We can accomodate infinite amount of people into an Hotel with infinite rooms. Next, we can add extra infinitude of people. (See Hilbert's paradox of the Grand Hotel.) Obviously, the trick is just to postpone the solution. Instead of providing the result the method just creates an infinite (i.e. never-ending) process: you shift all people right one room, accomodate newcomer and shift the rest in the next round. I see it as infinite, never-ending process. I do not see what you get in result. I do not see infinity-hotel accomodating 2 infinities of individuals. We say: ok, we will never terminate. That is why we can make accomodate 2n in the Hotel of size n. Good cheat.
Cantor says that we can enumerate all sets of integers but this list of sets will be incomplete because we can always scan it and build another set, Δ, which is different from all the sets in the list. Despite the process does not terminate either, the math community yet consider the enumeration of Δ impossible.
I see that it would be an absurd if you could accomplish building the Cantor antidiagonal set. But, accomodating 2n people into n rooms is identically absurdish. What makes the Hilbert conclusion less absurdish? Why cannot we be happy with existence of antidiagonal, which contradicts to the common sense? The process of diagonalization is never ending, as is the acommodation of guests in the Grand Hotel anyway.