MoL-2008-06: Models of the Polymodal Provability Logic

MoL-2008-06: Icard, Thomas (2008) Models of the Polymodal Provability Logic. [Report]

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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.

Item Type: Report
Report Nr: MoL-2008-06
Series Name: Master of Logic Thesis (MoL) Series
Year: 2008
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38

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