A Formally Verified Abstract Account of Gödel’s Incompleteness Theorems

  • Andrei PopescuEmail author
  • Dmitriy TraytelEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11716)


We present an abstract development of Gödel’s incompleteness theorems, performed with the help of the Isabelle/HOL theorem prover. We analyze sufficient conditions for the theorems’ applicability to a partially specified logic. In addition to the usual benefits of generality, our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosser’s variation of the first theorem, Jeroslow’s variation of the second theorem, and the Świerczkowski–Paulson semantics-based approach. As part of our framework’s validation, we upgrade Paulson’s Isabelle proof to produce a mechanization of the second theorem that does not assume soundness in the standard model, and in fact does not rely on any notion of model or semantic interpretation.



We thank Bernd Buldt for his patient explanations on material in his monograph, and the reviewers for insightful comments and suggestions.


  1. 1.
    Ammon, K.: An automatic proof of Gödel’s incompleteness theorem. Artif. Intell. 61(2), 291–306 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Auerbach, D.: Intensionality and the Gödel theorems. Philos. Stud. Int. J. Philos. Anal. Tradit. 48(3), 337–351 (1985)CrossRefGoogle Scholar
  3. 3.
    Blanchette, J.C., Popescu, A., Traytel, D.: Unified classical logic completeness. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 46–60. Springer, Cham (2014). Scholar
  4. 4.
    Boolos, G.: The Logic of Provability. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  5. 5.
    Buldt, B.: The scope of Gödel’s first incompleteness theorem. Log. Univers. 8(3), 499–552 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bundy, A., Giunchiglia, F., Villafiorita, A., Walsh, T.: An incompleteness theorem via abstraction. Technical report, Istituto per la Ricerca Scientifica e Tecnologica, Trento (1996)Google Scholar
  7. 7.
    Carnap, R.: Logische syntax der sprache. Philos. Rev. 44(4), 394–397 (1935)CrossRefGoogle Scholar
  8. 8.
    Davis, M.: The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems, and Computable Functions. Dover Publication, Mineola (1965)zbMATHGoogle Scholar
  9. 9.
    Diaconescu, R.: Institution-Independent Model Theory, 1st edn. Birkhäuser, Basel (2008)zbMATHGoogle Scholar
  10. 10.
    Feferman, S., Dawson Jr., J.W., Kleene, S.C., Moore, G.H., Solovay, R.M., van Heijenoort, J. (eds.): Kurt Gödel: Collected Works. Vol. 1: Publications 1929–1936. Oxford University Press, Oxford (1986)Google Scholar
  11. 11.
    Fiore, M.P., Plotkin, G.D., Turi, D.: Abstract syntax and variable binding. In: Logic in Computer Science (LICS) 1999, pp. 193–202. IEEE Computer Society (1999)Google Scholar
  12. 12.
    Gabbay, M.J., Mathijssen, A.: Nominal (universal) algebra: equational logic with names and binding. J. Log. Comput. 19(6), 1455–1508 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38(1), 173–198 (1931)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Goguen, J.A., Burstall, R.M.: Institutions: abstract model theory for specification and programming. J. ACM 39(1), 95–146 (1992)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Harrison, J.: HOL light proof of Gödel’s first incompleteness theorem., directory Arithmetic
  16. 16.
    Hilbert, D., Bernays, P.: Grundlagen der Mathematik, vol. II. Springer, Heidelberg (1939)zbMATHGoogle Scholar
  17. 17.
    Jeroslow, R.G.: Redundancies in the Hilbert-Bernays derivability conditions for Gödel’s second incompleteness theorem. J. Symb. Log. 38(3), 359–367 (1973)CrossRefGoogle Scholar
  18. 18.
    Kaliszyk, C., Urban, J.: HOL(y)Hammer: online ATP service for HOL light. Math. Comput. Sci. 9(1), 5–22 (2015)CrossRefGoogle Scholar
  19. 19.
    Kikuchi, M., Kurahashi, T.: Generalizations of Gödel’s incompleteness theorems for \(\sum \) n-definable theories of arithmetic. Rew. Symb. Logic 10(4), 603–616 (2017)CrossRefGoogle Scholar
  20. 20.
    Kossak, R.: Mathematical Logic. SGTP, vol. 3. Springer, Cham (2018). Scholar
  21. 21.
    Kunčar, O., Popescu, A.: A consistent foundation for Isabelle/HOL. In: Urban, C., Zhang, X. (eds.) ITP 2015. LNCS, vol. 9236, pp. 234–252. Springer, Cham (2015). Scholar
  22. 22.
    Kunčar, O., Popescu, A.: Comprehending Isabelle/HOL’s consistency. In: Yang, H. (ed.) ESOP 2017. LNCS, vol. 10201, pp. 724–749. Springer, Heidelberg (2017). Scholar
  23. 23.
    Löb, M.: Solution of a problem of Leon Henkin. J. Symb. Log. 20(2), 115–118 (1955)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nipkow, T., Wenzel, M., Paulson, L.C. (eds.): Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002). Scholar
  25. 25.
    O’Connor, R.: Essential incompleteness of arithmetic verified by Coq. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 245–260. Springer, Heidelberg (2005). Scholar
  26. 26.
    Paulson, L.C.: A machine-assisted proof of Gödel’s incompleteness theorems for the theory of hereditarily finite sets. Rew. Symb. Logic 7(3), 484–498 (2014)CrossRefGoogle Scholar
  27. 27.
    Paulson, L.C.: A mechanised proof of Gödel’s incompleteness theorems using Nominal Isabelle. J. Autom. Reason. 55(1), 1–37 (2015)CrossRefGoogle Scholar
  28. 28.
    Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, a practical link between automatic and interactive theorem provers. In: The 8th International Workshop on the Implementation of Logics, IWIL 2010, Yogyakarta, Indonesia, 9 October 2011, pp. 1–11 (2010)Google Scholar
  29. 29.
    Popescu, A., Roşu, G.: Term-generic logic. Theor. Comput. Sci. 577, 1–24 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Popescu, A., Trayel, D.: A formally verified abstract account of Gödel’s incompleteness theorems (extended report) (2019).
  31. 31.
    Popescu, A., Traytel, D.: Formalization associated with this paper (2019).
  32. 32.
    Quaife, A.: Automated proofs of Löb’s theorem and Gödel’s two incompleteness theorems. J. Autom. Reason. 4(2), 219–231 (1988)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Raatikainen, P.: Gödel’s incompleteness theorems. In: The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University (2018)Google Scholar
  34. 34.
    Schlichtkrull, A., Blanchette, J.C., Traytel, D., Waldmann, U.: Formalizing Bachmair and Ganzinger’s ordered resolution prover. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS (LNAI), vol. 10900, pp. 89–107. Springer, Cham (2018). Scholar
  35. 35.
    Shankar, N.: Metamathematics, Machines, and Gödel Proof. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  36. 36.
    Sieg, W.: Elementary proof theory. Technical report, Institute for Mathematical Studies in the Social Sciences, Stanford (1978)Google Scholar
  37. 37.
    Sieg, W., Field, C.: Automated search for Gödel’s proofs. Ann. Pure Appl. Logic 133(1–3), 319–338 (2005)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Smith, P.: An Introduction to Gödel’s Incompleteness Theorems. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  39. 39.
    Smorynski, C.: The incompleteness theorems. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 821–865. North-Holland, Amsterdam (1977)CrossRefGoogle Scholar
  40. 40.
    Świerczkowski, S.: Finite sets and Gödel incompleteness theorems. Diss. Math. 422, 1–58 (2003)zbMATHGoogle Scholar
  41. 41.
    Tarski, A., Mostowski, A., Robinson, R.: Undecidable Theories. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1953). 3rd edn. 1971CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceMiddlesex UniversityLondonUK
  2. 2.Institute of Information Security, Department of Computer ScienceETH ZürichZurichSwitzerland

Personalised recommendations