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 known as
It is a common expression to refer to a finite number when finite set is meant.

That is, a finite number of can usually more precisely be worded a finite set of.

However, it is often the case that finite number works better, so on both forms will be found.

Similarly, the term finitely many can also be seen in a similar context.

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