為什麼我們考慮不超過$ \ omega $的枚舉而不是將其保留為盡可能多的序數?


1

幾分鐘前,我問了一個關於"證明"的問題,該證明證明了

我被告知要研究序數,在越過 $ \ omega $ 之後,我們不再考慮將其作為枚舉。

為什麼會這樣?如果不設置此限制,會帶來負面影響嗎?

編輯:我一直認為 $ \ mathbb {N} $ 是"計數數字"-但是...當我們跨過像 $ \ omega $ $ \ omega + 1 $ 等,我們是否仍在有效地計數?

2

A set $S$ is enumerable (or, countable) if we can enumerate it: $$ S = \{s_1,s_2,s_3,\ldots\} $$ In other words, there is a mapping from $\mathbb{N}$ onto $S$.

Cantor showed that $\mathbb{R}$ isn't enumerable.

We can consider more relaxed notions of enumeration. For example, a set $S$ is well-orderable if there is a linear order $<$ on $S$ such that any non-empty subset of $S$ has a minimum. This encompasses your examples, and much more.

The axiom of choice is equivalent to the well-ordering principle, which states that every set can be well-ordered. Hence if you assume the axiom of choice, every set can be enumerated in this sense.