Notre Dame Journal of Formal Logic

A Simple Proof of Arithmetical Completeness for $\Pi_1$-Conservativity Logic

Giorgi Japaridze

Abstract

Hájek and Montagna proved that the modal propositional logic ILM is the logic of $\Pi_1$-conservativity over sound theories containing I$\Sigma_1$ (PA with induction restricted to $\Sigma_1$ formulas). I give a simpler proof of the same fact.

Article information

Source
Notre Dame J. Formal Logic, Volume 35, Number 3 (1994), 346-354.

Dates
First available in Project Euclid: 21 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1040511342

Digital Object Identifier
doi:10.1305/ndjfl/1040511342

Mathematical Reviews number (MathSciNet)
MR1326118

Zentralblatt MATH identifier
0822.03013

Subjects
Primary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}
Secondary: 03F30: First-order arithmetic and fragments 03F40: Gödel numberings and issues of incompleteness

Citation

Japaridze, Giorgi. A Simple Proof of Arithmetical Completeness for $\Pi_1$-Conservativity Logic. Notre Dame J. Formal Logic 35 (1994), no. 3, 346--354. doi:10.1305/ndjfl/1040511342. https://projecteuclid.org/euclid.ndjfl/1040511342


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References

  • Berarducci, A., “The interpretability logic of Peano Arithmetic," The Journal of Symbolic Logic, vol. 55 (1990), pp. 1059–1089. Zbl 0725.03037 MR 92f:03066
  • Boolos, G., The Logic of Provability, Cambridge University Press, Cambridge, 1993. Zbl 0891.03004 MR 95c:03038
  • Dzhaparidze (Japaridze), G., “The logic of linear tolerance," Studia Logica, vol. 51 (1992), pp. 249–277. Zbl 0769.03010 MR 94e:03061
  • Dzhaparidze (Japaridze), G., “A generalized notion of weak interpretability and the corresponding modal logic," Annals of Pure and Applied Logic, vol. 61 (1993), pp. 113–160. Zbl 0791.03032 MR 94d:03030
  • de Jongh, D., and F. Veltman, “Provability logics for relative interpretability," pp. 31–42 in Mathematical Logic, edited by P. Petkov, Plenum Press, New York, 1990. Zbl 0794.03026 MR 92d:03011
  • Hájek, P., and F. Montagna, “The logic of $\Pi_1$-conservativity," Archive for Mathematical Logic, vol. 30 (1990), pp. 113–123. Zbl 0713.03007 MR 92a:03024
  • Shavrukov, V., The logic of relative interpretability over Peano Arithmetic (in Russian). Preprint No. 5, Steklov Mathematical Institute, Moscow, 1988.
  • Smorynski, C., “The incompleteness theorems," pp. 821–865 in Handbook of Mathematical Logic, edited by J. Barwise, North-Holland, Amsterdam, 1977. Zbl 0443.03001 MR 56:15351
  • Solovay, R., “Provability interpretations of modal logic," Israel Journal of Mathematics, vol. 25 (1976), pp. 287–304. Zbl 0352.02019 MR 56:15369
  • Visser, A., “Interpretability logic," pp. 175–209 in Mathematical Logic, edited by P. Petkov, Plenum Press, New York, 1990. Zbl 0793.03064 MR 93k:03022
  • Zambella, D., “On the proofs of arithmetical completeness of interpretability logic," Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 542–551. Zbl 0788.03027 MR 93k:03022