Strong Completeness of Provability Logic for Uncountable Languages

For an ordinal $\lambda >0$, we use the Erdős–Rado partition theorem to prove the failure of strong completeness of $\mathsf{G}\mathsf{L}$ for modal languages of cardinality $(2^{\mathrm{\mid }\lambda \mathrm{\mid }+{\mathrm{\aleph }}_0}{)}^{+}$ 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 $\mathsf{G}\mathsf{L}$-consistent set of formulas having neither $(\mathrm{\Theta },{\mathcal{I}}_{\lambda })$-model nor $(\mathrm{\Theta },{{\tau }_c}_{+\lambda })$-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 $\mathsf{G}\mathsf{L}$ and $\mathsf{G}\mathsf{L}.3$ respectively for languages of cardinality $\lambda$.