Hilbert's hotel: why a full infinite hotel has room for more

Hilbert’s hotel is a thought experiment in which a hotel with infinitely many rooms, all occupied, can always take more guests. Mathematician David Hilbert used it in a lecture in January 1924 to show how strangely infinite sets behave.

For one new guest, everyone moves from room n to room n + 1: there is no last room, so nobody is stranded and room 1 comes free. For infinitely many new guests, everyone moves from room n to room 2n, which empties every odd-numbered room.

Can infinitely many coaches of guests fit?

Yes. After the existing guests move to the even rooms, the passenger in seat n of coach m takes room pⁿ, where p is the (m + 1)th prime: coach 1 uses powers of 3, coach 2 powers of 5, and so on. Because every whole number factors into primes in only one way, no two guests ever share a room. The scheme stretches to any finite number of layers, such as ships of coaches, but not always to infinitely many.

The hard limit is a bigger infinity. In 1891 Georg Cantor proved there are more real numbers than natural numbers, so a coach with one passenger for every real number between 0 and 1 can never be fitted in.

Was the hotel really Hilbert’s?

Yes, though for a long time nobody knew. Hilbert never published the hotel, and historian Helge Kragh at first concluded it was really George Gamow’s. He later retracted that after tracing it to Hilbert’s 1924 lecture. The hotel only became widely known when Gamow described it in his 1947 book, and it took nearly three more decades to become a staple of arguments in cosmology, philosophy and theology.

The same shuffling of countable sets appears in the Banach–Tarski paradox, also from 1924, which shows how a ball can in theory be cut up and reassembled into two copies of itself.