{"paper":{"title":"Analytic equivalence relations satisfying hyperarithmetic-is-recursive","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Antonio Montalb\\'an","submitted_at":"2013-06-06T19:10:07Z","abstract_excerpt":"We prove, in ZF+$\\bf\\Sigma^1_2$-determinacy, that for any analytic equivalence relation $E$, the following three statements are equivalent: (1) $E$ does not have perfectly many classes, (2) $E$ satisfies hyperarithmetic-is-recursive on a cone, and (3) relative to some oracle, for every equivalence class $[Y]_E$ we have that a real $X$ computes a member of the equivalence class if and only if $\\om_1^X\\geq\\om_1^{[Y]}$.\n  We also show that the implication from (1) to (2) is equivalent to the existence of sharps over $ZF$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1513","kind":"arxiv","version":2},"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"}