  In 1994 Jech gave a model theoretic proof of G\"odel'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 $\Sigma^0_2$ 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 with
the modal operator interpreted as truth in every $\Sigma^0_2$-model.
