{"paper":{"title":"Universal classes near $\\aleph_1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Marcos Mazari-Armida, Sebastien Vasey","submitted_at":"2017-12-07T22:47:50Z","abstract_excerpt":"Shelah has provided sufficient conditions for an $L_{\\omega_1, \\omega}$-sentence $\\psi$ to have arbitrarily large models and for a Morley-like theorem to hold of $\\psi$. These conditions involve structural and set-theoretic assumptions on all the $\\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\\aleph_0$ and $\\aleph_1$ which suffice when $\\psi$ is restricted to be universal:\n  $\\mathbf{Theorem}$ Assume $2^{\\aleph_{0}} < 2 ^{\\aleph_{1}}$. Let $\\psi$ be a universal $L_{\\omega_{1}, \\omega}$-sentence.\n  - If $\\psi$ is categorical in $\\aleph_{0}$ and $1 \\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02880","kind":"arxiv","version":5},"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"}