Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-20T02:21:53.187Z Has data issue: false hasContentIssue false
  • English
  • Français

GÖDEL’S SECOND INCOMPLETENESS THEOREM: HOW IT IS DERIVED AND WHAT IT DELIVERS

Part of: Theory of computing Proof theory and constructive mathematics Computability and recursion theory

Published online by Cambridge University Press:  10 June 2020

SAEED SALEHI
SAEED SALEHI*
Affiliation:
RESEARCH INSTITUTE FOR FUNDAMENTAL SCIENCES UNIVERSITY OF TABRIZ 29 BAHMAN BOULEVARD, P.O. BOX 51666-17766, TABRIZ, IRAN and SCHOOL OF MATHEMATICS INSTITUTE FOR RESEARCH IN FUNDAMENTAL SCIENCES P.O. BOX 19395-5746, TEHRAN, IRAN E-mail: [email protected] URL: http://saeedsalehi.ir/
Get access Rights & Permissions [Opens in a new window]

Abstract

The proofs of Gödel (1931), Rosser (1936), Kleene (first 1936 and second 1950), Chaitin (1970), and Boolos (1989) for the first incompleteness theorem are compared with each other, especially from the viewpoint of the second incompleteness theorem. It is shown that Gödel’s (first incompleteness theorem) and Kleene’s first theorems are equivalent with the second incompleteness theorem, Rosser’s and Kleene’s second theorems do deliver the second incompleteness theorem, and Boolos’ theorem is derived from the second incompleteness theorem in the standard way. It is also shown that none of Rosser’s, Kleene’s second, or Boolos’ theorems is equivalent with the second incompleteness theorem, and Chaitin’s incompleteness theorem neither delivers nor is derived from the second incompleteness theorem. We compare (the strength of) these six proofs with one another.

Keywords

first incompleteness theoremsecond incompleteness theoremGödel’s proofRosser’s proofKleene’s proofChaitin’s proofBoolos’ proof

MSC classification

Primary: 03F40: Gödel numberings and issues of incompleteness
Secondary: 03F30: First-order arithmetic and fragments 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) 03D32: Algorithmic randomness and dimension
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 26 , Issue 3-4 , December 2020 , pp. 241 - 256
Check for updates
Copyright
© The Association for Symbolic Logic 2020

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.)

References

REFERENCES

Artemov, S. and Beklemishev, L., Provability logic, Handbook of Philosophical Logic (Gabbay, D. and Guenthner, F., editors), second ed., Springer, New York, 2005, pp. 189–360.Google Scholar
Avron, A., A note of provability, truth and existence. Journal of Philosophical Logic, vol. 20 (1991), no. 4, pp. 403–409.CrossRefGoogle Scholar
Beklemishev, L., Reflection principles and provability algebras in formal arithmetic. Russian Mathematical Surveys, vol. 60 (2005), no. 2, pp. 197–268.CrossRefGoogle Scholar
Blanck, R., On Rosser sentences and proof predicates, M.A. thesis, Department of Philosophy, University of Göteborg, Sweden, 2006. Available at http://bit.do/fxUqf.Google Scholar
Boolos, G., A new proof of the Gödel incompleteness theorem. Notices of the American Mathematical Society, vol. 36 (1989), no. 4, pp. 388–390. A Letter from George Boolos, ibid 36 (1989), no. 6, p. 676.Google Scholar
Boolos, G., The Logic of Provability, Cambridge University Press, Cambridge, 1994.CrossRefGoogle Scholar
Chaitin, G., Computational complexity and Gödel’s incompleteness theorem. SIGACT News, vol. 9 (1971), pp. 11–12. Abstract in Notices of the American Mathematical Society, vol. 17 (1970), no. 6, p. 672.CrossRefGoogle Scholar
Craig, W., On axiomatizability within a system. The Journal of Symbolic Logic, vol. 18 (1953), no. 1, pp. 30–32.CrossRefGoogle Scholar
Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I., Monatshefte für Mathematik und Physik, vol. 38 (1931), no. 1, pp. 173–198 (in German). Translated into English as: On formally undecidable propositions of Principia Mathematica and Related Systems, I, Kurt Gödel, Collected Works, Volume I: Publications 1929–1936 (S. Feferman, J. W. Dawson Jr., S. C. Kleene, G. H. Moore, R. M. Solovay, and J. van Heijenoort, editors), Oxford University Press, Oxford, 1986, pp. 135–152.CrossRefGoogle Scholar
Guaspari, D. and Solovay, R., Rosser sentences. Annals of Mathematical Logic, vol. 16 (1979), no. 1, pp. 81–99.CrossRefGoogle Scholar
Hájek, P. and Pudlák, P., Metamathematics of First-Order Arithmetic, second ed., Springer-Verlag, New York, 1998.Google Scholar
Isaacson, D., Necessary and sufficient conditions for undecidability of the Gödel sentence and its truth, Logic, Mathematics, Philosophy: Vintage Enthusiasms—Essays in Honour of John L. Bell (DeVidi, D., Hallett, M., and Clarke, P., editors), Springer, New York, 2011, pp. 135–152.CrossRefGoogle Scholar
Kaye, R., Models of Peano Arithmetic, Oxford University Press, Oxford, 1991.Google Scholar
Kikuchi, M., Kurahashi, T., and Sakai, H., On proofs of the incompleteness theorems based on Berry’s paradox by Vopěnka, Chaitin, and Boolos. Mathematical Logic Quarterly, vol. 58 (2012), no. 4–5, pp. 307–316.CrossRefGoogle Scholar
Kleene, S., General recursive functions of natural numbers. Mathematische Annalen, vol. 112 (1936), no. 1, pp. 727–742.CrossRefGoogle Scholar
Kleene, S., A symmetric form of Gödel’s theorem. Indagationes Mathematicae, vol. 12 (1950), pp. 244–246.Google Scholar
Kreisel, G., On weak completeness of intuitionistic predicate logic. The Journal of Symbolic Logic, vol. 27 (1962), no. 2, pp. 139–158.CrossRefGoogle Scholar
Kreisel, G. and Takeuti, G., Formally self-referential propositions for cut free classical analysis and related systems. Dissertationes Mathematicæ, vol. 118 (1974), pp. 1–50.Google Scholar
Macintyre, A. and Simmons, H., Gödel’s diagonalization technique and related properties of theories. Colloquium Mathematicum, vol. 28 (1973), pp. 165–180.CrossRefGoogle Scholar
Maehara, S., Boolos Shi No Genkou Wo Mite. Gendai Shisou (December 1989), pp. 80–92.Google Scholar
Putnam, H., Nonstandard models and Kripke’s proof of the Gödel theorem. Notre Dame Journal of Formal Logic, vol. 41 (2000), no. 1, pp. 53–58.CrossRefGoogle Scholar
Rosser, B., Extensions of some theorems of Gödel and Church. The Journal of Symbolic Logic, vol. 1 (1936), no. 3, pp. 87–91.CrossRefGoogle Scholar
Salehi, S., Gödel’s incompleteness phenomenon—Computationally. Philosophia Scientiæ, vol. 18 (2014), no. 3, pp. 22–37.Google Scholar
Salehi, S. and Seraji, P., Gödel–Rosser’s incompleteness theorem, generalized and optimized for definable theories. Journal of Logic and Computation, vol. 27 (2017), no. 5, pp. 1391–1397.Google Scholar
Salehi, S. and Seraji, P., On constructivity and the Rosser property: A closer look at some Gödelean proofs. Annals of Pure and Applied Logic, vol. 169 (2018), no. 10, pp. 971–980.CrossRefGoogle Scholar
Slaman, T. A., $\Sigma_n\hspace{-1.6pt}\!\!\!\!$-bounding and ∆n-induction. Proceedings of The American Mathematical Society, vol. 132 (2004), no. 8, pp. 2449–2456.CrossRefGoogle Scholar
Smoryński, C., Self-Reference and Modal Logic, Springer, New York, 1985.CrossRefGoogle Scholar
Tarski, A., Mostowski, A., and Robinson, R. M., Undecidable Theories, North-Holland, Netherlands, 1953. Reprinted by Dover Publications, 2010.Google Scholar
Visser, A., Another look at the second incompleteness theorem. The Review of Symbolic Logic, vol. 13 (2020) no. 2, pp. 269–295.CrossRefGoogle Scholar
von Bülow, C., A remark on equivalent Rosser sentences. Annals of Pure and Applied Logic, vol. 151 (2008), no. 1, pp. 62–67.CrossRefGoogle Scholar