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.*
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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 |
---|---|

Report Nr: | PP-2005-16 |

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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