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Metaphysics: Sizes of Infinity, Part 2 (Getting Real)

Video transcript

(Intro music) Hi, I'm Agustin Rayo. I'm an associate professor[br]of philosophy at MIT. And today I want to talk about how there are some infinities[br]that are bigger than others. So we've seen that there[br]are a bunch of infinities, which are really the same size, in the sense that they can all be put in one-one correspondence with one another. So one might be tempted to think that all infinities are the same size. That any two infinities can be put in one-one correspondence with each other. But in fact that is not true, and that is the second big[br]theorem that Cantor proved. What he showed is that[br]there are more real numbers than natural numbers. So if as many new guests as real numbers showed up to the hotel, we could not accommodate that. In fact, we couldn't even accommodate them if the hotel was empty to begin with. So here's how to do it. What we're gonna prove is that there are more real numbers between zero and one than there are natural numbers. And the way we're gonna prove it is by reductio. We're going to assume the opposite of what we want to prove and derive a contradiction[br]from that assumption. Because the opposite[br]entails a contradiction, it can't be true. So the thing we wanted[br]to prove must be true. So what's the opposite of[br]what we wanted to prove? Well, it's the idea that the real numbers between zero and one are in one-one correspondence[br]with the natural numbers. So it means in particular, that we can assign a[br]different natural number to each real number between zero and one. So assume that's true. Here's a diagram representing them. Each real number between zero and one can be represented as a decimal expansion. You can write it as "0.", and then an infinite sequence of digits. So suppose that we can assign a different natural number to each real[br]number between zero and one. So here's an example of how that might go. To the natural number zero we assigned this real number, and to[br]the natural number one we assigned this real[br]number, and so forth. Now, what we're gonna do, is we're gonna use our list to create an evil number. And here's how we do it. First, we consider the diagonal, which is just a result of[br]writing "0.", and then the sequence of digits which[br]we get from this diagonal here. And once we have that diagonal, we define its evil twin. The evil twin of the diagonal, is the number that you get by writing a seven whenever the diagonal had a three. And writing a three[br]whenever the diagonal had anything other than a three. So here's how it would go. So here we have a three, we write "7." Here we don't have a[br]three, we write "3." Here we have a three, we write "7." Not a three, we write "3." Not a three, we write "3." And so forth. Cantor's observation is that the evil twin cannot be on our list. Why can't it be on our list? Well, it can't be the[br]first member of our list, because the first member of our list has a three in its first position, but our evil number has[br]a seven in that position. What about the second one? Well it can't be the second one, because the second one has something other than a three in its second position. And our evil number has a three. And generally speaking, the evil number can't[br]be in the nth position, because whatever the number[br]in the nth position has as its nth digit, the evil number will[br]have something different. So here's what's happened. We've assumed for reductio that you can assign a[br]different natural number to each real number between zero and one. But then we found a real[br]number, the evil number, which isn't on that list. It contradicts our assumption[br]that we really had assigned a natural number to every real[br]number between zero and one. So our assumption must be false. And because what we[br]assumed is the negation of what we wanted to prove, it follows that what we[br]wanted to prove is true. There are more real numbers[br]between zero and one than natural numbers. So we've identified two[br]sizes of infinities so far. The size of the natural numbers, which we know is just as big as the size of the natural numbers plus one additional element, and just as big as the size of as many copies of the natural numbers as there are natural numbers. And we've identified a bigger size. The size of the real numbers[br]between zero and one. Are there other things[br]which are that bigger size? As it turns out, yes. There are exactly as many real numbers as there are real numbers[br]between zero and one. And there are exactly[br]as many points on a line as there are real numbers. And there are exactly as many points on a plane as there are real numbers. And there are exactly as many points on a cube as[br]there are real numbers. And there are exactly as many points in a hypercube as there are real numbers. So, is the infinity of the real numbers the biggest infinity there is? As it turns out, there are infinitely many sizes of infinity. Another thing Cantor proved is that whenever you have a set, the set's powerset (in other words, the set of all subsets[br]of the original set) is bigger. So, the powerset of the[br]set of natural numbers is bigger than the set of natural numbers. And the powerset of the powerset is bigger than the powerset. And the powerset of the[br]powerset of the powerset is bigger than that. And so forth with no end. Subtitles by the Amara.org community