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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 showthe three classes of transitive Kripke frames, the class which stronglyvalidates the $\omega$-rule, the class which weakly validates the$\omega$-rule, and the class which is defined by the L\"{o}b formula, aremutually different, while all of them characterize $\mathbf{GL}$. This gives anexample of a proof system $P$ and a class $C$ of Kripke frames such that $P$ issound with respect to $C$ but the soundness cannot be proved by simpleinduction on the height of the derivations in $P$. We also show Kripkecompleteness of $\mathsf{NGL}$ in an algebraic manner. As a corollary, we showthat the class of modal algebras which is defined by equations $\Boxx\leq\Box\Box x$ and $\bigwedge_{n\in\omega}\Diamond^{n}1=0$ is not a variety.

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