Dedekind-infinite set

In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.

Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentieth century Zermelo–Fraenkel set theory, today the most commonly used form of axiomatic set theory, was proposed as an axiomatic system to formulate a theory of sets free of paradoxes such as Russell's paradox. Using the axioms of Zermelo–Fraenkel set theory with the originally highly controversial axiom of choice included (ZFC) one can show that a set is Dedekind-finite if and only if it is finite in the sense of having a finite number of elements. However, there exists a model of Zermelo–Fraenkel set theory without the axiom of choice (ZF) in which there exists an infinite, Dedekind-finite set, showing that the axioms of ZF are not strong enough to prove that every set that is Dedekind-finite has a finite number of elements. There are definitions of finiteness and infiniteness of sets besides the one given by Dedekind that do not depend on the axiom of choice.

...
Wikipedia