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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 $ω^ω$ equipped with the interval topology. Icard later introduced a family of topologies $\mathcal I_λ$ for $λ< ω$, with the purpose of providing semantics for Japaridze&#39;s polymodal logic $\sf{GLP}$ $_ω$. Icard&#39;s construction was later extended by Joosten and the second author to arbitrary ordinals $λ\geq ω$. We further generalize Icard topologies in this article. Given a scattered space $\mathfrak X = (X, τ)$ and an ordinal $λ$, we define a topology $τ_{+λ}$ in such a way that $τ_{+0}$ is the original topology $τ$ and $τ_{+λ}$ coincides with $\mathcal I_λ$ when $\mathfrak X$ is an ordinal endowed with the left topology. We then prove that, given any scattered space $\mathfrak X$ and any ordinal $λ>0$ such that the rank of $(X, τ)$ is large enough, $\sf{GL}$ is strongly complete for $τ_{+λ}$. One obtains the original Abashidze-Blass theorem as a consequence of the special case where $\mathfrak X=ω^ω$ and $λ=1$.