On the classification of propositional provability logics.

*(English. Russian original)*Zbl 0704.03005
Math. USSR, Izv. 35, No. 2, 247-275 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 915-943 (1989).

A modal version of a general problem is considered: which laws of provability in a given theory T can be demonstrated by means of a given theory U? [See the reviewer, Semiotika Inf. 14, 115-133 (1980; Zbl 0463.03006).] Thus let [T,U] be the set of modal formulae provable in U under each interpretation such that \(\square F\) is interpreted as the formal assertion of the provability of F in the r.e. theory T. All such [T,U] for T, U extending Peano arithmetic PA we call provability logics. After the papers of A. Visser [J. Philos. Logic 13, 97-113 (1984; Zbl 0581.03009)], the reviewer [Izv. Akad. Nauk SSSR, Ser. Mat. 49, 1123- 1154 (1985; Zbl 0598.03012)] and G. Dzhaparidze [Modal logical tools of studying provability (Russian), Ph. D. Dissertation, Moscow University (1986)] four series of provability logics have been described. The present paper accomplishes the final step in the Classification Theorem proving that there are no more logics of provability. This Classification Theorem is a wonderful and rather technical result, which required a deep study of extensions of PA.

Reviewer: S.Artemov