Mathematics: Sizes of infinity part 2: Getting real

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Mathematics: Sizes of infinity part 2: Getting real발음듣기

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(Intro music) Hi, I'm Agustin Rayo.발음듣기

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I'm an associate professor of philosophy at MIT.발음듣기

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And today I want to talk about how there are some infinities that are bigger than others.발음듣기

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So we've seen that there 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.발음듣기

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So one might be tempted to think that all infinities are the same size.발음듣기

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That any two infinities can be put in one-one correspondence with each other.발음듣기

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But in fact that is not true, and that is the second big theorem that Cantor proved.발음듣기

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What he showed is that there are more real numbers than natural numbers.발음듣기

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So if as many new guests as real numbers showed up to the hotel, we could not accommodate that.발음듣기

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In fact, we couldn't even accommodate them if the hotel was empty to begin with.발음듣기

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What we're gonna prove is that there are more real numbers between zero and one than there are natural numbers.발음듣기

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And the way we're gonna prove it is by reductio.발음듣기

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We're going to assume the opposite of what we want to prove and derive a contradiction from that assumption.발음듣기

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Because the opposite entails a contradiction, it can't be true.발음듣기

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So the thing we wanted to prove must be true.발음듣기

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So what's the opposite of what we wanted to prove?발음듣기

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Well, it's the idea that the real numbers between zero and one are in one-one correspondence with the natural numbers.발음듣기

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So it means in particular, that we can assign a different natural number to each real number between zero and one.발음듣기

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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.발음듣기

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You can write it as "0.", and then an infinite sequence of digits.발음듣기

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So suppose that we can assign a different natural number to each real number between zero and one.발음듣기

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So here's an example of how that might go.발음듣기

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To the natural number zero we assigned this real number, and to the natural number one we assigned this real number, and so forth.발음듣기

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Now, what we're gonna do, is we're gonna use our list to create an evil number.발음듣기

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And here's how we do it.발음듣기

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First, we consider the diagonal, which is just a result of writing "0.", and then the sequence of digits which we get from this diagonal here.발음듣기

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And once we have that diagonal, we define its evil twin.발음듣기

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The evil twin of the diagonal, is the number that you get by writing a seven whenever the diagonal had a three.발음듣기

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And writing a three whenever the diagonal had anything other than a three.발음듣기

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So here's how it would go.발음듣기

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So here we have a three, we write "7."발음듣기

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Here we don't have a three, we write "3."발음듣기

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Here we have a three, we write "7."발음듣기

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Not a three, we write "3."발음듣기

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Not a three, we write "3." And so forth.발음듣기

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Cantor's observation is that the evil twin cannot be on our list.발음듣기

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Why can't it be on our list?발음듣기

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Well, it can't be the first member of our list, because the first member of our list has a three in its first position, but our evil number has a seven in that position.발음듣기

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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.발음듣기

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And our evil number has a three.발음듣기

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And generally speaking, the evil number can't be in the nth position, because whatever the number in the nth position has as its nth digit, the evil number will have something different.발음듣기

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So here's what's happened. We've assumed for reductio that you can assign a different natural number to each real number between zero and one.발음듣기

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But then we found a real number, the evil number, which isn't on that list.발음듣기

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It contradicts our assumption that we really had assigned a natural number to every real number between zero and one.발음듣기

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So our assumption must be false.발음듣기

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And because what we assumed is the negation of what we wanted to prove, it follows that what we wanted to prove is true.발음듣기

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There are more real numbers between zero and one than natural numbers.발음듣기

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So we've identified two sizes of infinities so far.발음듣기

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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.발음듣기

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And we've identified a bigger size.발음듣기

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The size of the real numbers between zero and one.발음듣기

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Are there other things which are that bigger size?발음듣기

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As it turns out, yes. There are exactly as many real numbers as there are real numbers between zero and one.발음듣기

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And there are exactly as many points on a line as there are real numbers.발음듣기

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And there are exactly as many points on a plane as there are real numbers.발음듣기

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And there are exactly as many points on a cube as there are real numbers.발음듣기

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And there are exactly as many points in a hypercube as there are real numbers.발음듣기

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So, is the infinity of the real numbers the biggest infinity there is?발음듣기

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As it turns out, there are infinitely many sizes of infinity.발음듣기

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Another thing Cantor proved is that whenever you have a set, the set's powerset (in other words, the set of all subsets of the original set) is bigger.발음듣기

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So, the powerset of the set of natural numbers is bigger than the set of natural numbers.발음듣기

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And the powerset of the powerset is bigger than the powerset.발음듣기

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And the powerset of the powerset of the powerset is bigger than that.발음듣기

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And so forth with no end.발음듣기

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