Suppose you’re a hotel manager and your hotel is full. That’s great, of course, but there’s always the temptation to squeeze in more guests. In real life, this might mean clearing out the broom cupboards and getting bad reviews on Trip Advisor, but in the world of maths it’s no problem. As long as your hotel has infinitely many rooms, that is.
The idea goes back to the German mathematician David Hilbert, who used the example of a hotel to demonstrate the counter-intuitive games you can play with infinity. Suppose that your hotel has infinitely many rooms, numbered 1, 2, 3, etc. All rooms are occupied, when a new guest arrives and asks to be put up. What do you do? It’s easy. Ask the guest in room 1 to move to room 2, the guest in room 2 to move into room 3, the guest in room 3 to move into room 4, and so on. If there were only finitely many rooms, the guest in the last room would have nowhere to go, but since there are infinitely many, everybody will find a new abode. You’ll have to ask the guests to move simultaneously though, because if you ask them to move one after the other, the move might take an infinite amount of time, since infinitely many guests have to move. … (+Plus Magazine)