Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.
[edit] Finitely many new guests
Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.
[edit] Infinitely many new guests
It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room n to room 2n, and all the odd-numbered rooms will be free for the new guests.
[edit] Infinitely many coaches with infinitely many guests each
It is even possible to accommodate countably infinitely many coach-loads of countably infinite passengers each. The possibility of doing so depends on the seats in the coaches being already numbered (alternatively, the hotel manager must have the axiom of choice at his or her disposal). First empty the odd numbered rooms as above, then put the first coach's load in rooms 3n for n = 1, 2, 3, ..., the second coach's load in rooms 5n for n = 1, 2, ... and so on; for coach number i we use the rooms pn where p is the (i + 1)-th prime number. You can also solve the problem by looking at the license plate numbers on the coaches and the seat numbers for the passengers (if the seats are not numbered, number them). Regard the hotel as coach #0, and the initial room numbers as the seat numbers on this coach. Interleave the digits of the coach numbers and the seat numbers to get the room numbers for the guests. The hotel (coach #0) guest in seat (original room) number 1729 moves to room 01070209 (i.e, room 1,070,209.) The passenger on seat 4935 of coach 198 goes to room 4199385 of the hotel.
These cases demonstrate the 'paradox', by which we mean not that it is contradictory, but rather that a counter-intuitive result is provably true: The situations "every room is occupied" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.
Some find this state of affairs profoundly counterintuitive. The properties of infinite "collections of things" are quite different from those of finite "collections of things". In an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is as many as the total quantity of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets, this cardinality is called ![\aleph_0]()
(aleph-null).
Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers - those numbers which can be written as a quotient of integers - contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: There is a bijection from the naturals to the rationals.
Another story regarding the Grand Hotel can be used to show that mathematical induction only works from an induction basis.
Suppose that the Grand Hotel does not allow smoking, and no cigars may be taken into the Hotel. Despite this, the guest in room 1 goes to the guest in room 2 to get a cigar. The guest in room 2 goes to room 3 to get two cigars - one for himself and one for the guest in room 1. In general, the guest in room N goes to room (N+1) to get N cigars. They each return, smoke one cigar and give the rest to the guest from room (N-1). Thus despite the fact no cigars have been brought into the hotel, each guest can smoke a cigar inside the property.
The fallacy of this story derives from the fact that there is no inductive point (base-case) from which the induction can derive. Although it is shown that if the guest from room N has (N+1) cigars then both he and all guests in lower-numbered rooms can smoke, it is never proved that any of the guests actually have cigars. The fact that the story mentions that cigars are not allowed into the hotel is designed to highlight the fallacy. However, unless it is shown that in the limit there is a guest with infinitely many cigars, the proof is flawed regardless of whether or not cigars are allowed in the hotel.