Arithmetical Soundness and Completeness for $$\varvec{\Sigma }_{\varvec{2}}$$ Numerations

Studia Logica 106 (6):1181-1196 (2018)
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Abstract

We prove that for each recursively axiomatized consistent extension T of Peano Arithmetic and \, there exists a \ numeration \\) of T such that the provability logic of the provability predicate \\) naturally constructed from \\) is exactly \ \rightarrow \Box p\). This settles Sacchetti’s problem affirmatively.

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Citations of this work

Current Research on Gödel’s Incompleteness Theorems.Yong Cheng - 2021 - Bulletin of Symbolic Logic 27 (2):113-167.
Another look at the second incompleteness theorem.Albert Visser - 2020 - Review of Symbolic Logic 13 (2):269-295.
Rosser Provability and Normal Modal Logics.Taishi Kurahashi - 2020 - Studia Logica 108 (3):597-617.
Arithmetical completeness theorems for monotonic modal logics.Haruka Kogure & Taishi Kurahashi - 2023 - Annals of Pure and Applied Logic 174 (7):103271.

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References found in this work

The Logic of Provability.George Boolos - 1993 - Cambridge and New York: Cambridge University Press.
Arithmetization of Metamathematics in a General Setting.Solomon Feferman - 1960 - Journal of Symbolic Logic 31 (2):269-270.
Provability Interpretations of Modal Logic.Robert M. Solovay - 1981 - Journal of Symbolic Logic 46 (3):661-662.
Four valued semantics and the liar.Albert Visser - 1984 - Journal of Philosophical Logic 13 (2):181 - 212.

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