# Definition:Finite

From ProofWiki

## 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** iff it is a real number.