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. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/