{"paper":{"title":"Categoricity and Universal Classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Kaisa Kangas, Tapani Hyttinen","submitted_at":"2017-12-22T15:49:18Z","abstract_excerpt":"Let $(\\mathcal{K} ,\\subseteq )$ be a universal class with $LS(\\mathcal{K})=\\lambda$ categorical in regular $\\kappa >\\lambda^+$ with arbitrarily large models, and let $\\mathcal{K}^*$ be the class of all $\\mathcal{A}\\in\\mathcal{K}_{>\\lambda}$ for which there is $\\mathcal{B} \\in \\mathcal{K}_{\\ge\\kappa}$ such that $\\mathcal{A}\\subseteq\\mathcal{B}$. We prove that $\\mathcal{K}^*$ is categorical in every $\\xi >\\lambda^+$, $\\mathcal{K}_{\\ge\\beth_{(2^{\\lambda^+})^+}} \\subseteq \\mathcal{K}^{*}$, and the models of $\\mathcal{K}^*_{>\\lambda^+}$ are essentially vector spaces (or trivial i.e. disintegrated)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08532","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}