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.