"Dedekind cut" meaning in English

{
  "etymology_text": "Named after German mathematician Richard Dedekind (1831–1916), who introduced the concept (although a similar construction was used in Euclid's Elements to define proportional segments).",
  "forms": [
    {
      "form": "Dedekind cuts",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Dedekind cut (plural Dedekind cuts)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
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          "source": "w"
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          "parents": [
            "Entries with language name categories using raw markup",
            "Entry maintenance"
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          "source": "w"
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          "kind": "other",
          "name": "English terms with non-redundant non-automated sortkeys",
          "parents": [
            "Terms with non-redundant non-automated sortkeys",
            "Entry maintenance"
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          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1990, Judith Roitman, Introduction to Modern Set Theory, Wiley, page 70",
          "text": "More than one formal solution was offered; the one which is easiest to work with (all of the solutions are provably equivalent) is the method of Dedekind cuts.",
          "type": "quotation"
        },
        {
          "ref": "1997, Reuben Hersh, What is Mathematics, Really?, Oxford University Press, page 274",
          "text": "To identify Dedekind cuts as the sought-for \"real number system,\" we must show that they include all the rationals and irrationals—all the numbers that can be approximated with arbitrary accuracy by rationals.",
          "type": "quotation"
        },
        {
          "ref": "2011, Alexander R. Pruss, Actuality, Possibility, and Worlds, Continuum Books, page 55",
          "text": "But there are, of course, many set theoretic ways of expressing Riemannian manifolds, just as there are many set theoretic ways of expressing real numbers (one can express them as pairs of lower and upper Dedekind cuts, or as lower Dedekind cuts, or as upper Dedekind cuts, or as equivalence classes of Cauchy sequences, and so on).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any partition of the set of rational numbers into non-empty sets A and B such that all elements of A are less than all elements of B and A contains no greatest element; intended as a construction of a real number."
      ],
      "id": "en-Dedekind_cut-en-noun-sc0aG-PD",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "partition",
          "partition"
        ],
        [
          "rational numbers",
          "rational numbers"
        ],
        [
          "non-empty",
          "non-empty"
        ],
        [
          "real number",
          "real number"
        ]
      ],
      "raw_glosses": [
        "(mathematics) Any partition of the set of rational numbers into non-empty sets A and B such that all elements of A are less than all elements of B and A contains no greatest element; intended as a construction of a real number."
      ],
      "topics": [
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fi",
          "lang": "Finnish",
          "sense": "partition of rational numbers",
          "word": "Dedekindin leikkaus"
        },
        {
          "code": "fr",
          "lang": "French",
          "sense": "partition of rational numbers",
          "tags": [
            "feminine"
          ],
          "word": "coupure de Dedekind"
        },
        {
          "code": "pl",
          "lang": "Polish",
          "sense": "partition of rational numbers",
          "tags": [
            "masculine"
          ],
          "word": "przekrój Dedekinda"
        }
      ],
      "wikipedia": [
        "Dedekind cut",
        "Richard Dedekind"
      ]
    }
  ],
  "word": "Dedekind cut"
}

{
  "etymology_text": "Named after German mathematician Richard Dedekind (1831–1916), who introduced the concept (although a similar construction was used in Euclid's Elements to define proportional segments).",
  "forms": [
    {
      "form": "Dedekind cuts",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Dedekind cut (plural Dedekind cuts)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
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        "English countable nouns",
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        "English entries with language name categories using raw markup",
        "English eponyms",
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        "English terms with quotations",
        "en:Mathematics"
      ],
      "examples": [
        {
          "ref": "1990, Judith Roitman, Introduction to Modern Set Theory, Wiley, page 70",
          "text": "More than one formal solution was offered; the one which is easiest to work with (all of the solutions are provably equivalent) is the method of Dedekind cuts.",
          "type": "quotation"
        },
        {
          "ref": "1997, Reuben Hersh, What is Mathematics, Really?, Oxford University Press, page 274",
          "text": "To identify Dedekind cuts as the sought-for \"real number system,\" we must show that they include all the rationals and irrationals—all the numbers that can be approximated with arbitrary accuracy by rationals.",
          "type": "quotation"
        },
        {
          "ref": "2011, Alexander R. Pruss, Actuality, Possibility, and Worlds, Continuum Books, page 55",
          "text": "But there are, of course, many set theoretic ways of expressing Riemannian manifolds, just as there are many set theoretic ways of expressing real numbers (one can express them as pairs of lower and upper Dedekind cuts, or as lower Dedekind cuts, or as upper Dedekind cuts, or as equivalence classes of Cauchy sequences, and so on).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any partition of the set of rational numbers into non-empty sets A and B such that all elements of A are less than all elements of B and A contains no greatest element; intended as a construction of a real number."
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "partition",
          "partition"
        ],
        [
          "rational numbers",
          "rational numbers"
        ],
        [
          "non-empty",
          "non-empty"
        ],
        [
          "real number",
          "real number"
        ]
      ],
      "raw_glosses": [
        "(mathematics) Any partition of the set of rational numbers into non-empty sets A and B such that all elements of A are less than all elements of B and A contains no greatest element; intended as a construction of a real number."
      ],
      "topics": [
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Dedekind cut",
        "Richard Dedekind"
      ]
    }
  ],
  "translations": [
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "partition of rational numbers",
      "word": "Dedekindin leikkaus"
    },
    {
      "code": "fr",
      "lang": "French",
      "sense": "partition of rational numbers",
      "tags": [
        "feminine"
      ],
      "word": "coupure de Dedekind"
    },
    {
      "code": "pl",
      "lang": "Polish",
      "sense": "partition of rational numbers",
      "tags": [
        "masculine"
      ],
      "word": "przekrój Dedekinda"
    }
  ],
  "word": "Dedekind cut"
}