Hilbert's hotel

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From the unpublished, and even never written, but widely circulating volume: "The Complete Collection of Hilbert Stories" by R. Courant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.