An axiom in set theory: Given a collection (even an infinite collection) of non-empty sets, it is possible to construct a new set by arbitrarily choosing one element from each set.
This is a controversial axiom, though less so today than it used to be (most mathematicians accept it). Essentially the problem is: By what mathematical mechanism can an arbitrary choice be made? Accepting the axiom of choice leads to some unintuitive conclusions, particularly for constructivists, but who said this shit had to be intuitive anyway.