# 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？

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.