Hotel Hilbert 

home  courses  topics  theorems  starters  worksheets  timeline  KS3  KS4  KS5 

David Hilbert had this idea: if you have a hotel with a denumerable (infinite) set of rooms, numbered 1, 2, 3, and so on, then any number of guests can be accommodated without evicting the current guests by moving the current guests from room in which they are into the room which is the next one up on a number scale. For example, from room 1 to room 2, from room 2 to room 3, from room a to room a +1. Think of a situation such as this one: you are a hotel manager in a hotel with an infinite number of rooms and infinite number of people occupying these rooms. If you get another infinite number of people who all want to stay in your hotel what will you do? Will you turn these new people away and tell them that you are full? And how will you deal with that kind of chaos? Well, because you know about Hilbert, you can accommodate everyone! You can employ the system described above, or you can invent a new system  another example: move a person from room 1 into room 2; person from room 2 into room 4, person from room 3 into room 6 and so on  generally moving everyone who is already in the hotel with a room number a into a room number 2a. Then another infinite set of people will be able to get into the odd numbered rooms. What conclusions can you gain from thinking about this? What properties of infinity can you think of? Can infinity 'stretch'? Can you add infinities? Can you multiply them? Discuss with some friends... 
You can find more about Hilbert himself here. He is famous, among other things, for giving mathematicians a number of problems in 1900. See which ones are solved and which ones are not here and try for yourself to solve the yet unsolved ones! See more about infinity here. See Euclid's proof on the infinity of prime numbers here.


artefacts  numerals  concepts  people  places  pythagoreans  egyptians  babylonians
_____________________________________________________________________________________________________________________ Acknowledgements  Copyright  Contact  Mission Statement  Tell a friend about this site 
