The Independence of the Axiom of Choice

Ever since its introduction into mathematics the Axiom of Choice has been regarded as a principle which in some sense is not as intuitively
obvious as the other axioms of set theory. Many mathematicians have gone so far as to reject its use altogether, and in some cases great pains
have been taken in order to avoid using it. In his fundamental papers, Godel, showed that if one is willing to accept the fact that no
contradiction can be obtained using the other axioms, then no contradiction can be obtained by using the Axiom of Choice. This result is sometimes
referred to as the Consistency of the Axiom of Choice, though more correctly it should be called the relative consistency since the consistency
of set theory itself cannot be proved without appealing to more powerful mathematical ideas than set theory itself. Though quite complicated,
Godel' s proof of the relative consistency can be written down in elementary number theory.