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June 2013 Models of transfinite provability logic
David Fernández-Duque, Joost J. Joosten
J. Symbolic Logic 78(2): 543-561 (June 2013). DOI: 10.2178/jsl.7802110

Abstract

For any ordinal $\Lambda$, we can define a polymodal logic $\mathsf{GLP}_\Lambda$, with a modality $[\xi]$ for each $\xi < \Lambda$. These represent provability predicates of increasing strength. Although $\mathsf{GLP}_\Lambda$ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted $\mathsf{GLP}^0_\omega$. Later, Icard defined a topological model for $\mathsf{GLP}^0_\omega$ which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary $\Lambda$. More generally, for each $\Theta,\Lambda$ we build a Kripke model $\mathfrak I^\Theta_\Lambda$ and a topological model $\mathfrak T^\Theta_\Lambda$, and show that $\mathsf{GLP}^0_\Lambda$ is sound for both of these structures, as well as complete, provided $\Theta$ is large enough.

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David Fernández-Duque. Joost J. Joosten. "Models of transfinite provability logic." J. Symbolic Logic 78 (2) 543 - 561, June 2013. https://doi.org/10.2178/jsl.7802110

Information

Published: June 2013
First available in Project Euclid: 15 May 2013

zbMATH: 1275.03158
MathSciNet: MR3145195
Digital Object Identifier: 10.2178/jsl.7802110

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 2 • June 2013
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