An $\omega$-rule for the logic of provability and its models

In this paper, we discuss a proof system $\mathsf{NGL}$ for the logic $\mathbf{GL}$ of provability, which is equipped with an $\omega$-rule. We show the three classes of transitive Kripke frames, the class which strongly validates the $\omega$-rule, the class which weakly validates the $\omega$-rule, and the class which is defined by the L\"{o}b formula, are mutually different, while all of them characterize $\mathbf{GL}$. This gives an example of a proof system $P$ and a class $C$ of Kripke frames such that $P$ is sound with respect to $C$ but the soundness cannot be proved by simple induction on the height of the derivations in $P$. We also show Kripke completeness of $\mathsf{NGL}$ in an algebraic manner. As a corollary, we show that the class of modal algebras which is defined by equations $\Box x\leq\Box\Box x$ and $\bigwedge_{n\in\omega}\Diamond^{n}1=0$ is not a variety.