{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:4MVWZ225QHH5RFYDAO3R2RY7JJ","short_pith_number":"pith:4MVWZ225","schema_version":"1.0","canonical_sha256":"e32b6ceb5d81cfd8970303b71d471f4a604cdca04cff0be436fe3f257aa76e9b","source":{"kind":"arxiv","id":"2007.07560","version":6},"attestation_state":"computed","paper":{"title":"On the uncountability of $\\mathbb{R}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Dag Normann, Sam Sanders","submitted_at":"2020-07-15T09:19:01Z","abstract_excerpt":"Cantor's first set theory paper (1874) establishes the uncountability of $\\mathbb{R}$. We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of NIN (resp. NBI), i.e. the third-order statement: there is no injection (resp. bijection) from $[0,1]$ to $\\mathbb{N}$. Working in Kohlenbach's higher-order Reverse Mathematics, we show that NIN and NBI are hard to prove in terms of (conventional) comprehension axioms, while many basic theorems, like Arzela's convergence theorem for the Rie"},"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":"2007.07560","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2020-07-15T09:19:01Z","cross_cats_sorted":[],"title_canon_sha256":"ae9338bd57a0ecfd4d32116bf4d7a75755dd2bfb000732681b72b638e646ca54","abstract_canon_sha256":"906e630d57e1484367ef44f2a70aef0799d8754fb8629d2182fe868152c05890"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:11:04.181238Z","signature_b64":"W8y/9a38vcANrP5BHeZSZzemYokZYylHSWAYM2ZEZnt8M6/zFo4D55XyBxn4+rdo3l3HExzlv6z6NznRIWrCBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e32b6ceb5d81cfd8970303b71d471f4a604cdca04cff0be436fe3f257aa76e9b","last_reissued_at":"2026-07-05T04:11:04.180788Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:11:04.180788Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the uncountability of $\\mathbb{R}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Dag Normann, Sam Sanders","submitted_at":"2020-07-15T09:19:01Z","abstract_excerpt":"Cantor's first set theory paper (1874) establishes the uncountability of $\\mathbb{R}$. We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of NIN (resp. NBI), i.e. the third-order statement: there is no injection (resp. bijection) from $[0,1]$ to $\\mathbb{N}$. Working in Kohlenbach's higher-order Reverse Mathematics, we show that NIN and NBI are hard to prove in terms of (conventional) comprehension axioms, while many basic theorems, like Arzela's convergence theorem for the Rie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.07560","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2007.07560/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2007.07560","created_at":"2026-07-05T04:11:04.180854+00:00"},{"alias_kind":"arxiv_version","alias_value":"2007.07560v6","created_at":"2026-07-05T04:11:04.180854+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.07560","created_at":"2026-07-05T04:11:04.180854+00:00"},{"alias_kind":"pith_short_12","alias_value":"4MVWZ225QHH5","created_at":"2026-07-05T04:11:04.180854+00:00"},{"alias_kind":"pith_short_16","alias_value":"4MVWZ225QHH5RFYD","created_at":"2026-07-05T04:11:04.180854+00:00"},{"alias_kind":"pith_short_8","alias_value":"4MVWZ225","created_at":"2026-07-05T04:11:04.180854+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.05676","citing_title":"Plato and the foundations of mathematics","ref_index":65,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4MVWZ225QHH5RFYDAO3R2RY7JJ","json":"https://pith.science/pith/4MVWZ225QHH5RFYDAO3R2RY7JJ.json","graph_json":"https://pith.science/api/pith-number/4MVWZ225QHH5RFYDAO3R2RY7JJ/graph.json","events_json":"https://pith.science/api/pith-number/4MVWZ225QHH5RFYDAO3R2RY7JJ/events.json","paper":"https://pith.science/paper/4MVWZ225"},"agent_actions":{"view_html":"https://pith.science/pith/4MVWZ225QHH5RFYDAO3R2RY7JJ","download_json":"https://pith.science/pith/4MVWZ225QHH5RFYDAO3R2RY7JJ.json","view_paper":"https://pith.science/paper/4MVWZ225","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2007.07560&json=true","fetch_graph":"https://pith.science/api/pith-number/4MVWZ225QHH5RFYDAO3R2RY7JJ/graph.json","fetch_events":"https://pith.science/api/pith-number/4MVWZ225QHH5RFYDAO3R2RY7JJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4MVWZ225QHH5RFYDAO3R2RY7JJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4MVWZ225QHH5RFYDAO3R2RY7JJ/action/storage_attestation","attest_author":"https://pith.science/pith/4MVWZ225QHH5RFYDAO3R2RY7JJ/action/author_attestation","sign_citation":"https://pith.science/pith/4MVWZ225QHH5RFYDAO3R2RY7JJ/action/citation_signature","submit_replication":"https://pith.science/pith/4MVWZ225QHH5RFYDAO3R2RY7JJ/action/replication_record"}},"created_at":"2026-07-05T04:11:04.180854+00:00","updated_at":"2026-07-05T04:11:04.180854+00:00"}