Stay at Hilbert's Grand Hotel, the paradoxical hotel with infinite roomsS

Find out about an old paradox dreamed up by mathematicians in the roaring twenties. It takes a simple (and hypothetical) hotel and uses it to peer into the eyes of the infinite.

After a long, tedious journey, you head to a hotel and try to check in. It's a twenty-room hotel, and all the rooms are full. You're informed that you can't stay there. You walk to the next hotel. It's a hundred-room hotel. All the rooms are full, so you can't stay there. The next hotel has 2000 rooms, and they're all full. You can't stay there.

The last hotel in town is Hilbert's Grand Hotel. All the rooms are full. The clerk at the desk informs you that indeed, you can stay there. Why? Because it's an infinite-room hotel. But really, you argue, it doesn't matter how many rooms are in the hotel. What matters is that all the rooms are full. An infinite number of rooms, all of which are occupied still can't accommodate you.

It's true that the rooms are all occupied, but infinity has a strange characteristic. In most tallies, the amount of a certain group of numbers (say, the odd numbers) is smaller than the overall amount of numbers. For every group of odd-numbered rooms (1, 3, 5, 7...), there have to be a larger number of overall rooms (unless the total number of rooms is one). For every group of even numbered rooms, there have to be a larger number of total rooms (even if there are only two rooms, only one of them will have an even number). But infinity is different. Take away one, or a hundred, or a million, and it's still infinity. Add one, or a hundred, or a million, and it's still infinity. Divide it by anything you want. It's still infinity.

You can't take the "last" room, since by definition, all the rooms are occupied. But if the clerk were to tell the guest in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, every guest in the hotel could move, and you would just take over room one. (Although if anyone else came along, you wouldn't stay there for long.)

Want to get freakier? (I knew you did.) What if an infinite number of guests came? Sure, the clerk could repeat the process for checking in one guest an infinite number of times, but that's tedious. Instead, to house the first guest, the clerk would tell the guest in room 1 to move to room 2, just like before. Instead of moving the guest in room 2 to the next room over, room 2 would be moved to room 4, and room 3 would be moved to room 6. Each room 'n' would be moved to room '2n.' This would leave an infinite number of rooms free, for an infinite number of guests.

[Photo: Cezary P. Via Swarthmore]