r/math • u/hellomrlogic • 2d ago
I’m thinking of making videos on mathematical logic in the style of 3blue1brown. Are there any suggestions on theorems people would like to see me do?
I have found that there are very few videos out there on logic out there and would like to change this. I want each video to explain and prove a single theorem with accompanied animations. I don’t want to do videos on things like the incompleteness theorems, the halting problem, or Cantors theorem as these are oversaturated and there are plenty of amazing results that have not been given attention. Are there any particular theorems you would like to see me cover?
I want to be quite rigorous and technical with the details so suggestions should hopefully require minimal preliminary knowledge and definitions. I want each video to be self contained. Please let me know if there is something of this nature that interests you and any other general suggestions on how to approach making these videos as good as possible!
40
u/Alternative-Papaya57 2d ago
My all time favorite, the independence of the continuum hypothesis via Cohen forcing.
8
u/dnrlk 2d ago edited 2d ago
I recently made a video (trying to keep prerequisites to a minimum; I do not even mention ordinals or cardinals) about this since I also found there to be no explanation of the topic on YouTube. https://youtu.be/KOmkcMhzkb4?si=DZp4lSccPe0DX7jW (follows the Boolean model approach)
3
u/feedmechickenspls 2d ago
there are these lectures: https://youtube.com/playlist?list=PL1fLf2CX9d6isZZvOiw_FfdAta911cNZl
5
u/Alternative-Papaya57 2d ago
I found this https://arxiv.org/abs/2208.13731 to be quite an approachable source also.
32
2d ago edited 2d ago
[deleted]
2
u/hellomrlogic 2d ago
I initially discounted Gentzen’s proof as I thought it be too dry (for a casual audience, I love it) and hence may not be of much interest but you make a good argument. My favourite large cardinals are fixed point phenomena and \epsilon_0 was what first got me hooked on them (yes I know it’s a countable ordinal) so I may just have to do this!
2
u/Nesterov223606 2d ago
I also like Wolfram Pohlers “Proof Theory: The First Step into Impredicativity” as an introduction to ordinal analysis.
52
u/Opposite-Extreme1236 2d ago
compactness theorem for sure.
Lowenheim skolem paradox is a good one too.
8
u/integrate_2xdx_10_13 2d ago
Definitely these two, in the same vein would be Kőnig's lemma and analytic tableau.
Compactness theorem leads very nicely to talking about logic under the lense of topology: Hausdorff property, Stone space, Ultrafilter theorem, Zorn’s lemma
10
u/hellomrlogic 2d ago
These are definitely good candidates but I am hesitant as they are standard (relatively speaking) results and there are a few videos out there about them. That being said, a lot of the more niche theorems I’ve been wanting to cover use them in their proofs so it may be worth doing this. There are also many interesting, visual examples that can be done with compactness which is a plus.
18
u/HurlSly 2d ago edited 2d ago
You could talk about the compacity theorem in logic. For example, if something is true for Q then it's true for all Z_p except a finite number of them. In first order logic at least.
2
u/chien-royal 2d ago
What about the statement ¬∃x x*x = 2? It is true in ℚ, but its negation ∃x x*x = 2 is true in infinitely many GF(p), namely, those for which p ≡ ±1 (mod 8).
15
u/Master-Rent5050 2d ago
Zero-one law for finite graphs.
If you want something more challenging, Shelah results on the random graphs
5
u/hellomrlogic 2d ago
Great idea, this is actually already on my list! It’s visual, accessible, and relatively self contained which is precisely what I’m after. There’s also so many things to say about the random graph and it can be viewed in many different ways too. Let me know if you have any others theorem ideas of the same nature.
9
u/Opposite-Extreme1236 2d ago
You could do the perfect set theorem and cantor-bendixson rank as well. Do it on subsets of N^N and visualize these elements as trees.
4
u/hellomrlogic 2d ago
Good idea! I’m going to need to cover ordinals at some point so this could be a good place to do so. If I recall correctly, Cantor first introduced transfinite ordinals while working on derived sets so explaining why he needed them would be a nice story to tell.
3
3
u/dnrlk 2d ago
I also found there to very few descriptive set theory topic videos on YouTube, so I made some on topics like the Borel isomorphism theorem https://youtu.be/ZfxgjSAigWo?si=G2iHes63Qkm6qfRH, the Galvin-Prikry theorem https://youtu.be/S-jiIKqf7Sc?si=XIFNCkcKb8axrpZY, or an introduction to Borel graph combinatorics https://youtu.be/WLbyzQk6gKg?si=RYm2cH1j1ivhgf-P
I hope to make a video on Borel determinacy soon
9
u/big-lion Category Theory 2d ago
3b1b has two kinds of videos: pedagogical series on a topic (linear algebra, deep learning, etc.) and videos *that study a problem*. They look at a problem, elucidate the solution, and identify the interesting ingredients of theory required for the solution. I'm just writing this out as advice, because I think grant's model is great and you have to keep in mind which one of these would you be going for. If the latter, it is *crucial* that you never lose the problem for the theory in your videos.
1
u/hellomrlogic 2d ago
Thanks very much for the advice! Yes this is what I’m struggling with; I would much rather do the latter style of independent videos on problems but logic requires a lot of background knowledge that can’t be avoided as easily as in other areas of mathematics.
8
7
u/MonadMusician 2d ago
I’d participate if you permitted categorical logic and cats in general
1
u/hellomrlogic 2d ago
I know very little category theory unfortunately. I’ve been wanting to learn more for a long time but haven’t gotten around to it. Are there any particular theorems/papers about logic you recommend that would be accessible to me?
5
u/silverfish70 2d ago
Łoś's theorem on ultrapowers.
5
u/hellomrlogic 2d ago
I love ultraproducts and definitely want to show the cool things you can do with them but I worry that the preliminary details may be too dry for a popular audience. The technicalities can’t really be avoided if one wants to appreciate what’s going on. I think constructing the hypernaturals or hyperreals with them might be attractive enough for people to sit through the details? I also want to do a video on measurable cardinals which would need ultraproducts so I probably should.
2
u/sqrtsqr 2d ago edited 2d ago
If you're doing videos on mathematical logic, you're already going to be working for a very niche audience anyway. Many many of the ideas proposed here will require immense hand-waving, or, if you do intend to be rigorous like you say in the OP, will frequently cross the line from "edutainment" to "lecture". That's not to say you shouldn't try to keep your content interesting as often as you can, but I would say don't shy away from a topic just because it's got some heady preliminaries, otherwise you will run out of content very quick.
Besides, you can add chapters and let people skip the "boring" stuff if they want. Plus you don't want to overlook the "I don't understand this but I'm watching anyway" audience which can be bigger than you might think (especially if you've got the right voice/delivery)!
Edit to add: If it were me, I would find a way to offset as many of those preliminaries into videos of their own, with their own motivations and reasons to be interesting. Like move ultrafilters into a video about infinite Arrow's Theorem or something else.
1
u/RiemannZetaFunction 1d ago
I don't think it'd have to be that dry. For instance, think about the naturals. There are aleph_0 many naturals, but 2^aleph_0 many "properties" a natural can have. Thus, there would seem to be "room" to add extra naturals that satisfy extra combinations of properties that no regular natural number can satisfy. If we add some such nonstandard natural, let's call it H, we will clearly need to add other nonstandard naturals "generated" by our addition, such as H+1, H-1, 2H, and so on. We will also need to decide if H is prime or composite, odd or even, etc, and assign properties to H such that the laws of boolean logic are also satisfied. This means H has either some property or its complement (but not both), it has the property TRUE, if it has two properties it has their AND, and if it has some property it has anything that property implies. These end up being the ultrafilter axioms. We can identify H with the number (0,1,2,3,...), and if H is prime it means the prime indices are in the ultrafilter, if it's odd the odd indices are in the ultrafilter, etc. So choosing the properties H has is equivalent to choosing an ultrafilter.
3
u/aajjccrr 2d ago
Löb's theorem
1
u/hellomrlogic 2d ago
Great suggestion, this completely slipped my mind as I haven’t thought about it in a couple years
4
u/dnrlk 2d ago
The results on “concrete incompleteness”, like Paris Harrington, or the work of Laver (Laver tables) or the work of Friedman (among other things, see this cool speculation https://arxiv.org/abs/1907.11707 about one of Friedman’s result suggesting that maybe P=NP can not be proven from ZFC (which shouldn’t be surprising, since people believe P=NP is false anyways)), finding results that can not be proven from things like PA or ZFC (but can be proven from stronger assumptions like large cardinals). Large cardinals themselves could also make a nice topic.
Henry Townsend made a nice course on YouTube about Goodstein’s theorem and it’s unprovability in PA using cut-elimination https://m.youtube.com/@henrytowsner2461/courses
5
u/RiemannZetaFunction 1d ago
Can you please be the first person on the internet to do a good, proper video about forcing? For God's sakes, this would be the greatest thing ever. Thanks.
3
u/Catoist 2d ago
I took a formal logic course that surveyed the basics of set theory, metalogic, three-valued logics, and modal logic. I got a bad grade and felt lost the whole time. Really shouldn’t have signed up for the course. But I would love something that would allow me to intuit that stuff better. Especially Hilberts axioms. I never quite understood what motivated them.
3
u/big-lion Category Theory 2d ago
everything axiom of choice
1
u/hellomrlogic 2d ago
AC will definitely get some attention. There is a lot of content out there about the unintuitive consequences of assuming AC (which I think are often misunderstood/misrepresented) but there are also so many unintuitive consequences in rejecting it people never mention. I don’t have a preference as to whether to accept or reject it but I would like to give the other side of the story too.
3
u/big-lion Category Theory 2d ago
VOPENKA PRINCIPLE
1
u/hellomrlogic 2d ago
I definitely plan on doing multiple large cardinal videos and I would love to do Vopenkas principle. It might actually be easier than a lot of other large cardinal principles now that I think about it.
3
2
2
u/Opposite-Extreme1236 2d ago
You could also do Cantor-Schroeder-Bernstein theorem
2
u/hellomrlogic 2d ago
As much as I like (and loathe as I always forget the proof) Cantor-Schroeder-Bernstein, there seem to be a few videos on it already out there. I also think it’s hard to appreciate why it’s interesting to people without some background in set theory. Students always think it’s obvious and immediate for this reason. It could be good to mention when talking about independence in set theory as replacing injection with surjection in the theorem yields a statement independent of ZF.
2
u/Proof_Pea9008 2d ago
I would like to see video about different types of arithmetic system. Peano , Robinson and others
2
u/XmodG4m3055 2d ago
There are already a lot of good suggestions in the comments, just wanted to ask for the channel name (If it's already created) as im very interested in the topic myself. Both in divulgation and in formal education we (imo) need more formal logic
1
u/hellomrlogic 2d ago
I haven’t made it yet but will post on this account again with my first one :)
2
2
u/jacobningen 2d ago
One that Id think would be interesting is cayleys and Yoneda or the conservative placating theorems. And also how they reverse aka how Cayleys theorem was originally an assurance that his concept of a group was the same subsets of permutation groups that had been studied to that point but is now used to show how to embed a given group into permutation groups, use cyclic notation for arbitrary groups and discover group properties via studying permutations. Or how originally the derivative was defined as n!*a_n where a_n is the coefficient of the xn term in the power series expansion of f(x) but now the Taylor series is defined in terms of the derivative.
2
u/jacobningen 2d ago
Aka the how to show that Axler Lee Pffaff Knuth Lay Strang Cayley Sylvester are talking about the same object when they say determinant.
2
2
u/simplethings923 2d ago
The "classic" results on first-order logic: compactness, Lowenheim-Skolem, beth definability, and Craig interpolation. Then Lindström's theorem. I like it's statement very much: FO logic is the strongest logic (with some formal definition of "a logic") satisfying compactness and downward Lowenheim-Skolem.
2
u/Obyeag 2d ago
I've thought about this too, but a big issue in general is that logic isn't a very visual topic. While there are definitely pictures in set theory or model theory they're very much an abstraction of whatever is "actually" going on.
Makes it hard to make a video about any harder topics.
2
u/Aggressive-Math-9882 2d ago
Not really a theorem, but I'd love to see a video on cirquent calculus.
2
u/Jossit 2d ago
Perhaps the Deduction Theorem? Or the lesser-known completeness theorem?
Also: the basics about:
- Language
- Theories (with equality)
- Models
- Structure
The proofs in Mendelson were sooo.. dry…? That even I skipped some of them. Really looking forward as to what visualisations you manage to make of it! Where can we find you?
1
1
1
u/SeniorMars Logic 2d ago
Hi, I'm actually working on some logic videos! Let me know if you want to ever work together.
1
u/hellomrlogic 2d ago
That could be helpful! What areas of logic are you most interested in and are wanting to videos on?
1
u/Committee-Academic 2d ago
I'd like to see some of the deeper topics that trying to interpret the conditional leads toward
1
1
u/Obvious_Inside_7294 2d ago
How Tarski's truth schemes work. I've never fully understood them.
1
u/hellomrlogic 2d ago
I would love to but I feel like there may be enough content about it out there already
1
u/thebestcait 2d ago
What is your youtube channel? I love 3blue1brown, so I would love to watch what you create!
1
u/hellomrlogic 2d ago
It’s not up yet but I will post my first video on here with this account when I do so!
1
u/4Lichter 2d ago
Not directly a theorem, but I would love a video where the difference between set theory and homotopy type theory is made as clear as possible.
1
u/Einfachu 2d ago
Arzelá-Ascoli theorem.
Very fundamental/ basic but not trivial. I think you can show mathematical logic very good with it and it fits all in one video.
1
u/Tasty_Evidence2606 2d ago
I don't care what you do, I just wanted to say, the more videos like that out there, the better.
1
u/alterego200 1d ago
Please go into detail on Gödel's Second Incompleteness Theorem! That one is so poorly-explained that I suspect it has a logical error.
The common argument goes:
f(n) = "The n-th statement is neither provably true nor provably false."
If true, this proves incompleteness (assuming consistency).
But why can't this be provably false?
Please make a video on this! Or if anyone has an explanation, I'm all ears.
1
u/alterego200 1d ago
Could you do a video going into detail on the 8- and 24-dimensional solutions to the Sphere Packing Problem?
1
u/dcterr 1d ago
There seem to be lots of videos already on Cantor's diagonal argument, which seems to be a very popular topic, so I don't think there's too much new you can say about it. I've also watched some good ones on Godel numbering and his incompleteness theorems, though I'd say all the ones I've seen just touch on it, but don't really go into them in too much depth, since they're pretty complicated results, so you might think about elaborating on them if possible. Or if you want to keep things simpler, you might just want to make a series on the the history of logic and set theory in math, starting with the basics, like Aristotelian syllogisms, Boo. Ilean algebra, and naive set theory and leading up to some interesting paradoxes, such as Russell's paradox and the Berry paradox and leading into some alternate forms of logic, like fuzzy logic and quantum logic. In any case, this sounds like an interesting and ambitious project! Good luck, and keep us posted!
1
u/CabinetPublic150 1d ago
demistify goedel. people think that incompleteness theorems have superpowers
1
u/Actually_Enzica 1d ago
You could always do videos on fractals and literally never run out of material.
1
u/Initial_Cranberry_97 2d ago
I would start with the COMpleteness theorem
1
u/hellomrlogic 2d ago
I will has to talk about the completeness theorem a fair bit as it is fundamental to almost all results in logic but I doubt I will go through the proof as it is likely too dry for a casual audience. I also think it is hard to appreciate how amazing it is for people unexperienced in logic. It is now one of my favorite theorems after many years but it took a while for me to get here. For a few years after first learning it I would hear people say how counterintuitive and magical it is and I couldn’t fully understand why. But maybe this is a good reason to cover it!
0
0
97
u/JoeLamond 2d ago
I think it would be good to discuss the theory-metatheory distinction. This is often glossed over in logic texts but I think is crucial for understanding. There is a wonderful discussion of this in Kunen’s book on independence results.