Hilbertís Hotel is a hotel with an infinite number of rooms, each one of which is occupied. The hotel gives rise to a paradox: the hotel is full, and yet it has vacancies.
That the hotel is full is obvious. It has an infinite number of rooms, and an infinite of guests; every room is occupied. That the hotel has vacancies is a little more difficult to demonstrate.
Suppose that a new visitor arrives; can he be accommodated? At first it seems that he cannot, but then the hotel clerk has an idea: He moves the guest in Room 1 to Room 2, and the guest in Room 2 to Room 3, and so on. Every guest is moved to the next room along.
For every guest, in every room, there is another room into which they can be moved. This leaves Room 1 vacant for the new visitor. Although the hotel is full, then, the new guest can be accommodated in Room 1.
It is not only one new guest that can be accommodated; in fact, Hilbertís Hotel has an infinite number of vacancies. By moving every guest to the room the number of which is double the number of their current room, all of the even numbered rooms can be vacated for new guests. There are, of course, an infinite number of even numbered rooms, and so an infinite number of new guests can be accommodated.