In the universe Roomeron, there are infinitely many planets. Each planet has an infinite number of hotels, and each hotel has an infinite number of rooms. Since the business is so great, you decide to build a hotel of your own, also with an infinite number of rooms. To keep track of the rooms, each is numbered starting at 1. The hotels and planets are similarly numbered.
During the current tourist season, every room of every hotel, (including yours) on every planet is full. A freak catastrophe occurs in every other hotel besides yours and their rooms become trashed. The guests from those hotels ask to stay in your unwrecked hotel.
How can you put the infinitely many guests from infinitely many hotels from infinitely many planets in your already full hotel?
I am not a genius at infinity but i think the solution of multiplying each room no by two will only accomodate guests from one other hotel.
and the pairing solution(p,h,r) also has the same fault.
lets say you are h0. other hotels are h1, h2 etc.
if you multiply h0 by two you have free 1,3,5,.... infinite numbers which will only accomodate hotel h1's guests!
for accomodating h2's guests you will need to multiply each room no of h0 by three, and so for the infinite hotels on your planet you will need to multiply room nos of h0 by infinity. which will make a second order infinity, im not too sure if that is defined !
Even if you do that you still have only one planet accomodated!
for incorporating the other infinite number of planets you have to muttiply that with another infinity.
Another way could be to split each room to infinite number of smaller rooms and each of those rooms have to be split into another infinite number of rooms.
I know its physically paradoxical but i think this is the only way.