Why is choosing elements in equivalence classes not a choice?

Question

This is Asaf's answer from this link: How do we know an $ \aleph_1 $ exists at all?

I don't understand this sentence that is;

From each equivalence class choose the representative which is an ordinal (which does not require any form of choice, as the equivalence classes can be described without the axiom of choice, as well as being an ordinal). The set of representatives is a set of ordinals, we take its union.

Here's what I think this means. Please tell me I'm following this argument correctly.

Let $X$ be the class of all the well orderings of $\omega$

Let $[G]$={$F \in X$|$F$ is isomorphic with $G$} for every $G\in X$.

Then we 'choose' representatives from each $[G]$ and take a union.

I see this is definitely a choice since there might be infinitely many [G]'s. Why is this not a choice??

Answer 1

Since your choice comes from the collection of ordinals, you can specify that you want the least such ordinal that satisfies the condition. This ordinal exists and is unique, so there is no choice involved. (Of course as Asaf's answer says it turns out that there is only one ordinal that satisfies the property, but you can also make this selection without knowing that fact)

Answer 2

There is a theorem which states that every well-ordered set is isomorphic to a unique ordinal.

Since the equivalence relation is that of order isomorphism all the elements in a given isomorphism class are isomorphic to a unique ordinal. We thus have eliminated the need to choose, since there is only one possible choice.

The full point I the sentence was that the statements saying that two ordered sets are isomorphic, and that a certain set is an ordinal do not require the axiom of choice. Thus this makes sense to do these constructions in ZF.