one track mind, one track heart
consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied:
- a new guest arrives and wishes to be accommodated in the hotel. because the hotel has infinitely many rooms, we move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. by repeating this procedure, it is possible to make room for any finite number of new guests.
- it is also possible to accommodate a countably infinite number of new guests: in general, the guest occupying room n to room 2n, and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.
these cases constitute a paradox not in the sense that they entail a logical contradiction, but in that they demonstrate a counter-intuitive result that is provably true: the statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.
- a new guest arrives and wishes to be accommodated in the hotel. because the hotel has infinitely many rooms, we move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. by repeating this procedure, it is possible to make room for any finite number of new guests.
- it is also possible to accommodate a countably infinite number of new guests: in general, the guest occupying room n to room 2n, and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.
these cases constitute a paradox not in the sense that they entail a logical contradiction, but in that they demonstrate a counter-intuitive result that is provably true: the statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.