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J. Aguilera Ozuna, D. Fernández-Duque:
"Strong completeness of provability logic for ordinal spaces";
Journal of Symbolic Logic, 82 (2017), 2; S. 608 - 628.

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 $\lambda=1$.

Strong Completeness of Provability Logic for Ordinal Spaces (PDF Download Available). Available from: https://www.researchgate.net/publication/284219338_Strong_Completeness_of_Provability_Logic_for_Ordinal_Spaces [accessed Jan 15 2018].

http://dx.doi.org/10.1017/jsl.2017.3