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 squared equals a positive number, but remember that infinity is not exactly a number as such.
Also known as
When Georg Cantor did his original work on his development of set theory, the concepts were considered alien and difficult to accept.
To sweeten the pill slightly, he coined the word transfinite, which he used instead of infinite, so as not to scare mathematicians who already had a conception of the meaning of infinite, and were having difficulty accepting the challenge to their notions.
Hence the concept of a transfinite set, which is exactly the same as an infinite set.
Contemporary mathematics does not bother much with the term transfinite, except inasmuch as it applies to the concept of a transfinite ordinal.
Also see
- Results about infinity can be found here.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): infinite