A Topological Completeness Theorem for Transfinite Provability Logic

We prove a topological completeness theorem for the modal logic GLP containing operators $\langle\lambda\rangle$ for $\lambda \in$ Ord intended to capture progressively stronger notions of consistency in mathematical theories. We show that, given a scattered space $X$ of large-enough rank, any sentence $\phi$ consistent with GLP can be satisfied in a polytopological space based on the finitely many Icard topologies over $X$ that correspond to the finitely many modalities appearing in $\phi$.