Hilbert's paradox of the Grand Hotel
 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.