infinite

A set $S$ is infinite if it is not finite (http://planetmath.org/Finite); that is, there is no $n\in \mathbb{N}$ for which there is a bijection between $n$ and $S$.

Assuming the Axiom of Choice (http://planetmath.org/AxiomOfChoice) (or the Axiom of Countable Choice), this definition of infinite sets is equivalent to that of Dedekind-infinite sets (http://planetmath.org/DedekindInfinite).

Some examples of finite sets:

• The empty set: $\{\}$.

• $\{0,1\}$

• $\{1,2,3,4,5\}$

• $\{1,1.5,e,\pi \}$

Some examples of infinite sets:

• $\{1,2,3,4,\mathrm{\dots }\}$.

• The primes: $\{2,3,5,7,11,\mathrm{\dots }\}$.

• The rational numbers: $\mathbb{Q}$.

• An interval of the reals: $(0,1)$.

The first three examples are countable, but the last is uncountable.