Countability of a Set

All sets can be classified as either countable or uncountable.

Countable Sets

Such a bijection gives a natural way of listing the elements of $S$, hence the name countable. Similarly, the ability to list the elements of $S$ gives such a bijection.

This idea is demonstrated explicitly in the proof that rational numbers are countable.

Using the cardinality in terms of surjections result (with axiom of choice), we know that $S$ is countable if there is a function $f:\mathbb{N}\to S$ such that

Uncountable Sets