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In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, $${\displaystyle \{2,4,6,8,10\}}$$is a finite set with five elements. The number of elements of a finite set is a natural … See more
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 moreAny 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 and the infinite lies at the core of set … See more
1920Kuratowski finiteness is defined1924Tarski presents definitions of finiteness1954Tarski presents theorems on the existence of successors of cardinals, and the axiom of choice1958Lévy publishes a paper on the independence of various definitions of finiteness1963Dedekind's Essays on the Theory of Numbers is published1972Suppes publishes Axiomatic Set Theory1974Apostol publishes Mathematical Analysis1976Rudin publishes Principles Of Mathematical Analysis1981Cohn publishes Universal Algebra2006de la Cruz, Dzhafarov, and Hall publish a paper on definitions of finiteness based on order propertiesIn 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 correspondence with the set of those natural numbers less than some specific … See moreIn 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
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