# Hilbert's Hotel

Hilbert's Hotel is a paradox or thought experiment devised by mathematician David Hilbert to illustrate the astonishing consequences of using the concept of infinity in mathematics . This shows that the sets of natural numbers , whole numbers and rational numbers are of equal power .

## A hotel with an infinite number of rooms

In a hotel with a finite number of rooms, no more guests can be accepted as soon as all rooms are occupied ( drawer principle ). Hilbert's Hotel now has an infinite number of rooms (numbered with natural numbers starting with 1). One could assume that the same problem would also arise here, namely when all rooms are occupied by (infinitely many) guests.

However, there is a way to make room for an additional guest even though all rooms are occupied. The guest from room 1 goes to room 2, the guest from room 2 goes to room 3, the guest from room 3 to room 4, etc. This makes room 1 free for the new guest. Since the number of rooms is infinite, there is no “last” guest who cannot move into another room. If you repeat this, you get space for any but finite number of new guests. It is even possible to make room for an infinite number of new guests: the guest from room 1 goes into room 2 as before, the guest from room 2 but in room 4, the guest from room 3 in room 6, etc. In short, everyone Guest multiplies their room number by 2 to get the new one. This leaves all rooms with an odd number free for the countably infinite number of newcomers. With this approach it is important that all guests change rooms at the same time, for example when a gong triggered by the porter. If this were done one after the other, with an infinite number of guests and an infinite number of rooms, it would take an infinite amount of time.

If an infinite number of buses, each with an infinite number of guests, arrive, these guests can all be accommodated in the already full hotel. This can be done, for example, by clearing the rooms with odd numbers as just described and then sending the guests from bus 1 to rooms 3, 9, 27, ... (i.e. to those rooms that are numbered with powers of 3; 3 = 3 1 , 9 = 3 2 , 27 = 3 3 , ...), the guests from bus 2 to rooms 5, 25, 125, 625 etc., i.e. the guests from bus to rooms etc., whereby the -te Is prime . As a result, all the guests who have arrived are accommodated in the hotel and even an infinite number of rooms (such as room 15, whose number is not a power of a prime number) are free. Another, more efficient option would be to have the hotel guests move from their rooms to their rooms , so that all of the even rooms are free. Then the new guests from the bus can use the number to occupy the rooms whose room numbers are divisible by , but not by , so that no room remains free. Cantor's diagonal method is another way of accommodating the guests . ${\ displaystyle i}$${\ displaystyle p_ {i}, p_ {i} ^ {2}, p_ {i} ^ {3}}$${\ displaystyle p_ {i}}$${\ displaystyle i + 1}$${\ displaystyle n}$${\ displaystyle 2n-1}$${\ displaystyle n}$${\ displaystyle 2 ^ {n}}$${\ displaystyle 2 ^ {n + 1}}$

## Power of infinite sets

All of these possibilities are not really paradoxical, they just contradict intuition . It is difficult to get an idea of ​​infinite "summaries of things" because their properties are very different from those of ordinary, finite "summaries of things". In a hotel with a finite number of rooms, the number of rooms with an odd number is obviously smaller than the number of all rooms as soon as there is at least one room with an even number. In Hilbert's Hotel, which is aptly called the “ Grand Hotel ”, the “number” of rooms with an odd number is, in a certain sense, “just as large” as the “number” of all rooms. Expressed mathematically this is how it is: The thickness of the subset of rooms with an odd number is equal to the thickness of the set of all rooms. One can define infinite sets via the property of having a real subset of equal power. The power of countably infinite sets is called (" Aleph 0"). ${\ displaystyle \ aleph _ {0}}$

## Movie

Hilbert's Hotel found its way into several short films, including a. "Hotel Hilbert" (director: Caroline Ross-Pirie, Great Britain, 1996), awarded at the VideoMath Festival 1998 in Berlin and "Hilbert's Grand Hotel" (director: Djenaba Davis-Eyo, Great Britain, 2018).

## literature

• Francis Casiro: The Hilbert Hotel . In infinity (plus one). Hilbert Hotel, Russells Barbier, Peanos Himmelsleiter, Cantors Diagonale, Planck's Constant (= Spectrum of Science , Special 2, 2005). Spectrum of Science Verlagsgesellschaft, Heidelberg, ISBN 3-938639-08-3 , pp. 76–80