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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 $n$.


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