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 exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.^[1]

A simple example is ${\displaystyle \mathbb {N} }$, the set of natural numbers. From Galileo's paradox, there exists a bijection that maps every natural number n to its square n². Since the set of squares is a proper subset of ${\displaystyle \mathbb {N} }$, ${\displaystyle \mathbb {N} }$ is Dedekind-infinite.

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 usual sense. 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 is finite.^[2][1] There are definitions of finiteness and infiniteness of sets besides the one given by Dedekind that do not depend on the axiom of choice.

A vaguely related notion is that of a Dedekind-finite ring.