In the universe Roomeron, there are infinitely many planets. Each planet has an infinite number of hotels, and each hotel has an infinite number of rooms. Since the business is so great, you decide to build a hotel of your own, also with an infinite number of rooms. To keep track of the rooms, each is numbered starting at 1. The hotels and planets are similarly numbered.

During the current tourist season, every room of every hotel, (including yours) on every planet is full. A freak catastrophe occurs in every other hotel besides yours and their rooms become trashed. The guests from those hotels ask to stay in your unwrecked hotel.

How can you put the infinitely many guests from infinitely many hotels from infinitely many planets in your already full hotel?

To gain one more room you just move each guest to a room number one greater than their current room. This leaves Room 1 vacant for your guest.

You could do this an infinite number of times to cater for an infinite number of newcomers.

Or to create an infinite number of rooms in one action -
move each guest into a room with a room number double their current room number (these would all be even) leaving an infinite number of odd numbered rooms vacant.

Posted by Vernon Lewis on 2007-01-25 09:11:08