Ordinal completeness of bimodal provability logic GLB

Bimodal provability logic GLB, introduced by G. Japaridze, currently plays an important role in the applications of provability logic to proof-theoretic analysis. Its topological semantics interprets diamond modalities as derived set operators on a scattered bitopological space. We study the question of completeness of this logic w.r.t. the most natural space of this kind, that is, w.r.t. an ordinal α equipped with the interval topology and with the so-called club topology. We show that, assuming the axiom of constructibility, GLB is complete for any α≥אω . On the other hand, from the results of A. Blass it follows that, assuming the consistency of “there is a Mahlo cardinal,” it is consistent with ZFC that GLB is incomplete w.r.t. any such space. Thus, the question of completeness of GLB w.r.t. natural ordinal spaces turns out to be independent of ZFC