Hostname: page-component-7dd5485656-kp629 Total loading time: 0 Render date: 2025-10-30T19:18:52.725Z Has data issue: false hasContentIssue false
  • English
  • Français

AN ANALYSIS OF THE CONSTRUCTIVE CONTENT OF HENKIN’S PROOF OF GÖDEL’S COMPLETENESS THEOREM

Part of: Model theory General logic Mathematical Logic and Foundations Proof theory and constructive mathematics

Published online by Cambridge University Press:  10 September 2025

HUGO HERBELIN Open the ORCID record for HUGO HERBELIN [Opens in a new window]  and
DANKO ILIK Open the ORCID record for DANKO ILIK [Opens in a new window]
HUGO HERBELIN*
Affiliation:
IRIF, CNRS, INRIA, UNIVERSITÉ PARIS CITÉ PARIS 75013, FRANCE
DANKO ILIK
Affiliation:
CENTRE NATIONAL D’ÉTUDES SPATIALES PARIS 75039, FRANCE E-mail: danko.ilik@cnes.fr
*
E-mail: hugo.herbelin@inria.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Gödel’s completeness theorem for classical first-order logic is one of the most basic theorems of logic. Central to any foundational course in logic, it connects the notion of valid formula, i.e., a formula satisfied in all models, to the notion of provable formula.

We survey a few standard formulations and proofs of the completeness theorem before focusing on the formal description of a slight modification of Henkin’s proof within intuitionistic second-order arithmetic.

It is usual, in the context of the completeness of intuitionistic logic with respect to various semantics, such as Kripke or Beth semantics, to follow the Curry–Howard correspondence and to interpret the proofs of completeness as programs which turn proofs of validity for these semantics into proofs of derivability.

We apply this approach to Henkin’s proof to phrase it as a program which transforms any proof of validity with respect to Tarski semantics into a proof of derivability.

By doing so, we hope to shed some “effective” light on the relation between Tarski semantics and syntax: proofs of validity are syntactic objects with which we can compute.

Keywords

Tarski semanticsHenkin’s proof of Gödel’s completenesscomputational contents of proofsreverse mathematics

MSC classification

Primary: 03B10: Classical first-order logic 03B30: Foundations of classical theories (including reverse mathematics) 03B35: Mechanization of proofs and logical operations 03C07: Basic properties of first-order languages and structures 03F65: Other constructive mathematics
Secondary: 03-03: Historical (must also be assigned at least one classification number from Section 01) 03-04: Explicit machine computation and programs (not the theory of computation or programming)

Information

Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 43
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abel, A. and Sattler, C., Normalization by evaluation for call-by-push-value and polarized lambda calculus , Proceedings of the 21st International Symposium on Principles and Practice of Declarative Programming PPDP ’19 (E. Komendantskaya, editor), Association for Computing Machinery, New York, 2019.Google Scholar
Altenkirch, T., Dybjer, P., Hofmann, M., and Scott, P. J., Normalization by evaluation for typed lambda calculus with coproducts , LICS ’01: Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science (J. Halpern, editor), IEEE Computer Society, Washington, 2001, pp. 303–310.Google Scholar
Altenkirch, T., Hofmann, M., and Streicher, T., Reduction-free normalisation for a polymorphic system , Proceedings, 11th Annual IEEE Symposium on Logic in Computer Science, New Brunswick, New Jersey, USA, July 27-30, 1996 (E. Clarke, editor), IEEE Computer Society Press, New Brunswick, New Jersey, 1996, pp. 98–106.Google Scholar
Ariola, Z. M. and Herbelin, H., Minimal classical logic and control operators , Thirtieth International Colloquium on Automata, Languages and Programming, ICALP’03, Eindhoven, the Netherlands, June 30—July 4, 2003 (J. C. M. Baeten, J. K. Lenstra, J. Parrow, and G. Woeginger, editors), Lecture Notes in Computer Science, 2719, Springer-Verlag, Berlin, 2003, pp. 871–885.Google Scholar
Asperti, A., Ricciotti, W., Coen, C. S., and Tassi, E., Formal metatheory of programming languages in the Matita interactive theorem prover . Journal of Automated Reasoning , vol. 49 (2012), no. 3, pp. 427–451.Google Scholar
Barwise, J., An introduction to first-order logic , Handbook of Mathematical Logic (Barwise, J., editor), Studies in Logic and the Foundations of Mathematics, 90, Elsevier, Amsterdam, 1977, pp. 5–46.Google Scholar
Berardi, S., Intuitionistic completeness for first order classical logic . Journal of Symbolic Logic , vol. 64 (1999), no. 1, pp. 304–312.Google Scholar
Berardi, S. and Valentini, S., Krivine’s intuitionistic proof of classical completeness (for countable languages) . Annals of Pure and Applied Logic , vol. 129 (2004), nos. 1–3, pp. 93–106.Google Scholar
Berger, J., A decomposition of Brouwer’s fan theorem . Journal of Logic and Analysis , vol. 1 (2009), pp. 1–8.Google Scholar
Berger, U. and Schwichtenberg, H., An inverse of the evaluation functional for typed lambda-calculus , Proceedings, Sixth Annual IEEE Symposium on Logic in Computer Science, 15-18 July, 1991, Amsterdam, the Netherlands (G. Kahn, editor), IEEE Computer Society, Los Alamitos, CA, 1991, pp. 203–211.Google Scholar
Beth, E. W., Semantic entailment and formal derivability , Mededeelingen der Koninklijke Nederlandsche Akademie van Wetenschappen, Afd. Letterkunde , vol. 18, Noord-Hollandsche, Amsterdam, 1955, pp. 309–342.Google Scholar
Blanchette, J. C., Popescu, A., and Traytel, D., Unified classical logic completeness , Automated Reasoning (Demri, S., Kapur, D., and Weidenbach, C., editors), Springer International Publishing, Cham, 2014, pp. 46–60.Google Scholar
Brede, N. and Herbelin, H., On the logical structure of choice and bar induction principles , Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) (L. Libkin, editor), IEEE, Los Alamitos, CA, 2021, pp. 1–13.Google Scholar
Bridges, D., Palmgren, E., and Ishihara, H. Constructive mathematics , The Stanford Encyclopedia of Philosophy (Zalta, E. N. and Nodelman, U., editors), Metaphysics Research Lab, Stanford University, Stanford, CA, 2022. ed. 2022.Google Scholar
Buss, S. R., An introduction to proof theory , Handbook of Proof Theory (Buss, S. R., editor), North-Holland, North-Holland, Amsterdam, 1998, pp. 1–78.Google Scholar
Church, A., Introduction to Mathematical Logic , vol. 1, The Princeton University Press, Princeton, New Jersey, 1956.Google Scholar
Cohen, L., Forster, Y., Kirst, D., Paiva, B. D. R., and Rahli, V., Separating markov’s principles , Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science (U. D. Lago and J. Esparza, editors), Association for Computing Machinery, New York, 2024.Google Scholar
The Coq Development Team , The Coq reference manual, version 8.20. 2024, Distributed electronically at http://coq.inria.fr/refman Google Scholar
\bysame, The Coq standard library, version 8.20. Technical Report, 2024, Distributed electronically at http://coq.inria.fr/stdlib Google Scholar
Coquand, C., A formalised proof of the soundness and completeness of a simply typed lambda-calculus with explicit substitutions . Higher-Order and Symbolic Computation , vol. 15 (2002), no. 1, pp. 57–90.Google Scholar
Coquand, T. and Dybjer, P., Intuitionistic model constructions and normalization proofs . Mathematical Structures in Computer Science , vol. 7 (1997), no. 1, pp. 75–94.Google Scholar
Coquand, T. and Paulin-Mohring, C., Inductively defined types , Proceedings of colog’88 (Martin-Löf, P. and Mints, G., editors), Lecture Notes in Computer Science, 417, Springer-Verlag, Berlin, Heidelberg, 1990.Google Scholar
Coquand, T. and Smith, J. M., An application of constructive completeness , Types for Proofs and Programs (S. Berardi, editor), Lecture Notes in Computer Science, 1158, Springer, Berlin, Heidelberg, 1995, pp. 76–84.Google Scholar
Danvy, O., Type-directed partial evaluation , Proceedings of the 23rd ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (H.-J. Boehm and G. Steele, editors), Association for Computing Machinery, New York, NY, 1996, pp. 242–257.Google Scholar
Espíndola, C., Achieving completeness: From constructive set theory to large cardinals , Ph.D. thesis, Stockholm University, 2016.Google Scholar
Espíndola, C., Semantic completeness of first-order theories in constructive reverse mathematics . Notre Dame Journal of Formal Logic , vol. 57 (2016), no. 2, pp. 281–286.Google Scholar
Feferman, S., Dawson, J. W. Jr, Kleene, S. C., Moore, G. H., Solovay, R. M., and van Heijenoort, J., Kurt Gödel: Collected Works. Vol. 1: Publications 1929–1936 , Oxford University Press, New York, 1986.Google Scholar
Forssell, H. and Espíndola, C., Constructive completeness and non-discrete languages, preprint, 2017, arXiv:1709.05817v1.Google Scholar
Forster, Y., Kirst, D., and Wehr, D., Completeness theorems for first-order logic analysed in constructive type theory , Logical Foundations of Computer Science (S. Artemov and A. Nerode, editors), Lecture Notes in Computer Science, 11972, Springer, Cham, 2020, pp. 47–74.Google Scholar
Forster, Y., Kirst, D., and Wehr, D., Completeness theorems for first-order logic analysed in constructive type theory: Extended version . Journal of Logic and Computation , vol. 31 (2021), no. 1, pp. 112–151.Google Scholar
Frege, G., Begriffschrift, Eine der Arithmetischen Nachgebildete Formelsprache Des Reinen Denkens , Halle, Halle, 1879. English translation e.g. in [32] or [86].Google Scholar
Frege, G., Begriffschrift, Eine der Arithmetischen Nachgebildete Formelsprache Des Reinen Denkens , Halle, 1879. English translation e.g. in [86].Google Scholar
Friedman, H., Intuitionistic completeness of heyting’s predicate calculus . Notices of the American Mathematical Society , vol. 22 (1975), pp. A–648.Google Scholar
Friedman, H., Classically and intuitionistically provably recursive functions , Higher Set Theory (Scott, D. S. and Muller, G. H., editors), Lecture Notes in Mathematics, 669, Springer, Berlin, 1978, pp. 21–27.Google Scholar
Gentzen, G., Untersuchungen über das logische Schließen . Mathematische Zeitschrift , vol. 39 (1935), pp. 176–210. 405–431. English Translation in [83], “Investigations into logical deduction”, pages 68–131.Google Scholar
Gödel, K., Über die vollständigkeit des logikkalküls, Ph.D. thesis, University of Vienna, 1929. English translation in [86] or [27].Google Scholar
Gödel, K., Die vollständigkeit des axiome des logischen functionenkalküls . Monatshefte für Mathematik und Physik , vol. 37 (1930), pp. 349–360. Reprinted in [27].Google Scholar
Gödel, K., Über formal unentscheidbare sätze der principia mathematica und verwandter systeme . Monatshefte für Mathematik und Physik , vol. 38 (1931), pp. 173–198.Google Scholar
Griffin, T. G., The formulae-as-types notion of control , Conf. Record 17th Annual ACM Symp. On Principles of Programming Languages, POPL ’90, San Francisco, CA, USA, 17-19 Jan 1990 (F. E. Allen, editor), ACM Press, New York,1990, pp. 47–57.Google Scholar
Hasenjaeger, G., Eine bemerkung zu henkin’s beweis für die vollständigkeit des prädikatenkalküls der ersten stufe . Journal of Symbolic Logic , vol. 18 (1953), pp. 42–48.Google Scholar
Henkin, L., The completeness of the first-order functional calculus . Journal of Symbolic Logic , vol. 14 (1949), no. 3, pp. 159–166.Google Scholar
Henkin, L., The discovery of my completeness proofs . The Bulletin of Symbolic Logic , vol. 2 (1996), no. 2, pp. 127–158.Google Scholar
Herbelin, H., An intuitionistic logic that proves Markov’s principle , Proceedings of the 25th Annual Ieee Symposium on Logic in Computer Science, LICS 2010, 11-14 July 2010, Edinburgh, United Kingdom (J.-P. Jouannaud, editor), IEEE Computer Society, Los Alamitos, CA, 2010, pp. 50–56.Google Scholar
Herbelin, H. and Lee, G., Forcing-based cut-elimination for gentzen-style intuitionistic sequent calculus , Logic, Language, Information and Computation, 16th International Workshop, Wollic 2009, Tokyo, Japan, June 21-25, 2009, Proceedings (Ono, H., Kanazama, M., and de Queiroz, R., editors), Lecture Notes in Computer Science, 5514, Springer, Berlin, Heidelberg, 2009, pp. 209–217.Google Scholar
Hermant, O. Semantic cut elimination in the intuitionistic sequent calculus , Typed Lambda Calculi and Applications, 7th International Conference, TLCA 2005, Nara, Japan, April 21-23, 2005, Proceedings (Urzyczyn, P., editor), Lecture Notes in Computer Science, 3461, Springer, Berlin, Heidelberg, 2005, pp. 221–233.Google Scholar
Hermant, O. and Lipton, J., Completeness and cut-elimination in the intuitionistic theory of types—Part 2 . Journal of Logic and Computation , vol. 20 (2010), no. 2, pp. 597–602.Google Scholar
Hilbert, D. and Ackermann, W., Grundzüge der Theoretischen Logik , Springer Verlag, Berlin, 1928. English translation of the 1938 edition in [48].Google Scholar
Hilbert, D. and Ackermann, W., Principles of Mathematical Logic , Oxford University Press, New York, 1928.Google Scholar
Hintikka, J., Form and content in quantification theory . Acta Philosophica Fennica , vol. 8 (1955), pp. 11–55.Google Scholar
Hintikka, J., Notes on the quantification theory . Societas Scientiarum Fennica, Commentationes Physico-Mathematicae , vol. 17 (1955), no. 1.Google Scholar
Ilik, D., Constructive ultrafilter theorem and completeness for classical predicate logic (formalisation). 2008. https://iaddg.net/danko/boolean_completeness.zip Google Scholar
Ilik, D., Delimited control operators prove double-negation shift . Annals of Pure and Applied Logic , vol. 163 (2011), no. 11, pp. 1549–1559. In press.Google Scholar
Ilik, D., Continuation-passing style models complete for intuitionistic logic . Annals of Pure and Applied Logic , vol. 164 (2013), no. 6, pp. 651–662.Google Scholar
Ilik, D., The exp-log normal form of types: Decomposing extensional equality and representing terms compactly . ACM SIGPLAN Notices , vol. 52 (2017), no. 1, pp. 387–399.Google Scholar
Ilik, D., Lee, G., and Herbelin, H., Kripke models for classical logic . Annals of Pure and Applied Logic , vol. 161 (2010), no. 11, pp. 1367–1378. Special Issue on Classical Logic and Computation.Google Scholar
Jaškowski, S., On the rules of suppositions in formal logic . Studia Logica , vol. 1 (1934), pp. 5–32. Reprinted in S. McCall (1967) Polish Logic 1920-1939, Oxford: Oxford Univ. Press, pp. 232–258.Google Scholar
Jech, T. J., The Axiom of Choice , Dover Books on Mathematics Series, American Elsevier Publishing Company, New York, 1973.Google Scholar
Johansson, I., Der minimalkalkül, ein reduzierter intuitionistischer formalismus . Compositio Mathematica , vol. 4 (1937), pp. 119–136.Google Scholar
Kanger, S., Provability in Logic , Acta Universitatis Stockholmiensis, Almqvist & Wiksell, Stockholm, 1957.Google Scholar
Kirst, D., Mechanised metamathematics: An investigation of first-order logic and set theory in constructive type theory, Ph.D. thesis, University of Sarrbrücken, 2023. private communication.Google Scholar
Kleene, S. C., Recursive functions and intuitionistic mathematics , Proceedings of the International Congress of Mathemaiticans (Graves, L. M., Hille, E., Smith, P. A., and Zariski, O., editors), American Mathematical Society, Providence, 1952, pp. 679–685.Google Scholar
Kreisel, G., Mathematical significance of consistency proofs . Journal of Symbolic Logic , vol. 23 (1958), no. 2, pp. 155–182.Google Scholar
Kreisel, G., On weak completeness of intuitionistic predicate logic . Journal of Symbolic Logic , vol. 27 (1962), no. 2, pp. 139–158.Google Scholar
Krivine, J.-L., Lambda-Calcul, Types et modèles , Masson, Paris, 1990.Google Scholar
Krivine, J.-L., Opérateurs de mise en mémoire et traduction de Gödel . Archive for Mathematical Logic , vol. 30 (1990), no. 4, pp. 241–267. (French).Google Scholar
Krivine, J.-L., Lambda-Calculus, Types and Models , Ellis Horwood, Hemel Hempstead, 1993.Google Scholar
Krivine, J.-L., Classical logic, storage operators and second-order lambda-calculus . Annals of Pure and Applied Logic , vol. 68 (1994), no. 1, pp. 53–78.Google Scholar
Krivine, J.-L., Une preuve formelle et intuitionniste du théorème de complétude de la logique classique . The Bulletin of Symbolic Logic , vol. 2 (1996), no. 4, pp. 405–421.Google Scholar
Krivtsov, V. N., An intuitionistic completeness theorem for classical predicate logic . Studia Logica: An International Journal for Symbolic Logic , vol. 96 (2010), no. 1, pp. 109–115.Google Scholar
Krivtsov, V. N., Semantical completeness of first-order predicate logic and the weak fan theorem . Studia Logica , vol. 103 (2015), no. 3, pp. 623–638.Google Scholar
Martin-Löf, P., An intuitionistic theory of types, predicative part , Logic Colloquium (H. E. Rose and J. C. Shepherdson, editors), North Holland, Amsterdam, 1975, pp. 73–118.Google Scholar
McCarty, C., Completeness and incompleteness for intuitionistic logic . Journal of Symbolic Logic , vol. 73 (2008), no. 4, pp. 1315–1327.Google Scholar
Okada, M., A uniform semantic proof for cut-elimination and completeness of various first and higher order logics . Theoretical Computer Science , vol. 281 (2002), nos. 1–2, pp. 471–498.Google Scholar
Parigot, M., Lambda-mu-calculus: An algorithmic interpretation of classical natural deduction , Logic Programming and Automated Reasoning: International Conference LPAR ’92 Proceedings, St. Petersburg, Russia (A. Voronkov, editor), Springer-Verlag, Berlin, Heidelberg, 1992, pp. 190–201.Google Scholar
Rasiowa, H. and Sikorski, R., A proof of the completeness theorem of Grödel . Fundamenta Mathematicae , vol. 37 (1950), no. 1, pp. 193–200.Google Scholar
Sambin, G., Pretopologies and completeness proofs . Journal of Symbolic Logic , vol. 60 (1995), no. 3, pp. 861–878.Google Scholar
Schwichtenberg, H. and Wainer, S. S., Proofs and Computations , Cambridge University Press, Cambridge, 2011.Google Scholar
Schütte, K., Ein system des verknüpfenden schliessens . Archiv für mathematische Logik und Grundlagenforschung , vol. 2 (1956), nos. 2–4, pp. 55–67. (German).Google Scholar
Shillito, I. and Kirst, D., A mechanised and constructive reverse analysis of soundness and completeness of bi-intuitionistic logic , Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2024, London, Uk, January 15–16, 2024 (Timany, A., Traytel, D., Pientka, B., and Blazy, S., editors), ACM, New York, NY, 2024, pp. 218–229.Google Scholar
Simpson, S. G., Subsystems of Second Order Arithmetic , Springer Seminars in Immunopathology, Springer, Berlin, Heidelberg, 1999.Google Scholar
Smullyan, R. M., First-Order Logic , Ergebnisse der Mathematik und ihrer Grenzgebiete, 43, Springer-Verlag, New York, 1968.Google Scholar
Spector, C., Provably recursive functionals of analysis: A consistency proof of analysis by an extension of principles in current intuitionistic mathematics , Recursive Function Theory: Proceedings of Symposia in Pure Mathematics , vol. 5, (Dekker, F. D. E., editor), American Mathematical Society, Providence, 1962, pp. 1–27.Google Scholar
Szabo, M. E., The Collected Works of Gerhard Gentzen , North Holland, Amsterdam, 1969.Google Scholar
Troelstra, A. S., Metamathematical Investigation of Intuitionistic Arithmetic and Analysis , Lecture Notes in Mathematics, 344, Springer-Verlag, Berlin, 1973.Google Scholar
Troelstra, A.S. and van Dalen, D. Constructivism in Mathematics: An Introduction , Studies in Logic and the Foundations of Mathematics, 121, Elsevier, Amsterdam, 1988.Google Scholar
van Heijenoort, J., From Frege to Gödel, a Source Book in Mathematical Logic, 1879–1931 , Harvard University Press, New York, 1967.Google Scholar
Veldman, W., An intuitionistic completeness theorem for intuitionistic predicate logic . Journal of Symbolic Logic , vol. 41 (1976), no. 1, pp. 159–166.Google Scholar
Veldman, W., Intuitionism: An inspiration? Jahresbericht der Deutschen Mathematiker-Vereinigung , vol. 123 (2021), no. 4, pp. 221–284.Google Scholar