{"paper":{"title":"Strong Completeness of Provability Logic for Uncountable Languages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Provability logic GL fails strong completeness for modal languages of cardinality (2^|λ| + ℵ₀)^+ on ordinals with generalized Icard topologies.","cross_cats":[],"primary_cat":"math.LO","authors_text":"Grigorii Stepanov, Mohammad Golshani, Reihane Zoghifard","submitted_at":"2026-02-10T07:06:59Z","abstract_excerpt":"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 s"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For an ordinal λ>0, there exists a GL-consistent set of formulas having neither (Θ, I_λ)-model nor (Θ, τ_c +λ)-model when the language has cardinality (2^|λ|+ℵ₀)^+; λ-bouquet spaces yield strong completeness of GL for languages of cardinality λ.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The generalized Icard topologies I_λ and τ_c +λ are the right semantics for testing strong completeness; if other topologies or different model classes are intended, the failure result may not apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Strong completeness of GL fails for modal languages of size (2^|λ|+ℵ₀)^+ on generalized Icard topologies but holds for GL and GL.3 on λ-bouquet and ultralinear λ-bouquet spaces.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Provability logic GL fails strong completeness for modal languages of cardinality (2^|λ| + ℵ₀)^+ on ordinals with generalized Icard topologies.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f68d19a9f33ae2c016d9f118456c83ae47fe84ffec08b072e3ee0bbfaa85db10"},"source":{"id":"2602.09470","kind":"arxiv","version":2},"verdict":{"id":"6eeb50e7-3f6d-46b8-8aed-352f4121fd3d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T05:57:44.527062Z","strongest_claim":"For an ordinal λ>0, there exists a GL-consistent set of formulas having neither (Θ, I_λ)-model nor (Θ, τ_c +λ)-model when the language has cardinality (2^|λ|+ℵ₀)^+; λ-bouquet spaces yield strong completeness of GL for languages of cardinality λ.","one_line_summary":"Strong completeness of GL fails for modal languages of size (2^|λ|+ℵ₀)^+ on generalized Icard topologies but holds for GL and GL.3 on λ-bouquet and ultralinear λ-bouquet spaces.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The generalized Icard topologies I_λ and τ_c +λ are the right semantics for testing strong completeness; if other topologies or different model classes are intended, the failure result may not apply.","pith_extraction_headline":"Provability logic GL fails strong completeness for modal languages of cardinality (2^|λ| + ℵ₀)^+ on ordinals with generalized Icard topologies."},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}