How Infinity Works (And How It Breaks Math)
Вставка
- Опубліковано 4 лип 2023
- 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.
Some sounds from freesound.org:
freesound.org/people/newlockn...
freesound.org/people/Lukeo135...
freesound.org/people/OwlStorm...
freesound.org/people/Kane5312...
freesound.org/people/ngruber/...
freesound.org/people/deleted_...
#SOME3
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
+1
"functional analysis" or in just undergrad analysis
I find it unnecessary and pedantic to talk about polynomial for sqrt its not even right if I'm nit picky p(t) is the associated polynomial function with P(X)=X^2-x, polynomials are not useful here. Sqrt(x) when x is positive is just the positive number t such that t^2=x period. And of course functional analysis is a totally different field than real analysis which is more appropriate for the subject here.
The square root thing also caught my attention and from my very brief research I managed to gather that while square root of 4 is ±2, the symbol √4 is not really used to indicate a square root but rather a principal square root. So basically what is said in the video is correct but what is showed isn't, at least if I understand correctly but I would love for someone to correct me if I'm wrong
That is a misperception of the definition of a square root. Every nonzero complex number has exactly two square roots. If the argument is a positive real, then the positive square root is called the principal square root. The *radical symbol*, by convention, denotes the principal square root. But the phrase "square root" applies to both roots. en.m.wikipedia.org/wiki/Square_root
I think I remember from far too long ago that this very point was dealt with by defining the sqrt of a positive real to be indeed positive - but that was the 'principal square root'; without prejudice as to whether any other number could also square to the original number. And of course, in the Reals, the negative of that positive root also squares to the original number. In the complex plane, life is far more interesting!
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
The alephs map the ordinals onto cardinals one to one, so that makes it less surprising, yet still perplexing given how "sparse" cardinals are within the ordinals.
The cardinality of a countable infinite set (e.g. all natural numbers) is not a natural number, but Beth_0, whereas the set of ordinal numbers of a countable set is the set of natural numbers. The set of cardinality numbers of all non-empty finite sets is the set of natural numbers. The set of cardinality numbers of all infi ite sets seems to be the set of Beth_n where n is a natural number. Now the question is why all possible cardinslity numbers wouldn't form a set.
@@micknamens8659 Because if you union a set of cardinals, you get a cardinal greater or equal to all of the cardinals in the set. So if the cardinals are a set, you can union them all to get a cardinal greater than or equal to all cardinals, a maximum cardinal. But there is no maximum cardinal, a contradiction. Presumably you knew that already, though.
@@stevenfallinge7149 What do you mean by "to union a set"?
@@micknamens8659 It's the union of the elements of a set. Formally, the union of a set A is the set S such that x∈S if and only if there exists Y such that x∈Y and Y∈A. For example, the union of the set {{1,2},{2,3,6},{7}} is the set {1,2,3,6,7}. Here, it's also understood that each cardinality is a set. This is possible with the axiom of choice, so that each cardinal can be defined as a least equinumerous ordinal, or with the axiom of regularity, so that each cardinal can be defined as a set of least rank.
I love the sound design in your videos, it reinfprces the visuals and makes everything feel more real.
what is the song at the beginning and the end?
@ Cyber Crime Story by < E S C P >
I agree
... real, and not at all complex.
@@davidwright8432 haha nice one
Dude, you are bound to have 100k by the end of this year. Your content its like no other. Keep up the awesome work 👍
i dont think he reads numbers
comments*
@@whannabi fr
@@whannabiyes sir
@@agooddoctorfan651anyway, you had laid Jesus Christ in a manger, as the inn was full, but you will soon find out if the King (👑) is out of the manger, and into the water to be baptized, as then you will not be able to call it Christmas anymore, and so a New Year begins.
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.
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.
∞
I think that's cultural, here in Denmark it is always considered to be multivalued and if you don't give both answers on a test you would loose points for it.
Love the direction and editing of your videos. A ton of effort that works perfectly with the information being taught.
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.
Things I specifically liked about this video:
- The provision of a proof at 6:42. It shows that you're not only concerned with providing facts, but also with the justifications for provided facts.
- Those animations are fine af.
- I never had a good understanding about cardinal numbers. This video explained the relation between all aleph- and bet-cardinals really well to me.
- The moment you said: "We're gonna do a little history lesson", my mind went crazy, because I knew exactly what you were gonna talk about: ZFC and Gödels incompleteness theorem. The following minutes I was amazed by the precision by which you explained some basic logic concepts, after which you even talked for a moment about the axiom of choice!
I was already familiar with most of the topics mentioned in this video, mostly because of Vsauce's video 'How To Count Past Infinity', which I saw several years ago, and partially because I studied mathematics for a year. However, in Vsauce's video, I think he didn't explain cardinal numbers very well because I didn't understand it at the time.
All in all, you made a masterpiece of a video. Thank you.
Vsauce's video: m.ua-cam.com/video/SrU9YDoXE88/v-deo.html&pp=ygUPdnNhdWNlIGluZmluaXR5
Woah this is some super high quality content, keep up the good work man. Can't wait for your channel to blow up!
This channel is going to blow up the next couple of months! The video quality is so high, very under appreciated atm
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.
Well, yes. But this video is aimed at someone not yet familiar with quite how slippery 'obvious' math notions can be. Explanation has to start somewhere. Initial explanations of this sort can be neither completely comprehensive nor completely correct - from a (in your case, far) more advanced standpoint. But they can be 'good enough' to get a clear initial idea across. With the understanding that refinements will follow if you continue studying. I didn't get to model theory til grad school. All Hail, Chiang & Keisler (Model Theory, now 3rd ed.)
I used to be against the axiom of choice, but that changed when I learned that the generalized continuum hypothesis implies it. It is simplest to operate in the system of ZF+GCH.
@@viliml2763 Yeah I think it's fair if you commit yourself to a "restricted" model of set theory, e.g. L or if you just want CH to be true. Of course, adopting choice, even if true in those frameworks, still means you end up proving theorems that state non-realizable things. Of course, if you're not working constructively then this will happen already in arithmetic also.
@@viliml2763
I find it interesting that you hold GCH as more normal than choice... I don't believe in believing in axioms, I think of them more as "how nice do you want sets to be", and GHC is one of the nicest ways to think of sets
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
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
Dude this video is so good!!! It helped me understand much more than I already knew! And the animation and teaching styles are phenomenal! Keep it up man!
I don't understand how this has so little views. Such an interesting video, and so much work put into this. Thank you!
Yoo, congrats on the honorable mention in SoME3. Totally deserved.
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
The definition of the square root at 9:45 is a little iffy. The square root symbol is used to represent the principal square root function, which only produces one number. Therefore, if x^2 = y, then x can be two values, but rewriting this equation as x = sqrt(y) makes x only equal one value.
That is, if x is a positive real number. When it comes to complex numbers, we usually take all the roots
Every complex number (hence every real) has two square roots (except 0, which has a single square root). By convention, the square root *symbol* refers to the positive root (in case the operand is a positive real).
The symbol interpretation is not the same as the definition of "square root" -- the map is not the territory.
This is such a good, high quality video. The fact it doesn't have a million views already is a crime
The visuals and even the sounds and details in this are wonderful. Great content keep it up I love it
Wow this is an amazingly accessible video that covers nontrivial subjects like ZFC and CH too!
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.
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.
I like this style of video/animation and the sound effects are so satisfying imo
awesome content!!!! im actually sooo impressed by the editing and just everything in general
Your animation skills are AMAZING, gosh i love it, dog is really cute and highly expressive, very enjoyable!
I wish this had more views, this is an incredibly made video about stuff I love talking about.
Don’t let the view count right now get you down, ill always recommend your videos to others, and religiously watch them lol, I love this content.
The balance of simplifying but not oversimplifying is difficult to strike, but I think you nailed it.
Also, introducing so many concepts per minute without it getting overwhelming. Well done.
It's just amazing. I'm currently studying economics and we had almost no math taught. Such videos bring me both inspiration and sadness. Thank you for your work!
the doggo animations keep getting better and smoother, very nice, the mouth sync in particular is very ngood
Wow, such an incredible video. Love the pacing, how arguments built up on each other and how nicely all of this is animated and sounddesigned.
Even though I hate these kind of questions: What toolchain do you use to animate the 2D parts? Based on the complexity of your animation with skewing and stretching involved and text fading in letter by letter and rotating, I'd assume After Effects or similar software. Do you render the LaTeX formulae as PDF, import it into After Effects and go on over there? If so, how do you manage to consistently have the "appearance skewing and fading in" effect at 11:20? Do you copy speed curves over to all other clips by hand? Generally, how can you achieve such a consistent look and feel regarding the animations throughout the video?
Wow...graphics are so top-notch! Not just artistically...but also instructionally, informatively. Kudos.
So who could imagine the shapeshifter that is infinity might drive someone mad who dared try explore it?
I love your way of explaining the concept!
This is by far one of the best explanations on this topic I have heard. Specifically how you laid out the difference between cardinality and size, which I can also then apply across my familiarity with ordinals to differentiate cardinality, from size, from order.
This itself has been very helpful in clearing things up for me. Of course I still desire a system which classifies infinity based on their actual size and have been working on trying to construct such a system which is no easy task. It requires a definition for infinity which is commutable when using operations in the same way a finite number would be, and this would mean examples like the hilbert hotel would not work the same either. I am hoping to call such infinities Terminal infinities as they will be exact in their description of size.
I am also hoping to come up with a system of infinities which measures the rate of increase between infinite values but this project is even more ambitious it would seem. These if I can manage them I would call Celerital Infinities.
Criminally underrated, great vid
Now I NEED a video about Frege's axioms and Russell's paradox. (subbed btw, cannot wait)
10:09 You have to be careful making statements about reals using their digits, since some reals have two digit representations, like 1.0 = 0.9999... So mapping from reals to pairs of reals by deinterleaving the digits doesn't give a unique mapping: 00.595959... and 01.505050... both map to the pair (0.555..., 1.0)!
I took some programming classes and this gives me the same energy. Just very similar rules
I have to watch it a few more times, but that was really excellent and I appreciate it.
Is this submitted into the #SOME3 challenge? If not, it absolutely needs to be
What’s that?
@@agooddoctorfan651 3Blue1Brown's Summer of Math Exposition 3. A competition of sorts that aims to encourage people to create educational math content.
@@agooddoctorfan651 3B1B contest
@@agooddoctorfan651 it stands for Summer Of Math Education, it was started by the UA-cam channel 3b1b (3blue1brown) to get more awareness of small math content creators or people educated in math to make new educational content on UA-cam.
@@agooddoctorfan651Summer Of Math Exposition
A competition thing setup by 3Blue1Brown...and perhaps other channels. Tries to promote smaller channels to...make math exposition videos.
this video was awesome, finally I understood and got insight to infinity and how it is talked about in math. Thank you so much now I am even more exciting to study math :D
I appreciate your use of the interrobang
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.
This is probably the best video I have seen on infinity.
Your videos are some of my favorites on the whole platform of UA-cam. You explain things from the bottom up, where many people will gloss over most things with large abstractions. Your style makes the information much more digestible and entertaining
Excellent explanation of cardinality concept.
THANK YOU for the breakdown of how the mathematically rigorous definition of cardinality isn't a perfect fit for plain english phrases like "how many," that is one of my biggest pet peeves about youtube math communication
I love your insistence on having a little "bing" or "brrt" audio sting whenever something pops up on screen!
Just examining the animations, you are an excellent example of quality over quantity. If you’re putting out lots of videos, then you’re the best UA-camr ever
Really good video, I loved it. This may be a bit stupid, but I really liked the dog at the start, too. He was cool. You're cool.
Awesome video, but there's one thing i'd like to clear up:
3:27 This impression isn't quite false. Cardinality is an intuitive notion of size that works on infinities. It's just that some of the intuition you get from the finite cases isn't correct, unless you specifically make the assumption that the sets are finite. "Your intuition won't always work here" should be clarified, but the full picture is more like "Your intuition won't always work here because it's formed from a special case", instead of "Your intuition won't always work here because this stuff is confusing" (which in my experience can scare people away), so maybe that should be clarified too.
This is just like with rational numbers. When we teach people about rational numbers, we keep using "x is smaller than y" and similar phrases to compare them, as if there's an intuitive notion of size for rational numbers, despite the fact that part of the intuition you get from integers breaks when you generalize (e.g. in the rationals, you can decrease a number forever without ever getting to 0 or below it, there is a number between each pair of numbers, every number is divisible by every nonzero number, etc.).
I understand that the (many) notions of size of infinite sets are not immediately natural to most people, but it is heartbreaking to keep hearing that infinities are unintuitive, when weirdly, almost no one ever says it about the rational numbers. Neither of these things are unintuitive, they're just generalizations.
Generalizations and abstractions shouldn't be seen as confusing, or covered with "beware: unintuitive math" signs. If anything, they're the opposite. It's about taking the essence of the original, and seeing what more can be done with it. It's about simplifying, removing the messy details of the specific case. Of course generalization does mean losing some properties, but in many cases, it's a small price to pay for beauty, if not a good thing by itself.
one of my favorite channel, i have become a fan of u!
Tangent: oh wow, I never even consciously noticed the switch from Greek letters to Hebrew ones in this branch of mathematics! Makes me wonder if there are other sections of math using other alphabets too. Maybe Cyrilic? Georgian? Hangul? (Hangul's system feels like it should b e a natural fit for *some* maths out there, no?)
We use several alphabets and still have contradictory conventions in which p might be momentum or a prime number or something else. I sometimes think that we should use Chinese characters.
Because of our limitations we tend to think of infinity as something really, really big, but it's not, it's infinite, which is a whole different concept. For the same reason children aren't easily convinced that 0,999...=1, they think it's a lot of nines but it's not, it's infinite nines and that makes all the difference. In a way it's like thinking about the 4th dimension, we can represent it, make calculations about it, but never imagine it.
9:46 If this were true, we would not need the “plus or minus” in the quadratic formula. No, the (real number) square root of “a” is defined to be the positive solution to x^2 = a, not both solutions.
For example, the golden ratio ≈ 1.618 is calculated from the expression (1+sqrt(5))/2. It is a single number, and while -0.618… has some things in common with the golden ratio, it is not the same number. That one is always written with a minus sign in front of the square root.
It is more common for mathematicians to want to use a single solution than both, so when they do want to use both, they explicitly include the plus or minus symbol. If we assumed that the plus or minus is built in to the square root itself, we make it difficult to extract a singular number out of it. I mean, when was the last time you put absolute value brackets around a square root?
It's worth noting that the definition of "infinity" in limits is not the same as the definition of the size of infinite sets, which is also not the same as infinite ordinals. Infinity as it is used in limits (both as an input and as an output) is more or less just a special placeholder symbol. It is often convenient to define it as greater than every other real number, and to define arithmetic operations on it based on how limits behave, in which case you're now working in the "extended real numbers", but it's distinct from the aleph infinities, and also from the omega infinities.
I know of 4 kinds of infinity in mathematics (and you covered most of them).
I think of the symbol at the opening (resembling an 8 on its side) as topological infinity. The space is non-compact but you want stopping points in all directions, so you throw in some extra points. For the real line those are +infinity & -infinity. There is no preferred ordering on the complex plane, so we just throw in one compactification point (unsigned infinity) to make the Riemann sphere. (In most cases there are also more complicated compactifications that we could use.)
Analytical infinity relates to the speed of approaching topological infinity. Thus e^x is 'more infinite' than x^9 as x tends to infinity.
Then there are infinite ordinal numbers and infinite cardinal numbers, which are different from each other and from topological & analytical infinity.
@@jorgenharmse4752 I'd never heard of analytical infinity (though the idea makes sense) before, neat.
Thank you. That term isn't officially used, but there are plenty of discussions in Calculus & Analysis courses of how fast something blows up (or approaches zero). I thought it belonged somewhere in the catalogue of infinities considered by mathematics.
@@jorgenharmse4752 I think it's generally just referred to as asymptotic behavior
Correct, but if we're talking about various infinities in mathematics then I think this is worth a mention.
awesome video, love to see these
all of your videos are amazing.
He is the perfect mix of funny and educational XD
This was wonderful
i love this style, algorithm bless this man
Fr
THIS CHANNEL IS A GOLDMINE WTFFF UR VIDS ARE SO GOOOOD SUBED
This channel's gotta blow up soon!
Mark our words haha
This is a really good video, Josh.
I especially enjoyed the sound effects.
Wake up josh has uploaded
GREAT VIDEO! Liked and subscribed ❤❤❤❤❤
This was a phenomenally concise video, clearing up the children's talk about the cardinalities. 👍 (Well, I am not qualified for anything, but anyways, just my opinion)
Hey wait a minute, this is really good!
Note that the definition of infinite cardinal numbers uses a notion of infinity that has nothing to do with the notion of infinity involved in limits. The former is called actual infinity, the latter potential infinity. It was once commonly accepted that only the latter notion makes sense, but since Cantor most mathematicians believe both exist, despite actual infinity being associated with many mathematical paradoxes, unlike potential infinity.
Mathematicians who reject the existence of actual infinity (of infinite cardinal numbers like aleph zero) are called finitists. They still accept the notion of infinity used in limits.
One way to clearly visualize why the set of all possible subsets of n, equivalent to a series of yes/no questions, actually leads to a bigger infinite cardinal when applied to the natural numbers, is to represent real numbers in base 2. If you write out a real number in binary, then each digit after the decimal place can represent one of the infinite series of yes/no questions. Every real between 0=0.000... and 1=0.111... represents one subset of the naturals, with each binary digit of the real number indicating whether the corresponding natural number is in the subset. Since there's a 1:1 mapping of the subsets of naturals and the reals from 0.0 to 1.0, they have the same cardinality. We can extend this 0.0 to 1.0 range to the whole set of reals by interweaving the digits to the left of the decimal place with the digits to the right of the decimal place, similarly to how you'd interleave the odd and even numbers. In the end, we arrive at the set of all reals being the same cardinality as the set of all subsets of the naturals.
Your vlog looks good. Perhaps you can talk about p-adic numbers or perplex numbers in your next vlog.
Love that video,
Here's something I thought while watching the video, it's inspired by cantor's diagonal argument :
Let's start with 0 and assign Naturals to Reals :
1. 0.0000...
2. 0.1000...
3. 0.2000...
...
10. 0.9000...
11. 0.0100...
12. 0.1100...
...
19. 0.8100...
20. 0.9100...
21. 0.0200...
and so on we can construct all real numbers between 0 and 1, (0.99999... = 1) using all the Naturals, so Bet0 is the cardinal of the Reals between 0 and 1
but it means that we can also construct all the reals between 1 and 2 using another bet0 and so on, so we're using bet0 times bet0 to construct all reals therefore bet0 * bet0 should be bet1
I realise that's what Josh meant by measuring the cardinality of all subsets, being 2^bet0 but why bet0^2 is not bet1 ?
The problem is that you have not actually assigned all the reals to naturals - you are missing all the irrational numbers. For example, which natural gives you pi? Because of this, beth_0^2 is actually still just beth_0.
@@JoshsHandle Thanks for you reply.
I thought that when we are at 4 times bet0 we're in the 3 to 4 range,
so all the naturals are assigned to all the reals between 3 and 4,
therefore the "infinite" natural being the fraction part of Pi + 1 (because I assign 1 to 3.000....) should represent pi, but that means that there is some sort of "infinitly" long natural number
@@maxim7718 And that is, unfortunately, the problem. There is no natural number that has infinitely many digits. This is why the set of all reals is "bigger" than the set of all naturals, because the way reals are defined, they are allowed to have infinite (potentially non-repeating) digits, while the definition of the natural numbers only allows for finite digits. If you want to see what natural numbers might look like if they had infinite digits, I recommend looking into a concept called p-adic numbers. Since those have infinite digits, there are again "more" of them than there are natural numbers.
@@JoshsHandleThank you very much, I understand it better now
I will definitely look into p-adics
Thanks ❤
What kind of infinity is used in algebra and calculus (like limits to infinity)? Is it ב0, or ב1, or something else?
God DAMN what an entry into #SoME3!!!
axiom of choice is needed for the existence of a basis in any vectorial space, which is not "niche, advanced, purely hypothtecal statement"
Alle Zahlen und Zeichen sind künstlichen Ursprungs und somit sind diese eine Fiction. Ändern sich die Zeichen ändert sich die Fiction. Das Längenmaß 1,00 m ist eine erfundene Größe welche wir benutzen um bestimmte Zustände zu berechnen. Dies verhält sich mit der Gewichtseinheit 1,00 kg in gleichem Maße welche es so nicht gibt da hier die Gravitation und der mit dieser in Verbindung stehende Luftdruck eine Rolle spielen, während der Strahlungsdruck oder Lichtdruck der Photonen ebenfalls einen Einfluss ausüben.
Man this is a good video because its causing that friction that comes with new ideas but its also good enough at exolaining that it overcomes it
Like once divorced from size technically the integers have the same cardinality as the rationals even though the integers are a subclass (subset?) Of the rationals
I'm not sure it's super accurate (or rather, not super clear) at 15:46 to say _"How true the continuum hypothesis depends on you feel that day."_ Rather, that claiming that mathematical statements are absolutely true as if they were some ideal platonic form in the ether, or talking about them as if they were entirely subjective, I think it's clearer to think of true/false as only make sense *within a system of axioms*.
Eg: it's true that angles in a triangle add up to 180 degrees in Euclidean geometry, but not in curved geometries. Thinking about truthfulness strictly in terms of consequences of axioms (and attached logical system) that has been picked by us is the way to frame things. Some choices of axioms are rather pointless (eg, equating truth and false); while some very closely match the maths derived from more intuitive concepts (eg, ZFC).
Putting things this way makes it obvious that it is meaningless to ask whether the continuum hypothesis (or the axiom of choice) is true or false, because that simply isn't a property that axioms can have. Claiming otherwise is equivalent to saying I am "true" for choosing pancakes over waffles for breakfast.
Aw man Gödel numbers and proofs.
Veritasium has a video about how we technically cant prove everything, and of course Vsauce has talked about infinity a few times, it was fun seeing the info in those videos come back to me while watching this.
Infninity is pretty mind boggling without context for sure lol, good video
Great content! A new 3b1b is rising to bless us
7:01 Theta changing its angle, k being springy, T being thermal, and mu experiencing friction. Great touch.
9:35 There are two square roots of 4, but "the (principal) square root of x" just has one value for any nonnegative x, or else it wouldn't be a function. Is the implication that subtraction of transfinite numbers isn't a function?
5:25
Should we consider set's density in place of its cardinality in that case ?
I don't think it's all that bad to think of infinity as a number. I mean, it's a point on the Riemann sphere, a point in projective geometry, and you can consider the extended reals where you do include with a few rules on how to use it (often the case in the definition of a measure).
When it comes to infinite cardinal "numbers", I'd rather consider them categories than actual numbers. The category below ℵ₀ being "Finite": Finite + Finite = Finite,
Finite × Finite = Finite.
I'm not sure whether "Zero" should be a category of its own, below "Finite".
Fun fact: There are infinities which _are_ numbers but very different from the infinitues this video went through: The nonstandard numbers.
There are different flavours of finite numbers (in order of increasing operational freedom): Natural, Integer, Rational, Real, Complex numbers, n×n Matrices of finite numbers. And there are embeddings of more restricted numbers into more free numbers (like Rationals into Reals, and (scalar) Reals into 1×1 matrix of Reals).
The cardinality of the set of Integers and the cardinality of the set of reals is different.
12:50 This is my favorite proof.
great video
You cover lots and lots of topics. I doubt that anyone can understand this without prior knowledge of these topics. It is important to note that the notion of a limit for n approaching infinity does not correspond to the cardinality of a set. But then, this concept of a limit is in and of itself very complex and it took mathematicians approximately 200 years to find a reasonably definition (essentially Weierstrass and Cauchy in the 19th century)
remember me when you are famous
dude you've GOT to submit this to the summer of math exposition 3 by 3blue1brown
unless you already have and just have no tags with it
I love this video! I found the animated mouth slightly behind the audio somewhat distracting... our brains are better at syncing mouths the other way around
I've a maths question about infinity; how many perfect squares are there? It feels like there's one perfect square for each cardinal number because squaring a cardinal number gives one perfect square. However, for any cardinal number n, there are √n perfect squares less than or equal to n. So, that suggests there are √∞ perfect squares - and thus ∞-√∞ irrational square roots.
dammnn good animation bro keep it up
Great video ! I'd be a bit more careful about terminology though ; integers != natural numbers (for example)
I'm going to criticize the video, hopefully in a way that is constructive, in case you or someone else wants to make a better one in the future.
There are multiple notions of infinite that appear in math. The ones I can think of off the top of my head have one of three flavors:
(i) infinities that appear when thinking about set theory: infinite cardinals, infinite ordinals (where omega +1 ≠ omega, surprise);
(ii) points that are added to spaces, often to make them easier to understand: ±infinity in the extended real line, the points at infinity in projective geometry (e.g., the infinite slope of a vertical line), the point added in Alexandroff extension;
(iii) infinities in number systems that contain them: hyperreal numbers, surreal numbers (where omega - omega = 0, big surprise!).
The video only covers a small subset of these, and what is covered in the video is imprecise at several points, which is unfortunate. The ±infinity at the ends of improper integrals are not the same infinity as any of the infinite cardinals, but 18:24 seems to confuse them. The limit of a function is described as "the closer you get to that input, the closer the function gets to that limit", which is just terrible; "You can make the function get as close as you want to that limit by getting close enough to that input" is much much better. Also, the limit of the partial sums used to define infinite series and the limit of a function are not the same thing. Again, something like "you can make the partial sum get as close as you want to that limit by adding enough terms" would be fine.
Rutgers taught me that there is no such thing as an infinity greater than another. maybe some publication has come out since then but I rather agree with the traditional view....AS infinity was defined by our calculus prof.
did the definition change?
thx
Wait I didn’t even realize you’re not like super famous. I thought the channel was huge
Your videos are so incredible. Holy shit.
It's just so frustrating that it's impossible to prove or disprove the continuum hypothesis... Like, is it true or not?
But at the same time, it's so interesting that the 2 incompleteness theorems are true. It makes math a rich tapestry of universes with different axioms and ensuing theorems
Almost died at 16:07 Could you make a video about the axiom of choice?