A short history of a long corridor

The hotel opened in 1924, the year the mathematician David Hilbert described it in a lecture at Göttingen on the nature of the infinite. He proposed a grand hotel with infinitely many rooms, numbered 1, 2, 3, and so on, without end, and every room occupied. Most establishments would call that full. Hilbert called it an opportunity.

When a single traveller arrives at a full hotel of this kind, the night manager does not turn them away. He asks every guest to move along one room: the guest in room 1 to room 2, the guest in room 2 to room 3, and so on. Every guest still has a room, since every room number has a successor, and room 1 stands ready for the newcomer. The hotel was full. The hotel had room. Both statements are true, and neither is an apology.

When a coach arrived carrying infinitely many passengers, the manager asked each guest to move to twice their room number. The occupied rooms became the even ones, and the odd-numbered rooms, of which there are infinitely many, received the coach party. The whole affair took one announcement.

The famous night was the night the fleet came: infinitely many coaches, each with infinitely many passengers. The manager, by then difficult to impress, sent the current guests to the powers of two and gave each coach its own prime number, lodging coach c, seat s in that prime raised to the power s. Since no two distinct primes share a power, no key opened two doors. It is recorded that some rooms, such as room 6, ended the night empty, the first vacancies in the hotel’s history. Management issued an apology.

The hotel has one honest limit, and the staff will tell you about it if you ask. The scheme works because the rooms can be counted, one, two, three, and so on, and because the guests, however many, can be put in a list. The night a party arrived whose members could not be listed, one guest for every point on a line, the night manager met the only defeat of his career. Some infinities are simply larger than others, and a hotel, even this one, is only countable.

The story spread the way good stories do. George Gamow put the hotel in a book in 1947, and in 2014 a TED-Ed film by Jeff Dekofsky checked a new generation in. This website is one of its guests.