Hilbert's Paradox of the Grand Hotel

Hilbert's paradox of the Grand Hotel is a thought experiment showing that a fully occupied hotel with infinitely many rooms can still take in more guests, even infinitely many more. The mathematician David Hilbert introduced the idea in a 1924-1925 lecture on the nature of the infinite, and it became widely known after George Gamow retold it in his 1947 book One Two Three... Infinity. In the scenario, a hotel has rooms numbered without any upper limit, so that it holds a countably infinite number of them; even when every room is occupied, the hotel can still welcome new arrivals by moving each existing guest to a higher numbered room, freeing space for the newcomers. The paradox is used to illustrate how the arithmetic of infinite sets defies intuitions built on finite experience.