Definition:Finite Set

Definition
A set $S$ is defined as finite :
 * $\exists n \in \N: S \sim \N_{< n}$

where $\sim$ denotes set equivalence.

That is, if there exists an element $n$ of the set of natural numbers $\N$ such that the set of all elements of $\N$ less than $n$ is equivalent to $S$.

Equivalently, a finite set is a set with a count.

Also see

 * Definition:Cardinality of Finite Set
 * Cardinality of Finite Set is Well-Defined
 * Definition:Countable Set
 * Definition:Uncountable Set
 * Definition:Infinite Set
 * Definition:Dedekind-Infinite