Abstract
Provability logics are modal or polymodal systems designed for modeling the behavior of Godel’s provability predicate and its natural extensions. If Λ is any ordinal, the Godel-Lob calculus GLP Λ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLP ω has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In this paper we generalize Beklemishev and Gabelaia’s result to GLP Λ for countable Λ. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLP Λ for the class of provability ambiances based on Icard polytopologies.
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