If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

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

Part 2 of a pair. After part 1, you might have thought that all different infinite collections of things are the same size. Not so! In this video, Agustin shows us another of Georg Cantor’s results: that for every size of infinity, there is a bigger one! An example: there are way more real numbers than there are natural numbers.

Speaker: Dr. Agustín Rayo, Professor of Philosophy, MIT.
Created by Gaurav Vazirani.

Want to join the conversation?

  • leaf blue style avatar for user Judd Nelson
    Im having a hard time understanding how infinity could be of different sizes. This seems like a contradiction in terms. If something is infinite it has no boundary, to claim a size difference implies a boundary or an endpoint. Where am I going wrong?
    (23 votes)
    Default Khan Academy avatar avatar for user
    • female robot grace style avatar for user J Helston
      The video sort of has this feel to it, but try putting the word "list" on it instead. I can list all natural numbers {1, 2, 3, ...}. I can list all integers {0, 1, -1, 2, -2, 3, ...}. I can list all rationals {this is a bit more challenging, but the Part 1 video gives an argument for this}. There are infinitely many numbers in each of these sets.

      The subject of this video, the reals, cannot be listed. The author shows that if you attempt to list them, you will always miss at least one. So there are more than what can be listed, i.e. more than the 'other' infinity from my previous paragraph.
      (19 votes)
  • male robot donald style avatar for user G.R.A.
    what is a natural number?
    (7 votes)
    Default Khan Academy avatar avatar for user
  • leaf green style avatar for user Dusty Footed Philosopher
    What is a power set? He talks about them at the end around .
    (5 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user rhouts
      The power set of any set, S, is the set of all subsets of S.

      The power set of a set always includes the empty set and the set itself as elements.

      And the power set of any set (finite or infinite) contains more elements than the given set.
      (9 votes)
  • starky ultimate style avatar for user Sofia
    At , I really don't get why the 'Evil Twin' can't be on the list. Can someone explain to me... Why?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Nicolas Posunko
      That's the way I picture it. We imagine all rational numbers between 0 and 1 in the form of decimals ordered and put in one-to-one correspondence with an ordered list of natural numbers. And then we artificially construct this "evil number" in such a way that it cannot be on the list by definition.
      We define the number based on all the decimals we have listed. Each digit in the evil number is dictated by some digit of a decimal on our list (and made different from it). We can then argue that it makes our number different from every single decimal on our list, because at least one digit is definitely not the same.
      Hope this helps!

      *—See comments on the comment for further details : )
      (7 votes)
  • starky ultimate style avatar for user Sofia
    So there are infinity sizes of infinity, and there is no biggest size of infinity?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user tuannb1997
    Somewhat this video is even more confusing than the first one. I assume that you're trying to find a real number that's not supposed to appear on our list, called the Evil Number so that the Contradiction is proved => Conclusion is what you claim in the beginning. But 1): Why do you construct the Diagonal the way you show me in the video ? I mean, why necessarily have to ? And 2): I really don't follow why it is not possible for the Evil Number to come up in the list; perharps at some point, you may find a Real Number as follow: 0.75964..... I'm very stuck at these two questions, so I'd be grateful if you may clarify this problem ! ! ! Thanks for the video anyway.
    (2 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user Evan
      For 2: The Evil Twin can't appear in the list because of the way we defined it. The way we build the Evil Twin guarantees it has some digit that's different from each other number in the list, by first picking a digit from each row in the list, then changing each one of them according to some rule.

      For 1: I think for any diagonal you choose (not starting from 0,0 say) you can do the same trick to build an Evil Twin, guaranteed to have at least one digit that's different from each number in the list.
      (2 votes)
  • male robot hal style avatar for user Narankhuu Ganbaatar
    I still don't understand why the 'evil twin' cannot be on the list. Can someone explain this to me further?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user Agent Smith
      Let us say that we have listed the real numbers as below:
      1) 128919314189037.009384141515...
      2) 304879162.30143806126415...
      3) 5048239418.038401348152...
      and so on
      Say the list is now "complete" in the sense for each number 1, 2, 3, 4,...and so on there is a corresponding real number.
      Now let us try and construct a real number not in the list. If this is possible then the infinity of the set of real numbers > the infinity of the set of natural numbers. We can do this by following the steps below:
      Look at the real number corresponding to 1). Our new real number will differ from this in the first digit (say we use 3 instead of 1). So our new number is 3......
      Now look at the real number corresponding to 2). Our new real number will differ from this in the second digit (say we use 7 instead of 0). So our new number is 37...
      Now look at the real number corresponding to 3). Our new real number will differ from this in the second digit (say we use 5 instead of 4). So, our new number will be 375...
      If we continue this way, our new real number 375...will be a real number which is not in the list. Therefore, the infinity of real numbers > the infinity of natural numbers.
      I hope this helped.
      (2 votes)
  • mr pants teal style avatar for user Ian
    The logic with the whole evil number affair seems a bit shaky to me. There are infinite combinations you can use to make up the "0.xxx..." decimals that can put between 0 and 1 (hence letting there be infinite numbers between them), and eventually the evil number will show up. It doesn't seem very sound just to say that "in any given situation a number is more likely not to be this one." While that is of course true, that number will have to show up at some point. Actually you could take any number and say that at any given point your selected number is more likely to not be it. Can someone explain to me why this theory (Not mine) works?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user junk4sush
    In the proof by contradiction, why is the contradiction(which the author assumes to be true) that there is a one-to-one relationship between Set of Natural Numbers and Set of Real Numbers? The contradiction may also be that the length of Set of Natural Numbers >= Set of Real Numbers? In this case, we would be able to accommodate the evil twin.
    (2 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Fabien Ninoles
      Is the size of the set of Natural Number can be greater than the size of the natural numbers ?

      The whole proof here is based on the following assumption (the Hilbert's Hotel present in the part 1):

      - If you can assign each element a of set A to at least one element of another set B exclusively, you know that the size of set A is at least the same size as set B.

      Since I can assign all natural number to at least one real number (for example, himself), I know that the set of real numbers can at least contains (or is at least the same size) of the set of natural numbers which contradict the '>' part of your hypothesis. The video proof that, in fact, the size of real numbers is not only 'at least the same size', but in fact far bigger since there is, for any list of real numbers between 0 and 1, at least one other real number between 0 and 1 that isn't part of the list (which is kind of mind blowing by itself).
      (1 vote)
  • aqualine tree style avatar for user CodeLoader
    Amazing videos about infinity, but does anybody else think it's more about math than philosophy?
    (2 votes)
    Default Khan Academy avatar avatar for user

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