Periodic frames

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.