Hilbert's Infinite Hotel

David Hilbert imagined a hotel with infinitely many rooms, all occupied. Could it accommodate a new guest? Yes — move every guest from room n to room n+1, freeing room 1. Infinitely many new guests? Move every guest from room n to room 2n, freeing all odd-numbered rooms. Infinitely many busloads of infinitely many new guests? Still possible. Hilbert used this thought experiment to make Cantor's counterintuitive set theory vivid and accessible. It remains the best introduction to cardinal arithmetic ever devised.