Strong Completeness of Provability Logic for Uncountable Languages

TL;DR

This paper investigates the limits of strong completeness in provability logic for uncountable languages, using advanced set-theoretic and topological methods to identify where completeness fails and succeeds.

Contribution

It demonstrates the failure of strong completeness for certain uncountable modal languages and introduces new classes of topological spaces that ensure completeness.

Findings

Strong completeness fails for languages of certain uncountable sizes.

Introduces $ extit{ extbf{$oldsymbol{ extlambda}$-bouquet spaces}}$ and ultralinear variants that achieve strong completeness.

Uses Erdős–Rado theorem to establish non-existence of models for some consistent sets.

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 $\lambda$.