Editing Choice function (section)

== History and importance ==
[[Ernst Zermelo]] (1904) introduced choice functions as well as the [[axiom of choice]] (AC) and proved the [[well-ordering theorem]],<ref name="Zermelo, 1904">{{cite journal| first=Ernst| last=Zermelo| year=1904| title=Beweis, dass jede Menge wohlgeordnet werden kann| journal=Mathematische Annalen| volume=59| issue=4| pages=514–16| doi=10.1007/BF01445300| url=http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=28526}}</ref> which states that every set can be [[well ordering|well-ordered]]. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the [[axiom of countable choice]] (AC<sub>ω</sub>) states that every [[countable set]] of nonempty sets has a choice function. However, in the absence of either AC or AC<sub>ω</sub>, some sets can still be shown to have a choice function.

*If <math>X</math> is a [[finite set|finite]] set of nonempty sets, then one can construct a choice function for <math>X</math> by picking one element from each member of <math>X.</math> This requires only finitely many choices, so neither AC or AC<sub>ω</sub> is needed.
*If every member of <math>X</math> is a nonempty set, and the [[union (set theory)|union]] <math>\bigcup X</math> is well-ordered, then one may choose the least element of each member of <math>X</math>. In this case, it was possible to simultaneously well-order every member of <math>X</math> by making just one choice of a well-order of the union, so neither AC nor AC<sub>ω</sub> was needed. (This example shows that the well-ordering theorem implies AC. The [[Converse (logic)|converse]] is also true, but less trivial.)