Definition 2.1.3: Countable and Uncountable
If a set A has the same cardinality as N (the natural numbers), then we say that A is countable. In other words, a set is countable if there is a bijection from that set to N.
An alternate way to define countable is: if there is a way to enumerate the elements of a set, then the set has the same cardinality as N and is called countable.
A set that is not countable (and not finite) is called uncountable.