Finite and inifite sets
Definition
A set $A$ is finite if $A = \emptyset$ or there is a bijection $f: A \to {1, 2, \cdots, n}.$
If $A$ is not finite, then $A$ is infinite.
Countability of sets
Definition
If there is a bijection $f: A \to B$, $A$ is cardinally equivalent to $B$.
Definition
If a set $A$ is cardinally equivalent to $\mathbb{N}$, then $A$ is countably inifinite.
If a set $A$ is either finite or countably infinite, then $A$ is countable.
If $A$ is not countable, then $A$ is uncountable.