Hilbert's hotel: Difference between revisions
imported>Peter Schmitt (→Introduction: rewriting (reduce to essentials)) |
imported>Peter Schmitt (→Introduction: continued rewriting) |
||
| Line 24: | Line 24: | ||
Thus room number '''1''' will become free for the new guest. | Thus room number '''1''' will become free for the new guest. | ||
Imagine now the arrival of a bus with infinitely many tourists. | |||
They still can be accomodated: This time the manager asks the guests to move from '''1''' to '''2''', | |||
from '''2''' to '''4''',from '''3''' to '''6''', and so on, namely from ''n'' to 2''n''. | |||
After this, only the rooms with even numbers are occupied, | |||
and the tourists can be put in the rooms with odd numbers. | |||
Revision as of 05:43, 17 June 2009
Hilbert's hotel is a popular illustration of some properties of infinite sets like the set of natural numbers (and other countably infinite sets).
The story — which is usually attributed to David Hilbert — appears in a book (One two three ... infinity, 1947) by George Gamow (in Chapter 1, Big numbers, pp.17-18) with the following footnote:
From the unpublished, and even never written, but widely circulating volume: "The Complete Collection of Hilbert Stories" by R. Courant
Introduction
Imagine a hotel with infinitly many rooms, the room numbers being all natural numbers. Assume further that the hotel is fully booked — all rooms are occupied.
Nevertheless, if a new guest arrives he need not be sent away because the manager can provide a room by asking all guests to move: the guest in room 1 into room 2, the guest in room 2 into room 3, the guest in 3 into 4, and so on, i.e., each guest moving from room number n to room number n+1. Thus room number 1 will become free for the new guest.
Imagine now the arrival of a bus with infinitely many tourists. They still can be accomodated: This time the manager asks the guests to move from 1 to 2, from 2 to 4,from 3 to 6, and so on, namely from n to 2n. After this, only the rooms with even numbers are occupied, and the tourists can be put in the rooms with odd numbers.