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Erschienen in:
2011 | OriginalPaper | Buchkapitel
Ordinal Completeness of Bimodal Provability Logic GLB
verfasst von : Lev Beklemishev
Erschienen in: Logic, Language, and Computation
Verlag: Springer Berlin Heidelberg
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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 bi-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
$\alpha \geq\aleph_\omega $
. 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
.