Definition: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.