Additive Group of Integers is Countably Infinite Abelian Group

Theorem
The set of integers under addition $\left({\Z, +}\right)$ forms a countably infinite abelian group.

Proof
From Integers under Addition form Abelian Group, $\left({\Z, +}\right)$ is an abelian group.

From Integers are Countably Infinite, the set of integers can be placed in one-to-one correspondence with the set of natural numbers.

Hence by definition, the underlying set of $\left({\Z, +}\right)$ is countably infinite.