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# Hilbert's paradox of the Grand Hotel

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

## The Paradox of the Grand Hotel

Consider a hypothetical hotel with infinitely many rooms, all of which are occupied - that is to say every room contains a guest. Suppose a new guest arrives and wishes to be accommodated in the hotel. If the hotel had only finitely many rooms, then it can be clearly seen that the request could not be fulfilled, but because the hotel has infinitely many rooms then if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, you can fit the newcomer into room 1. By extension it is possible to make room for a countably infinite number of new clients: 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.

It is even possible to accommodate countably infinitely many coach-loads of countably infinite passengers each - 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.) The passenger on seat 8234 of coach 56719 goes to room 5068721394 of the hotel.

This provides an important and non-intuitive result; 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.

## The Grand Hotel Cigar Mystery

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.

## The cosmological argument

Because the Hilbert's paradox is so counterintuitive, it has often been used as an argument against the existence of an actual infinity, for instance an argument for the existence of God posed by the Christian philosopher William Lane Craig is roughly as follows;

Although there is nothing mathematically impossible about the existence of such a hotel (or any other infinite object), intuitively no such object could ever exist, and this intuition is a specific case of the broader intuition that no actual infinite could exist. Since a temporal sequence receding infinitely into the past would constitute such an actual infinite, time must have "started" at some point. Since "time" cannot be started by any temporal thing, and every action must have a cause, this cause must be God.

It must be noted that Hilbert's hotel does not merely require a hotel of infinite magnitude to accommodate additional guests, but also involves the performing of supertasks. It could then be argued that it is unclear from Craig's argument whether this intuition of the fallacy of the hotel is really an indication of the physical impossibility of an actual infinite, or merely the practical impossibility of a supertask. A causal chain receding infinitely into the past need not involve supertasks.

Saint Thomas Aquinas' - arguably the most influential Christian theologian ever - made a well-known attempt to prove the existence of God through infinite regressions in his Summa Theologica.

## References in fiction

• The novel White Light by mathematician/science fiction writer Rudy Rucker includes a hotel based on Hilbert's paradox, and where the protagonist of the story meets Georg Cantor.
• 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 Høeg'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 concerns a hotel with an infinite number of rooms. Its slogan is "We're always full, but we always have room for you."