{"paper":{"title":"The Number of Atomic Models of Uncountable Theories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Douglas Ulrich","submitted_at":"2016-07-26T18:30:57Z","abstract_excerpt":"We show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with $ZFC + \\aleph_1 < 2^{\\aleph_0}$ that there is a complete theory in a language of size $\\aleph_1$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with $ZFC + \\aleph_1 < 2^{\\aleph_0}$ that for every complete theory $T$ in a language of size $\\aleph_1$, if $T$ has uncountable atomic models but no constructible models, then $T$ has $2^{\\aleph_1}$ atomic models of size $\\aleph_1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07833","kind":"arxiv","version":1},"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"}