"transfinite number" meaning in English

{
  "forms": [
    {
      "form": "transfinite numbers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "transfinite number (plural transfinite numbers)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
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          "kind": "other",
          "name": "English entries with incorrect language header",
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            "Entries with incorrect language header",
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          "kind": "other",
          "langcode": "en",
          "name": "Infinity",
          "orig": "en:Infinity",
          "parents": [],
          "source": "w"
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          "kind": "topical",
          "langcode": "en",
          "name": "Set theory",
          "orig": "en:Set theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1961, Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, Courier Dover Publications, page 228",
          "text": "It will be recalled that Cantor called the first transfinite number ℵ₀. He called the second transfinite number—the one describing the set of all real numbers— C. It has not been proved whether C is the next transfinite number after ℵ₀ or whether another number exists between them.",
          "type": "quotation"
        },
        {
          "ref": "1968, B. T. Levšenko, “Spaces of transfinite dimensionality”, in Fourteen Papers on Algebra, Topology, Algebraic and Differential Geometry, American Mathematical Soc., page 141",
          "text": "Let R be a bicompact of dimensionality #x5C;operatorname#x7B;ind#x7D;(R)#x5C;le#x5C;alpha. If #x5C;alpha is an isolated transfinite number, than at any point x#x5C;inR there exist arbitrarily small neighborhoods Vx with boundaries of dimensionality #x5C;operatorname#x7B;ind#x7D;#x5C;overline#x7B;Vx#x7D;#x5C;le#x5C;alpha-1.",
          "type": "quotation"
        },
        {
          "ref": "1990, Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press, page 180",
          "text": "After all, it was the ordinals that made precise definition of the transfinite cardinals possible. And until Cantor had introduced the order types of transfinite number classes, he could not define precisely any transfinite cardinal beyond the first power.",
          "type": "quotation"
        },
        {
          "ref": "2009, John Tabak, Numbers: Computers, Philosophers, and the Search for Meaning, Infobase Publishing, page 153",
          "text": "For example, does there exist a transfinite number that is strictly bigger than ℵ₀ and strictly smaller than ℵ₁? In this case an instance of this in between number is too big to be put into one-to-one correspondence with the set of natural numbers, and too small to be put into one-to-one correspondence with the set of real numbers.",
          "type": "quotation"
        },
        {
          "ref": "2012, Benjamin Wardhaugh, A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing, Princeton University Press, page 136",
          "text": "Having demonstrated the existence of a one-to-one correspondence, we can conclude that the class of the squares of all the natural numbers has the same transfinite number as the class of all the natural numbers! This result is not what might have been anticipated, seeing that the second class is a proper subset of the first.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any cardinal or ordinal number which is larger than any finite, i.e. natural number; often represented by the Hebrew letter aleph (ℵ) with a subscript 0, 1, etc."
      ],
      "hyponyms": [
        {
          "sense": "aleph number, beth number",
          "word": "aleph-null"
        },
        {
          "sense": "aleph number, beth number",
          "word": "aleph-one"
        },
        {
          "sense": "aleph number, beth number",
          "word": "ω"
        }
      ],
      "id": "en-transfinite_number-en-noun-03Ba1VoB",
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "cardinal",
          "cardinal number"
        ],
        [
          "ordinal number",
          "ordinal number"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "natural number",
          "natural number"
        ],
        [
          "Hebrew",
          "Hebrew"
        ],
        [
          "letter",
          "letter"
        ],
        [
          "aleph",
          "aleph"
        ],
        [
          "ℵ",
          "ℵ"
        ]
      ],
      "raw_glosses": [
        "(set theory) Any cardinal or ordinal number which is larger than any finite, i.e. natural number; often represented by the Hebrew letter aleph (ℵ) with a subscript 0, 1, etc."
      ],
      "related": [
        {
          "word": "hyperreal number"
        },
        {
          "word": "infinitesimal"
        }
      ],
      "topics": [
        "mathematics",
        "sciences",
        "set-theory"
      ],
      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "number larger than any finite number",
          "tags": [
            "masculine"
          ],
          "word": "numero transfinito"
        },
        {
          "code": "pt",
          "lang": "Portuguese",
          "sense": "number larger than any finite number",
          "tags": [
            "masculine"
          ],
          "word": "número transfinito"
        }
      ],
      "wikipedia": [
        "transfinite number"
      ]
    }
  ],
  "word": "transfinite number"
}

{
  "forms": [
    {
      "form": "transfinite numbers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "transfinite number (plural transfinite numbers)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "sense": "aleph number, beth number",
      "word": "aleph-null"
    },
    {
      "sense": "aleph number, beth number",
      "word": "aleph-one"
    },
    {
      "sense": "aleph number, beth number",
      "word": "ω"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "hyperreal number"
    },
    {
      "word": "infinitesimal"
    }
  ],
  "senses": [
    {
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        "English countable nouns",
        "English entries with incorrect language header",
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        "English multiword terms",
        "English nouns",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "en:Infinity",
        "en:Set theory"
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      "examples": [
        {
          "ref": "1961, Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, Courier Dover Publications, page 228",
          "text": "It will be recalled that Cantor called the first transfinite number ℵ₀. He called the second transfinite number—the one describing the set of all real numbers— C. It has not been proved whether C is the next transfinite number after ℵ₀ or whether another number exists between them.",
          "type": "quotation"
        },
        {
          "ref": "1968, B. T. Levšenko, “Spaces of transfinite dimensionality”, in Fourteen Papers on Algebra, Topology, Algebraic and Differential Geometry, American Mathematical Soc., page 141",
          "text": "Let R be a bicompact of dimensionality #x5C;operatorname#x7B;ind#x7D;(R)#x5C;le#x5C;alpha. If #x5C;alpha is an isolated transfinite number, than at any point x#x5C;inR there exist arbitrarily small neighborhoods Vx with boundaries of dimensionality #x5C;operatorname#x7B;ind#x7D;#x5C;overline#x7B;Vx#x7D;#x5C;le#x5C;alpha-1.",
          "type": "quotation"
        },
        {
          "ref": "1990, Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press, page 180",
          "text": "After all, it was the ordinals that made precise definition of the transfinite cardinals possible. And until Cantor had introduced the order types of transfinite number classes, he could not define precisely any transfinite cardinal beyond the first power.",
          "type": "quotation"
        },
        {
          "ref": "2009, John Tabak, Numbers: Computers, Philosophers, and the Search for Meaning, Infobase Publishing, page 153",
          "text": "For example, does there exist a transfinite number that is strictly bigger than ℵ₀ and strictly smaller than ℵ₁? In this case an instance of this in between number is too big to be put into one-to-one correspondence with the set of natural numbers, and too small to be put into one-to-one correspondence with the set of real numbers.",
          "type": "quotation"
        },
        {
          "ref": "2012, Benjamin Wardhaugh, A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing, Princeton University Press, page 136",
          "text": "Having demonstrated the existence of a one-to-one correspondence, we can conclude that the class of the squares of all the natural numbers has the same transfinite number as the class of all the natural numbers! This result is not what might have been anticipated, seeing that the second class is a proper subset of the first.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any cardinal or ordinal number which is larger than any finite, i.e. natural number; often represented by the Hebrew letter aleph (ℵ) with a subscript 0, 1, etc."
      ],
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "cardinal",
          "cardinal number"
        ],
        [
          "ordinal number",
          "ordinal number"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "natural number",
          "natural number"
        ],
        [
          "Hebrew",
          "Hebrew"
        ],
        [
          "letter",
          "letter"
        ],
        [
          "aleph",
          "aleph"
        ],
        [
          "ℵ",
          "ℵ"
        ]
      ],
      "raw_glosses": [
        "(set theory) Any cardinal or ordinal number which is larger than any finite, i.e. natural number; often represented by the Hebrew letter aleph (ℵ) with a subscript 0, 1, etc."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "set-theory"
      ],
      "wikipedia": [
        "transfinite number"
      ]
    }
  ],
  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "number larger than any finite number",
      "tags": [
        "masculine"
      ],
      "word": "numero transfinito"
    },
    {
      "code": "pt",
      "lang": "Portuguese",
      "sense": "number larger than any finite number",
      "tags": [
        "masculine"
      ],
      "word": "número transfinito"
    }
  ],
  "word": "transfinite number"
}