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arXiv · 2311.00581

The Modal Logic of Provability and Forcing

Taishi Kurahashi·Rihito Takase
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Abstract

Solovay's arithmetical completeness theorem states that the modal logic of provability coincides with the modal logic $\mathbf{GL}$. Hamkins and L\"owe studied the modal logical aspects of set theoretic multiverse and proved that the modal logic of forcing is exactly the modal logic $\mathbf{S4.2}$. We explore the interaction between the notions of provability and forcing in terms of modal logic. We introduce the bimodal logic $\mathbf{PF}$ and prove that the modal logic of provability and forcing is exactly $\mathbf{PF}$. We also introduce the bimodal logic $\mathbf{PF}^\omega$ and prove that $\mathbf{PF}^\omega$ is exactly the modal logic of provability and forcing true in $\omega$-models of set theory.

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Taishi Kurahashi, Rihito Takase. 2023-11-01. The Modal Logic of Provability and Forcing. https://arxiv.org/abs/2311.00581

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