What happens at infinity? - The Cantor set 

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

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

God: would you like to have length 0 or be uncountably infinite Cantor Set: *yes*

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

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

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

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

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.

12:03 I feel successfully intermissed.

Wow the intermission part really worked well

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

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.

Oh yes, time to get confused

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.

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.

im jens herro

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.

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 :)

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.

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 = LRRRRRRRRRRRRRRR = 0.0222222222222222 (base-3) = 0.33333333333 (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.

Bro, youre a hell of a teacher. I listen to your videos while driving (being a father and husband really limits the time frame i have to focus on myself, so i do what i can) and despite that, im able to grasp everything youre talking about clearly. I wish i had a teacher that was as good as you are, back when i was in school, maybe i would have pursued higher education.

This is great stuff. Hope you get much bigger in the Mathematics RU-vid sphere. This is good for showing one of the absurdities of infinite sets and real numbers. Personally, I like to refer to things like this to complain about the naming convention of "real numbers" and "imaginary numbers". The reals are completely nuts!

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!

Thanks for being an awesome channel

Lovely!! Thanks, Zach.

Awesome explanation! Thanks. I have watched both vsauce and numberphile. I personally like your channel better. Keep up the good work!

I read about this in Chaos: a new science. Nice!

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.

This is a wonderful video. Well-done.

if a set has dimension n

I doubt 0 has more L’s than my life

I had never heard of fractional dimensions before. Mind expanded. Thanks!

"The size of the Cantor set gets doubled when its side length is tripled." Could you please elaborate on how do you define double here? If you use cardinality, it is always that of R; if you use Lebesgue measure, it is always 0.

I just was researching fractional dimensions this morning, and this video comes out no more than 6 hours later. What a coincidence

You've made a very good video. Congrats!

Please.. Continue your efforts to increase curiosity in every student about science and math...I thank RU-vid for recommending your channel😍

The Cantor set (or Cantor discontinuum) is the set of all numbers in an interval whose base-3 representation does not contain the digit 1. Then, how about the set of all numbers whose base-3 representation contains finitely many digits 1? It can be seen that this set is a union of countably many scaled down copies of the Cantor set; and so it has measure 0. (Basically: every gap in the set is filled by another copy of the Cantor set.) It's also a dense set (and, of course, uncountably infinite like Cantor set itself). I don't know if this set has a name, or what are its other topological properties.

this and your last video were awesome dude -- really accessible introduction to Hausdorff's definition of dimensionality, and answered some of my musings about the Cantor set that I haven't had time to investigate. thanks!

Great video. I'd like to see another one that explores this topic even more At about 9:58 you said that the Cantor set has the same cardinality as the real numbers, but I think this statement may involve the assumption of the continuum hypothesis. You showed that the Cantor set is uncountable so it has greater cardinality than the countable rational numbers. I think the continuum hypothesis states that there are no orders of infinity between the cardinality of the integers and the real numbers, but CH has been shown to be independent of the other axioms of set theory.

This video should get love as much as the size of cantor set...❤

I remember seeing this is college. Fun times!

That's a really good way to think about dimensions!

Very simple explanation.. thanks a lot sir

very neat introduction to the properties of the Cantor set. wish it had existed when I was learning analysis for the first time.

This is soooo amazing 😍

Really amazing video

12:00 That's really nice man, I appreciate it

Your videos are so underrated.

11:50, never been mad before for having RU-vid premium

It is almost like Infinity raised to the infinity and that pattern itself also expands infinitely! Dang. This truly gives me a whole new perspective of numbers! :)

A genially simple way to bring our thinking to a higher level.

great video, thank you very much

I've just invented the non-inclusive cantor set, where instead of including the endpoints you exclude them. The full list of non-inclusive cantor numbers is as follows:

There is an error in the video about how to conclude the infinite size of the cantor set. The mapping of the endpoints to strings of ones and zeroes makes it a countable set which is much smaller than the set between 0 and 1. However the numbers between the endpoints remain uncountable and is therefore much larger.

Sounds like somebody's taking Intro to the Theory of Computation this semester lol (such as CS520 if you go to UW-Madison)

the intermition. great idea!! 👍❤️😃

We spend 1/3 of our lives sleeping, and the other 2/3 in the dark.

I’m a math teacher who spent way too long in college and graduate school. Congratulations, you finally got me to care about the Cantor set!! 🎉

Amazing dude👍🏻👍🏻👍🏻

Wow definitely many teachers should play this before a given class but it brings too many things together I guess hahaha yeah wow excellent illustration of fractal dimension calculation in particular! Thank you!!

Hi Zach, Another great video by you! This may be just semantics, but I'm a little bit unsure of the use of the term "fractional dimension" here. Sure, the dimension number is not an integer, but (ln(2))/(ln(3)) doesn't seem rational either... would it not be more appropriate to refer to it as "irrational dimension"? I think doing so would make it distinct from some of those other fractal-like examples that have "fractional dimension" that are rational numbers. Or have I just taken it way out of context?

Very fascinating! :)

Very nice video, I have a question as to what would the relationship be with a cantor set and an aleph number?

It's worth noting that the LRLRL... example you gave corresponds to 1/4, which is in the Cantor set. So, adding to the weirdness, although all the endpoints of the intervals have denominators that are powers of 3, in the limit, the Cantor set contains rationals whose denominators are not power of 3.

I think your argument for uncountability is missing something. If we assumed that only the endpoints are part of the set then it would definitely be possible to enumerate them: - On the i-th iteration (with i > 0) you are adding 2^i new endpoints, which you can easily enumerate from smallest to largest. - This means that before the i-th iteration you would have 2^i points already in the set. Then you could assign labels [2^i + 0, 2^i + 1, ..., 2^i + 2^i - 1 = 2^(i+1) - 1] to the new points. It is easy to show that this would be a bijection between the endpoints and the naturals, which would mean that the set would be countable. The reason why the cantor set is uncountable is because some points that are not endpoints are also part of the set. The actual proof uses a similar idea. First you look at the expansion of the number in base 3. Just like numbers can be written in base 10 like 0.1 or 0.9562, you can think of numbers in base 3 being written as 0.0. 0.1, 0.2, 1 and so forth. A number is only ever removed if a 1 appears in the numbers after the decimal point. 0.1 in base 3, the equivalent of 2/3 in base 10, gets removed in the first iteration and you can see that in the notation. This means that the cantor set is essentially the union of all numbers with only 0s and 2s after the "decimal point". Then you can use the same argument of having infinite lists of numbers.

YOU ARE THE BEST !

That’s beautiful

But this set has to be countable if we agree on the structure of the numbers, and we can easily show that every number in this set has to, at some point in the LR representation, become stable and exclusively have L's, or exclusively R's from that point on, that position would also denote the C_n on which it was added to the set. This also means that if you attempt to create a representation as shown in the counterexample of flipping the L/R at the specific position, you would never reach a point in which the tail stabilises at exclusively L's or exclusively R's, thus the created counterexample is not in the set (by definition we would never reach the C_n in which it would be created, because at some point it would flip positions from L to R or back.

Me at 2 Am: Casually studying about how Cantor Set has log_3(2) dimension

Coming to the realization that each split of the cantor set becomes its own cantor set without him having to tell me was a fun thing

Where do you have the zoom into the cantor set animation from? I can't find anything even close to it! It's mesmerizing

The length being zero isn't surprising at all if you realise how length is defined. It doesn't have much to do with the amount of points in the set. Those remain infinite throughout. It's just the sum of all the max - min of largest subsets that are continuous in nature, at each step. At first step, all points in the range [0,1] are included. So the largest subset that is continuous is [0,1] itself. 1-0 = 1. At second step, the largest subsets that are continuous are [0,1/3], [2/3,1]. 1/3 - 0 + 1 - 2/3 == 2/3. So it's no surprise that in the limiting case, when all the points become discrete and are no longer continuous, the largest length is the length of a single point itself. Which is x - x == 0.

i think you can use the LR explanation for finite numbers. personally i think rounding the number down is how this can work. for example, if i have 001, that's LLR which is any number between 2/9 and 1/3 so if you were to round down it can just be 2/9. EDIT: i am currently trying to make an encoder and decoder to turn strings into numbers and that number back into the string data. i have completed the encoder, but the decoder is a bit difficult. this, however, is possible.

I am absolutely scared shitless when it comes to arithmetic. I've never been good at it so never tried. As a college student with degrees I can honestly say I have calculators going back some years. Being one of the 1st in my area to carry a mobile, I did so because they had a built in calculator. So why do I love these types of video?? They more convoluted the better. I'm just starting to get it.

when a is an integer, a=3^n, and m

Bro this would have been nice to see last year in my Dyamics and Chaos theory class. That w/o a doubt the hardest math course in my life.

Hey Zach, can you do a video about Nuclear Engineering, or it's careers?

ok before watching the video heres my initial thought: any number between two rational numbers gets removed eventually... given any number we should be able to come up with two rational numbers whos denominator is a power of three and gets removed eventually. so all numbers get removed? (besides endpoints)

I had seen the 3Blue1Brown video on fractals but this explained the dimension thing way better

Finally! Someone that can include their mid roll ad spam at a break point that makes some sense

You can arrange every element of Cantor set to each of the elements of the set of numbers eliminated by only the first iteration. That means that what was removed from Cantor set is power to infinity larger than the Cantor set itself. How can we tell that the Cantor set size is the same as set of Rational numbers when we only considered a subset of Rational numbers?

What is the background music in the beginning of the video

That makes me wonder whether a function that maps elements in cantor set to 1 and those that are not to 0 would be riemann integrable since the set of points of discontinuity have length 0 and if so what such integral would equal

9:10 This however works only because you have a length of the number that is equally long or longer as the entries in the list of numbers you have. So in fact, you can never find any new number that way, as you would need an infinite amount of time to create one, i.e. your attempt to create a new number would never end, and as such no new number at all would be made.

I got a dominos ad right after staring at the screen while zooming into the cantor set and now I really want pizza.

Is this true that the Cantor set includes only multiples of powers one-third? I my understanding is that the Cantor set includes mostly irrational numbers and many rational numbers that are not multiples of integer powers of one-third. The usual example is one-fourth, which never gets removed. If it actually included only multiples of powers of one-third, there is no way it could be uncountably infinite because it would be a subset of the rationals, which are countably infinite. You'd would be able just count the rationals and skip the ones that are removed.

The most intriguing for me is that probably infinity just exists in math but not in the real world, assuming that the space-time is discrete rather than continuous, and maybe that could be the reason of the tiny errors in many mathematical models of physical systems

Very cool and interesting video

it's all the numbers inbetween 0 and 1, including 0 and 1, that, in ternary, don't have a 1 in them, or end with 1. ex: 1/3 is 0.1, 2/9 is 0.02, 0 is 0, and 1 is 1.

Amazing explanation...............

This is insane. Fractional dimensions? I’ve never even heard about that before!

Actually you can do the binary flipping thing with Integers as well. You could just add one to every digit and if you have a 9 you wrap it around to 0.

This looks like one of those cube fractals

My question is what makes any starting point for counting unique? Meaning how can we be sure that the location selected is really the value of 1? Could we not always move the starting point for counting over any arbitrary amount? Also could you not make a subsequent arrangement where we only take the endpoints, divide the length into 10 divisions , only taking each end point. And repeat that process again with just dividing the distance between two points into 10 units. Just keeping the end points each segment. And eventually if we kept doing this we would start with 2 points with a length of 0, and we would build up to the infinite number of points between 0 and 1 giving our line a length of 1?

Why did the youtube algorithm gods hide this channel from me for so long? Amazing content my dude!

Just some notation before I state my question. when a number ends with _3, that means base 3. Would the number 0.02222…_3 be in the Cantor set? What confuses me is 0.0222…_3 = 0.1_3 for the same reason 0.999… = 1. Even though its ternary value on the left never contains a 1, its value on the right does have a 1 in its expansion. So is it in the set or not?

This strikes me as being vaguely similar to the Zeno paradox...in a different context though of course...imagine Zeno trying to get his head around the idea of uncountable infinities...never mind not being able to make the final step, he would not even be able to make the first lol

that intermission music didn't make me stop wondering about infinity

I was just discussing how there are different sizes of infinity and actually used integers and fractions as my example, stating there are more fractions than whole numbers, going into detail about the infinite number of fractions in between 0-1, and then i see this video the next day😂😂 hell yeah

Great intermission lmao