{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "dedekind-continuity-irrational-cuts-1872",
    "title": "Dedekind: real numbers constructed through rational cuts",
    "summary": "Dedekind constructed irrational numbers from gaps in the rational order and organized the resulting real-number system around cut completeness. He connected that continuity property with convergence of bounded increasing quantities.\n\nA rational order may contain partitions that no rational number realizes. Dedekind introduces a real number for such a gap, distinguishes order-equivalent cuts at rational endpoints and argues that the completed number system has a unique boundary for every proper cut. This gives analysis an arithmetic continuity principle instead of borrowing geometric intuition. His later construction of natural-number arithmetic is a different contribution.",
    "boundary": "The admitted target is the cut construction, order-completeness statement and its analytic connection. Cuts are proper nonempty order partitions; equivalent rational-endpoint cuts represent the same number. The cited source is a university transcription of the 1912 fourth edition, with formula/glyph defects, rather than the 1872 facsimile. It does not prove the continuity of physical space, certify every arithmetic-operation proof or assert a modern well-order of the reals.",
    "area": "Foundations of real analysis",
    "topic": "Mathematics",
    "year": 1872,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "dedekind-continuity-irrational-cuts-1872",
    "target": {
      "record_id": "urn:uuid:3e6a4cc0-65ef-43c2-ad2e-6700d13037c0",
      "version": 1,
      "content_hash": "sha256:292f6f13bc78b0f976c30a5baee5fd122ab07f340093642b00cbc144a050249b"
    },
    "humans": [
      "Richard Dedekind"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed contributors to this scoped work. Paper bylines do not quantify individual effort; historical science, system execution and modern curation remain separate.",
    "agents": [],
    "related": [
      "dedekind-nature-meaning-numbers-1888",
      "cantor-uncountability-1874"
    ],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "wrs24-dedekind-continuity-1872",
        "title": "Richard Dedekind, Stetigkeit und irrationale Zahlen (1872); Radboud University literal German transcription of the fourth edition (1912)",
        "url": "https://www.math.ru.nl/werkgroepen/gmfw/bronnen/dedekind2.html"
      }
    ],
    "points": "280000",
    "published_grade": {
      "id": "dedekind-continuity-irrational-cuts-1872",
      "entry_id": "dedekind-continuity-irrational-cuts-1872",
      "band": "S",
      "modifier": null,
      "target": {
        "record_id": "urn:uuid:3e6a4cc0-65ef-43c2-ad2e-6700d13037c0",
        "version": 1,
        "content_hash": "sha256:292f6f13bc78b0f976c30a5baee5fd122ab07f340093642b00cbc144a050249b"
      },
      "what_changed": "Proper rational cuts define real numbers, equivalent rational-endpoint partitions are identified, and the completed order supports a cut-completeness principle linked to bounded monotone convergence.",
      "scope": "Proper rational cuts define real numbers, equivalent rational-endpoint partitions are identified, and the completed order supports a cut-completeness principle linked to bounded monotone convergence. The admitted target is the cut construction, order-completeness statement and its analytic connection. Cuts are proper nonempty order partitions; equivalent rational-endpoint cuts represent the same number. The cited source is a university transcription of the 1912 fourth edition, with formula/glyph defects, rather than the 1872 facsimile. It does not prove the continuity of physical space, certify every arithmetic-operation proof or assert a modern well-order of the reals.",
      "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:3e6a4cc0-65ef-43c2-ad2e-6700d13037c0",
        "version": 1,
        "content_hash": "sha256:292f6f13bc78b0f976c30a5baee5fd122ab07f340093642b00cbc144a050249b"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-03T00:00:00Z",
    "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": "Richard Dedekind",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "formulator"
          ],
          "source": {
            "citation": "Richard Dedekind, Stetigkeit und irrationale Zahlen (1872); Radboud University literal German transcription of the fourth edition (1912)",
            "url": "https://www.math.ru.nl/werkgroepen/gmfw/bronnen/dedekind2.html"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Richard Dedekind, Stetigkeit und irrationale Zahlen (1872); Radboud University literal German transcription of the fourth edition (1912)",
            "url": "https://www.math.ru.nl/werkgroepen/gmfw/bronnen/dedekind2.html"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The admitted target is the cut construction, order-completeness statement and its analytic connection. Cuts are proper nonempty order partitions; equivalent rational-endpoint cuts represent the same number. The cited source is a university transcription of the 1912 fourth edition, with formula/glyph defects, rather than the 1872 facsimile. It does not prove the continuity of physical space, certify every arithmetic-operation proof or assert a modern well-order of the reals.",
        "domain": "Foundations of real analysis"
      },
      "sources": [
        {
          "citation": "Richard Dedekind, Stetigkeit und irrationale Zahlen (1872); Radboud University literal German transcription of the fourth edition (1912)",
          "url": "https://www.math.ru.nl/werkgroepen/gmfw/bronnen/dedekind2.html"
        }
      ],
      "statement": "# Dedekind: real numbers constructed through rational cuts\n\nDedekind constructed irrational numbers from gaps in the rational order and organized the resulting real-number system around cut completeness. He connected that continuity property with convergence of bounded increasing quantities.\n\nA rational order may contain partitions that no rational number realizes. Dedekind introduces a real number for such a gap, distinguishes order-equivalent cuts at rational endpoints and argues that the completed number system has a unique boundary for every proper cut. This gives analysis an arithmetic continuity principle instead of borrowing geometric intuition. His later construction of natural-number arithmetic is a different contribution.\n\n## Scientific scope\n\nThe admitted target is the cut construction, order-completeness statement and its analytic connection. Cuts are proper nonempty order partitions; equivalent rational-endpoint cuts represent the same number. The cited source is a university transcription of the 1912 fourth edition, with formula/glyph defects, rather than the 1872 facsimile. It does not prove the continuity of physical space, certify every arithmetic-operation proof or assert a modern well-order of the reals.\n\n## People and systems\n\nRichard Dedekind\n\n\n\n## Original sources\n\n- [Richard Dedekind, Stetigkeit und irrationale Zahlen (1872); Radboud University literal German transcription of the fourth edition (1912)](https://www.math.ru.nl/werkgroepen/gmfw/bronnen/dedekind2.html)\n",
      "summary": "Dedekind constructed irrational numbers from gaps in the rational order and organized the resulting real-number system around cut completeness. He connected that continuity property with convergence of bounded increasing quantities.\n\nA rational order may contain partitions that no rational number realizes. Dedekind introduces a real number for such a gap, distinguishes order-equivalent cuts at rational endpoints and argues that the completed number system has a unique boundary for every proper cut. This gives analysis an arithmetic continuity principle instead of borrowing geometric intuition. His later construction of natural-number arithmetic is a different contribution.",
      "title": "Dedekind: real numbers constructed through rational cuts"
    },
    "record_id": "urn:uuid:3e6a4cc0-65ef-43c2-ad2e-6700d13037c0",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:3e6a4cc0-65ef-43c2-ad2e-6700d13037c0",
      "version": 1,
      "content_hash": "sha256:292f6f13bc78b0f976c30a5baee5fd122ab07f340093642b00cbc144a050249b"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:f3acdd0b7724b9c27ac4cad0a19b57b64099904e67fe1358613c93915b29ff50"
  },
  "target": {
    "record_id": "urn:uuid:3e6a4cc0-65ef-43c2-ad2e-6700d13037c0",
    "version": 1,
    "content_hash": "sha256:292f6f13bc78b0f976c30a5baee5fd122ab07f340093642b00cbc144a050249b"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "dedekind-continuity-irrational-cuts-1872",
    "entry_id": "dedekind-continuity-irrational-cuts-1872",
    "revision": 1,
    "title": "Dedekind: real numbers constructed through rational cuts",
    "context": "Foundations of real analysis",
    "track": "Foundations of real analysis",
    "band": "S",
    "modifier": null,
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-03",
    "evidence_cutoff": "2026-10-03",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Proper rational cuts define real numbers, equivalent rational-endpoint partitions are identified, and the completed order supports a cut-completeness principle linked to bounded monotone convergence.",
    "scope": "Proper rational cuts define real numbers, equivalent rational-endpoint partitions are identified, and the completed order supports a cut-completeness principle linked to bounded monotone convergence. The admitted target is the cut construction, order-completeness statement and its analytic connection. Cuts are proper nonempty order partitions; equivalent rational-endpoint cuts represent the same number. The cited source is a university transcription of the 1912 fourth edition, with formula/glyph defects, rather than the 1872 facsimile. It does not prove the continuity of physical space, certify every arithmetic-operation proof or assert a modern well-order of the reals.",
    "evidence_summary": "Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.",
    "references": [
      {
        "title": "Richard Dedekind, Stetigkeit und irrationale Zahlen (1872); Radboud University literal German transcription of the fourth edition (1912)",
        "url": "https://www.math.ru.nl/werkgroepen/gmfw/bronnen/dedekind2.html",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      }
    ],
    "roles": [
      {
        "who": "Richard Dedekind",
        "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.",
        "credit": "SCIENCE"
      }
    ],
    "target": {
      "record_id": "urn:uuid:3e6a4cc0-65ef-43c2-ad2e-6700d13037c0",
      "version": 1,
      "content_hash": "sha256:292f6f13bc78b0f976c30a5baee5fd122ab07f340093642b00cbc144a050249b"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "280000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": null,
  "reading_status": "SOURCE_LINKED_NOTE"
}