A new perspective on the arithmetical completeness of GL ∗

Solovay’s proof of the arithmetical completeness of the provability logic GL proceeds by simulating a finite Kripke model inside the theory of Peano Arithmetic (PA). In this article, a new perspective on the proof of GL’s arithmetical completeness will be given. Instead of simulating a Kripke structure inside the theory of PA, it will be embedded into an arithmetically defined Kripke structure. We will examine the relation of strong interpretability, which will turn out to have exactly the suitable properties for assuming the role of the accessibility relation in a Kripke structure whose domain consists of models of PA. Given any finite Kripke model for GL, we can then find a bisimilar model whose nodes are certain nonstandard models of PA. The arithmetical completeness of GL is an immediate consequence of this result. In order to define the bisimulation, however, and to prove its existence, the most crucial and ingenious ingredients of Solovay’s original proof are needed. The main result of the current work is thus not so much a new proof as a new perspective on an already known proof.