Provability Logic: models within models in Peano Arithmetic

In 1994 Jech gave a model-theoretic proof of Gödel’s second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech’s proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without passing through the arithmetized completeness theorem, that the existence of a model of PA of complexity Σ2 is independent of PA, where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic where lφ is defined as the formalization of “φ is true in every Σ2-model”.