# What does infinity mean in mathematics?

Oct 18, 2016

See explanation...

#### Explanation:

Various concepts of infinity appear in different places in mathematics in different ways.

One of the basic axioms of standard set theory is that there is an infinite set. In some formulations of set theory you would use the set of natural numbers $\mathbb{N}$ as the standard example of an infinite set.

Once you start talking about "the set of Natural numbers", you are effectively talking about "completed infinities". Once you start to entertain that idea, then you may find more than you expected.

An infinite sequence of elements from a set $A$ is a mapping from $\mathbb{N} \to A$. For example, we could define a mapping that for each Natural number $n$ gives us a Real number ${a}_{n}$. We would then have a sequence of Real numbers:

${a}_{1} , {a}_{2} , {a}_{3} , \ldots$

For example, the formula:

${a}_{n} = 3 - \frac{1}{n}$

defines a mapping from $\mathbb{N}$ to $\mathbb{R}$ taking the values:

$2 , \frac{5}{2} , \frac{8}{3} , \frac{11}{4} , \frac{14}{5} , \frac{17}{6} , \ldots$

The terms of this monotonically increasing sequence are all numbers between $2$ and $3$. The limit of the sequence is $3$ which is also known as the least upper bound of the sequence.

Is it possible to enumerate all of the numbers between (say) $0$ and $1$ in one sequence?

The answer is a fairly definitive "No".

A mathematician called Georg Cantor showed that if you had such a sequence of numbers then you could always construct a number that was not in the list.

This implies that there are infinities larger than the number of Natural numbers.

This subject is much larger than you might think.