In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where $n$ is a natural number. (The value n=0 is allowed; that is, the empty set is finite.) All finite sets are countable Some authors use "countable" to mean "countably infinite", and thus do not consider finite sets to be countable., but not all countable sets are finite.

Equivalently, a set is finite if its cardinality, i.e. the number of its elements, is a natural number. For instance, the set of integers between -15 and 3 (excluding the end points) is finite, since it has 17 elements. The set of all prime numbers is not finite. Infinite sets are sets which are not finite.

A set is called Dedekind finite if there exists no bijection between the set and any of its proper subsets. If the axiom of choice holds, a set is finite if and only if it is Dedekind finite.

Necessary and sufficient conditions for finiteness

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If the axiom of choice holds, then 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 natural number.
2. S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time.
3. (Paul Stäckel) S can be given a total ordering which is both well-ordered forwards and backwards. That is, every non-empty subset of S has both a least and a greatest element in the subset.
4. (Richard Dedekind) Every function from S one-to-one into itself is onto.
5. Every function from S onto itself is one-to-one.
6. (Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion.
7. S can be well-ordered and any two well-orderings on it are order isomorphic. In other words, the well-orderings on S have exactly one order type.

Footnotes

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See also

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• infinity
• hereditarily finite set

References

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• Patrick Suppes, Axiomatic Set Theory, D. Van Nostrand Company, Inc., 1960

Discrete mathematics | Set theory | Mathematical terminology | Cardinal numbers

Endliche und unendliche Menge | Ensemble fini | Insieme finito | Eindig | Zbiór skończony | Äärellinen joukko | Скінченна множина | 有限集合