STRONG COMPLETENESS OF PROVABILITY LOGIC FOR ORDINAL SPACES

Abstract

AbstractGiven a scattered space $\mathfrak{X} = \left( {X,\tau } \right)$ and an ordinal λ, we define a topology $\tau _{ + \lambda } $ in such a way that τ+0 = τ and, when $\mathfrak{X}$ is an ordinal with the initial segment topology, the resulting sequence {τ+λ}λ∈Ord coincides with the family of topologies $\left\{ {\mathcal{I}_\lambda } \right\}_{\lambda \in Ord} $ used by Icard, Joosten, and the second author to provide semantics for polymodal provability logics.We prove that given any scattered space $\mathfrak{X}$ of large-enough rank and any ordinal λ > 0, GL is strongly complete for τ+λ. The special case where $\mathfrak{X} = \omega ^\omega + 1$ and λ = 1 yields a strengthening of a theorem of Abashidze and Blass.