Editing Dedekind-infinite set (section)

==Comparison with the usual definition of infinite set==
This definition of "[[infinite set]]" should be compared with the usual definition: a set ''A'' is [[finite set|infinite]] when it cannot be put in bijection with a finite [[ordinal number|ordinal]], namely a set of the form {{nowrap|{0, 1, 2, ..., ''n''&minus;1}{{null}}}} for some natural number ''n'' – an infinite set is one that is literally "not finite", in the sense of bijection.

During the latter half of the 19th century, most [[mathematician]]s simply assumed that a set is infinite [[iff|if and only if]] it is Dedekind-infinite. However, this equivalence cannot be proved with the [[axiomatic set theory|axioms]] of [[Zermelo–Fraenkel set theory]] without the [[axiom of choice]] (AC) (usually denoted "'''ZF'''"). The full strength of AC is not needed to prove the equivalence; in fact, the equivalence of the two definitions is [[strictly]] weaker than the [[axiom of countable choice]] (CC). (See the references below.)