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TagKasper Müller


Math Fun: Hilbert’s Hotel Manager Copes With Infinity With Poise!

What, exactly, happens when a would-be guest shows up at a fully booked hotel — with infinite rooms?

Hilbert’s Hotel is a thought experiment that the great mathematician David Hilbert (1862–1943) developed to help us see the “counterintuitiveness of infinity.” He asks us to imagine a hotel which is “full” — except that because it is infinite, it can always create one more room. Mathematician Marianne Freiberger explains: Suppose that your hotel has infinitely many rooms, numbered 1, 2, 3, etc. All rooms are occupied, when a new guest arrives and asks to be put up. What do you do? It’s easy. Ask the guest in room 1 to move to room 2, the guest in room 2 to move into room 3, the guest in room 3 to move into room 4, and so on. If there…