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## Main Question or Discussion Point

We all know that the axiom of choice is equivalent to the existence of a well-ordering for any set. And, this of course implies that [tex]\mathbb{R}[/tex] can be well-ordered, in particular. However, how do we know that the axiom of choice is actually

*needed*in the case of the reals? That is, if we remove the axiom of choice, do the reals become a set that cannot be well-ordered? Furthermore, is the axiom of choice needed for every uncountable set?