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

  • Taishi Kurahashi


We prove that for each recursively axiomatized consistent extension T of Peano Arithmetic and \(n \ge 2\), there exists a \(\Sigma _2\) numeration \(\tau (u)\) of T such that the provability logic of the provability predicate \(\mathsf{Pr}_\tau (x)\) naturally constructed from \(\tau (u)\) is exactly \(\mathsf{K}+ \Box (\Box ^n p \rightarrow p) \rightarrow \Box p\). This settles Sacchetti’s problem affirmatively.


Provability logic Sacchetti’s logics Arithmetical completeness theorem Formalized arithmetic Numerations 


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This work was supported by JSPS KAKENHI Grant Number 16K17653.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Natural ScienceNational Institute of Technology, Kisarazu CollegeKisarazuJapan

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