The Hilbert Hotel
From Prep08Wiki
You may have heard of the Hilbert hotel? This hotel has an infinite number of rooms. We will assume here that we are dealing with countably infinite. For students: that means we have 1, 2, 3, 4, .... (no end).
The Hilbert hotel has the funny property that there is always more room. Suppose that a new guest shows up. We can just move guest 1 to room 2, guest 2 to room 3, etc. So guests 1, 2, 3, 4, 5 .... are now in rooms 2, 3, 4, 5, 6, ....
I.e. we just opened up room 1 and we can put our new guest there.
Here are some standard questions - and some not so standard questions - about the Hilbert hotel:
1. If a Hilbert schoolbus (infinite number of seats) pulls up, can we accomodate the infinite number of guests? If so, describe how.
2. The cleaning crew do not like the infinite key-chains very much (often leads to bad backs). How could the keys be distributed so that the cleaning crew only has to hold a minimal number of keys? Explain carefully.
3. Some of Escher's prints show (or suggest) an infinite number of objects. Which ones could be drawings of the Hilbert hotel? What part of the print would correspond to a room? See Escher Artwork
4. Draw your own Hilbert hotel.
