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ANOTHER LOOK AT THE SECOND INCOMPLETENESS THEOREM

Published online by Cambridge University Press:  10 June 2019

ALBERT VISSER
ALBERT VISSER*
Affiliation:
Department of Philosophy, Utrecht University
*
*DEPARTMENT OF PHILOSOPHY UTRECHT UNIVERSITY JANSKERKHOF 13, 3512BL UTRECHT, THE NETHERLANDS E-mail: a.visser@uu.nl
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Abstract

In this article we study proofs of some general forms of the Second Incompleteness Theorem. These forms conform to the Feferman format, where the proof predicate is fixed and the representation of the set of axioms varies. We extend the Feferman framework in one important point: we allow the interpretation of number theory to vary.

Keywords

03A0503B2503F2503F3003F45Second Incompleteness Theoremformal theories
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 2 , June 2020 , pp. 269 - 295
Copyright
Copyright © Association for Symbolic Logic 2019 

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References

BIBLIOGRAPHY

Artemov, S. & Beklemishev, L. (2004). Provability logic. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic, Second Edition, Vol. 13. Dordrecht: Springer, pp. 229–403.Google Scholar
Beklemishev, L. (2006). The worm principle. In Chatzidakis, Z., Koepke, P., and Pohlers, W., editors. Logic Colloquium ’02, Lecture Notes in Logic, Vol. 27. Natick, Massachusetts: A.K. Peters and CRC Press, pp. 75–95.Google Scholar
Boolos, G. (1993). The Logic of Provability. Cambridge: Cambridge University Press.Google Scholar
Buss, S. (1986). Bounded Arithmetic. Napoli: Bibliopolis.Google Scholar
Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49, 35–92.CrossRefGoogle Scholar
Hájek, P. & Pudlák, P. (1993). Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic. Berlin: Springer.CrossRefGoogle Scholar
Japaridze, G. (1985). The polymodal logic of provability. Intensional Logics and Logical Structure of Theories: Material from the fourth Soviet–Finnish Symposium on Logic, Telavi, May 20–24, 1985, Metsniereba, Tbilisi, pp. 16–48.Google Scholar
Japaridze, G. & de Jongh, D. (1998). The logic of provability. In Buss, S., editor. Handbook of Proof Theory, Amsterdam: North-Holland, pp. 475–546.CrossRefGoogle Scholar
Kaye, R. (1991). Models of Peano Arithmetic. Oxford Logic Guides. Oxford: Oxford University Press.Google Scholar
Kripke, S. (1962). “Flexible” predicates of formal number theory. Proceedings of the American Mathematical Society, 13(4), 647–650.Google Scholar
Kurahashi, T. (2018a). Arithmetical completeness theorem for modal logic K. Studia Logica, 106(2), 219–235.CrossRefGoogle Scholar
Kurahashi, T. (2018b). Arithmetical soundness and completeness for ${{\rm{\Sigma }}_2}$-numerations. Studia Logica, 106(6), 1181–1196.CrossRefGoogle Scholar
Montagna, F. (1978). On the algebraization of a Feferman’s predicate (the algebraization of theories which express Theor; X). Studia Logica, 37, 221–236.CrossRefGoogle Scholar
Paris, J. & Kirby, L. (1978). ${{\rm{\Sigma }}_n}$-collection schemas in arithmetic. In Macintyre, A., Pacholski, L., and Paris, J., editors. Logic Colloquium ’77. Studies in Logic and the Foundations of Mathematics, Vol. 96. Amsterdam: North–Holland, pp. 199–209.CrossRefGoogle Scholar
Pudlák, P. (1985). Cuts, consistency statements and interpretations. The Journal of Symbolic Logic, 50(2), 423–441.CrossRefGoogle Scholar
Shavrukov, V. (1994). A smart child of Peano’s. Notre Dame Journal of Formal Logic, 35, 161–185.CrossRefGoogle Scholar
Shavrukov, V. Y. & Visser, A. (2014). Uniform density in lindenbaum algebras. Notre Dame Journal of Formal Logic , 55(4), 569–582.CrossRefGoogle Scholar
Smoryński, C. (1985). Self-Reference and Modal Logic. Universitext. New York: Springer.CrossRefGoogle Scholar
Smoryński, C. (1989). Arithmetic analogues of McAloon’s unique Rosser sentences. Archive for Mathematical logic 28, 1–21.CrossRefGoogle Scholar
Solovay, R. (1976). Provability interpretations of modal logic. Israel Journal of Mathematics, 25, 287–304.CrossRefGoogle Scholar
Tarski, A., Mostowski, A., & Robinson, R. (1953). Undecidable Theories. Amsterdam: North–Holland.Google Scholar
Visser, A. (1989). Peano’s smart children: A provability logical study of systems with built-in consistency. Notre Dame Journal of Formal Logic, 30(2), 161–196.CrossRefGoogle Scholar
Visser, A. (1992). An inside view of EXP. The Journal of Symbolic Logic, 57(1), 131–165.CrossRefGoogle Scholar
Visser, A. (2005). Faith & Falsity: A study of faithful interpretations and false ${\rm{\Sigma }}_1^0$-sentences. Annals of Pure and Applied Logic, 131(1–3), 103–131.CrossRefGoogle Scholar
Visser, A. (2011). Can we make the Second Incompleteness Theorem coordinate free. Journal of Logic and Computation, 21(4), 543–560. First published online August 12, 2009, doi: 10.1093/logcom/exp048.CrossRefGoogle Scholar
Visser, A. (2014). Peano Corto and Peano Basso: A study of local induction in the context of weak theories. Mathematical Logic Quarterly, 60(1–2), 92–117.CrossRefGoogle Scholar
Visser, A. (2015). Oracle bites theory. In Gosh, S. and Szymanik, J., editors. The Facts Matter. Essays on Logic and Cognition in Honour of Rineke Verbrugge. London: College Publications, pp. 133–147.Google Scholar
Visser, A. (2018). The interpretation existence lemma. Feferman on Foundations, Outstanding Contributions to Logic, Vol. 13. New York: Springer, pp. 101–144.CrossRefGoogle Scholar
Zambella, D. (1996). Notes on Polynomially Bounded Arithmetic. The Journal of Symbolic Logic, 61(3), 942–966.CrossRefGoogle Scholar