Hilbert's paradox of the Grand Hotel

Hilbert's Hotel is an imaginary by the mathematician David Hilbert paradox or thought experiment to illustrate startling consequences of the use of the concept of infinity in mathematics.

A hotel with infinitely many rooms

In a hotel with a finite number of rooms, no more guests can be accommodated as soon as all rooms are occupied ( pigeonhole principle ). Hilbert's Hotel infinitely many rooms does now ( numbered with integers beginning with 1 ). One might assume that the same problem would occur again when all the rooms are occupied by ( infinitely many ) guests.

However, there is a way to make room for another guest, even though all the rooms are occupied. The guest from room 1 goes to room 2, the guest of Room 2 goes to room 3, which is of Rooms 3 Rooms 4 after etc. This room 1 free for the new guest. Since the number of rooms is infinite, there is no "last" guest who could not move to another room. Repeating this, we obtain space for an arbitrary but finite number of new guests. It is even possible to make room for a countable infinite number of new guests: The guest from room 1 goes as before in room 2, the guest rooms of 2, but in room 4, which are of three rooms in Room 6, etc. This means that all rooms odd-numbered free for countably infinite number of new arrivals.

Now, if countably infinitely many ancestors busses with a countable number of guests, these guests can all be housed in the already full hotel. This is, for example, by making the rooms with odd numbers as just described free and then the guests from bus 1 in the rooms 3, 9, 27, ... posted ( ie in those rooms that are numbered with powers of 3, 3 = 31, 9 = 32, 27 = 33, ... ) and guests from bus 2 in the rooms 5, 25, 125, 625, etc., etc., so the guests from the bus in the rooms etc., where the - th prime number. This all arrived guests are accommodated in the hotel and even infinitely many rooms (such as the room 15, the number is not a power of a prime number is ) free. Another, more efficient option would be to allow the guests to move each of the rooms in the room, so all rooms are even free. Then the new guests out of the bus can occupy the room with the number, their room numbers by, but not divisible by, so that no room would remain free. Another possibility, which accommodate guests, Cantor's diagonal method provides.

Cardinality of infinite sets

All these possibilities are not really a paradox, but only contradict intuition. It is difficult to get an idea of the infinite "Summaries of things ", since their properties are very different from those of ordinary, finite "Summaries of things ." In a hotel with a finite number of rooms, number of rooms with an odd number is lesser than the total number of rooms. In Hilbert's Hotel, which is aptly named " Grand Hotel ", is the "number " of rooms with odd numbers in a certain sense " the same size " as the "number " of all rooms. In mathematical terms, is this: The cardinality of the subset of the room with an odd number is equal to the cardinality of the set of all rooms. One can define infinite sets over the property to have an equally powerful proper subset. The cardinality of countable sets is called ( " Aleph 0").

Film

Hilbert's Hotel and even mentioned in a short film: "Hotel Hilbert " (production team: John Jaworski, Anne -Marie Gallen, 30 min, UK, 1996), inter alia, awarded on the VideoMath Festival in Berlin in 1998.