"Hartogs number" meaning in All languages combined

{
  "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"
        },
        {
          "kind": "other",
          "name": "Entries with translation boxes",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Set theory",
          "orig": "en:Set theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "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": "quote"
        }
      ],
      "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 eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "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": "quote"
        }
      ],
      "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"
}