You may be surprised to know there are many kinds of infinity. The first infinity you're likely to come across is in the counting numbers. The numbers 1, 2, 3, 4, 5, . . . go on forever; there is no biggest number. (Infinity isn't a number.) So we say that the set of counting numbers is infinite. It might seem like that's as big as infinity gets. But there are even bigger infinities. For example, the infinity that represents the real numbers between 0 and 1 is bigger than the infinity representing the counting numbers. There are more real numbers between 0 and 1 than you could count, even if you had an infinitely long time to count them. (An unsolved conjecture in mathematics is that there is no infinity between 0 and 1.) You might be surprised to know there are just as many rational numbers (numbers that can be written as fractions) between D and 1 as there are counting numbers. It's possible to count the rational numbers between 0 and 1 (although you could never finish). Can you think of a way to do it? Galileo's paradox: There are just as many square numbers as there are counting numbers. You can see that because you can match the squares up with the counting numbers (1 with 1, 4 with 2, 9 with 3, 16 with 4, and so on), and you never run out of either. But doesn't it seem that there must be more counting numbers than squares, since not every number is a square? Can you add up an infinite number of things and get a finite number? Yes, you can. Look at the sum 1/2 + 1/4 + 1/8 + 1/16 + You can try playing with this on a calculator or you can find the answer easily by drawing the right picture. (Start with a square, divide it in half and color in part of it. Then what should you do?) Achilles and the tortoise: This is one form of Zeno's Paradox. Achilles and a tortoise are racing. Because Achilles is so swift, the tortoise is given a head start. For Achilles to catch the tortoise, he must first get to where the tortoise is. But by that time, the tortoise has moved on, and Achilles is still behind. In fact, Achilles can never catch up because every time he gets to where the tortoise was, the tortoise has moved forward. (You can use the same basic argument to prove that motion is impossible.) Russell's Paradox: You can identify a set by a common characteristic of all its elements, for example {even counting numbers}. Consider the set {all sets that don't contain themselves as elements}. This set contains many ordinary sets, for example {blue automobiles}. The set of blue automobiles isn't a blue automobile itself, so it isn't a member of that set. A tricky paradox comes about however if we consider sets that DO contain themselves as elements of the set. One example that illustrates what can happen is the "Barber of Seville." Suppose that some small town has a barber, and that this barber will shave those and only those citizens of the town who don't shave themselves. Then the question is, who shaves the barber? If he shaves himself, then he does not shave himself, and similarly if he does not shave himself, then he shaves himself. Paradox!!! Think about the questions raised in the examples and prepare responses to one or more of them