About Hilbert's countably infinite guests In Hilbert's hotel if we get each guest to his next room (guest in the room $n$ will get in room $(n + 1)$) then the first room will be empty. About this method Wikipedia says:
"By repeating this procedure, it is possible to make room for any finite number of new guests."
But why only finite? Why can't we add countably infinite number of new guests this way?
 A: Recall that infinite hotels do not actually exist; the whole story is just a metaphor for constructing functions between various sets. It would be futile to attempt to motivate the conditions in the story by thinking about how infinite hotels "ought to" work if they did exist; at the end of the day which kind of solutions the story should allow is measured purely by whether or not they can be interpreted as valid mathematics.
When one guest arrives, what the story tries to illuminate is that we can construct an bijective function, for example $f: (\mathbb N \cup \{\frac12\})\to\mathbb N$ by defining
$$ f(\frac12) = 0,\ f(0) = 1, \ f(1) = 2, \ f(2) = 3, \ \ldots $$
We can run this a finite number of times to get for example $g: (\mathbb N\cup\{\frac12,\frac13,\frac14\})\to\mathbb N$:
$$ g(\tfrac14)=0,\ g(\tfrac13)=1,\ g(\tfrac12) = 2,\ g(0) = 3, \ g(1) = 4, \ g(2) = 5, \ \ldots $$
But if we try to make space for an infinite number of new elements by simply repeating the process, trying to make a $h:(\mathbb N\cup\{\frac12,\frac13,\frac14,\frac15,\ldots\}) \to\mathbb N$, then it doesn't work. Nothing ends up mapping to anything in particular, and we don't have any function to look at in the end.
Back in the hotel metaphor we can interpret this problem by demanding that every guest must end up in some room where they can go to sleep undisturbed. Requiring every guest to keep moving for an infinity of times does not achieve that.
A: :) Indeed. This can be accomplished all at once, too: for example, by sending guests from room $n$ to room $2n$, all the odd-numbered rooms will be emptied. :)
