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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 Regular cardinal ： ウィキペディア英語版 Regular cardinal In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one that cannot be broken into a smaller collection of smaller parts. If the axiom of choice holds (so that any cardinal number can be well-ordered), an infinite cardinal $\backslash kappa$ is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than $\backslash kappa$, the elements of which are cardinals less than $\backslash kappa$. (The situation is slightly more complicated in contexts where the axiom of choice might fail; in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above definition is restricted to well-orderable cardinals only.) An infinite ordinal $\backslash alpha$ is regular if and only if it is a limit ordinal which is not the limit of a set of smaller ordinals which set has order type less than $\backslash alpha$. A regular ordinal is always an initial ordinal, though some initial ordinals are not regular. Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular. __NOTOC__ == Examples == The ordinals less than $\backslash omega$ are finite. A finite sequence of finite ordinals always has a finite maximum, so $\backslash omega$ cannot be the limit of any sequence of type less than $\backslash omega$ whose elements are ordinals less than $\backslash omega$, and is therefore a regular ordinal. $\backslash aleph\_0$ (aleph-null) is a regular cardinal because its initial ordinal, $\backslash omega$, is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite. $\backslash omega+1$ is the next ordinal number greater than $\backslash omega$. It is singular, since it is not a limit ordinal. $\backslash omega+\backslash omega$ is the next limit ordinal after $\backslash omega$. It can be written as the limit of the sequence $\backslash omega$, $\backslash omega+1$, $\backslash omega+2$, $\backslash omega+3$, and so on. This sequence has order type $\backslash omega$, so $\backslash omega+\backslash omega$ is the limit of a sequence of type less than $\backslash omega+\backslash omega$ whose elements are ordinals less than $\backslash omega+\backslash omega$, therefore it is singular. $\backslash aleph\_1$ is the next cardinal number greater than $\backslash aleph\_0$, so the cardinals less than $\backslash aleph\_1$ are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So $\backslash aleph\_1$ cannot be written as the sum of a countable set of countable cardinal numbers, and is regular. $\backslash aleph\_\backslash omega$ is the next cardinal number after the sequence $\backslash aleph\_0$, $\backslash aleph\_1$, $\backslash aleph\_2$, $\backslash aleph\_3$, and so on. Its initial ordinal $\backslash omega\_\backslash omega$ is the limit of the sequence $\backslash omega$, $\backslash omega\_1$, $\backslash omega\_2$, $\backslash omega\_3$, and so on, which has order type $\backslash omega$, so $\backslash omega\_\backslash omega$ is singular, and so is $\backslash aleph\_\backslash omega$. Assuming the axiom of choice, $\backslash aleph\_\backslash omega$ is the first infinite cardinal which is singular (the first infinite ''ordinal'' which is singular is $\backslash omega+1$). Proving the existence of singular cardinals requires the axiom of replacement, and in fact the inability to prove the existence of $\backslash aleph\_\backslash omega$ in Zermelo set theory is what led Fraenkel to postulate this axiom. 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「Regular cardinal」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.