Hilbert's Grand Hotel

The hotel is full, yet nevertheless can accommodate new guests, even infinitely many new guests. Thus the parable of Hilbert's Grand Hotel illuminates the nature of countable infinity.

Hilbert's Grand Hotel

Let us hear the parable of Hilbert's Grand Hotel. The hotel has infinitely many rooms, each a luxurious full-floor suite, numbered like the natural numbers: room 0, room 1, room 2, and so on without end. Every room is occupied—the hotel is completely full with infinitely many guests.

A new guest arrives seeking a room. She inquires at the reception desk—is there any possibility of accommodating her? The manager replies coolly, “No problem, just a minute.”