At first it seems obvious, but the more you think about it, the
stranger the deductions from this axiom seem to become; in the end you
cease to understand what is meant by it. (Bertrand Russell about the
Axiom of Choice)
[Naum Yakovlevich Vilenkin: "In search of infinity", Birkhäuser,
Boston (1995) p. 123]
The axiom of choice is obvious. But there are no uncountable sets.
Therefore the impossible task vanishes that elements must be well-
orderable without the possibility to distinguish and identify them.
It is quite possible to do a great deal of mathematics without the axiom of choice, but it is usually much more difficult, and the odd results that that axiom sometimes alows one to produce do not seem to effect anything essential.