Dedekind-infiniteDedekindInfinite2002-01-03 16:38:142009-01-27 07:37:45DefinitionDedekind-finiteDedekind finite\usepackage{amssymb}
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A set $A$ is said to be \emph{Dedekind-infinite}
if there is an injective function $f\colon\omega\to A$,
where $\omega$ denotes the set of natural numbers.
A set that is not Dedekind-infinite is said to be \emph{Dedekind-finite}.
A Dedekind-infinite set is clearly infinite,
and in ZFC it can be shown that
a set is Dedekind-infinite if and only if it is infinite.
It is consistent with ZF that
there is an infinite set that is not Dedekind-infinite.
However, the existence of such a set requires the failure
not just of the full Axiom of Choice, but even of the Axiom of Countable Choice.