When an Infinite Hotel Runs Out of Rooms
The infinite hotel is a useful setting for examining the counterintuitive properties of infinite sets. A hotel with countably infinitely many rooms can accommodate additional guests even when every room is occupied, because a countably infinite set can be placed in bijection with certain proper subsets of itself. This phenomenon does not, however, imply unlimited cardinal capacity. We formally distinguish accommodation from room occupancy, and show that a countably infinite hotel has capacity ℵ₀ and can accommodate any guest set of cardinality at most ℵ₀ under the one-guest-per-room rule. An uncountable guest set, such as one indexed by the real numbers, has the cardinality of the continuum, which Cantor's theorem shows to be strictly greater than ℵ₀ — hence no injection from the guests into the rooms exists, and complete accommodation is impossible. An infinite hotel can run out of rooms not because it has a final room, but because the arriving population has strictly greater cardinality than the room set.