# The Logic of Correct Models

## Fedor Pakhomov

**Authors:** Juan Aguilera and Fedor Pakhomov

In his well-known paper [4] Robert Solovay proved that the provability logic of \(\mathsf{PA}\) is the Gödel-Löb logic \(\mathsf{GL}\). Also in the same paper Solovay considered some set-theoretic interpretations of provability logic which lead to stronger provability logics.

Latter Japaridze [3] proved arithmetical completeness theorem for the polymodal provability logic \(\mathsf{GLP}\), where for a modernized version [2] of his result \([n]\phi\) is interpreted as “the sentence \(\phi\) is provable in \(\mathsf{PA}\) extended by all true \(\Pi_n\)-sentences.”

In our work [1] we investigated the set theoretic interpretation of \([n]\), where \([n]\phi\) mean “the sentence \(\phi\) is true in all \(\Sigma_{n+1}\)-correct transitive sets.” Assuming Gödel’s axiom \(V = L\) we prove that the set of polymodal formulas valid under this interpretation is precisely the set of theorems of the logic \(\mathsf{GLP}.3\). Here \(\mathsf{GLP}.3\) is the extension of \(\mathsf{GLP}\) by the axioms of linearity for all \([n]\). We also show that this result is not provable in \(\mathsf{ZFS}\), so the hypothesis \(V = L\) cannot be removed.

As part of the proof, we derive the following purely modal-logical results which are of independent interest: the logic \(\mathsf{GLP}.3\) coincides with the logic of closed substitutions of \(\mathsf{GLP}\), and is the maximal normal extension of \(\mathsf{GLP}\) that doesn’t prove \([n]\bot\), for any \(n\).

## Bibliography

- Juan P. Aguilera and Fedor Pakhomov,
*The Logic of Correct Models.*,,2402.15382, 20 pages, 2024.*arXiv preprints* - L.D. Beklemishev,
*A simplified proof of arithmetical completeness theorem for provability logic GLP.*,,274 (2011): 25-33.*Proceedings of the Steklov Institute of Mathematics* - G.K. Japaridze,
*The modal logical means of investigation of provability. Thesis in Philosophy*,Moscow (in Russian), 1986. - Robert M. Solovay,
*Provability interpretations of modal logic.*,,25 (1976): 287-304.*Israel Journal of Mathematics*