Hilbert's Hotel Is Always Open

What happens when an infinite hotel with no vacancies gets new guests? Joel David Hamkins walks through the famous Hilbert's Hotel thought experiment, starting with the problem of fitting one more guest into a completely full hotel. From there, the challenges escalate: a thousand new guests, an infinite bus, and even an infinite train carrying infinitely many passengers per car. Each scenario illustrates a deeper property of countable infinity. Hamkins then formalizes these ideas, explaining what it means for a set to be countable and proving that countable sets remain countable under unions, Cartesian products, and other operations. Along the way, he introduces the Cantor pairing function, shows why the rational numbers are countable, and notes the hidden role of the axiom of choice. The lecture closes with a tantalizing puzzle: Cantor's cruise ship arrives carrying passengers labeled by real numbers. Can Hilbert's Hotel accommodate them? The answer awaits in the next lecture.