"epsilon number" meaning in English

{
  "etymology_text": "From the Greek letter ε (“epsilon”), used to denote the numbers.",
  "forms": [
    {
      "form": "epsilon numbers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "epsilon number (plural epsilon 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": [],
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        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
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        },
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          "kind": "other",
          "name": "Pages with entries",
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        },
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          "kind": "other",
          "name": "Terms with Italian translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "langcode": "en",
          "name": "Infinity",
          "orig": "en:Infinity",
          "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": [
        {
          "ref": "1977, Herbert B. Enderton, Elements of Set Theory, Elsevier (Academic Press), page 240:",
          "text": "More generally, the epsilon numbers are the ordinals #x5C;varepsilon for which #x5C;varepsilon#x3D;#x5C;omega#x5C;varepsilon. The smallest epsilon number is #x5C;varepsilon#x5F;0. It is a countable ordinal, being the countable union of countable sets. By the Veblen fixed-point theorem, the class of epsilon numbers is unbounded.",
          "type": "quote"
        },
        {
          "ref": "1981, The Journal of Symbolic Logic, Association for Symbolic Logic, page 17:",
          "text": "We show that the associated ordinals are the #x5C;varepsilon#x5F;0th epsilon number and the first #x5C;varepsilon#x5F;0-critical number, respectively.",
          "type": "quote"
        },
        {
          "text": "2014, Charles C. Pinter, A Book of Set Theory, 2014, Dover, [Revision of 1971 Addison-Wesley edition], page 203,\nThus there is at least one epsilon number, namely ε₀; we can easily show, in fact, that ε₀ is the least epsilon number."
        }
      ],
      "glosses": [
        "Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ."
      ],
      "id": "en-epsilon_number-en-noun-emOOIA2T",
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "transfinite",
          "transfinite"
        ],
        [
          "ordinal number",
          "ordinal number"
        ],
        [
          "ω",
          "ω"
        ],
        [
          "surreal number",
          "surreal number"
        ],
        [
          "fixed point",
          "fixed point"
        ],
        [
          "exponential",
          "exponential"
        ],
        [
          "map",
          "map"
        ]
      ],
      "raw_glosses": [
        "(set theory) Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ."
      ],
      "related": [
        {
          "word": "delta number"
        },
        {
          "word": "gamma number"
        },
        {
          "word": "epsilon-induction"
        }
      ],
      "topics": [
        "mathematics",
        "sciences",
        "set-theory"
      ],
      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "type of transfinite or surreal number",
          "tags": [
            "masculine"
          ],
          "word": "numero epsilon"
        }
      ],
      "wikipedia": [
        "Epsilon numbers (mathematics)"
      ]
    }
  ],
  "word": "epsilon number"
}

{
  "etymology_text": "From the Greek letter ε (“epsilon”), used to denote the numbers.",
  "forms": [
    {
      "form": "epsilon numbers",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "epsilon number (plural epsilon numbers)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "delta number"
    },
    {
      "word": "gamma number"
    },
    {
      "word": "epsilon-induction"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Quotation templates to be cleaned",
        "Terms with Italian translations",
        "en:Infinity",
        "en:Set theory"
      ],
      "examples": [
        {
          "ref": "1977, Herbert B. Enderton, Elements of Set Theory, Elsevier (Academic Press), page 240:",
          "text": "More generally, the epsilon numbers are the ordinals #x5C;varepsilon for which #x5C;varepsilon#x3D;#x5C;omega#x5C;varepsilon. The smallest epsilon number is #x5C;varepsilon#x5F;0. It is a countable ordinal, being the countable union of countable sets. By the Veblen fixed-point theorem, the class of epsilon numbers is unbounded.",
          "type": "quote"
        },
        {
          "ref": "1981, The Journal of Symbolic Logic, Association for Symbolic Logic, page 17:",
          "text": "We show that the associated ordinals are the #x5C;varepsilon#x5F;0th epsilon number and the first #x5C;varepsilon#x5F;0-critical number, respectively.",
          "type": "quote"
        },
        {
          "text": "2014, Charles C. Pinter, A Book of Set Theory, 2014, Dover, [Revision of 1971 Addison-Wesley edition], page 203,\nThus there is at least one epsilon number, namely ε₀; we can easily show, in fact, that ε₀ is the least epsilon number."
        }
      ],
      "glosses": [
        "Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ."
      ],
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "transfinite",
          "transfinite"
        ],
        [
          "ordinal number",
          "ordinal number"
        ],
        [
          "ω",
          "ω"
        ],
        [
          "surreal number",
          "surreal number"
        ],
        [
          "fixed point",
          "fixed point"
        ],
        [
          "exponential",
          "exponential"
        ],
        [
          "map",
          "map"
        ]
      ],
      "raw_glosses": [
        "(set theory) Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ."
      ],
      "topics": [
        "mathematics",
        "sciences",
        "set-theory"
      ],
      "wikipedia": [
        "Epsilon numbers (mathematics)"
      ]
    }
  ],
  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "type of transfinite or surreal number",
      "tags": [
        "masculine"
      ],
      "word": "numero epsilon"
    }
  ],
  "word": "epsilon number"
}