Summer of Math Exposition

I appreciated the topic of the continuoum hypothesis. Great ideas. A bit over my head but wonderful video

I do not have much background in number theory or set theory. I am somewhat aware there are different sizes of infinity. I know the words ordinal and cardinal appear in this context, but little more. I initially considered skipping this video. I think the topic choice is strong. The idea of a unique infinity certainly piques my interest. The video falls short on accessibility. For over an hour, the presentation is almost entirely dense blocks of formal set-theory notation and written paragraphs on a black background, with very little visual support for someone unfamiliar with the subject. I have a technical background and I found myself lost, which suggests a viewer with less technical background would struggle significantly more. The rigor seems to be there but it is unapproachable. I’d encourage the creator to think about where a diagram, an animation, or even just a worked visual example could replace slides of notation. The length could come down significantly once the concepts are shown rather than rigorously written out.

The topic is definitely interesting, and I’d be happy to follow it.

I would prefer it to be divided into several videos, so that they could be watched independently, perhaps also depending on the viewer’s level of prior knowledge.

The explanation is clear and the animation is simple. Some of the slides are excessively packed with formulas, which makes the listening experience heavier and reduces attention. A bit of background music would help.

Obviously too long to be of any use. And there is zero motivation. You start the video by saying “We construct a new universe using forcing” and you already lost me.

Starting a video with “Part I” is not really a good look. I rated this video as low as I did mainly because I only finished Part IX, and gave up at Part X. If I was following correctly, this means that I did not actually make it to the part where you talked about forcing. It is perhaps most problematic that, despite skimming the rest of the video, I’m not actually able to determine if that is true. For the remainder of this feedback, I will try to bracket this concern and review only what I did see.

At first I was a bit confused about your target audience, but it cleared up fairly quickly: by part III I decided this video is aimed at very patient undergraduate majors who had some abstract algebra experience. As much as I want to criticize the video for being long, it’s probably about the length that it needs to be. Still, this is maybe not unqualified praise: its length really limits the video’s reach.

Maybe more to the point, this video scans as a reading of a textbook. I don’t mean to downplay the effort you put into the preparation of the material, but i just say this to suggest I don’t think that the video format adds much to the exposition, over what I’d get from simply adding some connective tissue to the formulae you post on screen and publishing as a PDF. By the time I got to Part X, I was pausing frequently to understand the density of information on screen, and the format is actively working against us.

These criticisms notwithstanding, I did in fact learn something from this exposition. I had not heard of boolean-valued models before this, and the idea seems quite interesting. I had heard that posets/algebras got used in logic and foundations, but hadn’t understood how they entered into the picture, and I think I appreciate that now at least. The construction(s) of V^B is very cool and I’m glad you exposed me to it. This video was clearly a very ambitious labor of love, and I hope we see you putting that energy into other projects in the future.

The author of this video decided to take upon himself a daunting task: explain Cohen forcing from scratch.

Fortunately (or not?) for him, his video was assigned to a reviewer that already knows Cohen’s construction, so I can vouche that this video is surprisingly accurate.

That being said, it also seems much too challenging. For example, he basically condences five weeks of FOL and model theory into a six minutes exposition, and that’s before the video is even third of the way through! And not putting chapters on such a long video borders on crimes against humanity. You even have title screens! C’mon man, it takes two mintues!

I enjoyed this video a lot, as it provided a great revision of topics I used to love. I’m not sure how accessible it would be to someone who never learned axiomatic set theory, even if they have a B.Sc in mathematics.

To the author, I have a very specific recommendation: I bashed my head several times from several angles before I managed to break through to forcing. The approach you present is the highly abstract version based on boolean algebras (following Jech)? I found the combinatorial approach as presented in Helbaisen’s “Combinatorial Set Theory: With a Gentle Introduction to Forcing” much more pleasant as a way to get first bearing on Forcing. Obviously, you would have to learn the more modern approaches if you want to graduate to more advanced topics such as PCF theory or minimal extensions. But I think this is a good starting point for people who want to get a feeling for the subject without drowning in details.

Rare set-theory topic and hard-core. Could be a bit more entertainment.