Dedekindfinite

Dedekindfinite, also known as Dedekind-finite, refers to a property of sets in mathematics, particularly in the context of set theory and the foundations of mathematics. A set is said to be Dedekind-finite if it cannot be put into a one-to-one correspondence with any of its proper subsets. This concept was introduced by Richard Dedekind in the late 19th century as part of his work on the foundations of arithmetic.

The term "Dedekind-finite" is often contrasted with "Dedekind-infinite," which refers to sets that can be put into a one-to-one correspondence with a proper subset of themselves. For example, the set of natural numbers is Dedekind-infinite because it can be mapped onto its proper subset of even numbers.

Dedekind-finite sets have several important properties. One of the most notable is that any Dedekind-finite set is necessarily finite in the classical sense. This is because a set that is not finite must be Dedekind-infinite. Conversely, any finite set is Dedekind-finite.

The concept of Dedekind-finiteness is significant in the study of infinite sets and the development of axiomatic set theory. It provides a way to distinguish between finite and infinite sets without relying on the notion of cardinality, which is a more advanced concept. Dedekind-finite sets are also relevant in the context of the axiom of choice and the well-ordering principle, as they help to clarify the differences between various types of infinite sets.