Editing Zermelo–Fraenkel set theory (section)

=== Axiom of well-ordering (choice) ===
{{Main|Axiom of choice|Well-ordering theorem|Zorn's lemma}}

The last axiom, commonly known as the [[axiom of choice]], is presented here as a property about [[well-order]]s, as in {{harvtxt|Kunen|1980}}.
For any set <math>X</math>, there exists a [[binary relation]] <math>R</math> which [[well-order]]s <math>X</math>.  This means <math>R</math> is a [[linear order]] on <math>X</math> such that every nonempty [[subset]] of <math>X</math> has a [[least element]] under the order <math>R</math>.

<div style="margin-left:1.6em;"><math>\forall X \exists R ( R \;\mbox{well-orders}\; X).</math></div>

Given axioms ''1''&thinsp;–&thinsp;''8'', many statements are {{not a typo|provably}} equivalent to axiom ''9''. The most common of these goes as follows. Let <math>X</math> be a set whose members are all nonempty. Then there exists a function <math>f</math> from <math>X</math> to the union of the members of <math>X</math>, called a "[[choice function]]", such that for all <math>Y\in X</math> one has <math>f(Y)\in Y</math>. A third version of the axiom, also equivalent, is [[Zorn's lemma]].

Since the existence of a choice function when <math>X</math> is a [[finite set]] is easily proved from axioms ''1–8'', AC only matters for certain [[infinite set]]s. AC is characterized as [[constructive mathematics|nonconstructive]] because it asserts the existence of a choice function but says nothing about how this choice function is to be "constructed".