Strong Completeness of Provability Logic for Ordinal Spaces

Abashidze and Blass independently proved that the modal logic $\sf{GL}$ is complete for its topological interpretation over any ordinal greater than or equal to $\omega^\omega$ equipped with the interval topology. Icard later introduced a family of topologies $\mathcal I_\lambda$ for $\lambda < \omega$, with the purpose of providing semantics for Japaridze's polymodal logic $\sf{GLP}$ $_{\omega}$. Icard's construction was later extended by Joosten and the second author to arbitrary ordinals $\lambda \geq \omega$. We further generalize Icard topologies in this article. Given a scattered space $\mathfrak X = (X, \tau)$ and an ordinal $\lambda$, we define a topology $\tau_{+\lambda}$ in such a way that $\tau_{+0}$ is the original topology $\tau$ and $\tau_{+\lambda}$ coincides with $\mathcal I_\lambda$ when $\mathfrak X$ is an ordinal endowed with the left topology. We then prove that, given any scattered space $\mathfrak X$ and any ordinal $\lambda>0$ such that the rank of $(X, \tau)$ is large enough, $\sf{GL}$ is strongly complete for $\tau_{+\lambda}$. One obtains the original Abashidze-Blass theorem as a consequence of the special case where $\mathfrak X=\omega^\omega$ and $\lambda=1$.