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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Epsilon numbers (mathematics) ： ウィキペディア英語版 Epsilon numbers (mathematics) 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 :$\backslash varepsilon\; =\; \backslash omega^\backslash varepsilon,\; \backslash ,$ in which ω is the smallest infinite ordinal. Any solution to this equation has Cantor normal form $\backslash varepsilon\; =\; \backslash omega^$. The least such ordinal is ''ε''0 (pronounced epsilon nought or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: :$\backslash varepsilon\_0\; =\; \backslash omega^\}\; =\; \backslash sup\; \backslash \},\; \backslash omega^$ Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in $\backslash varepsilon\_1,\; \backslash varepsilon\_2,\backslash ldots,\backslash varepsilon\_\backslash omega,\; \backslash varepsilon\_,\; \backslash ldots,\; \backslash varepsilon\_,\; \backslash ldots,\; \backslash varepsilon\_,\; \backslash ldots,\; \backslash varepsilon\_\}\},\backslash ldots$. 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 is very important 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'' → ω''x''. defined gamma numbers (see additively indecomposable ordinal) to be numbers γ>0 such that α+γ=γ whenever α<γ, and delta numbers (see additively indecomposable ordinal#Multiplicatively indecomposable) 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. == Ordinal ε numbers == The standard definition of ordinal exponentiation with base α is: *$\backslash alpha^0\; =\; 1\; \backslash ,,$ *$\backslash alpha^\; =\; \backslash alpha^\backslash beta\; \backslash cdot\; \backslash alpha\; \backslash ,,$ *$\backslash alpha^\backslash kappa\; =\; \backslash limsup\_\; \backslash ,\; \backslash alpha^\backslash lambda$ for limit $\backslash kappa$. From this definition, it follows that for any fixed ordinal ''α'' > 1, the mapping $\backslash beta\; \backslash mapsto\; \backslash alpha^\backslash beta$ is a normal function, so it has arbitrarily large fixed points by the fixed-point lemma for normal functions. When $\backslash alpha\; =\; \backslash omega$, these fixed points are precisely the ordinal epsilon numbers. The smallest of these, ε₀, is the supremum of the sequence :$0,\; \backslash omega^0\; =\; 1,\; \backslash omega^1\; =\; \backslash omega,\; \backslash omega^\backslash omega,\; \backslash omega^,\; \backslash ldots,\; \backslash omega\; \backslash uparrow\; \backslash uparrow\; k,\; \backslash ldots$ in which every element is the image of its predecessor under the mapping $\backslash beta\; \backslash mapsto\; \backslash omega^\backslash beta$. (The general term is given using Knuth's up-arrow notation; the $\backslash uparrow\; \backslash uparrow$ operator is equivalent to tetration.) Just as ωω is defined as the supremum of for natural numbers ''k'', the smallest ordinal epsilon number ε₀ may also be denoted $\backslash omega\; \backslash uparrow\; \backslash uparrow\; \backslash omega$; this notation is much less common than ε₀. The next epsilon number after $\backslash varepsilon\_0$ is :$\backslash varepsilon\_1\; =\; \backslash sup\backslash \},\; \backslash omega^,\; \backslash dots\backslash \},$ in which the sequence is again constructed by repeated base ω exponentiation but starts at $\backslash varepsilon\_0\; +\; 1$ instead of at 0. Notice :$\backslash omega^\; =\; \backslash omega^\; \backslash cdot\; \backslash omega^1\; =\; \backslash varepsilon\_0\; \backslash cdot\; \backslash omega\; \backslash ,,$ :$\backslash omega^\; =\; ^\backslash omega\; =\; \backslash varepsilon\_0^\backslash omega\; \backslash ,,$ :$\backslash omega^\; =\; \backslash omega^,\; \backslash dots\backslash \}\; =\; \backslash sup\backslash \},\; \backslash dots\backslash \}$ An epsilon number indexed by a limit ordinal α is constructed differently. The number $\backslash varepsilon\_\backslash alpha$ is the supremum of the set of epsilon numbers $\backslash$. The first such number is $\backslash varepsilon\_\backslash omega$. Whether or not the index α is a limit ordinal, $\backslash varepsilon\_\backslash alpha$ is a fixed point not only of base ω exponentiation but also of base γ exponentiation for all ordinals . Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinal numbers themselves. For any ordinal number $\backslash alpha$, $\backslash varepsilon\_\backslash alpha$ is the least epsilon number (fixed point of the exponential map) not already in the set $\backslash$. It might appear that this is the non-constructive equivalent of the constructive definition using iterated exponentiation; but the two definitions are equally non-constructive at steps indexed by limit ordinals, which represent transfinite recursion of a higher order than taking the supremum of an exponential series. The following facts about epsilon numbers are very straightforward to prove: * Although it is quite a large number, $\backslash varepsilon\_0$ is still countable, being a countable union of countable ordinals; in fact, $\backslash varepsilon\_\backslash alpha$ is countable if and only if $\backslash alpha$ is countable. * The union (or supremum) of any nonempty set of epsilon numbers is an epsilon number; so for instance ::$\backslash varepsilon\_\backslash omega\; =\; \backslash sup\backslash$ : is an epsilon number. Thus, the mapping $n\; \backslash mapsto\; \backslash varepsilon\_n$ is a normal function. * Every uncountable cardinal number is an epsilon number. ::$1\; \backslash leq\; \backslash alpha\; \backslash rightarrow\; \backslash varepsilon\_\; \backslash ,.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Epsilon numbers (mathematics)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.