Epsilon number (original) (raw)
Liczba epsilonowa – liczba porządkowa o tej własności, że Najmniejszą liczbą epsilonową jest liczba Liczba jest przeliczalna – ma ona zastosowanie w wielu dowodach pozaskończonych, na przykład w dowodzie twierdzenia Goodsteina. Kolejne liczby epsilonowe indeksujemy kolejnymi liczbami porządkowymi, na przykład:
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| dbo:abstract | 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 in which ω is the smallest infinite ordinal. The least such ordinal is ε0 (pronounced epsilon nought or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: where sup is the supremum function, which is equivalent to set union in the case of the von Neumann representation of ordinals. Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in . The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal). The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic). Many larger epsilon numbers can be defined using the Veblen function. A more general class of epsilon numbers has been identified by John Horton Conway and Donald Knuth in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ωx. defined gamma numbers (see additively indecomposable ordinal) to be numbers γ>0 such that α+γ=γ whenever α<γ, and delta numbers (see multiplicatively indecomposable ordinals) to be numbers δ>1 such that αδ=δ whenever 0<α<δ, and epsilon numbers to be numbers ε>2 such that αε=ε whenever 1<α<ε. His gamma numbers are those of the form ωβ, and his delta numbers are those of the form ωωβ. (en) En mathématiques, les nombres epsilon sont une collection de nombres transfinis définis par la propriété d'être des points fixes d'une application exponentielle. Ils ne peuvent donc pas être atteints à partir de 0 et d'un nombre fini d'exponentiations (et d'opérations « plus faibles », comme l'addition et la multiplication). La forme de base fut introduite par Georg Cantor dans le contexte du calcul sur les ordinaux comme étant les ordinaux ε satisfaisant l'équation où ω est le plus petit ordinal infini ; une extension aux nombres surréels a été découverte par John Horton Conway. Le plus petit de ces ordinaux est ε0 (prononcé epsilon zero), « limite » (réunion) de la suite ; on a donc . (fr) 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. È dato da ovvero il limite della sequenza La sua è I numeri che hanno questa caratteristica (cioè i tali che ) sono detti numeri epsilon; il più piccolo di questi è appunto , mentre il -esimo è denotato da . L'ordinale ε0 è numerabile (esistono anche ordinali non numerabili). Questo ordinale è importante in molte dimostrazioni per induzione, in quanto in molti casi l'induzione transfinita è richiesta solamente fino a ε0 (come ad esempio nel teorema di Goodstein). È stato usato da Gerhard Gentzen per dimostrare la coerenza dell'aritmetica di Peano: insieme al secondo teorema di incompletezza di Gödel, questo dimostra che l'aritmetica di Peano non può provare la sua fondatezza (è l'ultimo ordinale con questa proprietà: per questo nell' è usata come misura della forza della teoria dell'aritmetica di Peano). Questo simbolo fu ideato dal matematico tedesco Georg Cantor. (it) Liczba epsilonowa – liczba porządkowa o tej własności, że Najmniejszą liczbą epsilonową jest liczba Liczba jest przeliczalna – ma ona zastosowanie w wielu dowodach pozaskończonych, na przykład w dowodzie twierdzenia Goodsteina. Kolejne liczby epsilonowe indeksujemy kolejnymi liczbami porządkowymi, na przykład: (pl) Числа эпсилон — ординалы, введенные немецким математиком Гергом Кантором и являющиеся неподвижными точками функции то есть удовлетворяющие равенству где — первый трансфинитный ординал. Числа эпсилон могут быть определены следующим образом (как супремумы трансфинитных последовательностей): * * * для предельного ординала Наименьший ординал, который является неподвижной точкой функции называется ординалом Кантора и обозначается как Впоследствии, в 1908 году, Освальд Веблен разработал более мощную ординальную нотацию — иерархию функций . В соответствии с нотацией Веблена . (ru) 艾普塞朗數ε乃是數學集合論中一系列的超限序數，其為指數映射的某些固定點。因此，它並不能透過較小序數有限次數的加法及乘法運算而獲得。康托爾原來引進的艾普塞朗數，乃以以下的方式定義:- ε乃是一個滿足以下式的序數，當中ω乃是最小的無限序數。 滿足上式的所有ε當中，最小的記為ε0。它可以透過以下的超限遞歸法獲得:- 其後如此類推， 更大的艾普塞朗數為。值得留意的是，ε0的基数，仍然為可數的。實際上，所有指標為可數的ε，其基數也是可數的。不可數的ε（意指滿足定義式的ε）存在，但其指標也是不可數的。 (zh) |
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| rdfs:comment | Liczba epsilonowa – liczba porządkowa o tej własności, że Najmniejszą liczbą epsilonową jest liczba Liczba jest przeliczalna – ma ona zastosowanie w wielu dowodach pozaskończonych, na przykład w dowodzie twierdzenia Goodsteina. Kolejne liczby epsilonowe indeksujemy kolejnymi liczbami porządkowymi, na przykład: (pl) Числа эпсилон — ординалы, введенные немецким математиком Гергом Кантором и являющиеся неподвижными точками функции то есть удовлетворяющие равенству где — первый трансфинитный ординал. Числа эпсилон могут быть определены следующим образом (как супремумы трансфинитных последовательностей): * * * для предельного ординала Наименьший ординал, который является неподвижной точкой функции называется ординалом Кантора и обозначается как Впоследствии, в 1908 году, Освальд Веблен разработал более мощную ординальную нотацию — иерархию функций . В соответствии с нотацией Веблена . (ru) 艾普塞朗數ε乃是數學集合論中一系列的超限序數，其為指數映射的某些固定點。因此，它並不能透過較小序數有限次數的加法及乘法運算而獲得。康托爾原來引進的艾普塞朗數，乃以以下的方式定義:- ε乃是一個滿足以下式的序數，當中ω乃是最小的無限序數。 滿足上式的所有ε當中，最小的記為ε0。它可以透過以下的超限遞歸法獲得:- 其後如此類推， 更大的艾普塞朗數為。值得留意的是，ε0的基数，仍然為可數的。實際上，所有指標為可數的ε，其基數也是可數的。不可數的ε（意指滿足定義式的ε）存在，但其指標也是不可數的。 (zh) 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 in which ω is the smallest infinite ordinal. Many larger epsilon numbers can be defined using the Veblen function. (en) En mathématiques, les nombres epsilon sont une collection de nombres transfinis définis par la propriété d'être des points fixes d'une application exponentielle. Ils ne peuvent donc pas être atteints à partir de 0 et d'un nombre fini d'exponentiations (et d'opérations « plus faibles », comme l'addition et la multiplication). La forme de base fut introduite par Georg Cantor dans le contexte du calcul sur les ordinaux comme étant les ordinaux ε satisfaisant l'équation où ω est le plus petit ordinal infini ; une extension aux nombres surréels a été découverte par John Horton Conway. (fr) 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. È dato da ovvero il limite della sequenza La sua è I numeri che hanno questa caratteristica (cioè i tali che ) sono detti numeri epsilon; il più piccolo di questi è appunto , mentre il -esimo è denotato da . L'ordinale ε0 è numerabile (esistono anche ordinali non numerabili). Questo simbolo fu ideato dal matematico tedesco Georg Cantor. (it) |
| rdfs:label | Epsilon number (en) Nombre epsilon (fr) Epsilon zero (it) Liczba epsilonowa (pl) Числа эпсилон (ru) 艾普塞朗數 (zh) |
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