Abstract
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.
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Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 16K17653.
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Kurahashi, T. Arithmetical Soundness and Completeness for \(\varvec{\Sigma }_{\varvec{2}}\) Numerations. Stud Logica 106, 1181–1196 (2018). https://doi.org/10.1007/s11225-017-9782-4
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DOI: https://doi.org/10.1007/s11225-017-9782-4