Models of the Polymodal Provability Logic
Thomas Icard
Abstract:
This thesis on the polymodal provability logic GLP is divides into
three main sections. In the first section, we investigate relational
models of GLP. After presenting a simplified treatment of
Beklemishev's blow-up model construction, we exploit completeness for
such models to obtain a new, purely semantic proof that GLP_0, the
closed fragment of GLP, is complete with respect to Ignatiev's frame
U. Following this, we investigate formula definable subsets of U in
anticipation of our work in the next two sections.
In the second section, we explore the connection between U and the
canonical frame of GLP_0. Using the theory of descriptive frames, we
extend U to a frame that is isomorphic to the canonical frame, thus
obtaining a detailed definition of this object in terms of a
coordinate system developed in the first section.
Finally, in the last section, we explore topological models of
GLP. The central result of this section is an analog of the
Abashidze-Blass Theorem for GL, to the effect that GLP_0 enjoys
topological completeness with respect to a simply defined polytopology
on the ordinal epsilon_0 +1. This space can be seen as a condensed,
and much simplified, version of the canonical frame of GLP_0. We then
consider the possibility of an extension of this theorem to full
GLP. However, such an extension would require large cardinal
assumptions beyond ZFC, so we leave this further question for future
work.
All of this work can be seen as an effort to overcome (and better
explain) the fact that GLP is frame incomplete.