Formally, a set S is called finite if there exists a bijection
$${\displaystyle f\colon S\to n}$$
for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). … See more

Any proper subset of a finite set S is finite and has fewer elements than S itself. As a consequence, there cannot exist a bijection between a finite set S and a proper subset of S. Any set with … See more

Georg Cantor initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Thus the distinction between the finite … See more

• FinSet
• Ordinal number
• Peano arithmetic See more

In Zermelo–Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent:
1. S is a finite set. That is, S can be placed into a one-to-one … See more

In contexts where the notion of natural number sits logically prior to any notion of set, one can define a set S as finite if S admits a bijection to some set of natural numbers of the form $${\displaystyle \{x\,|\,x<n\}}$$. Mathematicians more typically choose to … See more

• Barile, Margherita. "Finite Set". MathWorld. See more