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arXiv · 2608.07166

Strong completeness of the logic J

Juan P. Aguilera·Grigorii Stepanov
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Abstract

We prove that the polymodal logic $\mathsf{J}$ is strongly complete with respect to \textit{$\mathsf{J}$-bouquets}, a topological refinement of its Kripke semantics. In particular, it is strongly topologically complete. This yields the following completeness result for the provability logic $\mathsf{GLP}$: a countable set of formulae $\Gamma$ is consistent with $\mathsf{GLP}$ if and only if there is a $\mathsf{J}$-bouquet $B$ and $r\in B$ such that $B, r\Vdash \mathsf{GLP}$ and $B, r\Vdash\Gamma$. In contrast, we exhibit counterexamples showing that $\mathsf{GLP}$ is not strongly complete with respect to Beklemishev-Gabelaia spaces.

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Juan P. Aguilera, Grigorii Stepanov. 2026-08-07. Strong completeness of the logic J. https://arxiv.org/abs/2608.07166

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