The Logic of Correct Models

For each \(n\in\mathbb{N}\), let \([n]\phi\) mean "the sentence \(\phi\) is true in all \(\Sigma_{n+1}\)-correct transitive sets." Assuming G\"odel's axiom \(V = L\), we prove the following graded variant of Solovay's completeness theorem: the set of formulas valid under thi...

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Bibliographic Details
Published in:arXiv.org 2024-02
Main Authors: Juan Pablo Aguilera, Pakhomov, Fedor
Format: Article
Language:English
Subjects:
Logic
Theorems
Online Access:Get full text
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Summary:For each \(n\in\mathbb{N}\), let \([n]\phi\) mean "the sentence \(\phi\) is true in all \(\Sigma_{n+1}\)-correct transitive sets." Assuming G\"odel's axiom \(V = L\), we prove the following graded variant of Solovay's completeness theorem: the set of formulas valid under this interpretation is precisely the set of theorems of the linear provability logic GLP.3. We also show that this result is not provable in ZFC, so the hypothesis V = L cannot be removed. As part of the proof, we derive (in ZFC) the following purely modal-logical results which are of independent interest: the logic GLP.3 coincides with the logic of closed substitutions of GLP, and is the maximal non-degenerate, normal extension of GLP.
ISSN:2331-8422