Infinite set

Definition

A set $A$ is infinite when it is not finite, that is, when $A$ is not equinumerous with $\{1, \dots, n\}$ for any natural number $n$.

Remarks

Examples are $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$. With the axiom of choice, a set is infinite exactly when it is equinumerous with a proper subset of itself. This is the Dedekind definition. Infinite sets can have different sizes. See Cantor's theorem.

Explanations

L1 human

An infinite set never runs out of elements when you count them. The natural numbers are infinite: after each number there is a next one.