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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 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 Euclid's proof on the infinity of prime numbers here.

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