviewed and approved by Gödel himself.

  6. 6.

    See [8, p. 291].

  7. 7.

    Sometimes the term Henkin completeness theorem is used to make reference to any formulation of the type “every consistent set of formulas of first-order logic is satisfiable, that is, has some model.” This time, Church specifies that this consistent set is satisfiable in a model of cardinal enumerable.

  8. 8.

    See [8, pp. 244–245].

  9. 9.

    See [8, p. 307, nt. 510].

  10. 10.

    See [22, p. 389].

  11. 11.

    See [6, p. 173].

  12. 12.

    In 1979, a second edition appears.

  13. 13.

    See [26, p. 67]. See also [14]. Later we will discuss this change in relation to alternative proofs.

  14. 14.

    See, for example, [26].

  15. 15.

    See Gödel’s apologize in [11, p. 63].

  16. 16.

    See [3, p. 117].

  17. 17.

    See [3, p. 121].

  18. 18.

    See [31, p. 61].

  19. 19.

    See [25, p. 136].

  20. 20.

    See [10, p. 139].

  21. 21.

    See [22, p. 389].

  22. 22.

    See also [24, p. 192].

  23. 23.

    See [26, p. 67].

  24. 24.

    See [14].

  25. 25.

    See [32, p. viii].

  26. 26.

    Ibid.

  27. 27.

    See [8, p. 235] and [22, p. 394].

  28. 28.

    See [9, p. 354].

  29. 29.

    See [26, p. 69].

  30. 30.

    See [11, p. 63].

  31. 31.

    See [34, p. viii].

  32. 32.

    See [17, Preface].

  33. 33.

    See [27, p. 194].

  34. 34.

    See [1, p. 80].

  35. 35.

    See [33, p. 69].

  36. 36.

    See [3, p. 123].

  37. 37.

    See [12, p. 589].

  38. 38.

    See [22, p. 423].

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Authors and Affiliations

  1. Department of Logic and Philosophy of Science, Faculty of Philosophy, Avda. Tomás y Valiente s/n, Autonomous University of Madrid, Campus Canto Blanco, 28049, Madrid, Spain

    Enrique Alonso

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  1. Enrique Alonso
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Correspondence to Enrique Alonso .

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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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Alonso, E. (2014). Henkin’s Theorem in Textbooks. 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_11

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