Why does my answer sheet say the set of computable functions is uncountable? I'm trying to understand why I can't find room for the set of computable functions in the hotel of the Hilbert's Hotel Paradox.

I was thinking that, because Gödel numbering, I could consider the set of computable functions as numerable (and with cardinality equal to $\mathrm{card}(\mathbb N)$). However, I have an  answer sheet that says otherwise, and makes a proof of that I'm pretty sure that is wrong. At least I don't trust it: 


  Suppose that the computable functions set is enumerable, and $f_n$ is a function of the set. Consider $g(n)=f_n(n)+1$, its not in the set, therefore contradiction. Because that proof then the computable functions aren't countable, so they can't fit in the Hilbert Hotel.
There are countably many computable functions. If $f_n$ is a complete list of all (total) computable functions, then the function $g(n) = f_n(n)+1$ cannot be any of the functions on the list, and so is not computable. This is one way to construct an uncomputable function.