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

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- Dec 10th 2005, 03:57 PMThePerfectHackerAxiom of Choice
May someone explain the Axiom of Choice? I understand it as the

**only**axiom. From which all other theorems are deduced. - Jan 18th 2006, 11:49 AMTexasGirlThe basic...
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. - Jan 18th 2006, 02:52 PMThePerfectHacker
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?