Editing Well-ordering theorem (section)

==Proof from axiom of choice ==

The well-ordering theorem follows from the axiom of choice as follows.<ref>{{Cite book |last=Jech |first=Thomas |title=Set Theory |publisher=[[Springer Publishing|Springer]] |year=2002 |isbn=978-3-540-44085-7 |pages=48|edition=Third Millennium }}</ref><blockquote>Let the set we are trying to well-order be <math>A</math>, and let <math>f</math> be a choice function for the family of non-empty subsets of <math>A</math>. For every [[ordinal number|ordinal]] <math>\alpha</math>, define an element <math>a_\alpha</math> that is in <math>A</math> by setting <math>a_\alpha\ =\ f(A\smallsetminus\{a_\xi\mid\xi<\alpha\})</math> if this complement <math>A\smallsetminus\{a_\xi\mid\xi<\alpha\}</math> is nonempty, or leaves <math>a_\alpha</math> undefined if it is. That is, <math>a_\alpha</math> is chosen from the set of elements of <math>A</math> that have not yet been assigned a place in the ordering (or undefined if the entirety of <math>A</math> has been successfully enumerated). Then the order <math><</math> on <math>A</math> defined by <math>a_\alpha < a_\beta</math> if and only if <math>\alpha<\beta</math> (in the usual well-order of the ordinals) is a well-order of <math>A</math> as desired, of order type <math>\sup\{\alpha \mid a_\alpha\text{ is defined}\}+1</math>.</blockquote>