{"paper":{"title":"Ordinal Definability and Combinatorics of Equivalence Relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"William Chan","submitted_at":"2017-11-12T20:41:12Z","abstract_excerpt":"Assume $\\mathsf{ZF + AD^+ + V = L(\\mathscr{P}(\\mathbb{R}))}$. Let $E$ be a $\\mathbf{\\Sigma}^1_1$ equivalence relation coded in $\\mathrm{HOD}$. $E$ has an ordinal definable equivalence class without any ordinal definable elements if and only if $\\mathrm{HOD} \\models E$ is unpinned.\n  $\\mathsf{ZF + AD^+ + V = L(\\mathscr{P}(\\mathbb{R}))}$ proves $E$-class section uniformization when $E$ is a $\\mathbf{\\Sigma}^1_1$ equivalence relation on $\\mathbb{R}$ which is pinned in every transitive model of $\\mathsf{ZFC}$ containing the real which codes $E$: Suppose $R$ is a relation on $\\mathbb{R}$ such that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04353","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"}