# Why is choosing elements in equivalence classes not a choice?

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??

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 I think I will revise my answer on that thread to make this construction clearer. – Asaf Karagila Jul 22 '12 at 12:53 @Asaf Thank you. Excuse me keep asking different questions, but it's hard for me to understand the part deriving a contradiction too.(that is, i know that $\alpha$ is a supremum but why ordinals less than $\alpha$ is isomorphic to some order relation that well orders $\omega$? And why that $\alpha + 1$ is countable is a contradiction?) – Katlus Jul 22 '12 at 13:09 @Asaf Or did you mean this? Let $X$={$\beta \in OR$|$\alpha$ is equipotent with $\omega$}. Let $\gamma$ be the least element of $X$. Is this $\gamma$ same as $\alpha$ in your argument? – Katlus Jul 22 '12 at 13:24 I dunno why edit button is unavailable to me now.. I meant 'union' not 'least' above. – Katlus Jul 22 '12 at 13:26