{c}
{title2=$\mathbb N\not\hookrightarrow X$}

A <set> $X$ is Dedekind-finite when no <injective function> $\mathbb N\to X$ exists. In classical <Zermelo–Fraenkel set theory> this is equivalent to $X$ not being in <bijection> with a proper subset: an injective non-surjective self-map generates distinct iterates from an element outside its image, and a countably infinite subset permits a shift fixing its complement. A <finite set> is Dedekind-finite. Without the <axiom of choice>, some <infinite sets> can also be Dedekind-finite.


 Dedekind-finite set (source code)