Definition
Axiom of Choice
For every family
of non-empty sets indexed by a set , there exists a choice function The function
selects one element from each set simultaneously, even when is infinite and no selection rule is specified. The axiom asserts that such a function exists; it does not provide an algorithm for finding it. Adding this axiom to Zermelo–Fraenkel set theory (ZF) gives ZFC. Assuming ZF is consistent, the axiom can neither be proved nor disproved from ZF.