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## Wireless Philosophy

### Course: Wireless Philosophy>Unit 2

Lesson 1: Metaphysics

# 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?

• 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? • 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.
• what is a natural number? • What is a power set? He talks about them at the end around . • At , I really don't get why the 'Evil Twin' can't be on the list. Can someone explain to me... Why? • 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 : )
• So there are infinity sizes of infinity, and there is no biggest size of infinity? • 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. • 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.
• I still don't understand why the 'evil twin' cannot be on the list. Can someone explain this to me further? • 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.
• 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? • 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. • 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)
• Amazing videos about infinity, but does anybody else think it's more about math than philosophy? 