{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2002:77B4EBDEKJB77XIDJBJNG2FRBX","short_pith_number":"pith:77B4EBDE","schema_version":"1.0","canonical_sha256":"ffc3c204645243ffdd034852d368b10dcabef870082f83479d789296b490f502","source":{"kind":"arxiv","id":"math/0211433","version":1},"attestation_state":"computed","paper":{"title":"Pcf theory and Woodin cardinals","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Moti Gitik, Ralf Schindler, Saharon Shelah","submitted_at":"2002-11-27T22:49:34Z","abstract_excerpt":"We prove the following two results.\n  Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all bounded X subset aleph_{|alpha|^+}, M_n^#(X) exists.\n  Theorem B: Let kappa be a singular cardinal of uncountable cofinality. If {alpha<kappa| 2^alpha=alpha^+} is stationary as well as co-stationary then for all n< omega and for all bounded X subset kappa, M_n^#(X) exists.\n  Theorem A answers a question of Gitik and Mitchell, and Theorem B yields a lower bound f"},"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":"math/0211433","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"2002-11-27T22:49:34Z","cross_cats_sorted":[],"title_canon_sha256":"0199f6792cd9eff058a7f83ca2f1deddcea00497a46165d2d7ab97a5cfc4b89c","abstract_canon_sha256":"ed4ca5d700400ec1c81a43e2493da10a743e364eb39b95a2aa57bec9329f8296"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:00.515349Z","signature_b64":"9ACLQ+pqukdKrHDZ4RqE6dFIOshQ5GE3mTX6nZyviSQpUFJBMrSRFFZvffIFgCiIBczztgdQhwSxGqn12wTBBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ffc3c204645243ffdd034852d368b10dcabef870082f83479d789296b490f502","last_reissued_at":"2026-05-18T02:38:00.514838Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:00.514838Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pcf theory and Woodin cardinals","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Moti Gitik, Ralf Schindler, Saharon Shelah","submitted_at":"2002-11-27T22:49:34Z","abstract_excerpt":"We prove the following two results.\n  Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all bounded X subset aleph_{|alpha|^+}, M_n^#(X) exists.\n  Theorem B: Let kappa be a singular cardinal of uncountable cofinality. If {alpha<kappa| 2^alpha=alpha^+} is stationary as well as co-stationary then for all n< omega and for all bounded X subset kappa, M_n^#(X) exists.\n  Theorem A answers a question of Gitik and Mitchell, and Theorem B yields a lower bound f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0211433","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":"math/0211433","created_at":"2026-05-18T02:38:00.514921+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0211433v1","created_at":"2026-05-18T02:38:00.514921+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0211433","created_at":"2026-05-18T02:38:00.514921+00:00"},{"alias_kind":"pith_short_12","alias_value":"77B4EBDEKJB7","created_at":"2026-05-18T12:25:50.845339+00:00"},{"alias_kind":"pith_short_16","alias_value":"77B4EBDEKJB77XID","created_at":"2026-05-18T12:25:50.845339+00:00"},{"alias_kind":"pith_short_8","alias_value":"77B4EBDE","created_at":"2026-05-18T12:25:50.845339+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/77B4EBDEKJB77XIDJBJNG2FRBX","json":"https://pith.science/pith/77B4EBDEKJB77XIDJBJNG2FRBX.json","graph_json":"https://pith.science/api/pith-number/77B4EBDEKJB77XIDJBJNG2FRBX/graph.json","events_json":"https://pith.science/api/pith-number/77B4EBDEKJB77XIDJBJNG2FRBX/events.json","paper":"https://pith.science/paper/77B4EBDE"},"agent_actions":{"view_html":"https://pith.science/pith/77B4EBDEKJB77XIDJBJNG2FRBX","download_json":"https://pith.science/pith/77B4EBDEKJB77XIDJBJNG2FRBX.json","view_paper":"https://pith.science/paper/77B4EBDE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0211433&json=true","fetch_graph":"https://pith.science/api/pith-number/77B4EBDEKJB77XIDJBJNG2FRBX/graph.json","fetch_events":"https://pith.science/api/pith-number/77B4EBDEKJB77XIDJBJNG2FRBX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/77B4EBDEKJB77XIDJBJNG2FRBX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/77B4EBDEKJB77XIDJBJNG2FRBX/action/storage_attestation","attest_author":"https://pith.science/pith/77B4EBDEKJB77XIDJBJNG2FRBX/action/author_attestation","sign_citation":"https://pith.science/pith/77B4EBDEKJB77XIDJBJNG2FRBX/action/citation_signature","submit_replication":"https://pith.science/pith/77B4EBDEKJB77XIDJBJNG2FRBX/action/replication_record"}},"created_at":"2026-05-18T02:38:00.514921+00:00","updated_at":"2026-05-18T02:38:00.514921+00:00"}