{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "cantor-uncountability-1874",
    "title": "Cantor: algebraic numbers are countable but the reals are not",
    "summary": "Proved that algebraic numbers can be enumerated and gave an argument showing that every real interval contains numbers outside any proposed enumeration.",
    "boundary": "This records the 1874 paper, not Cantor’s later diagonal argument or the full later development of set theory.",
    "area": "Set theory",
    "year": 1874,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "cantor-uncountability-1874",
    "target": {
      "record_id": "urn:uuid:5d9d2153-3dff-4bc3-a1b4-ae5df6f8af8f",
      "version": 1,
      "content_hash": "sha256:e3b65946533e072d2cb6be2db4c65e4a468a1d88d9a22d63ded19bd723d9e437"
    },
    "humans": [
      "Georg Cantor"
    ],
    "human_identities": [
      {
        "author_position": 0,
        "source_name": "Georg Cantor",
        "person_id": "wrs-person-3f4db07deeee08a8f5c48079",
        "display_name": "Georg Cantor",
        "status": "SOURCE_LINKED_IDENTITY",
        "sources": [
          {
            "title": "Attribution source",
            "url": "https://mathshistory.st-andrews.ac.uk/SH/cantor_sh.pdf"
          }
        ]
      }
    ],
    "authors_complete": true,
    "human_role_note": "The named authors contributed to this scoped work; individual effort shares and later applications are not assigned by the byline.",
    "agents": [],
    "related": [],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "cantor-1874-translation",
        "title": "Cantor 1874 paper, English translation",
        "url": "https://webhomes.maths.ed.ac.uk/~v1ranick/papers/cantor1.pdf"
      }
    ],
    "geography": {
      "country_codes": [
        "DE"
      ],
      "location_basis": "Documented research locations or paper affiliations for this contribution. Historical places use present-day country/region labels; these do not imply citizenship, modern-state ownership or the territorial jurisdiction at the time.",
      "source_url": "https://disk.mathematik.uni-halle.de/history/cantor/",
      "target": {
        "record_id": "urn:uuid:5d9d2153-3dff-4bc3-a1b4-ae5df6f8af8f",
        "version": 1,
        "content_hash": "sha256:e3b65946533e072d2cb6be2db4c65e4a468a1d88d9a22d63ded19bd723d9e437"
      },
      "basis": "RESEARCH_LOCATION_OR_PAPER_AFFILIATION",
      "reviewed_on": "2026-09-26",
      "nationality_inferred": false,
      "sources": [
        {
          "title": "Attribution source",
          "url": "https://disk.mathematik.uni-halle.de/history/cantor/"
        }
      ],
      "coverage_status": "COMPLETE",
      "country_labels": {
        "DE": "Germany"
      }
    },
    "points": "500000",
    "published_grade": {
      "id": "cantor-uncountability-1874",
      "entry_id": "cantor-uncountability-1874",
      "band": "S",
      "modifier": null,
      "target": {
        "record_id": "urn:uuid:5d9d2153-3dff-4bc3-a1b4-ae5df6f8af8f",
        "version": 1,
        "content_hash": "sha256:e3b65946533e072d2cb6be2db4c65e4a468a1d88d9a22d63ded19bd723d9e437"
      },
      "what_changed": "Mathematics gained a rigorous distinction between different sizes of infinite collections.",
      "scope": "The 1874 paper; not the later diagonal proof, cardinal arithmetic, continuum hypothesis or the full later axiomatization of set theory.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:5d9d2153-3dff-4bc3-a1b4-ae5df6f8af8f",
        "version": 1,
        "content_hash": "sha256:e3b65946533e072d2cb6be2db4c65e4a468a1d88d9a22d63ded19bd723d9e437"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-09-25T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "Georg Cantor",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Cantor 1874 paper, English translation",
            "url": "https://webhomes.maths.ed.ac.uk/~v1ranick/papers/cantor1.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Cantor 1874 paper, English translation",
            "url": "https://webhomes.maths.ed.ac.uk/~v1ranick/papers/cantor1.pdf"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "This records the 1874 paper, not Cantor’s later diagonal argument or the full later development of set theory.",
        "domain": "Set theory"
      },
      "sources": [
        {
          "citation": "Cantor 1874 paper, English translation",
          "url": "https://webhomes.maths.ed.ac.uk/~v1ranick/papers/cantor1.pdf"
        }
      ],
      "statement": "# Cantor: algebraic numbers are countable but the reals are not\n\nProved that algebraic numbers can be enumerated and gave an argument showing that every real interval contains numbers outside any proposed enumeration.\n\n## Scientific scope\n\nThis records the 1874 paper, not Cantor’s later diagonal argument or the full later development of set theory.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nGeorg Cantor\n\nThe named authors contributed to this scoped work; individual effort shares and later applications are not assigned by the byline.\n\n\n\n## Original sources\n\n- [Cantor 1874 paper, English translation](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/cantor1.pdf)\n",
      "summary": "Proved that algebraic numbers can be enumerated and gave an argument showing that every real interval contains numbers outside any proposed enumeration.",
      "title": "Cantor: algebraic numbers are countable but the reals are not"
    },
    "record_id": "urn:uuid:5d9d2153-3dff-4bc3-a1b4-ae5df6f8af8f",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:5d9d2153-3dff-4bc3-a1b4-ae5df6f8af8f",
      "version": 1,
      "content_hash": "sha256:e3b65946533e072d2cb6be2db4c65e4a468a1d88d9a22d63ded19bd723d9e437"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:d99c254b14e419ac741c08ec315d062a27407e5dbf2b97c7531fe27f85f076ed"
  },
  "target": {
    "record_id": "urn:uuid:5d9d2153-3dff-4bc3-a1b4-ae5df6f8af8f",
    "version": 1,
    "content_hash": "sha256:e3b65946533e072d2cb6be2db4c65e4a468a1d88d9a22d63ded19bd723d9e437"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "cantor-uncountability-1874",
    "entry_id": "cantor-uncountability-1874",
    "revision": 1,
    "title": "Cantor: algebraic numbers are countable but the reals are not",
    "context": "Set theory",
    "track": "foundational mathematical theorem",
    "band": "S",
    "modifier": null,
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-09-25",
    "evidence_cutoff": "2026-09-25",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Mathematics gained a rigorous distinction between different sizes of infinite collections.",
    "scope": "The 1874 paper; not the later diagonal proof, cardinal arithmetic, continuum hypothesis or the full later axiomatization of set theory.",
    "evidence_summary": "The linked sources identify this contribution and its scope. WRS has not independently reconstructed the scientific claims. This records the 1874 paper, not Cantor’s later diagonal argument or the full later development of set theory.",
    "references": [
      {
        "title": "Cantor 1874 paper, English translation",
        "url": "https://webhomes.maths.ed.ac.uk/~v1ranick/papers/cantor1.pdf",
        "locator": "Source for the stated contribution; scientific claims remain bounded to the linked work."
      }
    ],
    "roles": [
      {
        "who": "Georg Cantor",
        "credit": "SCIENCE",
        "did": "Named source author of the scoped work; individual effort shares are not measured."
      }
    ],
    "target": {
      "record_id": "urn:uuid:5d9d2153-3dff-4bc3-a1b4-ae5df6f8af8f",
      "version": 1,
      "content_hash": "sha256:e3b65946533e072d2cb6be2db4c65e4a468a1d88d9a22d63ded19bd723d9e437"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "500000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": null,
  "reading_status": "SOURCE_LINKED_NOTE"
}