Grigorii Stepanov | Searcharxiv

arXiv2026

Strong completeness of the logic J

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.

math.LO↗

arXiv2026

Strong Completeness of Provability Logic for Uncountable Languages

For an ordinal $\lambda>0$, we use the Erd\H{o}s--Rado partition theorem to prove the failure of strong completeness of $\mathsf{GL}$ for modal languages of cardinality $(2^{|\lambda|+\aleph_0})^{+}$ with respect to models on ordinals equipped with the generalized Icard topologies $\mathcal{I}_{\lambda}$ and ${\tau_{c}}_{+\lambda}$. Specifically, we show that for such languages there exists a $\mathsf{GL}$-consistent set of formulas having neither $(\Theta, \mathcal{I}_{\lambda})$-model nor $(\Theta, {\tau_{c}}_{+\lambda})$-model. We also introduce two kinds of natural classes of topological spaces, called \emph{ $\lambda$-bouquet spaces} and \emph{ultralinear $\lambda$-bouquet spaces}, and prove that they yield strong completeness of $\mathsf{GL}$ and $\mathsf{GL}.3$ respectively for languages of cardinality $\lambda$.

math.LO↗