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Math Help - Axiom of Choice

  1. #1
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    Axiom of Choice

    May someone explain the Axiom of Choice? I understand it as the only axiom. From which all other theorems are deduced.
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  2. #2
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    The 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.
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  3. #3
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    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?
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