Adam: So last time we worked out that we measure size by pairing things up. And from that we worked out that the set of evens is the same size as the set of all whole numbers, that the set of square numbers is the same size as the set of whole numbers, etc.
Plato: Right. But isn't that obvious? Surely they're all the same size, they're infinite. We cant say anything more than that can we?
Adam: Yes we can, let's see how. Ok, so we're happy that the size of all even numbers is the same as the set of whole numbers because we can pair up numbers 1-2 2-4 3-6 4-8 etc.
Plato: Yes, that's fine.
Adam: And we can do this to all fractions as well. If we start with the smallest number on the bottom (1) and then go through all the numbers on the top that are smaller than it (only 1), then we put the next number on the bottom and go through all those possible fractions (½). We can carry on like this until we've covered all fractions. We have to leave out any repeats of course.
Plato. Let me see if I've understood this. So what you're saying is. We go up in two stages. First we fix the number on the bottom of the fraction, then we count through all the fractions that have that number on the bottom, then we go up to the next number on the bottom? So it goes (1/1) (½) (1/3, 2/3) (¼ ¾) (1/5 2/5 3/5 4/5) and so on?
Adam: Exactly, now we can go through our list in order and pair them up with each number. So all the fractions between 0 and 1 will be covered and none missed out. So there are as many fractions between 0 and 1 as there are whole numbers.
Plato: That's interesting, so there as many points on the number line between 0 and 1 as there are whole numbers.
Adam: Ah, there is a problem however, one that came a bit after your time Plato. The problem is that not every number can be represented as a fraction.
Plato: Nonsense, every number is either whole, or a fraction of a number surely?
Adam: Try this on for size, think of the square root of 2. The number which is 2 when you times it by itself. Are you happy that this is a real number?
Plato: Yes, you can construct it by the hypotenuse of a right angled triangle with side length one.
Adam: Indeed. Now you claim that this has length a/b for some whole numbers a and b right?
Plato: Indeed, you can put this in the diagram.
Adam: Now then you say that the square of side a has twice the size of square b. Because if a^2/b^2=2 then a^2=2b^2. Now any square with an even area must have an even side length. So we can draw the side length c, and we can see that this is a whole number.
Plato: Yes, so now the small triangle in the top left is a scaled down copy of the one we started with. All with integer sides.
Adam: Exactly. So you say that root 2 is a/b. We can also say it is b/c, this is smaller, and we can do the same thing again, and say it is c/d, each one smaller. And we can carry on infinitely.
Plato: What's your point.
Adam: Well each time we have whole numbers, and each time we're making it less. But we cant possibly keep reducing the number forever, that doesn't make sense. You cant take something away from a number infinitely many times. So you're wrong.
Plato: Wow *small mental break down*. Ok, so how can we talk about numbers then if they aren't fractions?
Adam: Well we can think about the decimal expansion. But we have to be happy that this expansion can be infinite. So for root two it goes 1.41421... for ever with no pattern or repetition.
Plato: ok, so how does this let us talk about two sorts of infinity?
Adam: Ok, so you said to me that there we as many numbers of whatever sort between 01 and 1 as there were whole numbers. So you say that there is some pairing, can you think of one for me?
Plato: Umm, ok. We can pair 1 with 0.1000000....., and 2 with 0.111111.... so it goes like this:
and so on forever, I'm not sure about the details but I could work that out.
Adam: No you couldn't. Think about the diagonal line running down the middle. In your arrangement spells out 11234... and so on in some pattern forever. Now consider adding 1 to each digit to make 22345... Now I want to ask you a question. In your pattern where does the number 0.22345... formed in this way fit?
Plato: Well it's not the first one, because it starts with a 2 not a 1. … and it's not the second one, it's second digit is a 2 not a 1, and it's not the third one, the third digit is wrong. In fact no matter where you put it, the digit that lies on that diagonal must be wrong. So it cant fit anywhere.
Adam: So you're wrong then. That sequence cant be right, you've missed one of the numbers out.
Plato: Ok, so that one is wrong. But be fair I just made it up off the top of my head. You cant expect it to be totally right. I've just got to put that number in there somewhere.
Adam: Ah, but here's the problem Plato. No matter how you do it, I can always make the same argument. I can always take the diagonal line and add one to each digit and then it will always not fit into any place in the sequence.
Plato: So no matter how the pairing is done you cant pair up all the numbers between 0 and 1 and all the whole numbers. So there must be more numbers between 0 and 1 than there are whole numbers.
Adam: So there really are two different kinds of infinity. Next time I'm going to suggest that we can think about where sequences end up “at infinity” and try and solve Zeno's paradox.