AN ESCAPE FROM VARDANYAN’S THEOREM

Abstract

Abstract Vardanyan’s Theorems [36, 37] state that $\mathsf {QPL}(\mathsf {PA})$ —the quantified provability logic of Peano Arithmetic—is $\Pi ^0_2$ complete, and in particular that this already holds when the language is restricted to a single unary predicate. Moreover, Visser and de Jonge [38] generalized this result to conclude that it is impossible to computably axiomatize the quantified provability logic of a wide class of theories. However, the proof of this fact cannot be performed in a strictly positive signature. The system $\mathsf {QRC_1}$ was previously introduced by the authors [1] as a candidate first-order provability logic. Here we generalize the previously available Kripke soundness and completeness proofs, obtaining constant domain completeness. Then we show that $\mathsf {QRC_1}$ is indeed complete with respect to arithmetical semantics. This is achieved via a Solovay-type construction applied to constant domain Kripke models. As corollaries, we see that $\mathsf {QRC_1}$ is the strictly positive fragment of $\mathsf {QGL}$ and a fragment of $\mathsf {QPL}(\mathsf {PA})$ .