Completeness of the G\"odel-L\"ob provability logic for the filter sequence of normal measures

TL;DR

This paper demonstrates the consistency of the G"odel-L"ob provability logic being complete for the filter sequence of normal measures under certain large cardinal assumptions, addressing longstanding open questions.

Contribution

It establishes the consistency of G"odel-L"ob logic's completeness for the filter sequence of normal measures, resolving questions posed since 1990.

Findings

G"odel-L"ob logic is consistent with respect to the filter sequence of normal measures

Addresses and answers questions from Blass (1990) and Beklemishev & Joosten

Shows the consistency results depend on large cardinal assumptions

Abstract

Assuming the existence of suitable large cardinals, we show it is consistent that the Provability logic $\mathbf{GL}$ is complete with respect to the filter sequence of normal measures. This result answers a question of Andreas Blass from 1990 and a related question of Beklemishev and Joosten.