Hilbert's paradox of the Grand Hotel
In a hotel with a finite number of rooms, once it is full, no more guests can be accommodated. Now imagine a hotel with an infinite number of rooms. You might assume that the same problem will arise when all the rooms are taken. However, there is a way to solve this: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1. Note that such a movement of guests would constitute a supertask.
It is even possible to make place for an infinite (countable) number of new clients: just move the person occupying room 1 to room 2, occupying room 2 to room 4, occupying room 3 to room 6, etc., and all the odd-numbered new rooms will be free for the new guests.
If an infinite (countable) number of coaches arrive, each with an infinite (countable) number of passengers, you can even deal with that: 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-st 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. Interleave the digits of the coach numbers and the seat numbers to get the room numbers for the guests. The guest in room number 1729 moves to room 1070209. The passenger on seat 8234 of coach 56719 goes to room 5068721394 of the hotel.
This state of affairs is not really paradoxical but just profoundly counterintuitive. It is difficult to come to grips with infinite 'collections of things', as their properties are quite different from the properties of ordinary 'collections of things'. In an ordinary hotel, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel the 'number' of odd-numbered rooms is as 'large' as the total 'number' of rooms. In mathematical terms, this would be expressed as follows: the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. In fact, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets this cardinality is called 
.
An even stranger story regarding this hotel shows that mathematical induction only works in one direction. No cigars may be brought into the hotel. Yet each of the guests (all rooms had guests at the time) got a cigar while in the hotel. How is this? The guest in Room 1 got a cigar from the guest in Room 2. The guest in Room 2 had previously received two cigars from the guest in Room 3. The guest in Room 3 had previously received three cigars from the guest in Room 4, etc. Each guest kept one cigar and passed the remainder to the guest in the next-lower-numbered room.
Application to the Cosmological Argument for the existence of God
A number of defenders of the cosmological argument, among others William Lane Craig, for the existence of God have attempted to use Hilbert's hotel as an argument for the physical impossibility of the existence of an actual infinity. Their argument is that, although there is nothing mathematically impossible about the existence of the hotel (or any other infinite object), intuitively (they claim) we know that no such hotel could ever actually exist in reality, and that this intuition is a specific case of the broader intuition that no actual infinite could exist. They argue that a temporal sequence receding infinitely into the past would constitute such an actual infinite.However, the paradox of Hilbert's hotel involves not just an actual infinite, but also supertasks; it is unclear whether this claimed intuition is really the physical impossibility of an actual infinite, or merely the physical impossibility of a supertask. A causal chain receding infinitely into the past need not involve any supertasks. See Thomas Aquinas' Summa Theologica for details about infinite regressions and the existence of God.
It should also be noted that the addition of guests to a full Hilbert's would require infinitely fast communication, in order for every guest to tell the next guest to move one up in a finite amount of time. Thus, a universe could contain an actual infinite hotel, but with a finite speed of light, and hence it would not be able to contain any more guests even if it was full. (Subtraction, however, does not suffer from this supposed "dilemma." Suppose that each odd guest spontaneously decided to step outside of his or her room at the same moment. Thus, half of the rooms would be cleared instantaneously. In fact, addition can be treated in the same way. Suppose that all of the even guests happened to be standing right outside of the doors to their rooms, and they all, independently of each other, entered at the same time. Thus, half of the rooms could be filled instantaneously. Thus, there is a dilemma only if infinitely fast communication is required in order for the guests to perform these actions, which it is not.)
This is debatable — if the infinite hotel was a single row of rooms, connected by a long hall, each new guest could be placed in the room nearest the front desk, and told to instruct the inhabitant of room 0 to move to room 1, and pass along a similar message, in a way similar to the working of an infinite systolic array. Then, more guests could always be added — but at a rate limited by the walking speed of the guests, their ability to gather their possessions, etc. However, in that case, the hotel would always be overfull (in an overfill, the number of rooms is exhausted, but guests have been forced to share rooms, or some are deprived of rooms). And of course, you can add more guests to a full finite hotel as well, making it overfull.
External links
- Welcome to the Hotel Infinity! — The paradox told as a humorous narrative, featuring a hotel owner and a building contractor based on the feuding 19th-century mathematicians Georg Cantor and Leopold Kronecker
- Argument that Hilbert's paradox is not really paradoxical; criticism of Craig's use of it attack the possibility of an actual infinity
