General Topological Frames for Polymodal Provability Logic
Yunsong Wang
Abstract:
The polymodal provability logic GLP is a system of propositional modal logic with infinitely many modalities having provability semantics. It was initially introduced by Japaridze in his PhD thesis [19]. GLP has significant applications in proof theory and arithmetic, however, it is well-known that GLP is Kripke incomplete. GLP is complete with respect to topological semantics [3], yet the relevant class of spaces is rather involved. Topological completeness of GLP under the natural class of ordinal spaces requires certain set-theoretic assumptions (the existence of large cardinals), however, it is still open whether it holds under these assumptions (see [4]). Therefore, it becomes crucial to search for some simpler models for GLP.
In this thesis, we define the concept of a general topological frame, that is, a topological space equipped with a distinguished set of admissible sets, akin to the notion of a general Kripke frame. Then, we describe a natural class of general topological frames on ordinals, that we call periodic frames. These frames are based on well-orderings equipped with some natural topologies introduced by Icard [18]. While GLP is known to be incomplete with respect to Icardâ€™s spaces, we show that the bimodal fragment of GLP is sound and complete with respect to the periodic frames. We hope that the results in this thesis will pave the way to further generalizations of this completeness to the whole system GLP.