Strong Completeness of Provability Logic for Uncountable Languages

Abstract

For an ordinal $\hat{I}»>0$, we use the Erdős--Rado partition theorem to prove the failure of strong completeness of $\mathsf{GL}$ for modal languages of cardinality $(2^{|\hat{I}»|+{\aleph }_0}{)}^{+}$ with respect to models on ordinals equipped with the generalized Icard topologies ${\mathcal{I}}_{\hat{I}}»$ and ${\ddot{I}{„}_c}_{+\hat{I}»}$. Specifically, we show that for such languages there exists a $\mathsf{GL}$-consistent set of formulas having neither $(\hat{I}˜,{\mathcal{I}}_{\hat{I}}»)$-model nor $(\hat{I}˜,{\ddot{I}{„}_c}_{+\hat{I}»})$-model. We also introduce two kinds of natural classes of topological spaces, called \emph{ $\hat{I}»$-bouquet spaces} and \emph{ultralinear $\hat{I}»$-bouquet spaces}, and prove that they yield strong completeness of $\mathsf{GL}$ and $\mathsf{GL}.3$ respectively for languages of cardinality $\hat{I}»$.