Consider $f:\mathbb N \to \mathcal P (\mathbb N)$ being a bijection from $\mathbb N$ to $\mathcal P (\mathbb N)$ such that every natural number is associated to a subset of $\mathbb N$ given by the binary representation of this natural number.
For example $$1=001\to\{1\},$$$$2=010\to\{2\},$$$$3=011\to\{2,1\},$$$$4=100\to\{3\},$$$$5=101\to\{3,1\},$$$$6=110\to\{3,2\},$$$$7=111\to\{3,2,1\}.$$
Every possible subset of $\mathbb N$ is then could an associated natural number be found by its binary representation.
My question now, I think this is an explicit bijection, how could we apply Cantor's theorem to this case and find a subset that is not associated by any binary representation of natural number?
This idea is explained in details in this article Building set and counting set, by Kuan Peng