# Dedekind-infinite

A set $A$ is said to be 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 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.

Title Dedekind-infinite Dedekindinfinite 2013-03-22 12:05:25 2013-03-22 12:05:25 yark (2760) yark (2760) 11 yark (2760) Definition msc 03E99 Dedekind infinite Cardinality Dedekind-finite Dedekind finite