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From the unpublished, and even never written, but widely circulating volume: "The Complete Collection of Hilbert Stories" by R. Courant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

The basic idea is that of a hotel with an infinite number of rooms - precisely one room for each positive integer.

It's sometimes visualized as an infinitely long corridor, with rooms numbered consecutively 1,2,3, ...

A stranger arriving at the reception when the hotel is full may get a room anyway. The management will simply send out an intercom asking every current guest to go out into the corridor, and then move into the room one step further down.

This way the first room will be left vacant for the new arrival.

By a similar procedure, any finite number of new arrivals may be accommodated.

If an infinite number of strangers arrive, they may still be accommodated. The procedure is similar to the finite case, except each current guest will be asked to move to the room with twice the current room number.

All odd-numbered rooms will then become vacant, so the first new guest may move into the first odd-numbered room (1), the second into the second odd-numbered room (3), and so on.