{"paper":{"title":"The uncountable spectra of countable theories","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Bradd Hart, Ehud Hrushovski, Michael C. Laskowski","submitted_at":"2000-07-01T00:00:00Z","abstract_excerpt":"Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T,kappa) be the number of isomorphism types of models of T of cardinality \\kappa. We denote by \\mu (respectively \\hat\\mu) the number of cardinals (respectively infinite cardinals) less than or equal to \\kappa. We prove that I(T,\\kappa), as a function of \\kappa > \\aleph_0, is the minimum of 2^{\\kappa} and one of the following functions:\n  1. 2^{\\kappa};\n  2. the constant function 1;\n  3. |\\hat\\mu^n/{\\sim_G}|-|(\\hat\\mu - 1)^n/{\\sim_G}| if \\hat\\mu<\\omega for some 1<n<\\omega and \\hat\\mu if \\hat\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0007199","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"}