Definition:Countably Infinite/Set

Definition
Let $S$ be a set.

Definition 2
An infinite set is countably infinite if it is countable, and is uncountable otherwise.

Also see

 * Definition:Countable Set


 * Infinite Set has Countably Infinite Subset: the Axiom of Countable Choice implies that $\aleph_0$ is the smallest possible cardinality of an infinite set.