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Axiom of choice

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In mathematics, the Axiom of Choice or AC is a fundamental principle in set theory which states that it is possible to choose an element out of each of infinitely many sets simultaneously. The validity of the axiom is not universally accepted among mathematicians and Kurt Gödel showed that it was independent of the other axioms of set theory.

The axiom states that if is a family of non-empty sets, there is a choice function such that for each we have : that is, "chooses" an element of each member of the family .

A closely related formulation of the axiom is that the Cartesian product of any family of non-empty sets is again non-empty.

Equivalent formulations

There are a number of statements equivalent to the Axiom of Choice.