= Solution
A set is <countable set>[countable] when it is finite or admits a bijection with a subset of $\mathbb N$; equivalently, its elements can be listed in a sequence. The integers are countable, for example in the order $0,1,-1,2,-2,\ldots$. Enumerating pairs by diagonals, or applying the <Cantor pairing function>, then shows that $\mathbb N\times\mathbb Z$ is countable.
Suppose instead that the <power set> $\mathcal P(\mathbb N)$ had an enumeration $A_1,A_2,\ldots$. The <Cantor diagonal argument> forms
$$
D=\{n\in\mathbb N:n\notin A_n\}.
$$
For every $n$, the set $D$ differs from $A_n$ at the element $n$, so it is absent from the alleged enumeration. Therefore
$$
\boxed{\mathcal P(\mathbb N)\text{ is uncountable}}.
$$
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