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Logic and math
Hilbert's hotel
David Hilbert, in a 1924 lecture on the infinite; the set theory behind it is Georg Cantor's, from the 1870s.
Imagine a hotel with infinitely many rooms, numbered 1, 2, 3 and so on without end, and every room occupied. A traveller arrives late and asks for a room. Then a coach pulls up carrying infinitely many more travellers, one for every counting number, and each of them wants a room too.
The manager cannot build new rooms and will not put two guests in one room. Can the hotel take in the single traveller, the infinite coachload, both, or neither?