| Information | |
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| has gloss | eng: In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation :\varepsilon = \omega^\varepsilon, in which ω is the smallest transfinite ordinal. Any solution to this equation has Cantor normal form \varepsilon = \omega^\varepsilon}. |
| lexicalization | eng: ε₀ |
| instance of | (noun) the number designating place in an ordered sequence ordinal number, no., ordinal |
| Meaning | |
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| Italian | |
| has gloss | ita: In matematica, ε0 è il più piccolo numero transfinito che non può essere raggiunto partendo da 0 ed eseguendo un numero finito di operazioni di addizioni di numeri ordinali più l'operazione α→ωα, dove ω è il numero ordinale transfinito più piccolo. |
| lexicalization | ita: Epsilon zero |
| Japanese | |
| has gloss | jpn: ε0（えぷしろん・のーと (Epsilon nought)、または、えぷしろん・ぜろ (Epsilon zero)）は、数学における超限順序数の一つ。ω（最小の超限順序数）から有限回の加算・乗算・冪乗では到達できない最小の超限順序数として定義される。従って極限順序数でもある。 :\epsilon_0 = \omega^\omega^\omega^\cdots}}}. で表すと次の通り。 :\epsilon_0 = \omega^\epsilon_0}. ただしこれは十分な定義ではない。α = ωα であるような γ 番目（0から数え始める）の順序数 α を εγ と書き、これらをエプシロン数と呼ぶ。この中で最小のものが ε0 である。 |
| lexicalization | jpn: エプシロン・ノート |
| Polish | |
| has gloss | pol: Liczba epsilonowa - liczba porządkowa \varepsilon o tej własności, że :\varepsilon=\omega^\varepsilon. Najmniejszą liczbą epsilonową jest liczba :\varepsilon_0 = \omega^\omega^\omega^\cdots}} = \sup \ \omega, \omega^\omega}, \omega^\omega^\omega}}, \omega^\omega^\omega^\omega}}, \dots \}. Liczba \varepsilon_0 jest przeliczalna - ma ona zastosowanie w wielu dowodach pozaskończonych, na przykład w dowodzie twierdzenia Goodsteina. Kolejne liczby epsilonowe indeksujemy kolejnymi liczbami porządkowymi, na przykład: :\varepsilon_0, \varepsilon_1, \varepsilon_2, \ldots, \varepsilon_\omega, \ldots, \varepsilon_\varepsilon_0}, \ldots, \varepsilon_\omega_1}, \ldots . :\varepsilon_1 = \sup\\varepsilon_0 + 1, \varepsilon_0 \cdot \omega, \varepsilon_0}^\omega, \varepsilon_0}^ |
| lexicalization | pol: Liczba epsilonowa |
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