Strong Completeness of Provability Logic for Uncountable Languages
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917490027659264 |
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| author | Golshani, Mohammad Stepanov, Grigorii Zoghifard, Reihane |
| author_facet | Golshani, Mohammad Stepanov, Grigorii Zoghifard, Reihane |
| contents | For an ordinal $λ>0$, we use the Erdős--Rado partition theorem to prove the failure of strong completeness of $\mathsf{GL}$ 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 $\mathsf{GL}$-consistent set of formulas having neither $(Θ, \mathcal{I}_λ)$-model nor $(Θ, {τ_{c}}_{+λ})$-model. We also introduce two kinds of natural classes of topological spaces, called \emph{ $λ$-bouquet spaces} and \emph{ultralinear $λ$-bouquet spaces}, and prove that they yield strong completeness of $\mathsf{GL}$ and $\mathsf{GL}.3$ respectively for languages of cardinality $λ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_09470 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong Completeness of Provability Logic for Uncountable Languages Golshani, Mohammad Stepanov, Grigorii Zoghifard, Reihane Logic For an ordinal $λ>0$, we use the Erdős--Rado partition theorem to prove the failure of strong completeness of $\mathsf{GL}$ 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 $\mathsf{GL}$-consistent set of formulas having neither $(Θ, \mathcal{I}_λ)$-model nor $(Θ, {τ_{c}}_{+λ})$-model. We also introduce two kinds of natural classes of topological spaces, called \emph{ $λ$-bouquet spaces} and \emph{ultralinear $λ$-bouquet spaces}, and prove that they yield strong completeness of $\mathsf{GL}$ and $\mathsf{GL}.3$ respectively for languages of cardinality $λ$. |
| title | Strong Completeness of Provability Logic for Uncountable Languages |
| topic | Logic |
| url | https://arxiv.org/abs/2602.09470 |