countability

We can distinguish infinite sets into countable infinity and uncountable infinity. An infinite set is countably infinite if it has the same cardinality as natural numbers . Otherwise, a set is uncountable.

A set that is either finite or countably infinite is sometimes called countable. Below are two equivalent definitions of countability:

Definition

a set is countable if there is injective (one-to-one) function from it to natural number .

Definition

Let be a set. An enumeration aka listing of is a surjective function . We say a set is countable if one can enumerate its elements