Strong Completeness of Provability Logic for Uncountable Languages
Mohammad Golshani, Reihane Zoghifard
TL;DR
This work investigates strong completeness of $\mathsf{GL}$ for languages with uncountably many propositional variables under ordinal topologies. It combines the Erdős–Rado partition theorem with generalized Icard topologies to establish strong incompleteness for languages of cardinality $(2^{|\lambda|+\aleph_0})^{+}$ with respect to $(\Theta, \mathcal{I}_\lambda)$ and $(\Theta, {\tau_c}_{+\lambda})$, and it introduces $\lambda$-bouquet spaces to recover strong completeness for languages of size $\lambda$. The main technical contributions include (i) negative incompleteness results for the specified topologies in the uncountable setting, (ii) the construction of $\lambda$-bouquet spaces that yield strong completeness of $\mathsf{GL}(\mathbb{P})$ for languages of cardinality $\lambda$, and (iii) a detailed analysis of how cofinality, hyperlogarithms, and rank interact in ordinal-topological semantics. Together, these results delineate the boundary between completeness and incompleteness in provability logic when moving to uncountable languages and provide a robust topological framework for achieving strong completeness in the uncountable regime.
Abstract
For an ordinal $λ>0$, we use the Erdős--Rado partition theorem to prove the failure of strong completeness for modal languages of cardinality $(2^{|λ|+\aleph_0})^{+}$ with respect to models on ordinals equipped with the generalized Icard topologies $\mathcal{I}_λ$ and ${τ_{c}}_{+λ}$. Specifically, we show that for such languages there exists a consistent set of formulas having neither $(Θ, \mathcal{I}_λ)$-model nor $(Θ, {τ_{c}}_{+λ})$-model. We also introduce a natural class of topological spaces, called $λ$-bouquet spaces, and prove that they yield strong completeness of $\mathsf{GL}$ for languages of cardinality $λ$.