P(Z+) is uncountable (Power set of Positive Integers)

Question

I have been trying to prove that the Power set of Positive Integers is uncountable using :

Is $i$ an element of the $i$-th subset

But I can not seem to prove it.

Answer 1

Given any set $X$, you can prove that no function $f : X \to \mathcal{P}(X)$ is surjective. To prove this this, take any such function $f$ and let $B_f = \{ a \in X \mid a \not \in f(a) \} \in \mathcal{P}(X)$; it follows that $B_f \ne f(x)$ for any $x \in X$. This result is known as Cantor's theorem.

A corollary of this is that no function $\mathbb{Z}^+ \to \mathcal{P}(\mathbb{Z}^+)$ is surjective. Can you see where to take it from here?