# Finite set

In mathematics, a set is called * finite* if and only if there is a bijection between the set and some set of the form {1, 2, ...,

*n*} where

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}**is a natural number. (The value

*n*=0 is allowed; that is, the empty set is finite.) All finite sets are countable

^{[1]}, 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 is finite, since it has 17 elements. The set of all prime numbers is not finite. Sets that are not finite are called infinite.

A set is called * Dedekind finite* if there exists no bijection between the set and any of its proper subsets. It is a theorem (assuming the axiom of choice) that a set is finite if and only if it is Dedekind finite.

## See also

## Note

**^**Some authors use "countable" to mean "countably infinite", and thus do not consider finite sets to be countable.

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