It is known that with a countable set of algorithms (characterised by a Gödel number), we cannot compute (build a binary algorithm which checks belonging) all subsets of N.
A proof could be summarized as: if we could, then the set of all subsets of N would be countable (we could associate to each subset the Gödel number of the algorithm which computes it). As this is false, it proves the result.
This is a proof I like as it shows that the problem is equivalent to the subsets of N not being countable.
Now I'd like to prove that the halting problem is not solvable using only this same result (uncountability of N subsets), because I guess those are very close problem. Is it possible to prove it this way ?