An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and states
that, given any set of mutually disjoint
nonempty sets, there exists at least one set
that contains exactly one element in common with each
of the nonempty sets. The axiom of choice is related to the
first of Hilbert's problems.
In Zermelo-Fraenkel set theory (in the form omitting the axiom of choice), Zorn's lemma,
the trichotomy law, and the well
ordering principle are equivalent to the axiom of choice (Mendelson 1997, p. 275).
In contexts sensitive to the axiom of choice, the notation "ZF" is often
used to denote Zermelo-Fraenkel without the axiom of choice, while "ZFC"
is used if the axiom of choice is included.