V2EX cofinality

Cofinality

Definition / 释义

cofinality(上确界性/共终性;通常译作“共终数”):在集合论与序理论中,用来衡量一个序结构(尤其是序数或偏序集)“需要多长的递增链(或有向集)才能逼近到顶端”。对序数/基数来说,一个序数的共终数是能在它里面找到的、其上确界等于它的最短严格递增序列的长度(序数)。

Pronunciation / 发音(IPA)

/kofnlti/

Examples / 例句

Cofinality tells us how “approachable” an ordinal is from below.
共终数告诉我们,一个序数从下方“逼近上去”有多容易。

In set theory, the cofinality of a limit ordinal is the least order type of an unbounded increasing sequence whose supremum is that ordinal.
在集合论中,极限序数的共终数是这样一种最小序型:存在一个无界的递增序列,其上确界正好等于该序数。

Etymology / 词源

**co-**(共同)+ final(终点、末端)+ -ity(名词后缀),字面意思是“共同到达终点的性质”。在数学语境里,“共同到达”指不同的元素/序列从下方逐步逼近并“到达”同一个上界(极限/顶端),于是引申为刻画这种逼近所需“长度”的概念。

Related Words / 相关词汇

Literary Works / 文献与著作中的出现

  • Set Theory Thomas Jech
  • Set Theory: An Introduction to Independence Proofs Kenneth Kunen
  • Cardinal Arithmetic Saharon Shelah
  • Handbook of Set Theory Matthew Foreman & Akihiro Kanamori (eds.)
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