Periodic frames

TL;DR

This paper introduces a natural class of countable topological frames on ordinals that establish the completeness of polymodal provability logic GLP, using algebras based on periodic sets of ordinals.

Contribution

It defines a new class of models for GLP completeness on ordinals using algebraic structures of periodic sets, bridging topological and set-theoretic approaches.

Findings

GLP is sound and complete for the new class of topological frames.

The algebra of subsets is based on generalizing periodic sets of ordinals.

The approach connects topological semantics with set-theoretic properties of ordinals.

Abstract

Polymodal provability logic GLP is incomplete w.r.t. Kripke frames. It is known to be complete w.r.t. topological semantics, where the diamond modalities correspond to topological derivative operations. However, the topologies needed for the completeness proof are highly non-constructive. The question of completeness of GLP w.r.t. natural scattered topologies on ordinals is dependent on large cardinal axioms of set theory and is still open. So far, we are lacking a useable class of models for which GLP is complete. In this paper we define a natural class of countable general topological frames on ordinals for which GLP is sound and complete. The associated topologies are the same as the ordinal topologies introduced by Thomas Icard. However, the key point is to specify a suitable algebra of subsets of an ordinal closed under the boolean and topological derivative operations. The algebras we define are based on the notion of a periodic set of ordinals generalizing that of an ultimately periodic binary omega-word.