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The axiom of choice is a somewhat controversial axiom in set theory, which is included in the axiomatic system of [[ZFC]]. It asserts that every family \(X\) of nonempty sets has a choice function, i.e. a map \(f: X \to \bigcup X\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). Essentially, for a collection of [[Set|sets]], it is possible to find a function which chooses one element from each "bag" in this collection of bags. In many cases, the axiom of choice is trivial and intuitive. For example, one can see that the axiom of choice is equivalent to the assertion that the [[Cartesian product]] of any collection of nonempty sets is nonempty. Note that the assertion that the Cartesian product of finitely many nonempty sets is nonempty is obvious, but it's possible to define Cartesian product of infinitely many sets. Despite these simple characteristics, the axiom of choice is not a theorem of ZF, and it has some consequences that may be counterintuitive.
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