Dedekindinfinite

Dedekind infinite refers to a property of sets in set theory that distinguishes certain infinite sets from finite ones. A set \(S\) is called Dedekind infinite if there exists a proper subset \(T \subset S\) that is in bijective correspondence with \(S\). In other words, one can pair each element of \(S\) with a distinct element of \(S\) leaving some elements unused, yet still cover the entire set. This concept was introduced by German mathematician Richard Dedekind in the late 19th century to formalize the idea of infinite collections that can be mapped onto themselves without being all of themselves.

The definition contrasts with finite sets, where no proper subset can be equipotent with the whole. For infinite sets, Dedekind’s criterion provides an algebraic condition equivalent to the existence of an infinite number of elements in many contexts. However, the equivalence between Dedekind infinity and standard infinity becomes subtle in the absence of the Axiom of Choice. With that axiom, every infinite set is Dedekind infinite; without it, there are models of set theory where an infinite set can fail to be Dedekind infinite, known as amorphous sets.

Dedekind infinite sets include all countably infinite sets such as the natural numbers, integers, and rational numbers, as well as many uncountable sets like the real numbers. The property plays a role in the Cantor–Bernstein–Schroeder theorem, which states that if two sets can be embedded into each other, they are in bijection. Dedekind infinity also underlies numerous constructions in analysis and topology, particularly in arguments that rely on the ability to choose countably many elements from a set.