On provability logics of Niebergall arithmetic
- Authors: Dvorkin L.V.1
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Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Issue: Vol 88, No 3 (2024)
- Pages: 61-100
- Section: Articles
- URL: https://bakhtiniada.ru/1607-0046/article/view/257716
- DOI: https://doi.org/10.4213/im9524
- ID: 257716
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Abstract
K. G. Niebergall suggested a simple example of a non-gödelean arithmeticaltheory $\mathrm{NA}$, in which a natural formalization of its consistencyis derivable. In the present paper we consider the provability logicof $\mathrm{NA}$ with respect to Peano arithmetic. We describe the class of itsfinite Kripke frames and establish the corresponding completeness theorem.For a conservative extension of this logic in the language with an additionalpropositional constant, we obtain a finite axiomatization. We also considerthe truth provability logic of $\mathrm{NA}$ and the provability logic of $\mathrm{NA}$ with respect to $\mathrm{NA}$ itself. We describe the classes of Kripkemodels with respect to which these logics are complete. We establish$\mathrm{PSpace}$-completeness of the derivability problem in these logicsand describe their variable free fragments. We also prove thatthe provability logic of $\mathrm{NA}$ with respect to Peano arithmeticdoes not have the Craig interpolation property.
Keywords
About the authors
Lev Veniaminovich Dvorkin
Steklov Mathematical Institute of Russian Academy of Sciences
ORCID iD: 0009-0001-3117-1318
without scientific degree, no status
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