Provability as a Modal Operator with the models of PA as the Worlds

This paper introduces a propositional modal model of Gödel-Löb Logic whose worlds are the models of Peano Arithemetic and whose box modality is equivalent to an operator satisfying the Hilbert Bernay’s conditions (e.g. provability.) The semantics of this model is extended to public announcement logic, and it is shown that announcing a formula is semantically equivalent to adding it as a new axiom. The graph structure of the model is also explored, and it is shown that if the descendants of a world are well-founded then they have finite depth.