V2EX limit ordinal

Limit Ordinal

Definition|释义

limit ordinal(极限序数):在集合论中,指既不是 0 也不是后继序数的序数。等价地说,一个序数 \(\lambda\) 是极限序数,当且仅当它非零且对任意 \(\alpha<\lambda\),都有 \(\alpha+1<\lambda\);直观上它“没有紧挨着的前一个序数”。常见例子:\(\omega\)(最小的无限序数)是极限序数。
(注:在不同语境下“limit”也可用于其他数学对象,但这里主要指序数的用法。)

Pronunciation|发音(IPA)

/lmt rdnl/

Examples|例句

A limit ordinal is not a successor ordinal.
极限序数不是后继序数。

In transfinite induction, the limit ordinal case is handled differently from the successor case.
在超限归纳中,极限序数情形的处理方式不同于后继情形。

Etymology|词源

limit 来自拉丁语 *limes/limit-*,原意与“边界、界限”相关,引申为“极限”。ordinal 来自拉丁语 ordinalis,意为“表示顺序的”。合在一起,limit ordinal 字面即“极限的顺序数”,在集合论里专指“非后继、处在‘极限位置’的序数”。

Related Words|相关词汇

Literary Works|文学/著作中的用例

  • Set Theory Thomas Jech(常在序数章节中系统使用“limit ordinal”区分后继与极限情形)
  • Set Theory: An Introduction to Independence Proofs Kenneth Kunen
  • Elements of Set Theory Herbert B. Enderton
  • Naive Set Theory Paul R. Halmos(入门性地介绍序数与相关术语)
  • Introduction to Set Theory Karel Hrbacek & Thomas Jech
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