"Hartogs number" meaning in English

{
  "etymology_text": "After German-Jewish mathematician Friedrich Hartogs (1874–1943).",
  "forms": [
    {
      "form": "Hartogs numbers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Hartogs number (plural Hartogs numbers)",
      "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"
          ],
          "source": "w"
        },
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          "kind": "other",
          "name": "English entries with language name categories using raw markup",
          "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": "Set theory",
          "orig": "en:Set theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "1973 [North-Holland], Thomas J. Jech, The Axiom of Choice, 2013, Dover, page 160,\nLet p be an infinite cardinal, |X|=p and let א=א(p) be the Hartogs number of p."
        },
        {
          "text": "1995, The Bulletin of Symbolic Logic, Volume 1, Association for Symbolic Logic, page 139,\nIf the Power Set Axiom is replaced by \"א(x) is bound for every x\" where\nא(x)={a|∃f(f is one-to-one function from a into x)},\nthen the theory is denoted by ZFH (H stands for Hartogs' Number)."
        },
        {
          "ref": "2014, Barnaby Sheppard, The Logic of Infinity, Cambridge University Press, page 341",
          "text": "Since the proof of Hartogs' Theorem does not appeal to the Axiom of Choice, the Hartogs number of a set X exists whether or not X has a well-ordering.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X."
      ],
      "id": "en-Hartogs_number-en-noun-75Dj8x5W",
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "set",
          "set"
        ],
        [
          "cardinality",
          "cardinality"
        ],
        [
          "ordinal number",
          "ordinal number"
        ],
        [
          "injection",
          "injection"
        ]
      ],
      "raw_glosses": [
        "(set theory) For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X."
      ],
      "related": [
        {
          "word": "equipotent"
        },
        {
          "word": "Hartogs function"
        }
      ],
      "synonyms": [
        {
          "word": "Hartogs' number"
        }
      ],
      "topics": [
        "mathematics",
        "sciences",
        "set-theory"
      ],
      "wikipedia": [
        "Friedrich Hartogs",
        "Hartogs number"
      ]
    }
  ],
  "word": "Hartogs number"
}

{
  "etymology_text": "After German-Jewish mathematician Friedrich Hartogs (1874–1943).",
  "forms": [
    {
      "form": "Hartogs numbers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Hartogs number (plural Hartogs numbers)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "equipotent"
    },
    {
      "word": "Hartogs function"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English entries with language name categories using raw markup",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "en:Set theory"
      ],
      "examples": [
        {
          "text": "1973 [North-Holland], Thomas J. Jech, The Axiom of Choice, 2013, Dover, page 160,\nLet p be an infinite cardinal, |X|=p and let א=א(p) be the Hartogs number of p."
        },
        {
          "text": "1995, The Bulletin of Symbolic Logic, Volume 1, Association for Symbolic Logic, page 139,\nIf the Power Set Axiom is replaced by \"א(x) is bound for every x\" where\nא(x)={a|∃f(f is one-to-one function from a into x)},\nthen the theory is denoted by ZFH (H stands for Hartogs' Number)."
        },
        {
          "ref": "2014, Barnaby Sheppard, The Logic of Infinity, Cambridge University Press, page 341",
          "text": "Since the proof of Hartogs' Theorem does not appeal to the Axiom of Choice, the Hartogs number of a set X exists whether or not X has a well-ordering.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X."
      ],
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "set",
          "set"
        ],
        [
          "cardinality",
          "cardinality"
        ],
        [
          "ordinal number",
          "ordinal number"
        ],
        [
          "injection",
          "injection"
        ]
      ],
      "raw_glosses": [
        "(set theory) For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "set-theory"
      ],
      "wikipedia": [
        "Friedrich Hartogs",
        "Hartogs number"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "Hartogs' number"
    }
  ],
  "word": "Hartogs number"
}