A topological completeness theorem for transfinite provability logic

Abstract

We prove a topological completeness theorem for the modal logic textsf{GLP} containing operators {langle xi rangle :xi in textsf{Ord}} intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence phi consistent with textsf{GLP} can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the finitely many modalities that occur in phi .