Provability logic: models within models in Peano Arithmetic
Alessandro Berarducci, Marcello Mamino · Bollettino Della Unione Matematica Italiana · 2022
Abstract In 1994 Jech gave a model-theoretic proof of Gödel’s second incompleteness theorem for Zermelo–Fraenkel set theory in the following form: $${{\,\mathrm{\mathrm {ZF}}\,}}$$ ZF does not prove that $${{\,\mathrm{\mathrm {ZF}}\,}}$$ 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 the need of formalizing the proof of the completeness theorem within $${{\,\mathrm{\mathrm {PA}}\,}}$$ PA , that the existence of a model of $${{\,\mathrm{\mathrm {PA}}\,}}$$ PA of complexity $$\Sigma ^0_2$$ Σ 2 0 is independent of $${{\,\mathrm{\mathrm {PA}}\,}}$$ 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 $$\Box \phi $$ □ ϕ is defined as the formalization of “ $$\phi $$ ϕ is true in every $$\Sigma ^0_2$$ Σ 2 0 -model”.