PP200516: 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 nonfinitary 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 prooftheoretic 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 

Report Nr:  PP200516 
Series Name:  Prepublication (PP) Series 
Year:  2005 
Uncontrolled Keywords:  provability logic, ordinal notations, consistency statements, provability algebras 
Subjects:  Logic 
Date Deposited:  12 Oct 2016 14:36 
Last Modified:  12 Oct 2016 14:36 
URI:  https://eprints.illc.uva.nl/id/eprint/161 
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