Editing Well-ordering theorem (section)

==Proof of axiom of choice==
The axiom of choice can be proven from the well-ordering theorem as follows.

:To make a choice function for a collection of non-empty sets, <math>E</math>, take the union of the sets in <math>E</math> and call it <math>X</math>. There exists a well-ordering of <math>X</math>; let <math>R</math> be such an ordering. The function that to each set <math>S</math> of <math>E</math> associates the smallest element of <math>S</math>, as ordered by (the restriction to <math>S</math> of) <math>R</math>, is a choice function for the collection <math>E</math>.

An essential point of this proof is that it involves only a single arbitrary choice, that of <math>R</math>; applying the well-ordering theorem to each member <math>S</math> of <math>E</math> separately would not work, since the theorem only asserts the existence of a well-ordering, and choosing for each <math>S</math> a well-ordering would require just as many choices as simply choosing an element from each <math>S</math>. Particularly, if <math>E</math> contains uncountably many sets, making all uncountably many choices is not allowed under the axioms of Zermelo-Fraenkel set theory without the axiom of choice.