X-2011-06:
Henk, Paula
(2011)
*A new perspective on the arithmetical completeness of GL.*
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## Abstract

Solovay’s proof of the arithmetical completeness of the provability

logic GL proceeds by simulating a finite Kripke model inside the

theory of Peano Arithmetic (PA). In this article, a new perspective on

the proof of GL’s arithmetical completeness will be given. Instead of

simulating a Kripke structure inside the theory of PA, it will be

embedded into an arithmetically defined Kripke structure. We will

examine the relation of strong interpretability, which will turn out

to have exactly the suitable properties for assuming the role of the

accessibility relation in a Kripke structure whose domain consists of

models of PA.

Given any finite Kripke model for GL, we can then find a bisimilar

model whose nodes are certain nonstandard models of PA. The arith-

metical completeness of GL is an immediate consequence of this result.

In order to define the bisimulation, however, and to prove its

existence, the most crucial and ingenious ingredients of Solovay’s

original proof are needed. The main result of the current work is thus

not so much a new proof as a new perspective on an already known

proof.

Item Type: | Report |
---|---|

Report Nr: | X-2011-06 |

Series Name: | Technical Notes (X) Series |

Year: | 2011 |

Uncontrolled Keywords: | Provability logic; Peano arithmetic; arithmetical completeness; nonstandard models |

Subjects: | Logic |

Date Deposited: | 12 Oct 2016 14:38 |

Last Modified: | 12 Oct 2016 14:38 |

URI: | https://eprints.illc.uva.nl/id/eprint/687 |

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