Why do we consider enumeration up to $\omega$ instead of leaving it to as many ordinal numbers?
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 $&lt;$ 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.