Abstract
Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof be replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic (the so-called Rosser logic of Gauspari-Solovay, 1979) so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in particular IΔ0+EXP. The method is adapted to obtain a similar completeness result for the Rosser logic.
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References
Guaspari, D. and Solovay, R.M. Rosser Sentences, Annals of Mathematical Logic 16, pp. 81–99, 1979.
de Jongh, D.H.J., A Simplification of a Completeness Proof of Guaspari and Solovay, Studia Logica 46, pp. 187–192, 1987.
Jumelet, M., On Solovay's Theorem, Master's Thesis, Department of Mathematics and Computer Science, University of Amsterdam, 1988.
Smoryński, C., Self-Reference and Modal Logic, Springer, New York, 1985.
Solovay, R.M., Provability Interpretations of Modal Logic, Israel Journal of Mathematics, vol. 25, pp. 287–304, 1976.
Verbrugge, L.C., Does Solovay's Completeness Theorem Extend to Bounded Arithmetic?, Master Thesis, Department of Mathematics and Computer Science, University of Amsterdam, 1988.
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de Jongh, D., Jumelet, M. & Montagna, F. On the proof of Solovay's theorem. Stud Logica 50, 51–69 (1991). https://doi.org/10.1007/BF00370387
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DOI: https://doi.org/10.1007/BF00370387