Research Article

Exploring the Foundational Significance of Gödelʼs Incompleteness Theorems

Yong Cheng [PDF]

Article information
Vol 2, No 1
RAP0012 – Research Article
Recieved: November 19, 2021
Accepted: June 21, 2022
Online Published: July 13, 2022
DOI: 10.18494/SAM.RAP.2022.0012
Cite this article
Cheng, Y. (2022). Exploring the Foundational Significance of Gödelʼs Incompleteness Theorems. The Review of Analytic Philosophy, 2(1), 1-30. Japan: MYU.


Gödelʼs incompleteness theorems, published in 1931, are important and
profound results in the foundations and philosophy of mathematics. On the
basis of new advances in research on incompleteness in the literature, we
discuss the correct interpretations of Gödelʼs incompleteness theorems, their
influence on various fields, and the limit of their applicability. The motivation
of this paper is threefold: to explore the foundational and philosophical
significance of new advances in research on incompleteness since Gödel, to
introduce new advances in research on incompleteness to the general philosophy
community, and to commemorate the 90th anniversary of the publication of
Gödelʼs incompleteness theorems.


The first incompleteness theorem, The second incompleteness theorem, The
limit of incompleteness, The intensional problem, Interpretation


  1. Baaz, M., Papadimitriou, C. H., Putnam, H. W., Scott, D. S., and Harper, C. L. (Eds.) (2014). Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press.
  2. Balthasar, G. (2020). On the Invariance of Gödel’s Second Theorem with Regard to Numberings. The Review of Symbolic Logic, 14, 51–84.
  3. Beklemishev, L. (2010). Gödel Incompleteness Theorems and the Limits of their Applicability. I. Russian Math Surveys, 65, 857–898.
  4. Benacerraf, P. (1967). God, the Devil and Gödel. The Monist, 51, 9–32.
  5. Boolos, George. (1993). The Logic of Provability. Cambridge University Press.
  6. Bovykin, Andrey. (2006). Brief Introduction to Unprovability. In Cooper, S. B., Geuvers, H., Pillay, A., and Väänänen, J. (Eds.), Logic Colloquium 2006, 38–64.
  7. Carlson, T. J. (2000). Knowledge, Machines, and the Consistency of Reinhardt’s Strong Mechanistic Thesis. Annals of Pure and Applied Logic, 105, 51–82.
  8. Chalmers, J. D. (1995). Minds, Machines, and Mathematics: A Review of Shadows of the Mind by Roger Penrose. Psyche, 2, 11–20.
  9. Cheng, Y. (2019). Incompleteness for Higher-Order Arithmetic: An Example Based on Harrington’s Principle. SpringerBriefs in Mathematics, Springer.
  10. Cheng, Y. (2020a). Gödel’s Incompleteness Theorem and the Anti-Mechanist Argument: Revisited. Studia Semiotyczne (a special issue titled ‘People, Machines and Gödel’), 34,159–182.
  11. Cheng, Y. (2020b). Finding the Limit of Incompleteness I. Bulletin of Symbolic Logic, 26, 268–286.
  12. Cheng, Y. (2021). Current Research on Gödel’s Incompleteness Theorems. Bulletin of Symbolic Logic, 27, 2, 113–167.
  13. Cheng, Y. (2022). On the Depth of Gödel’s Incompleteness Theorems. Philosophia Mathematica, 30, 2, 173–199.
  14. Detlefsen, M. (1979). On Interpreting Gödel’s Second Theorem. Journal of Philosophical Logic, 8, 297–313.
  15. Enderton, H. (2001). A Mathematical Introduction to Logic (2nd ed.). Academic Press.
  16. Feferman, S. (1960). Arithmetization of Mathematics in a General Setting. Journal of Symbolic Logic, 31, 269–270.
  17. Feferman, S. (1962). Transfinite Recursive Progressions of Axiomatic Theories. The Journal of Symbolic Logic, 27, 259–316.
  18. Feferman, S. (1988). Hilbert’s Program Relativized: Proof-Theoretical and Foundational Reductions. The Journal of Symbolic Logic, 53, 364–384.
  19. Feferman, S. (2006). The Impact of the Incompleteness Theorems on Mathematics. Notices of the AMS, 53, 434–439.
  20. Feferman, S. (2009). Gödel, Nagel, Minds, and Machines. In Jäger, G., Sieg, W. (Eds), Feferman on Foundations. Springer.
  21. Franzen, T. (2005). Gödel’s Theorem: an Incomplete Guide to its Use and Abuse. A. K. Peters.
  22. Friedman, H. (2009). My Forty Years On His Shoulders. In the Proceedings of the Goedel Centenary Meeting in Vienna, Horizons of Truth.
  23. Friedman, H. (2022) Boolean Relation Theory and Incompleteness. To appear in Lecture Notes in Logic, Association for Symbolic Logic.
  24. Gödel, K. (1931). Über Formal Unentscheidbare Sátze der Principia Mathematica und Verwandter Systeme I, Monatshefte für Mathematik und Physik, 38, 173–198.
  25. Gödel, K. (1932). On Completeness and Consistency. In Gödel 1986-2003, Vol. I, 235–237.
  26. Gödel, K. (1934). On Undecidable Propositions of Formal Mathematical Systems? In Gödel 1986-2003, Vol. I, 346–371.
  27. Gödel, K. (1951). Some Basic Theorems on the Foundations of Mathematics and their Implications. In Collected Works, Volume III: Unpublished Essays and Lectures. Oxford University Press, 304–323.
  28. Grzegorczyk, A. (2005). Undecidability without Arithmetization. Studia Logica: An International Journal for Symbolic Logic, 79, 163–230.
  29. Hájek, P. and Pudlák, P. (1993). Metamathematics of First-Order Arithmetic. Springer.
  30. Halbach, V. and Visser, A. (2014a). Self-Reference in Arithmetic I. Review of Symbolic Logic. 7, 671–691.
  31. Halbach, V. and Visser, A. (2014b). Self-Reference in Arithmetic II. Review of Symbolic Logic. 7, 692–712.
  32. Hilbert, D. and Bernays, P. (1939). Grundlagen der Mathematik, Vol. I, Vol II. Springer.
  33. Horsten, L. and Welch, P. (2016). Gödel’s Disjunction: The Scope and Limits of Mathematical Knowledge. Oxford University Press.
  34. Isaacson, D. (2011). Necessary and Sufficient Conditions for Undecidability of the Gödel Sentence and its Truth. In DeVidi, D., Hallett, M., and Clark, P. (Eds.), Logic, Mathematics, Philosophy: Vintage Enthusiasms. Springer, 135–152.
  35. Koellner, P. (2018a). On the Question of Whether the Mind can be Mechanized Part I: From Gödel to Penrose. Journal of Philosophy, 115, 337–360.
  36. Koellner, P. (2018b). On the Question of Whether the Mind can be Mechanized Part II: Penrose’s New Argument. Journal of Philosophy, 115, 453–484.
  37. Kotlarski, H. (2004). The Incompleteness Theorems after 70 Years. Annals of Pure and Applied Logic, 126, 125–138.
  38. Krajewski, S. (2020). On the Anti-Mechanist Arguments Based on Gödel Theorem. Studia Semiotyczne 34, 9–56.
  39. Lindström, P. (1997). Aspects of Incompleteness. Lecture Notes Logic, 10, 132.
  40. Lindström, P. (2006). Remarks on Penrose’s new argument. Journal of Philosophical Logic, 35, 231–237.
  41. Lucas, J. R. (1996). Minds, Machines, and Gödel: A Retrospect. In Millican, P. J. R. and Clark, A. (ed.), Machines and Thought: The Legacy of Alan Turing, Vol. 1. Oxford University Press.
  42. Murawski, R. (1999). Recursive Functions and Metamathematics: Problems of Completeness and Decidability, Gödel’s Theorems. Springer.
  43. Paris, J. and Harrington, L. (1977). A Mathematical Incompleteness. In Barwise, J. (ed.), Handbook of Mathematical Logic. North-Holland, 1133–1142.
  44. Penrose, R. (1989). The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press.
  45. Penrose, R. (1994). Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford University Press.
  46. Pudlák, P. (1999). A Note on Applicability of the Incompleteness Theorem to Human Mind. Annals of Pure and Applied Logic, 96, 335–342.
  47. Raatikainen, P. (2005). On the Philosophical Relevance of Gödel’s Incompleteness Theorems. Revue Internationale de Philosophie, 59, 513–534.
  48. Reinhardt, W. N. (1985a). Absolute Versions of Incompleteness Theorems. Noûs, 19, 317–346.
  49. Reinhardt, W. N. (1985b). The Consistency of a Variant of Church’s Thesis with an Axiomatic Theory of an Epistemic Notion. Revista Colombiana de Matematicas 19, 177–200.
  50. Reinhardt, W. N. (1986). Epistemic Theories and the Interpretation of Gödel’s Incompleteness Theorems. Journal of Philosophical Logic, 15, 427-474.
  51. Rosser, J. B. (1936). Extensions of Some Theorems of Gödel and Church. The Journal of Symbolic Logic, 1, 87–91.
  52. Salehi, S. and Seraji, P. (2017). Gödel–Rosser’s Incompleteness Theorem, Generalized and Optimized for Definable Theories. Journal of Logic and Computation, 27, 1391–1397.
  53. Shapiro, S. (1998). Incompleteness, Mechanism, and Optimism. The Bulletin of Symbolic Logic, 4, 273–302.
  54. Shapiro, S. (2003). Mechanism, Truth, and Penrose’s New Argument. Journal of Philosophical Logic, 32, 19–42.
  55. Smith, P. (2007). An Introduction to Gödel’s Theorems. Cambridge University Press.
  56. Smoryński, C. (1977). The Incompleteness Theorems. In Barwise, J. (Ed.), Handbook of Mathematical Logic. North-Holland, 821–865.
  57. Smullyan, M. R. (1992). Gödel’s Incompleteness Theorems. Oxford University Press.
  58. Stephen, G. S. (1988). Partial Realizations of Hilbert’s Program. The Journal of Symbolic Logic, 53, 349–363.
  59. Taishi, K. (2020). Rosser Provability and the Second Incompleteness Theorem. In Arai, T., Kikuchi, M., Kuroda, S., Okada M., and Yorioka, T. (eds), Advances in Mathematical Logic. SAML 2018. Springer Proceedings in Mathematics & Statistics. Springer, 369.
  60. Tarski, A., Mostowski, A., and Robinson, R. M. (1953). Undecidable Theories: Studies in Logic and the Foundations of Mathematics, North-Holland.
  61. Turing, A. (1939). Systems of Logic Based on Ordinals. Proceedings of the London Mathematical Society, 45, 161–228.
  62. Visser, A. (2011). Can We Make the Second Incompleteness Theorem Coordinate Free? Journal of Logic and Computation, 21, 543–560.
  63. Visser, A. (2016). The Second Incompleteness Theorem: Reflections and Ruminations. In Horsten Leon and Welch Philip (Eds.), Gödel’s Disjunction: The Scope and Limits of Mathematical Knowledge. Oxford University Press.
  64. Wang, H. (1990). Reflections on Kurt Gödel. The MIT Press.
  65. Wang, H. (1997). A Logical Journey: From Gödel to Philosophy. The MIT Press.
  66. Willard, E. D. (2001). Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles. Journal of Symbolic Logic, 66, 536–596.
  67. Willard, E. D. (2006). A Generalization of the Second Incompleteness Theorem and Some Exceptions to It. Annals of Pure and Applied Logic, 141, 472–496.
  68. Zach, R. (2007). Hilbert’s Program Then and Now. In Jacquette Dale (Ed.), Philosophy of Logic, A volume in Handbook of the Philosophy of Science, 411–447. Amsterdam: North Holland.
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