PROVABILITY LOGICS RELATIVE TO A FIXED EXTENSION OF PEANO ARITHMETIC

Let T and U be any consistent theories of arithmetic. If T is computably enumerable, then the provability predicate Pr-tau(x) of T is naturally obtained from each Sigma(1) definition tau(v) of T. The provability logic PL tau(U) of r relative to U is the set of all modal formulas which are provable in U under all arithmetical interpretations where square is interpreted by Pr-tau(x). It was proved by Beklemishev based on the previous studies by Artemov, Visser, and Japaridze that every PL tau(U) coincides with one of the logics DL alpha, D-beta, S-beta and GL(beta)(-), where a and beta are subsets of omega and beta is cofinite. We prove that if U is a computably enumerable consistent extension of Peano Arithmetic and L is one of GL(alpha), D-beta, S-beta, and GL(beta)(-), where a is computably enumerable and beta is cofinite, then there exists a Sigma(l) definition tau(v) of some extension of I Sigma(1) such that PL tau( U) is exactly L.