On provability logics of Niebergall arithmetic

Abstract

K. G. Niebergall suggested a simple example of a non-gödelean arithmetical theory $\mathrm{NA}$, in which a natural formalization of its consistency is derivable. In the present paper we consider the provability logic of $\mathrm{NA}$ with respect to Peano arithmetic. We describe the class of its finite Kripke frames and establish the corresponding completeness theorem. For a conservative extension of this logic in the language with an additional propositional constant, we obtain a finite axiomatization. We also consider the truth provability logic of $\mathrm{NA}$ and the provability logic of $\mathrm{NA}$ with respect to $\mathrm{NA}$ itself. We describe the classes of Kripke models with respect to which these logics are complete. We establish $\mathrm{PSpace}$-completeness of the derivability problem in these logics and describe their variable free fragments. We also prove that the provability logic of $\mathrm{NA}$ with respect to Peano arithmetic does not have the Craig interpolation property.