# PP-2005-16: A finitary treatment of the closed fragment of Japaridze's provability logic

PP-2005-16: Beklemishev, Lev D. and Joosten, Joost J. and Vervoort, Marco (2005) A finitary treatment of the closed fragment of Japaridze's provability logic. [Report]

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## Abstract

We study a propositional polymodal provability logic GLP introduced by G. Japaridze. The previous treatments of this logic, due to Japaridze and Ignatiev, heavily relied on some non-finitary principles such as transfinite induction up to \epsilon_0 or reflection principles. In fact, the closed fragment of GLP gives rise to a natural system of ordinal notation for \epsilon_0 that was used for a proof-theoretic analysis of Peano arithmetic and for constructing simple combinatorial independent statements. In this paper, we study Ignatiev's universal model for the closed fragment of this logic. Using bisimulation techniques, we show that several basic results on the closed fragment of GLP, including the normal form theorem, can be proved by purely finitary means formalizable in elementary arithmetic. As a corollary, the system of ordinal notation for \epsilon_0 based on the closed fragment of GLP is shown to be provably isomorphic to the standard system of ordinal notation up to \epsilon_0. We also settle negatively some conjectures by Ignatiev.

Item Type: Report PP-2005-16 Prepublication (PP) Series 2005 provability logic, ordinal notations, consistency statements, provability algebras Logic 12 Oct 2016 14:36 12 Oct 2016 14:36 https://eprints.illc.uva.nl/id/eprint/161