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Ever been to a Hotel only to find that it is full? The problem is
that it has only got a finite number of rooms, and so they can
quickly get full. However, Hilbert managed to build a hotel with an
infinite number of rooms. Below is the story of his hotel.
When the Hotel first opened, everything went fine. He had lots of
visitors. He gave the first person room 1, the second person room
2, and so on. This worked well until he found that one day after a
very busy week the Hotel was full (so there were now an infinite
number of people staying in the hotel).
However, he had a solution. He went on the loud speaker, and asked
everyone in the hotel if they could move to the next room number
after theirs. So the person in room 1 moved into room 2, the person
in room 100 moved into room 101, and so on. The new guests were
then given room 1.
Things were going well, until one day a tour company had an idea -
and started sending buses which contained an infinite number of
people. In fact, they sent an infinite number of buses (numbered 1,
2, 3 etc.) Struggling to work out what to do, he asked his friend
Cantor for some help:
"Simply ask everyone to move into
the room whose number is twice the number of that they are
currently in. Now all the odd numbered rooms are empty. Now assign
the first person in the first bus room 1, the second person in the
first bus room 3, then the first person in the second bus room 5,
and carry on in the same manner picking one person from each bus as
you walk up and down pass the row of buses, picking people from one
more bus each time."
The plan worked perfectly, soon everyone had got off the bus and
had a room in the hotel.
Hilbert was very pleased because he thought that he would be able
to use Cantor's method to allocate rooms to any number of visitors.
However, Cantor warned him that there might be situations when it
would be impossible to find a way of allocating rooms.
Hilbert found that difficult to believe. If you would like to read
more about Cantor's arguments, you might like to read the article
Infinity Is Not a Number - It's a Free Man