Full Idea
Axiom of Choice: For every set we can provide a mechanism for choosing one member of any non-empty subset of the set.
Gist of Idea
Choice: For every set a mechanism will choose one member of any non-empty subset
Source
Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15)
Book Reference
Clegg,Brian: 'Infinity' [Robinson 2003], p.206
A Reaction
This axiom is unusual because it makes the bold claim that such a 'mechanism' can always be found. Cohen showed that this axiom is separate. The tricky bit is choosing from an infinite subset.