{"paper":{"title":"Cardinality of Wellordered Disjoint Unions of Quotients of Smooth Equivalence Relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Stephen Jackson, William Chan","submitted_at":"2019-03-10T02:05:28Z","abstract_excerpt":"Assume $\\mathsf{ZF + AD^+ + V = L(\\mathscr{P}(\\mathbb{R}))}$. Let $\\approx$ denote the relation of being in bijection. Let $\\kappa \\in \\mathrm{ON}$ and $\\langle E_\\alpha : \\alpha < \\kappa\\rangle$ be a sequence of equivalence relations on $\\mathbb{R}$ with all classes countable and for all $\\alpha < \\kappa$, $\\mathbb{R} / E_\\alpha \\approx \\mathbb{R}$. Then the disjoint union $\\bigsqcup_{\\alpha < \\kappa} \\mathbb{R} / E_\\alpha$ is in bijection with $\\mathbb{R} \\times \\kappa$ and $\\bigsqcup_{\\alpha < \\kappa} \\mathbb{R} / E_\\alpha$ has the J\\'onsson property.\n  Assume $\\mathsf{ZF + AD^+ + V = L(\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.03902","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"}