Hilbert's paradox of the Grand Hotel

From Wikipedia, the free encyclopedia

Hilbert's paradox of the Grand Hotel is a mathematical paradox about infinite sets presented by German mathematician David Hilbert (1862–1943).

Contents

[edit] Situation

In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. Now, imagine a hotel with an infinite number of rooms. One might assume that the same problem will arise when a new guest comes along and all the rooms are occupied. However, in an infinite hotel, the situations "every room is occupied" and "no more guests can be accommodated" do not turn out to be equivalent. There is a way to solve the problem: 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.

It is also possible to make room for a countably infinite 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, and so on, and all the odd-numbered rooms will be free for the new guests.

Now imagine a countably infinite number of coaches arrive, each with a countably infinite number of passengers. Still, the hotel can accommodate them: 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. 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 01070209 (i.e, room 1,070,209. Leading zero added to clarify we take the first digit of the coach number first.) The passenger on seat 8234 of coach 56719 goes to room 5068721394 of the hotel.

Some find this state of affairs profoundly counterintuitive. The properties of infinite "collections of things" are quite different from those of ordinary "collections of things". In an ordinary 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 "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 ℵ0 (aleph-null).

An even stranger story regarding this hotel shows that mathematical induction only works from an induction basis. 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.

[edit] The cosmological argument

A number of defenders of the cosmological argument for the existence of God, such as Christian philosopher William Lane Craig, 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 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. Thomas Aquinas' Summa Theologica is a well-known attempt to prove the existence of God through infinite regressions.

[edit] References in fiction

  • Stephen Baxter's science fiction novel Transcendent has a brief discussion on the nature of infinity, with an explanation based on the paradox, modified to use starship troopers rather than hotels.
  • Geoffrey A. Landis' Nebula Award-winning short story "Ripples in the Dirac Sea" uses the Hilbert hotel as an explanation of why an infinitely-full Dirac sea can nevertheless still accept particles.
  • In Peter Hoeg's novel Smilla's Sense of Snow, the titular heroine reflects that it is admirable for the hotel's manager and guests to go to all that trouble so that the latecomer can have his own room and some privacy.
  • The booklet The Cat in Numberland by mathematician/philosopher Ivar Ekeland presents Hilbert’s paradox as a tale for children, in the tradition of Lewis Carroll. It is illustrated by John O’Brien.
  • Amanda Boyle's short film Hotel Infinity concerns a hotel with an infinite number of rooms. Its slogan is "We're always full, but we always have room for you."

[edit] See also

[edit] External links