Abstract:
Although Japaridze's polymodal logic GLP is known to be complete w.r.t. topological semantics, the topologies needed for the completeness proof are highly non-constructive. The question of completeness of GLP w.r.t. natural topologies on ordinals turns out to be dependent on large cardinal axioms of set theory. So, we are lacking a manageable 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 GLB is sound and complete. The associated topologies happen to be the same as the ordinal topologies introduced by Thomas Icard. However, the key point is to consider 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.