{"paper":{"title":"When an Equivalence Relation with All Borel Classes will be Borel Somewhere?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Menachem Magidor, William Chan","submitted_at":"2016-08-17T10:02:45Z","abstract_excerpt":"In $\\mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \\in L(\\mathbb{R})$ on $\\mathbb{R}$ with all $\\mathbf{\\Delta}_1^1$ classes and every $\\sigma$-ideal $I$ on $\\mathbb{R}$ so that the associated forcing $\\mathbb{P}_I$ of $I^+$ $\\mathbf{\\Delta}_1^1$ subsets is proper, there exists some $I^+$ $\\mathbf{\\Delta}_1^1$ set $C$ so that $E \\upharpoonright C$ is a $\\mathbf{\\Delta}_1^1$ equivalence relation. In $\\mathsf{ZF} + \\mathsf{DC} + \\mathsf{AD}_\\mathbb{R} + V = L(\\mathscr{P}(\\mathbb{R}))$, for every equivalence "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04913","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"}