艾禮富數

维基百科，自由的百科全书

艾禮富數正在翻译。欢迎您积极翻译与修订。
目前已翻译15%，原文在en:Aleph number。

在集合論這一數學分支裡，艾禮富數是一連串用來表示無限集合的勢(大小)的數。其標記符號為希伯來字母 $\aleph$ 。

自然數的勢標記為 $\aleph_0$ ，下一個較大的勢為 $\aleph_1$ ，再下一個是 $\aleph_2$ ，以此類推。一直繼續下來，便可以對任一序數 α 定義一個基數 $\aleph_\alpha$ ，下面將會詳細說明。

這一概念來自於格奧爾格·康托爾，他定義了勢並了解無限集合是可以有不同的勢的。

艾禮富數和一般在代數與微積分中出現的無限 (∞) 不同。艾禮富數會量測集合的大小，而無限只是定義成實數線上的最大的極限或擴展的實數軸上的端點。當某些艾禮富數會大於另一些艾禮富數時，無限只是無限而已。

[编辑] $\aleph_0$

Aleph-null ( $\aleph_0$ ) is by definition the cardinality of the set of all natural numbers, and (assuming, as usual, the axiom of choice) is the smallest of all infinite cardinalities. A set has cardinality $\aleph_0$ if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or "one-to-one correspondence", with the natural numbers. Such sets include the set of all prime numbers and the set of all rational numbers.

[编辑] $\aleph_1$

$\aleph_1$ is the cardinality of the set of all countable ordinal numbers, called ω₁ or Ω. Notice ω₁ is an uncountable set. This definition implies (already in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between $\aleph_0$ and $\aleph_1$ . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus $\aleph_1$ is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set Ω (the standard example of a set of size $\aleph_1$ ): any countable subset of Ω has an upper bound (with respect to the standard well-ordering of ordinals) in Ω (the proof is easy: a countable union of countable sets is countable; this is one of the most common applications of AC). This fact is analogous to the situation in $\aleph_0$ : any finite set of natural numbers (subset of ω) has a maximum which is also a natural number (has an upper bound in ω) — finite unions of finite sets are finite.

Ω is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the sigma-algebra generated by an arbitrary collection of subsets. This is harder than most explicit descriptions of "generation" in algebra (for example vector spaces, groups, etc.) because in those cases we only have to close with respect to finite operations — sums, products, and the like. The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of Ω.

[编辑] 連續統假設

The cardinality of the set of real numbers is $2^{\aleph_0}$ . It is not clear where this number fits in the aleph number hierarchy. It follows from ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that the celebrated continuum hypothesis, CH, is equivalent to the identity

$2^{\aleph_0}=\aleph_1.$

CH is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system. That it is consistent with ZFC was demonstrated by Kurt Gödel in 1940; that it is independent of ZFC was demonstrated by Paul Cohen in 1963.

[编辑] $\aleph_\omega$

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number $\aleph_\omega$ is the smallest upper bound of

$\left\{\,\aleph_n : n\in\left\{\,0,1,2,\dots\,\right\}\,\right\}.$

Aleph-ω is the first uncountable cardinal number that can be demonstrated within Zermelo-Fraenkel set theory not to be equal to the cardinality of the set of all real numbers; for any positive integer n we can consistently assume that $2^{\aleph_0} = \aleph_n$ , and moreover it is possible to assume $2^{\aleph_0}$ is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality $\aleph_0$ , meaning there is an unbounded function from $\aleph_0$ to it.

[编辑] 對普遍α的 $\aleph_\alpha$

To define aleph-α for arbitrary ordinal number α, we need the successor cardinal operation, which assigns to any cardinal number ρ the next bigger cardinal $ρ +$ .

We can then define the aleph numbers as follows

$\aleph_{0} = \omega$

$\aleph_{\alpha+1} = \aleph_{\alpha}^+$

and for λ, an infinite limit ordinal,

$\aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta.$

[编辑] Fixed points of aleph

For any ordinal α we have

$\alpha\leq\aleph_\alpha.$

In many cases $\aleph_{\alpha}$ is strictly greater than α. For example, for any successor ordinal α this holds. There are, however, some limit ordinals which are fixed points of the aleph function. The first such is the limit of the sequence

$\aleph_0, \aleph_{\aleph_0}, \aleph_{\aleph_{\aleph_0}},\ldots$

Any inaccessible cardinal is a fixed point of the aleph function as well.

[编辑] 大眾文化

In the Futurama episode Raging Bender, the movie theater's name was ( $\aleph_0$ )PLEX. This is an obvious continuation of The Simpsons movie theater joke, based on googol, or more specifically googolplex -- the Googleplex. This also seems to be a reference to the mathematical curiosity known as Hilbert's Hotel, or Hilbert's paradox of the Grand Hotel.

[编辑] 另見

Georg Cantor
基數
不可數集合
連續統假設
Beth number

[编辑] 外部連結

埃立克·魏爾斯史甸在MathWorld中所描述之Aleph-0。
Template:PlanetMath

来自“http://zh.wikipedia.org../../../%E8%89%BE/%E7%A6%AE/%E5%AF%8C/%E8%89%BE%E7%A6%AE%E5%AF%8C%E6%95%B8.html”

页面分类: 翻譯請求 | 集合论 | 基数