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# On the Proof of Solovay's Theorem

Dick de Jongh, Marc Jumelet and Franco Montagna
Studia Logica: An International Journal for Symbolic Logic
Vol. 50, No. 1, Provability Logic (Mar., 1991), pp. 51-69
Stable URL: http://www.jstor.org/stable/20015554
Page Count: 19
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## 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 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 $\text{I}\Delta _{0}+\text{EXP}$. The method is adapted to obtain a similar completeness result for the Rosser logic.

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