AI 中文总结
该研究证明多模态逻辑J相对于J-花束是强完备的，进而得到可证性逻辑GLP的完备性结果，同时通过反例说明GLP相对于Beklemishev-Gabelaia空间不具备强完备性。
AI 中文摘要
我们证明了多模态逻辑J相对于J-花束（其克里普克语义的拓扑细化）是强完备的，特别地，它是强拓扑完备的。这为可证性逻辑GLP得出如下完备性结果：可数公式集Γ与GLP一致，当且仅当存在一个J-花束B及r∈B，使得B,r⊩GLP且B,r⊩Γ。相比之下，我们给出反例表明GLP相对于Beklemishev-Gabelaia空间不是强完备的。
英文摘要
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 $Γ$ 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 $B, r\VdashΓ$. In contrast, we exhibit counterexamples showing that $\mathsf{GLP}$ is not strongly complete with respect to Beklemishev-Gabelaia spaces.
