How Infinity Works (And How It Breaks Math)

In which we learn about the cardinalities of sets, how they can be used to make a system of infinities, and how it all uncovered the limitations of modern mathematics.

Reuploaded due to a small technical error.

#SOME3

[francescomussin]

This video is amazing, really, but I've got a few little things to note:
- 9:37 I find this a bit misleading: formally the sqare root of a non negative real number x is defined to be the positive root of the polynomial p(t) = t^2 -x. Saying that sqrt(4) is equal to both -2 and 2 would not make sqrt a real function, as it gives off more than one real number.
- 17:30 I've heard a buch of times this explanation of limits, but I'd argue it is a bit off: imagine the real piecewise function f defined to be f(x) = -x for x > 0, f(x) = -x+1 for x <0, and f(x) = 1/2 for x = 0, then as x gets closer to 0, f(x) gets closer to 1/2. The thing is that it doesn't get arbitrarily close to 1/2. I would explain limits by saying (using your example) "f(x) can get as close as you want to 1 as long as x is close enough to 3". This, I think, better summarizes the formal definition of a limit, and it is what mathematicians think about when they prove most limits in Functional Analysis.

[fyu]

My favourite fact is about cardinal numbers is that you can prove there are so "many" of them, that the existance of the set containing all cardinal numbers leads to a contradiction simmilar to Russel's paradox, hence there is no set of all cardinal numbers

[decb.]

I love the sound design in your videos, it reinfprces the visuals and makes everything feel more real.

[VRX]

Dude, you are bound to have 100k by the end of this year. Your content its like no other. Keep up the awesome work 👍

[arbodox]

The presentation and visualizations in this video is absolutely phenomenal! This is probably the clearest explanation of the cardinal numbers I've come across yet.

[XT-N]

As a math student I already knew most of what was said in the video, but honestly I'm still very impressed by the editing and the amount of content you managed to fit into a 20min video.

As other people have pointed out though, the square root function is not usually defined as a multivalued function, and it only outputs positive numbers.

[Ashinle]

Love the direction and editing of your videos. A ton of effort that works perfectly with the information being taught.

[duukvanleeuwen]

This video is extremely well put together. Even though I was already familiar with most of the concepts in this video, I still really enjoying watching it, because it gives a very nice understanding of how each topic is related to one other. With those clean animations, this video encapsulates a variety of topics explained in the best way possible. Please make more videos like this.

[orisphera]

My notes:
I think of set cardinalities and limits as different meanings of “infinity”. How correct this is may be debatable
There are also ordinal numbers. These are ordinalities (if that's the term) of well-ordered sets. A well-ordered set is a set with a certain relation called order, and for two of them to have the same ordinality, they must have a one-to-one correspondence that preserves it. Normally, the order is defined as a relation that tells if one element comes after another one in some way. Two elements can't both come after each other, and a<b<c implies a<c. For it to be well-ordered, the order must be total, meaning that out of every two different elements, one comes after the other, and each subset must have a minimal element. You can also define well-ordering through the minimal elements, with the axiom that the minimum of the union of some subsets is the minimum of the minimums of them

[twixerclawford]

This is such a good, high quality video. The fact it doesn't have a million views already is a crime

[ilmorifajt]

Yoo, congrats on the honorable mention in SoME3. Totally deserved.

[emanuelbatalla]

Amazing content. I've watched a lot of math content and it's safe to say this is one of the best math videos I've ever seen

[tanchienhao]

Wow this is an amazingly accessible video that covers nontrivial subjects like ZFC and CH too!

[Dysiode]

I'm super stoned and I was able to follow along the whole time. Props!

Cardinality of infinite sets is super interesting, but it's so hard to visualize what sets represent higher cardinalities, similar to trying to visualize higher dimensions

[calmcat]

this is seriously the best video i've seen about the topic of infinity and its different "sizes". a lot of these sorts of things have a kind of "this is what is it is because i said so and you shouldn't worry about it" but this one really tells you why everything is the way it is instead of glossing over all the details. thanks for this.

[borahsilver]

Man, what a great video. I like the way you don't go for the low hanging fruit of "dramatic statements" like "There are as many fractions as there are integers!" just to wow people but instead decode that that statement is based on an abstraction of the idea of "as many". Jeez, there's a lot of pop sci writers who just want the wow factor and leave the audience impressed but still confused. Subscribed.

[QuantumHistorian]

The slight tangent on semantics and using old language in new situations at 2:57 is amazing. Briefly explaining that what words means changes on context is not only useful to keep in mind in many conversations, it's also a direct parallel between how languages grow and the process of generalisation in maths. And, in both cases, people often stumble and get so confused that they cannot proceed further. Yet a few short words explains clearly exactly whats happening. Kudos.

[driesclans]

This channel is going to blow up the next couple of months! The video quality is so high, very under appreciated atm

[DownDance]

I like this style of video/animation and the sound effects are so satisfying imo

[NikolajKuntner]

16:22 I feel like the common framing of "without assuming choice, math is harder" has the flavour of a self-fulfilling prophecy. Your theory T will generally have models that are not models of the theory with more axioms, T+{A}. Since choice is familiar from us from the finite realm and people developed math largely in choicy frameworks (e.g. in the guise of Zorn's lemma in algebra), much of the math you encounter at uni is that sort of math which feels lacking without choice. Now for that bias, the models that break choice are less well-investigated for it. In the sea of possible mathematics, what's truely the size of that in which full choice function existence is natural. If you ask your average software engineer at google, he will likely not even know or be able to come up with any mathematical problem or theorem that cannot be modeled in first-order arithmetic extended with finite types (N a type, function types A->B and disjoint sum types A+B, for all types A+B, iteratively) and maybe dependent choice. Even if admittedly a Fourier transform (R->R)->(R->R) implemented in Haskell is not a true reflection of the concept in measure theory. Big cardinals beyond a dozen powers of |R| are on nobodies radar. This was just a short rant about why choice is maybe more historic than a necessity for "most math". But while you were browsing though the 1930's results, pointing to Rice's theorem and the ilk could be used to connect it to some more impactful, in a "practical sense", issues. Nice video.