# What is Infinity?

##### 2 Answers

This can't be answered without context. Here are some of the uses in mathematics.

#### Explanation:

A set has infinite cardinality if it can be mapped one-to-one onto a proper subset of itself. This is not the use of infinity in calculus.

In Calculus, we use "infinity" in 3 ways.

**Interval notation:**

The symbols

The interval

**Infinite Limits**

If a limit fails to exist because as

Note that: the phrase "without bound" is significant. The nubers:

**Limits at Infinity**

The phrase "the limit at infinity" is used to indicate that we have asked what happens to

Examples include

The limit as

This is written

"The limit as

The limit

as

It depends on the context...

#### Explanation:

Consider the set of Real numbers

#AA x in RR, -oo < x < +oo#

Then we can write

We can also write expressions like:

#lim_(x->0+) 1/x = +oo#

#lim_(x->0-) 1/x = -oo#

...meaning that the value of

So in these contexts

**Infinity as a completion of #RR# or #CC#**

The projective line

We can then extend the definition of functions like

**Infinity in Set Theory**

The size (Cardinality) of the set of integers is infinite, known as countable infinity. Georg Cantor found that the number of Real numbers is strictly larger than this countable infinity. In set theory there are a whole plethora of infinities of increasing sizes.

**Infinity as a number**

Can we actually treat infinities as numbers? Yes, but things don't work as you expect all of the time. For example, we might happily say

There are number systems which include infinities and infinitesimals (infinitely small numbers). These provide an intuitive picture of the results of limit processes such as differentiation and can be treated rigorously, but there are quite a few pitfalls to avoid.