Set of Ordered Pairs of Integers is Countable Infinite

Theorem
The set of all ordered pairs of integers $\Z$ is countably infinite.

Proof
The set of all ordered pairs of a set $S$ is by definition the Cartesian product $S \times S$.

In this context we are determining the cardinality of $\Z \times \Z$.

From Integers are Countably Infinite, we have that $\Z$ is a countably infinite set.

The result then follows from Cartesian Product of Countable Sets is Countable.