 So we're saying that a set is countably infinite if it can be put in a one-to-one correspondence with the natural numbers. And, as you've seen, countably infinite sets include the natural numbers, the integers, the rational numbers, the computable numbers, provable statements, and so on. None of these results would be interesting, since, well, aren't these all infinite sets anyway, except that there are infinite sets that can't be put in a one-to-one correspondence with the natural numbers. And that has some very disturbing implications. So, let's consider the numbers with decimal expansions, numbers like 3.5, which has a finite expansion, or 1.3 repeating, which is rational with an infinite expansion, or something like this, which is irrational with an infinite expansion. Do these numbers form a countably infinite set? Well, let's find out. So, suppose we wanted to show that the real numbers are countably infinite. Then we'd have to assume we can put them in some order. And remember, you can always assume the antecedent of a conditional. So, let's assume we can list the elements of the real numbers in some order. So, suppose we list all real numbers between 0 and 1. So, the first number, I don't know, maybe it's this, the second, this, and so on. Now what? So, again, the important idea to keep in mind, if you don't find the flaw in your reasoning, someone else will. So, let's think about this. We're listing all the real numbers between 0 and 1. Well, let's see if we can find a real number not on our list. Let's think about that. Well, first of all, how do we know our numbers on our list? Well, suppose we have a number like this. We might make the following observations. We know that this number is not the first number on our list, since the first digit past the decimal point is different. It's not the second number on our list. Even though the first two digits are the same, that third digit past the decimal point is different. And again, it's not the third number on our list, since the first digit past the decimal point is different, and so on. So, we can determine which number this isn't by checking out the digits past the decimal point. Well, if this list includes all real numbers between 0 and 1, then every number is going to be someplace on that list. And so the question you've got to ask yourself is, self, can we find a number that differs from every number on the list in at least one decimal place? Or, because it's easier to build something specific than something generic, can we find a number that differs from the k-th number on our list in the k-th decimal place? And the answer to that is, yes. And we'll construct our number as follows. So, the first digit past the decimal place of the first number is 3, and so pick any digit besides 3. How about 5? Now, the second digit past the decimal point of the second number is 9, so pick any digit besides 9. How about 1? The third digit past the decimal point of the third number is 4, so pick any digit besides 4. How about, no, besides 4. That's better. And so on. And so if we construct our number this way, the number we produce in this way isn't on our list. It's not the first number on the list since its first decimal place is different. It's not the second number on the list since its second decimal place is different. It's not the third number since its third decimal place is different, and so on. And so because of the way we've produced it, this number is not on our list of real numbers. But we assumed we listed all the real numbers between 0 and 1, and since this is a contradiction, this means that we can't list all of our numbers this way. And the proceeding proves a very important theorem. The real numbers between 0 and 1 cannot be put in a 1-to-1 correspondence with a set of natural numbers. And this leads to a new definition, an infinite set that cannot be put in a 1-to-1 correspondence with a set of natural numbers, is uncountably infinite. Now, since the real numbers between 0 and 1 are a subset of all real numbers, this tells us the following, the set of real numbers is uncountably infinite. Or does it? There is one thing we do have to establish. It's possible that the set of real numbers isn't infinite, and so we do need to prove that we're dealing with an infinite set, in which case we have an uncountably infinite set. Now, I've taught this topic for many, many years, and the general reaction at this stunning result is, so what? I mean, so we have some sets that are infinite, and some sets that are countably infinite. Who cares? It doesn't really make any difference, because they're all infinite sets. And the answer to this is the existence of what we might call larger infinities, has several immediate consequences, most of which are very disturbing. So remember that we show that the number of numbers that can be written out is countably infinite. Since the number of numbers that can be written out is countably infinite, but the real numbers are uncountably infinite, this means there are real numbers that we cannot write out. Remember, we also show that the number of computable numbers is countably infinite, and since the number of computable numbers is countably infinite, but the real numbers are uncountably infinite, it follows there are real numbers whose value cannot be computed. Now, consider a statement like x is a real number, where x is some real number. This is a true statement that can be made for every real number, which means there are uncountably many such statements. But remember that we show the number of provable statements is countably infinite. And so the number of proofs is countably infinite. So this means there are true statements that can't be proven. And, actually, it's worse. Consider the statement x is a real number. Again, this is a true statement. We can put these statements in order by length and then alphabetize them within the length. That's what we did for being able to determine the number of numbers we can write or compute or the number of statements we can prove. And, in every case, we found that the number of such statements was countably infinite. And so that means the number of true statements we can write is countably infinite, but the number of true statements like this is uncountably infinite. And this means there are true statements we can't even write. And, again, you might decide that this really doesn't matter because we're talking about infinite sets anyway. But there are profound philosophical implications in the idea that there are real numbers whose values we can't compute, whose values we can't even write down, as well as true statements that can't be proven, and, in fact, true statements that can't even be written.