Maybe you know about this place. If not, you can imagine it. The Hotel Infinity, a hotel with an infinite number of rooms. Don’t snarl your mind with trying to imagine how it was built, or which room numbers can be found down which corridors. You don’t need to guess how deep a foundation it has to sit on, how many stories it is tall, or how many sub-basements it has. Don’t trouble your mind about the parking garage. Just imagine the hotel. With its infinite number of rooms. And it’s full.

A newcomer walks in. Asks for a room.

“No problem, says the desk clerk. We’ll fit you right in.”

How?

Easy. Everyone in the hotel is ordered to immediately pack their things, although nobody is asked to leave. Regardless of how grumpy it might make some of them, every guest will move to a different room. They are instructed to find the room whose number is the next number higher than the number of the room they occupy now. The guests in Room 99 move into Room 100. The folks in 240 head for 241. The people in Room 46 kazillion six thousand and seven set off to find Room 46 kazillion six thousand and eight. As soon as the people in Room 1 have left for Room 2, Room 1 will be empty. That’s where the bellhop is already leading the new guest.

If you can build one hotel that’s infinitely big, you can certainly build another. If you built it on wheels, it would be a bus. It could fill up with infinitely many people the same way the hotel did. When that busload of people pulls up to the front of the fully occupied Hotel Infinity, the desk clerk gets ready to make room for all the new guests.

How?

Once again, everyone must pack up and move. This time, everyone is sent to the room whose number is double the number of the room that they are occupying now. The people in Room 100 go straight to Room 200. The folks in Room 241 look for Room 482. The ones in 46 kazillion six thousand and eight quickly find their way to 92 kazillion 12 thousand and 16. The shuffling begins. With everyone now headed towards an even-numbered room, all the odd-numbered rooms become empty. The people begin filing in off the bus. They are led to their rooms one after another: Room 1, Room 3, Room 5…

That’s a famous story in math circles where the Hotel Infinity is known as Hilbert’s Hotel, named for David Hilbert who is credited with inventing the tale to help explain the bewildering mathematical theory of transfinite numbers described by George Cantor.

There are many who will say that Cantor’s theory was among the most important ideas of the 20^{th} century. The essential point is: once you get to infinity, arithmetic no longer operates the way we expect. There’s plenty of room inside of infinity to add things, even infinitely many of them. A number twice as big as infinity is still infinity, which itself is exactly the same size as half of infinity. No matter how much you take away from infinity, there will still be infinity left.

If the universe is infinite, what size would that be? If you shrink an infinite universe to half its size, it will still be infinite. Shrink it in half again and it won’t become any less infinite. Even if you shrink it infinitely many times, it will still be infinite.

Can you imagine how a Big Bang might occur in a universe like that?

Where is infinity, really? Is it *way out there, *or is it *somewhere else?*