Definition:Finite

Definition

Finite Cardinal

Let $\mathbf a$ be a cardinal.

Then $\mathbf a$ is described as finite if and only if:

$\mathbf a < \mathbf a + \mathbf 1$

where $\mathbf 1$ is (cardinal) one.

That is, such that $\mathbf a \ne \mathbf a + \mathbf 1$.

Finite Set

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.

Finite Extended Real Number

An extended real number is defined as finite if and only if it is a real number.

Also see

• Definition:Infinite

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): finite