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In 1994 Jech gave a model theoretic proof of G\"odel's second incompletenesstheorem for Zermelo-Fraenkel set theory in the following form: ZF does notprove that ZF has a model. Kotlarski showed that Jech's proof can be adapted toPeano Arithmetic with the role of models being taken by complete consistentextensions. In this note we take another step in the direction of replacingproof-theoretic by model-theoretic arguments. We show, without passing throughthe arithmetized completeness theorem, that the existence of a model of PA ofcomplexity $\Sigma^0_2$ is independent of PA, where a model is identified withthe set of formulas with parameters which hold in the model. Our approach isbased on a new interpretation of the provability logic of Peano Arithmetic withthe modal operator interpreted as truth in every $\Sigma^0_2$-model.

Provability Logic: models within models in Peano Arithmetic