Journal of Symbolic Logic

Models of transfinite provability logic

David Fernández-Duque and Joost J. Joosten

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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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J. Symbolic Logic, Volume 78, Issue 2 (2013), 543-561.

First available in Project Euclid: 15 May 2013

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Fernández-Duque, David; Joosten, Joost J. Models of transfinite provability logic. J. Symbolic Logic 78 (2013), no. 2, 543--561. doi:10.2178/jsl.7802110.

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