A Topological Completeness Theorem for Transfinite Provability Logic

TL;DR

This paper establishes a topological completeness theorem for a transfinite modal logic GLP, demonstrating that consistent sentences can be satisfied in specific polytopological spaces derived from scattered spaces with large rank.

Contribution

It introduces a topological completeness result for GLP with transfinite modalities, linking modal logic to polytopological spaces based on Icard topologies.

Findings

Proves completeness of GLP in certain topological spaces

Shows satisfaction of consistent sentences in finitely constructed polytopological spaces

Connects modal logic with topological and ordinal structures

Abstract

We prove a topological completeness theorem for the modal logic GLP containing operators $\langle\lambda\rangle$ for $\lambda \in$ Ord intended to capture progressively stronger notions of consistency in mathematical theories. We show that, given a scattered space $X$ of large-enough rank, any sentence $\phi$ consistent with GLP can be satisfied in a polytopological space based on the finitely many Icard topologies over $X$ that correspond to the finitely many modalities appearing in $\phi$.