$\mathbb{Z}$ is countable

Bijection. Define $f:\mathbb{N}\to\mathbb{Z}$ by
$$f(n)=\begin{cases}n/2&\text{if }n\text{ even},\\ -(n+1)/2&\text{if }n\text{ odd}.\end{cases}$$

Check. $0\mapsto0$, $1\mapsto-1$, $2\mapsto1$, $3\mapsto-2$, $4\mapsto2$, etc. Injective and surjective.

Conclusion. $|\mathbb{Z}|=\aleph_0$.