Tuesday, June 9, 2009

The Grand Hotel Paradox

Consider a hypothetical hotel with infinitely many rooms, all of which are occupied - that is to say every room contains a guest. Suppose a new guest arrives and wishes to be accommodated in the hotel.

If the hotel had only finitely many rooms, then it can be clearly seen that the request could not be fulfilled, but because the hotel has infinitely many rooms then if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, you can fit the newcomer into room 1.

By extension it is possible to make room for a countably infinite number of new clients: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room N to room 2*N, and all the odd-numbered rooms (a countably infinite number of rooms) will be free for the new guests.


Eliza said...

Wow Thad, that is way too deep for me.

MidSpeck said...

Room N to 2*N would not leave every odd room. 4 -> 8, for example.

Thaddeus said...

MidSpeck, the occupants in room 2 would fill room 4.

1 -> 2
2 -> 4
3 -> 6
4 -> 8
5 -> 10

Nobody is moving into odd rooms, leaving infinitely many vacancies.

Dorothy said...

Sounds like an odd hotel ...

MidSpeck said...

Thad, you're right -- I wasn't thinking straight.

Thaddeus said...

Can you even imagine a hotel with infinitely many rooms!? That would be sooo big! The First World Hotel in Malaysia is the largest hotel with 6,118 rooms, but that is still only finite! Infinite rooms would be at least ten times more!