Thread: Axiom of Choice

May someone explain the Axiom of Choice? I understand it as the only axiom. From which all other theorems are deduced.

You may already know this, as it is just a textbook definition, but here it is nevertheless:

For each surjective function f:X-->Y there is a function g: Y--> X such that for all y (which are elements of Y), f(g(y))=y.

With the aid of the axiom of choice, one can show that,

given two sets x and y, one of the following two statements holds:
(i) There is an injective function f:x-->y
(ii) there is an injective function f: y-->x

In other words, two sets can always be compared "in size," though this is not a simple matter.

I am guessing that there must exists a surjective map fact (I think it is called Trichtonomy theorem for sets) is fundamental in set theory?