Strong Completeness of Provability Logic for Uncountable Languages

Abstract

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 $\lambda$.