On the Craig interpolation and the fixed point property for GLP

Abstract

We prove the Craig interpolation and the fixed point property for GLP by finitary methods. Konstantin Ignatiev [4], among other things, established the Craig interpolation and the fixed point property for Japaridze's polymodal provability logic GLP. However, it remained open if these results could be established by finitary methods formalizable in Peano arithmetic PA. (The question concerning the Craig interpolation was stated e.g. in [3].) In this note we provide such proofs. These proofs are based on our previous paper [1] where a complete Kripke semantics for GLP is given. In that paper, using only finitary methods, the system GLP is reduced to a certain natural subsystem, denoted J.1 J is sound and complete w.r.t. a natural class of finite Kripke frames [1]. (It is well-known that GLP is not complete w.r.t. any class of Kripke frames.) We establish the Craig interpolation and the fixed point properties for J, which also enables us to extend them to GLP. Apart from the reduction of GLP to J established in ref. [1] our methods are very standard.