Axiom of choice In mathematics the axiom of choice, sometimes
called AC, is an axiom used in set theory. The axiom of choice
says that if you have a set of objects and you separate the set
into smaller sets, each containing at least one object, it is
possible to take one object out of each of these smaller sets
and make a new set. You do not always need to use the axiom of
choice to do this. You do not need to use the axiom of choice if
the starting set is finite, or if the starting set is infinite
and has a rule for how it can be divided built in. For example,
for any (infinite or finite) sets of pairs of shoes, one can
pick out the left shoe from each pair, but for an infinite
collection of pairs of socks, the axiom of choice is needed.