Definition:Finite Set

Definition

A set $S$ is defined as finite if and only if:

$\exists n \in \N: S \sim \N_{<n}$

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 $\mathsf{Pr} \infty \mathsf{fWiki}$ both forms will be found.

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