{"paper":{"title":"Axiomatizing logics of finite G\\\"odel-Kripke models","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"Natural axiomatic extensions fail to restore completeness for modal Gödel logics over finite Gödel-Kripke models.","cross_cats":[],"primary_cat":"math.LO","authors_text":"Amanda Vidal, Ricardo O. Rodriguez","submitted_at":"2026-05-15T10:04:29Z","abstract_excerpt":"We investigate completeness for modal G\\\"odel logics with respect to finite G\\\"odel-Kripke models, along with related aspects. It is well known that the logics studied in [4, 11] fail to be complete with respect to finite G\\\"odel-Kripke models. We show that the natural candidate axiomatic extensions do not restore completeness, thereby resolving a 15 year open problem posed in the aforementioned works. We then provide new axiomatizations that are complete for finite models and characterize intermediate witnessing conditions that hold for the basic logics."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The natural candidate axiomatic extensions do not restore completeness with respect to finite Gödel-Kripke models, thereby resolving a 15 year open problem. New axiomatizations are provided that are complete for finite models and characterize intermediate witnessing conditions that hold for the basic logics.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that the specific modal Gödel logics studied in references [4,11] are the right starting point and that finite Gödel-Kripke models form the intended class of structures for which completeness is sought.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Resolves open problem by proving natural extensions of modal Gödel logics are incomplete for finite models and supplies new complete axiomatizations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Natural axiomatic extensions fail to restore completeness for modal Gödel logics over finite Gödel-Kripke models.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fe2e9f8981dbf7bba4151557a061f244ff7604094c555d3f83f6a56883abb716"},"source":{"id":"2605.15810","kind":"arxiv","version":1},"verdict":{"id":"7f50c853-b48a-4d32-9382-34284b7f6e19","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:46:42.267359Z","strongest_claim":"The natural candidate axiomatic extensions do not restore completeness with respect to finite Gödel-Kripke models, thereby resolving a 15 year open problem. New axiomatizations are provided that are complete for finite models and characterize intermediate witnessing conditions that hold for the basic logics.","one_line_summary":"Resolves open problem by proving natural extensions of modal Gödel logics are incomplete for finite models and supplies new complete axiomatizations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that the specific modal Gödel logics studied in references [4,11] are the right starting point and that finite Gödel-Kripke models form the intended class of structures for which completeness is sought.","pith_extraction_headline":"Natural axiomatic extensions fail to restore completeness for modal Gödel logics over finite Gödel-Kripke models."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15810/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T20:01:19.140612Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:01:13.801490Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:48.732665Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:21:55.888928Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c42c49d64f9027157f7d730b41b12ca3db307273f39558b6816982a6cb958e77"},"references":{"count":15,"sample":[{"doi":"","year":2016,"title":"M. Baaz and R. Iemhoff. Skolemization in intermediate logics with the finite model property.Logic Journal of the IGPL, 24(3):224–237, 2016","work_id":"3619f300-5fed-47e2-9e51-c3000c467c91","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"F. Bou, F. Esteva, L. Godo, and R.O. Rodriguez. Possibilistic semantics for a modal KD45 extension of G¨ odel fuzzy logic. In J. P. et. at. Carvalho, editor,IPMU, pages 123–135, Cham, 2016. Springer I","work_id":"c608fa3b-9348-4959-bb40-fccb2b4528f5","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"X. Caicedo, G. Metcalfe, R.O. Rodriguez, and J. Rogger. Decidability of order-based modal logics. Journal of Computer and System Sciences, 88:53 – 74, 2017","work_id":"3765efa3-4cbf-4b0d-9f31-7b6342a60a67","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"X. Caicedo and R.O. Rodriguez. Standard G¨ odel modal logics.Studia Logica, 94(2):189–214, 2010","work_id":"1b1f4f1c-cac3-4482-bea5-089ae3a49814","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"X. Caicedo and R.O. Rodriguez. Bi-modal G¨ odel logic over [0, 1]-valued Kripke frames.Journal of Logic and Computation, 25(1):37–55, 2015","work_id":"01b7c30a-141a-4feb-8bef-e3d97e2f0be0","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":15,"snapshot_sha256":"e6b5e1b7254f45fc5e27a503982e353000373764ef475d74124425772ec8a6e5","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"fad8c5e44ec3144449e04fb830025598e333fcc801ef38513bc9bb4214ad7616"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}