Calibrating Provability Logic: From Modal Logic to Re ection Calculus

Several interesting applications of provability logic in proof theory made use of a polymodal logic GLP due to Giorgi Japaridze. This system, although decidable, is not very easy to handle. In particular, it is not Kripke complete. It is complete w.r.t. neighborhood semantics, however this could only be established recently by rather complicated techniques [1]. In this talk we will advocate the use of a weaker system, called Reflection Calculus, which is much simpler than GLP, yet expressive enough to regain its main proof-theoretic applications, and more. From the point of view of modal logic, RC can be seen as a fragment of polymodal logic consisting of implications of the form A→ B, where A and B are formulas built-up from > and the variables using just ∧ and the diamond modalities. In this paper we formulate it in a somewhat more succinct self-contained format. Further, we state its arithmetical interpretation, and provide some evidence that RC is much simpler than GLP. We then outline a consistency proof for Peano arithmetic based on RC and state a simple combinatorial statement, the so-called Worm principle, that was suggested by the use of GLP but is even more directly related to the Reflection Calculus.