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.
1 · Accommodation versus room occupancy
The argument rests on separating three notions that are easy to conflate. Accommodation — giving every guest a distinct room — is represented by an injection from guests to rooms. Full occupancy — leaving no room vacant — is represented by a surjection. A bijection represents both conditions holding simultaneously. Throughout, the rule that each room holds at most one guest is a modeling assumption about the hotel, not a theorem of set theory; set theory only decides, once that assumption is fixed, whether the required injection exists.
2 · Hilbert's Hotel: one and countably many new guests
When one new guest arrives at a fully occupied countably infinite hotel, shift every existing occupant from room rₙ to room rₙ₊₁ and seat the newcomer in r₁ — a bijection, so no guest is turned away: ℵ₀ + 1 = ℵ₀. For countably many newcomers, move the occupant of rₙ into r₂ₙ and seat newcomer n in r₂ₙ₋₁: the even rooms take the original guests, the odd rooms the newcomers, and again the assignment is a bijection: ℵ₀ + ℵ₀ = ℵ₀.
3 · Strict inequality and running out of rooms
The turning point is cardinal, not physical: if the guest set G has strictly greater cardinality than the room set R, then no injection G ↪ R exists at all, and consequently no complete one-guest-per-room accommodation is possible. This is stronger than saying one particular assignment fails — every possible assignment must fail, because the failure is a fact about the cardinalities of G and R, not about any specific attempted pairing. For a guest population indexed by the real numbers, |G| equals the cardinality of the continuum, strictly greater than ℵ₀ = |R|, and the countably infinite hotel necessarily runs out of rooms.
4 · Cantor's theorem and the capacity principle
The strict inequality comes from Cantor's theorem: for every set A, the power set P(A) satisfies |P(A)| > |A|, via the diagonal argument on D = {a ∈ A : a ∉ f(a)}. Taking A = ℕ gives |P(ℕ)| strictly greater than ℵ₀, and the reals share this cardinality. The general capacity principle follows: under the one-guest-per-room rule, all guests are accommodated if and only if |G| ≤ |R|. The precise meaning of an infinite hotel 'running out of rooms' is therefore cardinal rather than physical.