Incompleteness, Undecidability and Automated Proofs

(Invited Talk)
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9890)


Incompleteness and undecidability have been used for many years as arguments against automatising the practice of mathematics. The advent of powerful computers and proof-assistants – programs that assist the development of formal proofs by human-machine collaboration – has revived the interest in formal proofs and diminished considerably the value of these arguments.

In this paper we discuss some challenges proof-assistants face in handling undecidable problems – the very results cited above – using for illustrations the generic proof-assistant Isabelle.


  1. 1.
  2. 2.
    Archive of formal proofs. Accessed 18 May 2016
  3. 3.
    Coq homepage. Accessed 25 Oct 2014
  4. 4.
    HOL4 homepage. Accessed 25 Oct 2014
  5. 5.
    Isabelle homepage. Accessed 20 Oct 2014
  6. 6.
    Matita hompage. Accessed 25 Oct 2014
  7. 7.
  8. 8.
    Asperti, A., Ricciotti, W.: Formalizing turing machines. In: Ong, L., de Queiroz, R. (eds.) WoLLIC 2012. LNCS, vol. 7456, pp. 1–25. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Benzmüller, C., Woltzenlogel Paleo, B.: Automating Gödel’s ontological proof of God’s existence with higher-order automated theorem provers. In: Schaub, T., Friedrich, G., O’Sullivan, B. (eds.) ECAI 2014, Frontiers in Artificial Intelligence and Applications, vol. 263, pp. 93–98. IOS Press (2014)Google Scholar
  10. 10.
    Du Bois-Reymond, E.H.: Über die Grenzen des Naturerkennens; Die sieben Welträthsel, zwei Vorträge. Von Veit, Leipzig (1898)Google Scholar
  11. 11.
    Bourbaki, N.: Theory of Sets. Elements of Mathematics. Springer, Heidelberg (1968)MATHGoogle Scholar
  12. 12.
    Calude, C.: Theories of Computational Complexity, North-Holland, Amsterdam (1988)Google Scholar
  13. 13.
    Calude, C.S., Calude, E., Marcus, S.: Passages of proof. Bull. Eur. Assoc. Theor. Comput. Sci. 84, 167–188 (2004)MathSciNetMATHGoogle Scholar
  14. 14.
    Calude, C.S., Hay, N.J.: Every computably enumerable random real is provably computably enumerable random. Logic J. IGPL 17, 325–350 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Calude, C.S., Müller, C.: Formal proof: reconciling correctness and understanding. In: Carette, J., Dixon, L., Coen, C.S., Watt, S.M. (eds.) MKM 2009, Held as Part of CICM 2009. LNCS, vol. 5625, pp. 217–232. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Cooper, S.B.: Computability Theory. Chapman Hall/CRC, London (2004)MATHGoogle Scholar
  17. 17.
    Edwards, C.: Automated proofs. Math struggles with the usability of formal proofs. Commun. ACM 59(4), 13–15 (2016)CrossRefGoogle Scholar
  18. 18.
    Feferman, S.: Are there absolutely unsolvable problems? Gödel’s dichotomy. Philosophia Math. 14(2), 134–152 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gödel, K.: Some basic theorems on the foundations of mathematics and their implications. In: Feferman, S., Dawson Jr., J.W., Goldfarb, W., Parsons, C., Solovay, R.M. (eds.) Collected Works. Unpublished Essays and Lectures. vol. III, pp. 304–323. Oxford University Press (1995)Google Scholar
  20. 20.
    Gordon, M.: From LCF to HOL: a short history. In: Proof, Language, and Interaction, pp. 169–186 (2000)Google Scholar
  21. 21.
    Hales, T.C.: A proof of the Kepler conjecture. Ann. Math. 162(3), 1065–1185 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hernández-Orozco, S., Hernández-Quiroz, F., Zenil, H., Sieg, W.: Rare speed-up in automatic theorem proving reveals tradeoff between computational time and information value (2015).
  23. 23.
    Heule, M.J.H., Kullmann, O., Marek, V.W.: Solving and verifying the boolean Pythagorean triples problem via cube-and-conquer (2016). [cs.DM]
  24. 24.
    Hilbert, D.: Hilbert’s 1930 radio speech.
  25. 25.
    Kleene, S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1952)Google Scholar
  26. 26.
    Konev, B., Lisitsa, A.: A SAT attack on the Erdös discrepancy conjecture (2014).
  27. 27.
    Martin-Löf, P.: Verification then and now. In: De Pauli-Schimanovich, W., Koehler, E., Stadler, F. (eds.) The Foundational Debate, Complexity and Constructivity in Mathematics and Physics, pp. 187–196. Kluwer, Dordrecht (1995)Google Scholar
  28. 28.
    Norrish, M.: Mechanised computability theory. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) ITP 2011. LNCS, vol. 6898, pp. 297–311. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  29. 29.
    Soare, R.I.: Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer, Heidelberg (1987)CrossRefMATHGoogle Scholar
  30. 30.
    Szasz, N.: A machine checked proof that Ackermann’s function is not primitive recursive. In: Huet, G. (ed.) Logical Environments, pp. 31–7. University Press (1991)Google Scholar
  31. 31.
    Tao, T.: The Erdös discrepancy problem (2015).
  32. 32.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry. University of California Press, Berkeley and Los Angeles (1951)MATHGoogle Scholar
  33. 33.
    Thompson, D.: Formalisation vs. understanding. In: Calude, C.S., Dinneen, M.J. (eds.) UCNC 2015. LNCS, vol. 9252, pp. 290–300. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  34. 34.
    Xu, J., Zhang, X., Urban, C.: Mechanising turing machines and computability theory in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 147–162. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  35. 35.
    Zammit, V.: A mechanisation of computability theory in HOL. In: von Wright, J., Harrison, J., Grundy, J. (eds.) TPHOLs 1996. LNCS, vol. 1125. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  36. 36.
    Zammit, V.: On the Readability of Machine Checkable Formal Proofs. Ph.D. Thesis, University of Kent, March 1999Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand
  2. 2.Department of PhilosophyStanford UniversityStanfordUSA

Personalised recommendations