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Step by Recursive Step: Church's Analysis of Effective Calculability

  • Wilfried Sieg (a1)
  • DOI:
  • Published online: 15 January 2014

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was “Church's Thesis” put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ-definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity.

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[2]Alonzo Church , Alternatives to Zermelo's assumption, Transactions of the American Mathematical Society, vol. 29 (1927), pp. 178208, Ph.D. thesis, Princeton.

[3]Alonzo Church , On the law of the excluded middle, Bulletin of the American Mathematical Society, vol. 34 (1928), pp. 7578.

[4]Alonzo Church , A set of postulates for the foundation of logic, part I, Annals of Mathematics, vol. 33 (1932), no. 2, pp. 346–66.

[5]Alonzo Church , A set of postulates for the foundation of logic, part II, Annals of Mathematics, vol. 34 (1933), no. 2, pp. 839–64.

[6]Alonzo Church , The Richard paradox, American Mathematical Monthly, vol. 41 (1934), pp. 356–61.

[7]Alonzo Church , A proof of freedom from contradiction, Proceedings of the National Academy of Sciences, vol. 21 (1935), pp. 275–81.

[9]Alonzo Church , A note on the Entscheidungsproblem, Journal of Symbolic Logic, vol. 1 (1936), pp. 40–1, Corrections, Journal of Symbolic Logic, vol. 1 (1936), pp. 101–2.

[10]Alonzo Church , An unsolvable problem of elementary number theory, American Journal of Mathematics, vol. 58 (1936), pp. 345–63.

[11]Alonzo Church , Review of [54], Journal of Symbolic Logic, vol. 2 (1937), no. 1, p. 43.

[12]Alonzo Church , Review of [63], Journal of Symbolic Logic, vol. 2 (1937), no. 1, pp. 42–3.

[17]Alonzo Church and J. Barkley Rosser , Some properties of conversion, Transactions of the American Mathematical Society, vol. 39 (1936), pp. 472–82.

[19]Martin Davis , Why Gödel didn't have Church's thesis, Information and Control, vol. 54 (1982), pp. 324.

[39]Stephen C. Kleene , Proof by cases in formal logic, Annals of Mathematics, vol. 35 (1934), pp. 529–44.

[42]Stephen C. Kleene , A theory of positive integers in formal logic, American Journal of Mathematics, vol. 57 (1935), pp. 153–73, 219–44, the story behind this paper is described in [47, pp. 57–58]; it was first submitted on October 9, 1933; its revised version was re-submitted on June 13, 1934.

[43]Stephen C. Kleene , General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1936), pp. 727–42.

[44]Stephen C. Kleene , λ-definability and recursiveness, Duke Mathematics Journal, vol. 2 (1936), pp. 340–53, [40] and [41] are abstracts of these two papers and were received by the American Mathematical Society on 07 1, 1935.

[45]Stephen C. Kleene , A note on recursive functions, Bulletin of the American Mathematical Society, vol. 42 (1936), pp. 544–6.

[47]Stephen C. Kleene , Origins of recursive function theory, Annals of Historical Computing, vol. 3 (1981), no. 1, pp. 5266.

[49]Stephen C. Kleene and J. Barkley Rosser , The inconsistency of certain formal logics, Annals of Mathematics, vol. 36 (1935), no. 2, pp. 630–6.

[50]Elliott Mendelson , Second thoughts about Church's thesis and mathematical proofs, The Journal of Philosophy, vol. 87 (1990), no. 5, pp. 225–33.

[51]Daniele Mundici and Wilfried Sieg , Paper machines, Philosophia Mathematica, vol. 3 (1995), pp. 530.

[55]J. Barkley Rosser , A mathematical logic without variables, Annals of Mathematics, vol. 36 (1935), no. 2, pp. 127–50, also Duke Mathematics Journal, vol. 1, pp. 328–55.

[56]J. Barkley Rosser , Highlights of the history of the lambda-calculus, Annals of Historical Computing, vol. 6 (1984), no. 4, pp. 337–49.

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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
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