“They woke me up three times to change rooms.”
A guest, room 2Three minutes about infinity. Captions on.
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The hotel holds as many guests as there are counting numbers. The endless names outnumber them, and that is Cantor's theorem.
The hotel. David Hilbert described it in his lectures on the infinite at Göttingen in the winter of 1924 to 1925; George Gamow made it famous in One Two Three… Infinity (1947). The history follows Helge Kragh, “The True (?) Story of Hilbert's Infinite Hotel” (2014).
The arithmetic. One more guest: room n moves to n + 1. One bus: n moves to 2n. Infinitely many buses: guests move to powers of 2, and each bus takes the powers of its own prime. Rooms that are not a power of a prime, such as 6, stay empty.
The last bus. Georg Cantor's diagonal argument, published in 1891, which itself used endless strings of two letters.
Pictures. Drawn for this film.
Sound. Any voices, music and effects are generated with ElevenLabs. The narrators are synthetic voices, not people.
Type. Fraunces and Space Grotesk, both under the SIL Open Font License.