Hilbert's paradox of the Grand Hotel
Hilbert's paradox of the Grand Hotel is a mathematical paradox named after the German mathematician David Hilbert. Hilbert used it as an example to show how infinity does not act in the same way as regular numbers do. Normal hotels have a set number of rooms. This number is finite. Once every room has been assigned to a guest, any new guest that wants a room and does not have one yet cannot be served - in other words, the hotel is fully booked. Now suppose that there is a hotel that has an infinite number of rooms. As a convenience, the rooms have numbers, the first room has the number 1, the second has number 2, and so on. If all the rooms are filled, it might appear that no more guests can be taken in, as in a hotel with a finite number of rooms. This is wrong, though. A room can be provided for another guest. This can be done by moving the guest in room 1 to room 2, the guest in room 2 to room 3, and so on. In the general case, the guest in room n will be moved to room n+1. After all guests have moved, room 1 is empty, and the new guest now has a room to occupy. This shows how we can find a room for a new guest even if the hotel is already full, something that could not happen in any hotel with a finite number of rooms.
Another thing that can be done with this imaginary hotel is to double the number of people inside, again when all the rooms are already full. This is done by asking each guest to multiply their room number by two and move to that room. (If their previous room number was n, this time they would move to room number 2n.) This would send the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, the guest in room 4 to room 8, and so on. After finishing, we find that all the rooms with the odd numbers are empty. Then we can put an infinite number of guests into these empty rooms. Now the number of guests in the hotel has been doubled without making the hotel bigger.
This is not really a paradox, it is only counterintuitive. In a normal hotel, with a finite number of rooms, the number of odd-numbered rooms, is smaller than the total number of rooms. In Hilbert's Hotel this does not seem to be the case.