{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:525IIALQDZQQOFSSHNH2WMO6YG","short_pith_number":"pith:525IIALQ","schema_version":"1.0","canonical_sha256":"eeba8401701e610716523b4fab31dec1ad3f414e32caa74f3dc27c34898b7111","source":{"kind":"arxiv","id":"1903.03902","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1903.03902","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-03-10T02:05:28Z","cross_cats_sorted":[],"title_canon_sha256":"4e18b73eaf173ae90dda3fc15cf014e98dbbe9067f8a8aa5145d789f4fbbbb63","abstract_canon_sha256":"0b3322c444c7a22103fbbc5b68189921cf5cc0317b8d33fe33bf607bd954e470"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:40.657476Z","signature_b64":"g1axQGUHiQZpQmvTbdrAIdzrMmJvjYzRghrZgYWJ/XSIh7N0siD9D66hD5n+7JapqIFGpt2PUdlIMSf/hop8AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eeba8401701e610716523b4fab31dec1ad3f414e32caa74f3dc27c34898b7111","last_reissued_at":"2026-05-17T23:51:40.656944Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:40.656944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1903.03902","created_at":"2026-05-17T23:51:40.657024+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.03902v1","created_at":"2026-05-17T23:51:40.657024+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.03902","created_at":"2026-05-17T23:51:40.657024+00:00"},{"alias_kind":"pith_short_12","alias_value":"525IIALQDZQQ","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_16","alias_value":"525IIALQDZQQOFSS","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_8","alias_value":"525IIALQ","created_at":"2026-05-18T12:33:10.108867+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/525IIALQDZQQOFSSHNH2WMO6YG","json":"https://pith.science/pith/525IIALQDZQQOFSSHNH2WMO6YG.json","graph_json":"https://pith.science/api/pith-number/525IIALQDZQQOFSSHNH2WMO6YG/graph.json","events_json":"https://pith.science/api/pith-number/525IIALQDZQQOFSSHNH2WMO6YG/events.json","paper":"https://pith.science/paper/525IIALQ"},"agent_actions":{"view_html":"https://pith.science/pith/525IIALQDZQQOFSSHNH2WMO6YG","download_json":"https://pith.science/pith/525IIALQDZQQOFSSHNH2WMO6YG.json","view_paper":"https://pith.science/paper/525IIALQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.03902&json=true","fetch_graph":"https://pith.science/api/pith-number/525IIALQDZQQOFSSHNH2WMO6YG/graph.json","fetch_events":"https://pith.science/api/pith-number/525IIALQDZQQOFSSHNH2WMO6YG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/525IIALQDZQQOFSSHNH2WMO6YG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/525IIALQDZQQOFSSHNH2WMO6YG/action/storage_attestation","attest_author":"https://pith.science/pith/525IIALQDZQQOFSSHNH2WMO6YG/action/author_attestation","sign_citation":"https://pith.science/pith/525IIALQDZQQOFSSHNH2WMO6YG/action/citation_signature","submit_replication":"https://pith.science/pith/525IIALQDZQQOFSSHNH2WMO6YG/action/replication_record"}},"created_at":"2026-05-17T23:51:40.657024+00:00","updated_at":"2026-05-17T23:51:40.657024+00:00"}