In set theory a set S is Dedekind-infinite if there is a bijective function from S onto some proper subset of S, or equivalently if there is an injective function 
from the natural numbers into S. In the absence of choice, Dedekind-infinite is a stronger condition than merely infinite, where an infinite set is defined as one which does not have a bijective mapping onto a finite set--in other words, is not a finite set. Given the axiom of choice, a set is infinite iff it is Dedekind-infinite, but without choice it is consistent that a set could be infinite but not Dedekind-infinite. This can be taken as an argument in favor of the axiom of choice.
Named after the German mathematician Richard Dedekind.
