Models of transfinite provability logic

Abstract

AbstractFor any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted . Later, Icard defined a topological model for 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, Λ we build a Kripke model and a topological model , and show that is sound for both of these structures, as well as complete, provided Θ is large enough.