A set $S$ is infinite if it is not 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 (or the Axiom of Countable Choice), this definition of infinite sets is equivalent to that of Dedekind-infinite sets.

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:

• $\{1, 2, 3, 4, \ldots\}$
• The primes: $\{2, 3, 5, 7, 11, \ldots\}$
• The rational numbers: $\mathbb{Q}$
• An interval of the reals: $(0, 1)$

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