Dedekind-finite set ($\mathbb{N}/\hookrightarrow X$)

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.

If $X$ is a Dedekind-finite set, its set of finite repetition-free sequences is also Dedekind-finite. An injective sequence of distinct finite lists would, by countable union of explicitly ordered finite lists without choice, either produce an injection $\mathbb{N}\to X$ or use only finitely many entries. The latter possibility is impossible because a fixed finite pool supports only finitely many repetition-free lists. If $X$ is infinite, the one-entry lists also show that the resulting set is infinite.

An infinite Dedekind-finite set is an infinite set with no subset that is a countably infinite set. The name Dedekind set is sometimes used for this combination of properties. Under the axiom of choice no such set exists; results about these sets must avoid obtaining arbitrary enumerations of finite subsets by unstated choices.