Completeness of the Gödel--Löb Provability Logic for Finite Derived Ordinal Topologies

Abstract

We study the Gödel--Löb provability logic $\mathbf{GL}$ for finite derived topologies on the ordinals. Assuming GCH and the existence of a measurable cardinal $\kappa$ which carries a coherent sequence of normal measures $\vec{U}$ with $o^{\vec{U}}(\kappa )\ge \kappa$, we obtain a cofinality and GCH preserving generic extension in which ${\text{Log}}_d(\kappa ,{\tau }_n)=\mathbf{GL}$ for every $n<\omega$, where ${\tau }_n$ is the $n$-th derived topology. The result answers a well-know open question. Extending Beklemishev's theorem, we show that in the same model, for every $k<\omega$, ${\text{Log}}_d(\kappa ;{\tau }_{2k},{\tau }_{2k+1})={\mathbf{GLP}}_2,$ where the two modalities of ${\mathbf{GLP}}_2$ are interpreted respectively by $d_{2k}$ and $d_{2k+1}$.