Models of transfinite provability logic

Abstract

For any ordinal Λ, we can define a polymodal logic GLP(Λ), with a modality [ξ] for each ξ<Λ. These represent provability predicates of increasing strength. Although GLP(Λ) has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities. Later, Icard defined a topological model for the same fragment which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each Θ,Λwe build a Kripke model I(Θ,Λ) and a topological model T(Θ,Λ), and show that the closed fragment of GLP(Λ) is sound for both of these structures, as well as complete, provided Θis large enough.