Hilbert's Hotel
From Fermat's Enigma:
To help explain the mystery of infinity, Hilbert created an example of infinity, known as Hilbert's Hotel. This hypothetical hotel has the desirable attribute of having an infinite number of rooms. One day a new guest arrives and is dissapointed to learn that, despite the hotel's infinite size, all the rooms are occupied. Hilbert, the clerk, thinks for a while and then reassures the new arrival that he will find an empty room. He asks all his current guests to move to the next room, so that the guest in room 1 moves to room 2, the guest in room 2 moves to room 3 and so on. Everybody who is in the hotel still has a room, which allows the new arrival to slip into the vacant room 1. This shows that infinity plus one equals infinity. Similarly, infinity subtract one is infinity, and indeed infinity subtract one million is still infinity.
The following night Hilbert has to deal with a much greater problem. The hotel is still full when an infinitely large bus arrives with an infinite number of new guests. Hilbert remains unperterbed and rubs his hands at the thought of infinitely more hotel bills. He asks all his current guests to move to the room that is double the number of their current room. So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, and so on. Everybody who was in the hotel still has a room and yet an infinite number of rooms, all the odd ones, have ben vacated for the new arrivals. This shows that double infinity is still infinity.
Hilbert's Hotel seems to suggest that all infinities are as large as each other, because various infinities seem to be able to squeeze into the same infinite hotel - the infinity of even numbers can be matched up and compared with the infinity of all counting numbers. However, some infinities are indeed bigger than others. For example, any attempt to pair every rational number with every irrational number ends in failure, and in fact it can be proved that the infinite set of irrational numbers is larger than the infinite set of rational numbers. Mathematicians have had to develop a whole system of nomenclature to deal with the varying scales of infinity, and conjuring with these concepts is one of today's hottest topics.
Some infinities are bigger than others? Watch my head explode. This book is great, by the way, I finally have a clear definition of magic imaginary numbers, which I didn't really grok before (the square root of -1 times whatever number you want).
-Russ