The provability logic for Σ 1-interpolability

Abstract

We say that two arithmetical formulas A, B have the Σ 1- interpolation property if they have an ‘interpolant’ σ, i.e., a Σ 1 formula such that the formulas A→σ and σ→ B are provable in Peano Arithmetic PA. The Σ 1- interpolability predicate is just a formalization of this property in the language of arithmetic. Using a standard idea of Gödel, we can associate with this predicate its provability logic, which is the set of all formulas that express arithmetically valid principles in the modal language with two modal operators, □ for provability in PA and ↠ for Σ 1-interpolability. In this paper we determine this provability logic (denoted by ELH). It turns out that this problem appears to be quite similar to the problem of such modal description of relative interpretability, a problem, solved (independently) by Berarducci and Shavrukov. However, the present case is sometimes easier. In particular, we prove in this paper the fixed point property and the Craig interpolation property for our modal system ELH.