What happens at infinity? - The Cantor set

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[zachstar]

This one came out much longer than expected as it was a more technical video, because of that I had to omit some things I'm going to include in this comment.

1) Since the cantor set is uncountable, that means there are points in it that are NOT endpoints in any of the C_n sets (I brought this up but never acknowledged the answer). In fact, the set being uncountable means there must be irrational numbers in the set since rational numbers are countable. 
2) One example of a point in the cantor set which is not an endpoint is 1/4, if you put a dot at 1/4 and move it down, it'll always be in the next C_n set but it will never be an endpoint.
3) 1/4 is known to be in the set because it has a ternary (base 3) form that does not include the number 1 (1/4 = .020202020.... in base 3). I never discussed this in the video but the cantor set consists ONLY of numbers in [0, 1] that can be written in base 3 form, without the number 1. (Note: Base 2 = binary, base 3 = ternary).
4) There's a cool property about the cantor set that can be proved graphically, and if you want a challenge try to prove it. Property: For ANY number between 0 and 2 (call it p), there exists two numbers in the cantor set (call them a and b), such that a+b=p. 
5) Sorry for anything that wasn't to scale, told the animator to make everything as proportional as possible but he needed room to write everything

[RCSmiths]

This is what makes science and math simply unforgettable for me. To learn about things like this is magnificent and surreal. Amazing!

[thetntsheep]

God: would you like to have length 0 or be uncountably infinite

Cantor Set: *yes*

[Adix_Null]

Teacher: this exam will be straight forward.
The exam:

[samuelthecamel]

Fun fact: LRLRLRLR... = 1/4 so it's not just numbers that have a denominator that is a power of 3

[victorscarpes]

The best way to think about the length is that every C_n has 2/3 the length of C_n-1. If you start with 1 and multiply by 2/3 on every step, C_n will have length (2/3)^n. Since the cantor set is the limit of C_n as n approaches infinity, the length is the limit of (2/3)^n as n approaches infinity, wich is also know as zero.

[helloitsme]

I like that this shows that a set can be uncountable and still have length 0, cause I've always wondered whether that was possible

[cobalt]

Fun fact: at 7:13 you mention that you could think of the elements of the Cantor set as binary numbers, with 0's in place of the L's and 1's in place of the R's. However, if you instead replace the R's with 2's, you actually get the element of the Cantor set written in base 3. That's another way to think of how to construct the Cantor set: you take all the numbers from 0-1 written in base 3, and at each step you remove all numbers with a 1 in that digit after the decimal point.

[alteskonto]

Wow the intermission part really worked well

[morkovija]

"No one shall expel us from the Paradise that Cantor has created" - David Hilbert 1925

[NoorquackerInd]

I can't believe binge watching 3Blue1Brown got me so ahead that I was completely ok with everything in this video

[sunimod]

12:03 I feel successfully intermissed.

[benweieneth]

3:17
Fun fact: The set of all fractions with denominators that are whole powers of three (i.e. the endpoints of the Cantor set) is a countable set. This means that *almost all* the points left behind are not endpoints.

[Andrewzero]

One takeaway from this is that "infinity" doesn't necessarily mean "all". Merely, infinity is an indicator that there is some sort of unending procedure that can be used to generate numbers, and this procedure can be used unendingly.

If you remove an infinite number of numbers between 0 and 1, there will still be infinitely many numbers left over.

And correct me if I'm wrong here, but the distance between negative infinity and positive infinity is the same as the distance from zero to infinity.

[kuchenzwiebel]

This testing if a number is in the cantor set reminds me of the "The terrible sound you never want to hear when working on turbine engines" meme.

[jimnewton]

One curious thing about the Cantor set, is that, yes, it contains all the endpoints of the C0, C1, C2, ... sequence, but it does not ONLY contain those endpoints. This is clear, because there are countably many such endpoints---they can be enumerated. The Cantor set contains uncountably many points which are not endpoints of any such interval.

[inverse_of_zero]

Mate, the end part on the fractional dimension was absolutely gold. I learnt the "box counting dimension" at university but it really glossed over my head when I was an undergrad. Your analogy is so simple to understand, that I'll probably use it to explain it to my students. Thank you :)

[sturmifan]

you are only the second one that explained the "I can always find a number thats not on your list" in an understandable way, thanks!

[alansmithee]

9:00
What baffles me is you could do the same thing with infinitely long decimal integers and discover that there are uncountably many of them.
Despite the fact that there are countably many decimal integers total.

[karan_jain]

It is easy to find a number written in the LRLR form. Just replace all the L's with 0's and all the R's with 2's, and you will get the number in base-3.
For example, 1/3 = = (base-3) = (base-10)
You could also go the other way around to find the LRLR form from the number.
For example, 1/4 = 0.25 (base-10) = 0.02020202020202020202... (base-3) = LRLRLRLRLRLRLRLRLRLR... in the cantor set.