Studia Logica

, Volume 107, Issue 1, pp 145–165 | Cite as

Is There a “Hilbert Thesis”?

  • Reinhard KahleEmail author


In his introductory paper to first-order logic, Jon Barwise writes in the Handbook of Mathematical Logic (1977):

[T]he informal notion of provable used in mathematics is made precise by the formal notion provable in first-order logic. Following a sug[g]estion of Martin Davis, we refer to this view as Hilbert’s Thesis.

This paper reviews the discussion of (different variations of) Hilbert’s Thesis in the literature. In addition to the question whether it is justifiable to use Hilbert’s name here, the arguments for this thesis are compared with those for Church’s Thesis concerning computability. This leads to the question whether one could provide an analogue for proofs of the concept of partial recursive function.


David Hilbert Formal proofs Church’s Thesis Diagonalization 


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This work is partially supported by the Portuguese Science Foundation, FCT, through the projects UID/MAT/00297/2013 (Centro de Matemática e Aplicações), PTDC/FIL-FCI/109991/2009, The Notion of Mathematical Proof, and PTDC/MHC-FIL/2583/2014, Hilbert’s 24th Problem.


  1. 1.
    Ackermann, W., Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit, Mathematische Annalen 93:1–36, 1925.CrossRefGoogle Scholar
  2. 2.
    Azzouni, J., The derivation-indicator view of mathematical practice, Philosophia Mathematica 12(3):81–105, 2004.Google Scholar
  3. 3.
    Barwise, J., An introduction to first-order logic, in J. Barwise, (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 5–46.Google Scholar
  4. 4.
    Beklemishev, L., and A. Visser, Problems in the logic of provability, in D. Gabbay, S. Goncharov, and M. Zakharyaschev, (eds.), Mathematical Problems from Applied Logic I: Logics for the XXIst Century, vol. 4 of International Mathematical Series, Springer, Berlin, 2005, pp. 77–136.Google Scholar
  5. 5.
    Berk, L.A., Hilbert’s Thesis: Some Considerations about Formalizations of Mathematics, Ph.D. thesis, MIT, 1982.
  6. 6.
    Bernays, P., Abhandlungen zur Philosophie der Mathematik, Wissenschaftliche Buchgesellschaft, Darmstadt, 1976.Google Scholar
  7. 7.
    Boolos, G.S., J.P. Burgess, and R.C. Jeffrey, Computability and Logic, 4th edn., Cambridge University Press, Cambridge, 2003.Google Scholar
  8. 8.
    Boolos, G.S., and R.C. Jeffrey, Computability and Logic, 3rd edn., Cambridge University Press, Cambridge, 1989.Google Scholar
  9. 9.
    Davis, M., and W. Sieg, Conceptual confluence in 1936: Post and Turing, in G. Sommaruga, and T. Strahm, (eds.), Turing’s Revolution: The Impact of His Ideas about Computability, Springer, Berlin, 2015, pp. 3–27.Google Scholar
  10. 10.
    Ebbinghaus, H.-D., Ernst Zermelo, Springer, Berlin, 2007.Google Scholar
  11. 11.
    Feferman, S., H.M. Friedman, P. Maddy, and J.R. Steel, Does mathematics need new axioms?, Bulletin of Symbolic Logic 6(4):401–446, 2000.Google Scholar
  12. 12.
    Gentzen, G., Untersuchungen über das logische Schließen I, II, Mathematische Zeitschrift 39:176–210, 405–431, 1935.Google Scholar
  13. 13.
    Gentzen, G., The consistency of elementary number theory, in M.E. Szabo, (ed.), The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, 1969, pp. 132–201. English translation of Die Widerspruchsfreiheit der reinen Zahlentheorie originially published in 1939.Google Scholar
  14. 14.
    Gödel, K., On undecidable propositions of formal mathematical systems, in M. Davis, (ed.), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven Press, 1965, pp. 41–73.Google Scholar
  15. 15.
    Gödel, K., Vortrag über Vollständigkeit des Funktionenkalküls, in S. Feferman, et al., (eds.), Collected Works, vol. III of Unpublished Essays and Lectures, Oxford University Press, 1995, pp. 16–29. Lecture delivered on 6 September 1930 at the Conference on Epistemology of the Exact Sciences in Königsberg; German original and English translation.Google Scholar
  16. 16.
    Hales, T.C., Developments in formal proof, Séminaire Bourbaki, (2014), 1086, 66ème année, 2013–2014.Google Scholar
  17. 17.
    Hilbert, D., Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongreß zu Paris 1900, Nachrichten von der königl. Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse aus dem Jahre 1900, 1900, pp. 253–297.Google Scholar
  18. 18.
    Hilbert, D., Über den Zahlbegriff, Jahresbericht der Deutschen Mathematiker-Vereinigung 8:180–184, 1900.Google Scholar
  19. 19.
    Hilbert, D., Axiomatisches Denken, Mathematische Annalen 78(3/4):405–415, 1918. (Lecture delivered on 11 September 1917 at the Swiss Mathematical Society in Zurich).Google Scholar
  20. 20.
    Hilbert, D., Über das Unendliche, Mathematische Annalen 95:161–190, 1926.Google Scholar
  21. 21.
    Hilbert, D., Die Grundlagen der Mathematik, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 6(1/2):65–85, 1928.Google Scholar
  22. 22.
    Hilbert, D., Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen 104:485–494, 1931. (Lecture delivered in December 1930 in Hamburg).Google Scholar
  23. 23.
    Hilbert, D., The foundations of mathematics, in J. van Heijenoort, (ed.), From Frege to Gödel, Harvard University Press, 1967, pp. 464–479. English translation of [21].Google Scholar
  24. 24.
    Hilbert, D., On the infinite, in Jean van Heijenoort, (ed.), From Frege to Gödel, Harvard University Press, 1967, pp. 367–392. English translation of [20].Google Scholar
  25. 25.
    Hilbert, D., and W. Ackermann, Grundzüge der theoretischen Logik, vol. XXVII of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer, Berlin, 1928.Google Scholar
  26. 26.
    Hilbert, D., and W. Ackermann, Grundzüge der theoretischen Logik, vol. XXVII of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 2nd edn., Springer, Berlin, 1938.Google Scholar
  27. 27.
    Jäger, G., Inductive definitions and non-wellfounded proofs, Talk given in Tübingen in honor of Peter Schroeder-Heister’s 60th birthday, 2013.Google Scholar
  28. 28.
    Kahle, R., Von Dedekind zu Zermelo versus Peano zu Gödel, Mathematische Semesterberichte 64(2):159–167, 2017.Google Scholar
  29. 29.
    Kleene, S.C., Introduction to Metamathematics, North Holland, Amsterdam, 1952.Google Scholar
  30. 30.
    Kleene, S.C., Origins of recursive function theory, Annals of the History of Computing 3(1):52–67, 1981.Google Scholar
  31. 31.
    Kleene, S.C., Gödel’s impression on students of logic in the 1930s, in P. Weingartner, and L. Schmetterer, (eds.), Gödel Remembered, vol. IV of History of Logic, Bibliopolis, Berkeley, 1987, pp. 49–64.Google Scholar
  32. 32.
    Kreisel, G., Informal rigour and completeness proofs, in I. Lakatos, (ed.), Problems in the Philosophy of Mathematics, vol. 47 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 1967, pp. 138–186.Google Scholar
  33. 33.
    Kripke, S.A., The Church-Turing “Thesis” as a special corollary of Gödel’s completeness theorem, in B.J. Copeland, C.J. Posy, and O. Shagrir, (eds.), Computability, MIT Press, Cambridge, 2013, pp. 77–104.Google Scholar
  34. 34.
    Marfori, M.A., Informal proofs and mathematical rigour, Studia Logica 96:261–272, 2010.Google Scholar
  35. 35.
    Moschovakis, Y., Notes on Set Theory, 2nd edn., Undergraduate Texts in Mathematics, Springer, Berlin, 2006.Google Scholar
  36. 36.
    Naibo, A., M. Petrolo, and T. Seiller, On the computational meaning of axioms, in J. Redmond, O.P. Martins, and Á.N. Fernández, (eds.), Epistemology, Knowledge and the Impact of Interaction, Springer, Berlin, 2016, pp. 141–184.Google Scholar
  37. 37.
    Naibo, A., M. Petrolo, and T. Seiller, Verificationism and classical realizability, in C. Başkent, (ed.), Perspectives on Interrogative Models of Inquiry: Developments in Inquiry and Questions, Springer, Berlin, 2016, pp. 163–197.Google Scholar
  38. 38.
    Odifreddi, P., Classical Recursion Theory, vol. 125 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1989.Google Scholar
  39. 39.
    Parsons, C., Finitism and intuitive knowledge, in M. Schirn, (ed.), The Philosophy of Mathematics Today, Oxford University Press, Oxford, 1998, pp. 249–270.Google Scholar
  40. 40.
    Post, E.L., Finite combinatory processes—formulation 1, Journal of Symbolic Logic 1(3):103–105, 1936.Google Scholar
  41. 41.
    Prawitz, D., Natural Deduction, A Proof-Theoretical Study, Almquist and Wiksell, 1965.Google Scholar
  42. 42.
    Shapiro, S., The open texture of computability, in B.J. Copeland, C.J. Posy, and O. Shagrir, (eds.), Computability, MIT Press, Cambridge, 2013, pp. 153–181.Google Scholar
  43. 43.
    Shoenfield, J.R., Mathematical Logic, Addison-Wesley, 1967. Reprinted by ASL, AK Peters, 2000.Google Scholar
  44. 44.
    Sieg, W., In the shadow of incompletenss, in Hilbert’s Programs and Beyond, Oxford University Press, Oxford, 2013, pp. 155–192. First published in another collection in 2011.Google Scholar
  45. 45.
    Smullyan, R.M., Fixed points and self-reference, International Journal of Mathematics and Mathematical Sciences 7(2):283–289, 1984.Google Scholar
  46. 46.
    Soare, R.I., Interactive computing and relativized computability, in B.J. Copeland, C.J. Posy, and O. Shagrir, (eds.), Computability, MIT Press, Cambridge, 2013, pp. 203–260.Google Scholar
  47. 47.
    Streett, R.S., and E.A. Emerson, An automata theoretic decision procedure for the propositional mu-calculus, Information and Computation 81:249–264, 1989.Google Scholar
  48. 48.
    Tapp, C., An den Grenzen des Unendlichen, Mathematik im Kontext, Springer, Berlin, 2013.Google Scholar
  49. 49.
    Weir, A., Informal proof, formal proof, formalism, The Review of Symbolic Logic 9(1):23–43, 2016.Google Scholar

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Authors and Affiliations

  1. 1.CMA and DM, FCTUniversidade Nova de LisboaCaparicaPortugal

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