# Definition:Infinite

## Definition

### Infinite Cardinal

Let $\mathbf a$ be a cardinal.

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

- $\mathbf a = \mathbf a + \mathbf 1$

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

### Infinite Set

A set which is not finite is called **infinite**.

That is, it is a set for which there is no bijection between it and any $\N_n$, where $\N_n$ is the the set of all elements of $n$ less than $n$, no matter how big we make $n$.

### Infinity

Informally, the term **infinity** is used to mean **some infinite number**, but this concept falls very far short of a usable definition.

The symbol $\infty$ (supposedly invented by John Wallis) is often used in this context to mean **an infinite number**.

However, outside of its formal use in the definition of limits its use is strongly discouraged until you know what you're talking about.

It is defined as having the following properties:

\(\ds \forall n \in \Z: \, \) | \(\ds n\) | \(<\) | \(\ds \infty\) | |||||||||||

\(\ds \forall n \in \Z: \, \) | \(\ds n + \infty\) | \(=\) | \(\ds \infty\) | |||||||||||

\(\ds \forall n \in \Z: \, \) | \(\ds n \times \infty\) | \(=\) | \(\ds \infty\) | |||||||||||

\(\ds \infty^2\) | \(=\) | \(\ds \infty\) |

Similarly, the quantity written as $-\infty$ is defined as having the following properties:

\(\ds \forall n \in \Z: \, \) | \(\ds -\infty\) | \(<\) | \(\ds n\) | |||||||||||

\(\ds \forall n \in \Z: \, \) | \(\ds -\infty + n\) | \(=\) | \(\ds -\infty\) | |||||||||||

\(\ds \forall n \in \Z: \, \) | \(\ds -\infty \times n\) | \(=\) | \(\ds -\infty\) | |||||||||||

\(\ds \paren {-\infty}^2\) | \(=\) | \(\ds -\infty\) |

The latter result seems wrong when you think of the rule that a negative number square equals a positive one, but remember that infinity is not exactly a number as such.

## Also see

## Sources

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