# 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:

- $\forall n \in \Z: n < \infty$

- $\forall n \in \Z: n + \infty = \infty$

- $\forall n \in \Z: n \times \infty = \infty$

- $\infty^2 = \infty$

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

- $\forall n \in \Z: -\infty< n$

- $\forall n \in \Z: -\infty + n = -\infty$

- $\forall n \in \Z: -\infty \times n = -\infty$

- $\left({-\infty}\right)^2 = -\infty$

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.