Hilbert’s Hotel
David Hilbert · 1924, in lectures
The puzzle
A hotel with infinitely many rooms — numbered 1, 2, 3, … — is fully booked. A new guest arrives. Can it accommodate her?
Note
Yes. The manager asks every guest to move to the next room — guest 1 to room 2, guest 2 to room 3, and so on. Room 1 is now free. The hotel can also accommodate a countably infinite bus of new guests by moving everyone to twice their room number and putting the bus into the odd rooms. It can even handle infinitely many infinite buses. What it cannot do is accommodate a coachload of real numbers between 0 and 1 — Cantor proved that set is strictly larger than the integers. The hotel is meant to make infinite cardinalities concrete, and it succeeds: it shows that “fully booked” stops meaning what you thought it meant once the rooms stop being finite. There are sizes of infinity, and most of them are bigger than the one you started with.