Summer of Math Exposition

I originally found the cantor set rather confusing. Though I feel he explained each and every section very clearly, beautifully explained and woven together nicely.

Those who are new to or intrigued by real analysis. I would highly recommend giving this a watch.

I think this video has a lot of merits, but I will be that guy, and start with what I think is a big problem with this video:

One of the points of the video is that intuition is sometimes unreliable and that is the reason why we need to learn real analysis. The main example is this “number of times that the graph of a function crosses the x-axis”, for which intuition tells that it cannot be uncountable infinite but you provide the viewer with an example where it is the case.

However, I will argue that it is in fact you that led our intuition astray when you defined what you call “crossing the x-axis” for a curve. I think that, if you let reasonable person define what they mean by crossing the axis at $x$, they would say something like ”$f(x)=0$ and there exists $\delta>0$ such that $f(y)>0$ when $y\in [x-\delta,x)$ and $f(y)<0$ when $y\in (x,x+\delta)$, or vice-versa with the sign. For this definition of crossing, the intuition that you give at 26:31 that leads us to believe that this can happen at most countably many times is perfectly valid.

Instead, you gave a much more permissive definition for what crossing the axis means (which is still reasonable) BUT you failed to show in the visuals that this definition allowed for some more “pathological” behaviours to count as crossing (point out for example that with your definition, $f$ defined as $f(x) = 0$ if $x\leq 0$ and $f(x)=x\sin(1/x)$ for $x>0$ has a crossing at 0). Instead, you only showed cases that corresponded to the first (more natural in my opinion) definition of crossing.

I think that someone (for example a student, or even a mathematician) for whom you definition of crossing is clear would be way more cautious with their intuition, knowing that it does not capture well the very definition of crossing.

My fear is that somebody with the right intuition (for the naive understanding of crossing) who watched your video might start doubting their intuition when it is in fact valid.

Motivation The video is clearly motivated, although I feel like the actual question (“how many times can the graph of a function cross the x-axis ?”) that the video is answering could maybe have been stated in a clearer way, since you then spend 30 minutes building up to the answer. I feel like the bit about measure 0 sets was not strictly necessary to get to the answer, so it seems possible to me that a student lose track of what the goal was, at that point of the video.

Clarity One of the strong point of the video, I think that the explanations and visuals convey accurately the point that you are making throughout the video (except for the “crossing the axis” explanation).

Novelty I personally did not know the problem nor the construction of this particular function and I think that it is quite beautiful.

Memorability I think that the message of “the cantor set is weird and can be used to construct counter-examples to seemingly obvious facts” is what the reader will remember after watching the video, which is indeed a good thing to have in mind.

Other remarks The cardinal of the continuum is denoted by 2^{\Aleph_0}. The fact that 2^{\Aleph_0}=\Aleph_1 or not is actually independent of (cannot be proved or disproved from) the usual ZFC axioms that we use in standard mathematics. (You can look up “continuum hypothesis” to get more information about this).

I don’t think I can give much useful feedback as I am about to start my undergraduate journey and am not fluent enough yet with the topics involved in this video. However I think this video deserves a lot of praise. Despite not understanding every single concept I think that the presentation here is very clear and well executed, not to mention that the topics are fascinating and the flaw of one’s intuition at each moment is surprising and compelling. I also have great admiration for the length and detail of the video. You get the sense that a lot of hard work has been put into it.

Hey, nice job! This is a very nice piece of math, and I think you have some interesting ideas about how to communicate it, like how you avoid having to introduce the ternary expansion of a number between 0 and 1 for the Cantor set, talking instead about choices of “left” or “right” - I think both ways are good but the way you motivated that felt slick.

There are some things I might prefer to be different, for instance I think you rush through the idea that the existence of a bijection shows that sets have the same cardinality just a bit quickly, that’s a really key idea and I think it’s nice to include an example motivating why it’s true with some finite sets. (showing if our values all form neat pairs, then there can’t be more values left over in one set or the other) The decimal value you create for Cantor’s diagonal proof also doesn’t match the rule you defined - I understand it doesn’t really matter as long as you make it so the decimals disagree, but it could cause some slight confusion.

Anyway I think overall you’ve done very well, those were just a few things that got my goat a bit. Have a nice rest of your day

Despite lacking novelty, your video is a great recap of important results in real analysis. The explanations are very clear and the animations are also enlightening.

I like the title a lot.

The discussion of calculus as a prerequisite for analysis isn’t universal.

I don’t think it was totally necessary to define “set,” or “injective/surjective/bijective”. A lot of this stuff you can describe using informal words as needed. It took so long to get to the words in the title of the video, when you could have just started there.

At 11:45 you say “We call […] the size of the reals aleph 1,” which isn’t correct; that’s the continuum hypothesis and is independent of standard axioms.

The section directly discussing the cantor set was really well animated.

I think it would be nice to include a clearer brief definition up front of what it means for a function to “cross” the axis.

Overall I liked it! In real analysis, the proofs need to be carefully written to ensure their rigour. I liked the last Cantor set example that shows how a function can cross the axis an uncountable number of times. The one part that got me confused at first (but I figured it out later) was 13:21. I didn’t realize that the intervals could be split up. So inside my head I was thinking “Wouldn’t the set {0, 1} have measure 1 since you need an interval of at least size 1 to cover the two points?” But then I realized I was allowed to use 2 intervals.

I’m a few minutes into this video and I am already bothered by the audio. In the future, I strongly recommend investing in a real microphone. A good USB microphone costs less than $100 and will make the audio SOOOO much better!

I’m now 20 minutes into this video, and overall I’m finding it very interesting. However, I do note that all of the scenes are very similar…they have the same feel, the same type of visuals, and the same type of voiceover. I realize that this is, in some sense, the nature of the video, but I recommend that the creator try to find a way to create some “breaks,” or to include something that provides a little variety to the video. I fear that the target audience might feel like the goes on and on and become somewhat disengaged.

I’ve now finished the video, and my overall thought is that this is a really nice video. The topic is challenging, and the creator does a nice job of discussing the material in an engaging way. The visuals/animations are nicely done and really helped me understand what is going on. My critiques are listed above, and while they are relatively minor in the grand scheme of things, they are important enough that I am left feeling that this video is between good and great. But I want to encourage the creator to continue making videos because it seems like great videos are just around the corner.