A cut-free proof system for a predicate extension of the logic of provability

In this paper, we introduce a proof system $\mathsf{NQGL}$ for a Kripke complete predicate extension of the logic $\mathbf{GL}$, that is, the logic of provability, which is defned by $\mathbf{K}$ and the L\"{o}b formula $\Box(\Box p\supset p)\supset\Box p$. $\mathsf{NQGL}$ is a modal extension of Gentzen's sequent calculus $\mathsf{LK}$. Although the propositional fragment of $\mathsf{NQGL}$ axiomatizes $\mathbf{GL}$, it does not have the L\"{o}b formula as its axiom. Instead, it has a non-compact rule, that is, a derivation rule with countably many premises. We show that $\mathsf{NQGL}$ enjoys cut admissibility and is complete with respect to the class of Kripke frames such that for each world, the supremum of the length of the paths from the world is finite.