{"paper":{"title":"Modal group theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Embeddability among groups validates precisely the modal logic S4.2.","cross_cats":["math.GR"],"primary_cat":"math.LO","authors_text":"Wojciech Aleksander Wo{\\l}oszyn","submitted_at":"2026-05-13T23:28:19Z","abstract_excerpt":"I introduce modal group theory, in which we study the category of all groups, considering embeddability as providing a notion of modal possibility. Using HNN extensions and Britton's lemma, I demonstrate that the modal language of groups is more expressive than the first-order language of groups. I interpret the theory of true arithmetic in modal group theory, and show that, as sets of Goedel numbers, it is computably isomorphic to the modal theory of finitely presented groups. I answer an open question of Berger, Block, and Loewe by showing that the formulaic propositional modal validities of"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the formulaic propositional modal validities of groups under embeddings are precisely S4.2","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That embeddability in the category of groups provides a suitable accessibility relation for modal possibility, allowing HNN extensions and Britton's lemma to establish the claimed expressiveness and arithmetic interpretation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Modal group theory interprets true arithmetic and establishes that the propositional modal validities of groups under embeddings are exactly S4.2.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Embeddability among groups validates precisely the modal logic S4.2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1d2527565e1343976c8ad71314d184af78381b058955310fa8ec423d8dd5ccdc"},"source":{"id":"2605.14197","kind":"arxiv","version":1},"verdict":{"id":"a4765e6b-aef6-419f-899b-b9041336df44","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:36:05.896356Z","strongest_claim":"the formulaic propositional modal validities of groups under embeddings are precisely S4.2","one_line_summary":"Modal group theory interprets true arithmetic and establishes that the propositional modal validities of groups under embeddings are exactly S4.2.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That embeddability in the category of groups provides a suitable accessibility relation for modal possibility, allowing HNN extensions and Britton's lemma to establish the claimed expressiveness and arithmetic interpretation.","pith_extraction_headline":"Embeddability among groups validates precisely the modal logic S4.2."},"references":{"count":14,"sample":[{"doi":"","year":2023,"title":"The modal logic of abelian groups , journal =","work_id":"a09cb2f9-d12a-409d-9ef5-ea696212ee72","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Chang, C. C. and Keisler, H. J. , title =","work_id":"9c700e4b-861d-41a9-9f22-472f00f0c550","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1949,"title":"Higman, Graham and Neumann, B. H. and Neumann, Hanna , title =. Journal of the London Mathematical Society , volume =. 1949 , pages =","work_id":"38da33f5-fafa-42ff-9b18-05ec75ca3297","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1984,"title":"Models and Sets , series =","work_id":"7b232e84-4fbb-4e3e-8e93-6e5b9b406b83","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Modal model theory , journal =","work_id":"7fb7841b-3234-407b-885a-2233432ba264","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":14,"snapshot_sha256":"33f055301789a3c76e90ce727b110fd412119d95789a4d8d5d89a673e977b05a","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"7ef9bd606f94596276ff22933e8fd9bb05fbddbd52087f45ccea8f4b11f0a588"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}