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
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!
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.
@@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.
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
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
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
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.
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.
@@xinpingdonohoe3978 both the domain and range of the principal square root √x is x≥0 because it's useful to restrict polynomial roots where imaginary solutions wouldn't make sense, like in geometry. Cube roots don't matter because the domain and range are both all reals. No imaginaries involved.
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 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.
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.
@@agooddoctorfan651 3Blue1Brown's Summer of Math Exposition 3. A competition of sorts that aims to encourage people to create educational math content.
@@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.
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!
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
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.
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.
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.
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.
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)!
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.
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.
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
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.
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!
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?
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
(TCMS) Theory The term Transcendental Compact Multidimensional Set Theory refers to the study of sets that have the following properties: They are transcendental, meaning that they do not contain any algebraic numbers. They are compact, meaning that they are closed and bounded. They are multidimensional, meaning that they have more than two dimensions. This type of set theory is a newly invented field of study, but it has the potential to be very important for understanding the structure of complex data. It has the potential to shed light on some of the most fundamental questions in mathematics. For example, it could help us to understand the nature of infinity and the relationship between sets of different sizes. The study of transcendental numbers is already a complex and fascinating topic, and the addition of compactness and multidimensionality would only add to the challenge and the potential rewards. TCMS Theory has the potential to help us understand infinity in a way that we never have before. However, it is also possible that it could lead us down a rabbit hole of madness. Only time will tell what the true potential this field of study has. Known Sets: The Set of Limiting Meaning~
Thumbnail: ∞ is not a number Vsauce: Instead it's a "kind" of number. You need infinite numbers to talk about things that are unending. But some unending amounts--some infinities--are literally bigger than others. Let's visit some of them and count past them.
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.
6:46 I have a question: Since we assume the cardinality of the list of all real numbers is infinity, the process of the making this "new number" is never-ending, and so is the comparison of this "new number" with all real numbers in the list. Therefore we can never be sure the "new number" we are creating is truly unique. So why jump to the conclusion that the number is unique?
We can prove that the number is unique without having to manually check every case. This is analogous to how I could prove that every prime number above 2 is odd, without having to literally check that every such number satisfies that condition. In other words, we can use the tools of math to show that the conclusion is the only possible one to make, so we don't have to bother checking the entire number to make that conclusion.
When Josh said "We can't just slap this symbol on this set and just call it a day" me who completed a 4 year intensive course just by slapping infinity symbols on everything and telling my lecturer to use his imagination to fill in the gaps.
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.
0/0 ≈ the limit of x/x as x tends to zero ≈ 1 blew my mind. However, 1/0 ≈ limit of 1/x as x tends to zero shoots up to infinity. So we could probably define a function f(x) = x/0 ≈ limit of x/t as t tends to 0 Then f(x) ≈ 1 when x≈0 (As x and t both tend to zero, they could be equated) = undefined otherwise
"The limit" is only defined if both the positive and negative limit are defined and are equal to each other Basically, the limit from the right and the limit from the left have to be the same. And in your case, the positive limit of 1/x tends to infinity, while the negative limit of 1/x tends to negative infinity, so there is no "the limit of 1/x", such a thing is not defined.
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
The problem of all these different contortions and trying to figure out sets of numbers is that there is no such thing as infinity . When you realize that, then all these problems go away. Numbers originate from counting objects, real objects. Even in your imagination you are counting objects. You have stored pictures of what one is , two, three, up to about 12 and after that it gets kind of blurry. In order to prove a number exists , you really have to count up to that number and count one at a time. No short hand, as in multiplying or exponentiation can be allowed. The problem is induction. With all finite numbers you assume you can always add "one" to a number and get the next higher up number. You assume the next higher up number exists. If you think about it, there are only 10^80 particles in the universe. So you could not count higher than this. Before you reach 10^80 , you will run into other problems. If you are using a computer to keep track of your count, you will have to keep adding hard drives or memory to this computer to count higher. When you reach a certain point , the computer will be so massive , it will collapse into a black hole. There exists a number such that if you try to add "one" to it , you will find out that it will be impossible. That number is the largest number. It will be nonsense to speak of bigger numbers. This means infinity is just nonsense and so is zero. You can't have zero of anything. The empty set ( ) is supposed to have nothing between the brackets. But in the real world you can't do this. There are things tunneling in and out of the brackets. Things can appear between the brackets for short times and disappear. Empty space is teaming with activity. There are not infinitely many numbers between zero and 1 or between 1 and 2 etc. They are quantized. There are probably numbers that are just impossible. This is the real world. July like an electron in an atom. It can jump from one state to the next higher state without passing the intermediate energies. It does a quantum leap. So to figure out what one plus one really is, you have to add up all the possible ways , combinations and permutations there are to count things from one to two. Then divide. You will probably find out that it doesn't equal the classical two. It will probably be slightly less than two due to tunneling. The way Richard Feynman added up the electron paths was to take every possible path in the universe an electron could travel and then divide to get a distribution. This is how numbers should also be treated. Numbers represent collections of objects. Most mathematicians are stuck in the classical world. Newtons world does not exist.
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)
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
So if I understand correctly, we should use ∞ when we talk about a limit (like "what happened if this sum "goes" to infinity?) and we should use the hebrew cardinality symbols when we talk about infinity "as a number"? I think this should be taught in a lecture on infinite sets, it will clear the confusion of "an infinite set can be bigger than another infinite set" quite easily since this confusion comes from the use of the same symbol for the cardinality of those sets. If you just introduce the hebrew cardinality symbols, you have different 'numbers' for the cardinality of the set and there is no confusion anymore! Plus it does not take that much time to get a basic understanding as you've shown with this video, maybe 10-15 minutes of the lecture to introduce this new number systems and maybe even the associated operations. Maybe it should not even be in the final test associated with that lecture, but just here to clarify a part of the lecture that can be confusing when you meet it the first time. Anyway great video as always!!!
Infinity as a set. Mathematicians sidestep the question of what is a number by creating a new mathematical object called a set and then defining all numbers as a set. All numbers are sets but not all sets are numbers. For example the set {1, 3, 63876} is not a number it is a set of finite collection of numbers but not a number. Another example is {g, z, k, w} is a set of finite collection of the english alphabet and is not a number. Infinity is a set it is an infinite collection of numbers but not a number.
It basically depends on the context. It wouldn't make sense to write the limit as x goes to bet 0 because that is not what the limit is trying to convey. In calculus, there is only 1 infinity, so writing bet 0 or bet 1 is absolutely the same. So yes, it is very dependent on the context and the framework you are working under. They are refering to different ideas about infinity
I cant stand the fact that mathematicians have tried to rationalize infinity. The fact that it is commonly believed that there can be multiple infinities is astounding, and it is a critical misunderstanding of what infinity entails. An infinite series of ones is equal to an infinite series of twos, simply because infinity is not a number. In any other context, those two series of numbers would certainly create two different outcomes, but infinity doesnt work that way.
Mathematicians don't talk about 'multiple infinities'. They talk about infinite sets with different cardinality. You can't just talk about 'an infinite series of ones'. If you mean an infinite sequence, then the size of your (multi)set is aleph0. But there are other infinite sets with larger cardinalities
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.
I mean, for counting the real numbers, you don't have to count them in order, right? Like, you started at 0 when including negative numbers because you can't start at the largest negative number, since it doesn't exist. So, why do we have to start at the rational number directly after 0? What if we start at 0, but then skip to 1, and then go *back* to fill in the gap? So we start with 0, then 1, then go back to .1, .2, .3, .4, .5, .6, .7, .8, .9... then .01! We're counting all the numbers between 0 and one in a weird way, where it's like normal, but instead of going from .9 to 1, we put the 0 in *front* of the 1. Normally you go from 9 to 10, we go from 9 to 01, and we keep the decimal place in front of the number every time, .9 to .01. it's like we've treated the decimal place as a mirror. We can do this between every number, since again, we aren't constrained by where we have to start or go. 0, infinite numbers, 1, infinite numbers, 2, and we go on forever. Rational numbers have been made countable!
Canters diagonalization assumes that the decimal place is the same for every number, but in order to ignore that you'd have to assume that there aren't any numbers prior to the decimal place. If there are, you now have to do the diagonal in both directions at the same time. Otherwise, doing so points out a small issue. Lets say your table looks like this 2.5 1.6 Generally, you assume that the decimal places have an infinite number of zeros after them, because otherwise you'd run into an issue where you try to do this: .1 .2 Diagonal .X .2_ Instead, we do this .X000000 .2X00000 So now that we're doing this in both directions, it looks like this 000002.500000 000001.600000 And we read it diagonally 00000X.X00000 0000X1.6X0000 Ok. Lets actually just not include the decimals, then! And for fun, lets do the integers in order 000000 000001 000002 000003 Lets also have it so we add 1 to whatever number we read 00000X 0000X1 000X02 00X003 If we continue this forever, we know for certain that the resulting number will be ...111111111111111111111111 Allegedly, a brand new number! But it's just because the rate that our new number grows, and the rate that the integers grow, is different. Instead of numbers, lets say uts in binary. 00000 00001 00010 00011 00100 Each progressive number we count takes longer to change than the previous one, but if we let it go forever, we can assume it would reach ...111111111111111, right? But it would do so after the diagonal test does so because every digit takes the same time to change. In short, we can prove that decimals are countable, and the proof against it can be applied to real numbers, which we know for a fact is countable. It's proving something incorrectly
That’s creative, but unfortunately it doesn’t work. It’s actually a common thing people try. Sure, you can treat the decimal as a mirror. But then, which natural number is assigned to 0.333…? Or to pi?
I find it interesting that reality is filled with immensely complex situations. Assortments of abstractly connected thing's. Yet math can have a very rigid (watered down perspective) There's no way we can actually define every single aspect to a certain system. Math can definitely help find "approximate circumstances to a situation" but it's to linear to tell the entire story. It feels like math sketches a outline but isn't able to fill in all the blanks. Due to this paradox that this video covers.
I love your videos, but I feel like this video… kinda loses the plot on itself. It feels like a CGP Grey and Veritasium video, with the wrong ingredients. I suspect this is due to you getting too close to the “What is True?” dimension that Grey warns a LOT about. Even though I’ve seen other videos on this topic, I still finished the video confused. I still do like the video, Im sorry if this comes off negative. I feel its a hollow compliment but the visuals and audio are really good! Please keep making more videos 😊
6:44 What's stopping us from doing a similar proof for integer numbers? 1. Assume you have a list of integers. 2. Make a new integer that's one more than the "last" integer. (if "... and so on" is alllowed in the original proof, doesnt that imply that there is an ending to the list?) 3. This number is not in the list.
You're thinking of a list as a finite object. In the video, "list" is used as language for "a way to assign something to each of the infinite number of integers." So, we are "listing" infinite real numbers when I say to make a list of them. A real number can have an infinite number of digits, so it's reasonable to define a real number that has a digits for each of the infinite quantity of numbers in the list. The idea is that if you only have "as many" real numbers as integers, you haven't listed all the real numbers. In contrast, you can easily put all the integers in one of these infinite lists by just assigning each integer to itself. The more accurate way of putting all that that doesn't use language like "as many" to gloss over the details is that there's no way to match up integers to real numbers such that every possible real number gets a unique integer. What the proof is saying is that if we have made a way to assign integers to real numbers, there is a real number that is not matched up with any integer at all. This goes against the intuition that because integers are infinite, we must be able to pair them up one-to-one with other things we have infinitely many of. Because of this, we say there's "more" real numbers than integers.
There is one obvious infinity that exists and that’s space. Space cannot be finite because that would be something existing within a greater context of nonexistence. You can’t have something existing within something that doesn’t exist, that would be a structural paradox. Because space is a tangible construct, it’s infinite value can be given a numeric number. What would that number be? Well, since space has three tangible spatial properties being its height, width, and length, each of those (properties) value can be assigned “one,” or in other words, 1 cubed; and since it’s an infinity, it would have to be written as “one cubed times infinity. [That’s one height, times one width, times one length, times infinity.] One is the working number, and infinity is the constant. How can one be a working number you might ask; well, one can be divided to finity within the greater context of infinity…
Integers are not finite in length. They have an infinite amount of zeros preceding and following them. However, it would be overly burdensome to list all of the infinitely many zeros, so those zeroes are truncated, but they still exist. The idea of different sized infinites should therefore suggest that there can be different sized zeros. Instead think of the decimals as representing some full integer number of infinitesimals, and the whole number as representing how many times you have listed all of the infinitely abundant infinitesimals.
Imagine that surface area is Aleph_3 and volume is Bett_3; therefore, they are not the same size, but the cardinality of the categories about them is. So it is both, but only past Aleph_2.
14:48 It's not completely true that you need a more powerful system to proof that peano arithmetic is consistent. Gerhard Gentzen showed what you can proof Peano arithmetic is consistent, as long as a certain other system used in the proof does not contain any contradictions and the other system used in his proof is neither weaker nor stronger than the system of Peano axioms.
The segment on rational numbers (starting at 4:15) doesn't have me convinced that its cardinality is identical to integers. What about the infinite number of rational numbers between 0 and 1 plus the infinite number of rational numbers between 1 and 2 plus...the infinite number of rational numbers between any two integers x and x+1? Where do they fit in the unique pairings?
There are countless ways of showing the cardinality of rational numbers is the same as the cardinality of integers, you can relatively easily look them up
You maths dudes might be able to help out the quantum nerds by working out what number fits between zero and the next number up, this being the mass of an atom or the charge on a quark etc, and find which number when fitted into a set makes an illuminating change. This is just an experiment and may not yield results, but may be a way to break our present understanding and make a discovery. The presumption I have made is that the smallest possible physical quantity that exists in the physical world is the key to a puzzle. The overarching view is that numbers are a quantity, no matter what else we can do with them.
Mathematics can be related to & is often motivated by real-world problems, but it should not be defined by them. For example, you want the smallest positive mass that can exist, but it's not useful to make that the smallest positive number. You didn't specify the units, you might have to compute the gravitational attraction between two such masses, and numbers don't exist just for physics calculations.
I didn't specify anything. I've lost track of the video but essentially knowing myself I have probably tried to prompt a solution to a puzzle. I think I have suggested that which you say does not apply, that numbersMUST apply to the physical world.In the circumstance you suggest for example other calculations such as gravitational forces might really need to be faced up to as aspects of the original rather than numerical values of their own. Thanks for your reply. I am sure I am looking for what may be a new way to calculate,in order to help out the scientific community who are looking for a unifying principle.@@jorgenharmse4752
Arguably 0 isn't a number either for many of the same reasons, since a lot of exceptions have to be made for 0 0^0 is undefined, and so is n/0, among other examples. There's also no consensus on whether or not the natural numbers include 0. In many ways infinity and 0 are each other's inverses. 0 is the limit for endlessly shrinking values, while infinity is the limit for endlessly growing values
0 is pretty well defined though. It is as equally well defined as 1. 1 is the limit of endlessly shrinking values around 1, just like 0 is the limit of endlessly shrinking numbers around 0. There are also many operations that make exceptions for 1 as well, for example the logarithm with base 1. 0 is as much of a number as any other
It may not be a number, but in the world of the affinely extended real number system, and the real projective line (those are things go check them out on Wikipedia), it is.
I am not a mathematician or even a student of it. I do enjoy good explanations of it though. So thanks. One of the things that bugs me about infinity is that nothing we ever deal with has infinity. Whether infinite time or infinite numbers of things. I looked at ZFC axioms and in there is an axiom that basically magics infinity into existence. The axioms being things you take as truth without proof. In other words , they just magicked it into existence and all the "infinity" talk I suddenly justified. So I've always wondered if you take that little toy away from mathematicians or physicists,, do lots of the paradoxes disappear. Suddenly you can't infinitely divide up space. Or the infinities that plague the joining of QM and GR vanish. I don't know, but to me a lot to the worst nonsensical stuff disappears when you take away the infinity toys from maths/science and deal with reality.
I'm a little confused on counting rationals and reals. The cardinality of reals should be the same as irrationals. But between any 2 irrationals is a rational. How then can there be more irrationals than rationals. Pick sqrt(2) as a starting point. To get to the next irrational, either on the left or the right of it on the number line you necessary pass a rational. It seems impossible to start at an irrational number, travel along the number line any amount and count more irrationals than rationals? Is it maybe the case that, in the same way size is ill-posed when it comes to infinity, so to is counting? Or is there something else i'm missing here? edit: also fantastic video, I sincerely hope you make many more in the future.
The issue here is not necessarily counting, but that there is not a "next" irrational after some particular irrational. If you pick any two non-equal irrationals, they have an infinite quantity of rational and irrational numbers between them. It also happens that the cardinality of the set of rationals you skipped over is beth zero, while the cardinality of the set of irrationals you skipped over is beth one.
@@JoshsHandle I think I get it but will definitely have to digest this more. I'm trying to construct a way where I don't skip any numbers but I guess that is impossible? The statement that there exists a rational between any 2 irrationals, whilst true(?) says nothing about how many rationals and irrationals there are in this interval. And there is no method of zooming into the number line to notice a string of numbers i1,r,i2 with no other numbers between them, where i's are irrational and r is rational. Even at infinite magnification if that even makes sense. Infinity is weird! But I think I'm slowly getting there. Thanks!
"The cardinality of reals should be the same as irrationals." That happens to be true, although proving it without the axiom of choice requires a bit of work. (Given the axiom of choice, the cardinality of the irrationals is aleph_alpha for some alpha, so beth_1 = aleph_0 + aleph_alpha = aleph_alpha.)
18:12 I think it's misleading to say 'get closer to infinity'. Infinity means endlessness, which is why it's not a number which is after all a finite quantity - saying you're 'getting closer to infinity' is a contradiction in terms, because you can't reach infinity. If you did reach infinity, by counting long enough then the really big number you counted to would just be a big number with an endpoint on the counting line, not endlessness. I think it would be more accurate to say if you continue further and further or something. But that's a minor nitpick, I'm sure everybody knows what is meant, maybe just it encourages people to thinking about the concept the wrong way.
Excellent video! I just have a bit of constructive criticism regarding the thumbnail of the video, that says “This is not a number”. I think it is a bit misleading to say that infinity is or isn’t a number, since the word “number” is ambiguous to begin with. It is not a natural, real, or complex number. But it is a Hyperreal, transfinite cardinal, tranfinite ordinal and surreal number. So it depends on the context.
I actually thought about this when making the thumbnail and introduction. I very specifically chose the infinity symbol because that does not represent a number, that symbol is only used in calculus. I do agree that it is reasonable Aleph/Beth numbers numbers.
@@JoshsHandle perfect! Them I take back my criticism, I agree that ∞ is not a number (in the common way it’s used). I congratulate you again for the amazing video and animations!
Well when it comes to hyperreal numbers, we aren't using infinity per se. We are using ω and ε which can be thought of as infinitelly large and infinitely small, but they aren't exactly infinity. They are a simplification of the concept
@@AndresFirte well the difference is that infinity is just purely a concept lol. We use hyperreals to tame that concept. Hyperreals work with addition and all other operators as regular numbers. For example ω +1 > ω even though ω is thought of as infinitely large.
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?
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!
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.
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*
@@w花b fr
@@w花byes 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.
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
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.
Love the direction and editing of your videos. A ton of effort that works perfectly with the information being taught.
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
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
This is such a good, high quality video. The fact it doesn't have a million views already is a crime
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
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
This is probably the best video I have seen on infinity.
This channel is going to blow up the next couple of months! The video quality is so high, very under appreciated atm
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.
But how do you define the principal square root, and why? And what about other roots? Cube roots, for example.
@@xinpingdonohoe3978 both the domain and range of the principal square root √x is x≥0 because it's useful to restrict polynomial roots where imaginary solutions wouldn't make sense, like in geometry. Cube roots don't matter because the domain and range are both all reals. No imaginaries involved.
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 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.
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.
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!
Woah this is some super high quality content, keep up the good work man. Can't wait for your channel to blow up!
I like this style of video/animation and the sound effects are so satisfying imo
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
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.
Criminally underrated, great vid
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.
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.
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.
God DAMN what an entry into #SoME3!!!
It’s _so_ crazy how ♾️≠8 even though everyone knows that 8=8+1
*Visual Confusion*
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)!
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.
the doggo animations keep getting better and smoother, very nice, the mouth sync in particular is very ngood
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
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
Wake up josh has uploaded
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.
Your animation skills are AMAZING, gosh i love it, dog is really cute and highly expressive, very enjoyable!
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.
He is the perfect mix of funny and educational XD
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!
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?
THIS CHANNEL IS A GOLDMINE WTFFF UR VIDS ARE SO GOOOOD SUBED
axiom of choice is needed for the existence of a basis in any vectorial space, which is not "niche, advanced, purely hypothtecal statement"
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 style, algorithm bless this man
Fr
I appreciate your use of the interrobang
infiniti is like quantum mechanics in mathematics
I actually noticed some unintentional parallels of superposition during his explanations
Wait I didn’t even realize you’re not like super famous. I thought the channel was huge
awesome content!!!! im actually sooo impressed by the editing and just everything in general
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
12:50 This is my favorite proof.
(TCMS) Theory
The term Transcendental Compact Multidimensional Set Theory refers to the study of sets that have the following properties:
They are transcendental, meaning that they do not contain any algebraic numbers.
They are compact, meaning that they are closed and bounded.
They are multidimensional, meaning that they have more than two dimensions.
This type of set theory is a newly invented field of study, but it has the potential to be very important for understanding the structure of complex data. It has the potential to shed light on some of the most fundamental questions in mathematics. For example, it could help us to understand the nature of infinity and the relationship between sets of different sizes.
The study of transcendental numbers is already a complex and fascinating topic, and the addition of compactness and multidimensionality would only add to the challenge and the potential rewards.
TCMS Theory has the potential to help us understand infinity in a way that we never have before. However, it is also possible that it could lead us down a rabbit hole of madness. Only time will tell what the true potential this field of study has.
Known Sets:
The Set of Limiting Meaning~
UA-cam may have censored the set 😔
Thumbnail: ∞ is not a number
Vsauce: Instead it's a "kind" of number. You need infinite numbers to talk about things that are unending. But some unending amounts--some infinities--are literally bigger than others. Let's visit some of them and count past them.
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.
remember me when you are famous
Infinity is such a difficult concept to wrap your head around
6:46 I have a question:
Since we assume the cardinality of the list of all real numbers is infinity, the process of the making this "new number" is never-ending, and so is the comparison of this "new number" with all real numbers in the list. Therefore we can never be sure the "new number" we are creating is truly unique. So why jump to the conclusion that the number is unique?
We can prove that the number is unique without having to manually check every case. This is analogous to how I could prove that every prime number above 2 is odd, without having to literally check that every such number satisfies that condition. In other words, we can use the tools of math to show that the conclusion is the only possible one to make, so we don't have to bother checking the entire number to make that conclusion.
When Josh said "We can't just slap this symbol on this set and just call it a day" me who completed a 4 year intensive course just by slapping infinity symbols on everything and telling my lecturer to use his imagination to fill in the gaps.
Somebody now has to teach a dog to bark to infinity!
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.
0/0 ≈ the limit of x/x as x tends to zero ≈ 1 blew my mind.
However, 1/0 ≈ limit of 1/x as x tends to zero shoots up to infinity.
So we could probably define a function
f(x) = x/0 ≈ limit of x/t as t tends to 0
Then
f(x) ≈ 1 when x≈0 (As x and t both tend to zero, they could be equated)
= undefined otherwise
"The limit" is only defined if both the positive and negative limit are defined and are equal to each other
Basically, the limit from the right and the limit from the left have to be the same.
And in your case, the positive limit of 1/x tends to infinity, while the negative limit of 1/x tends to negative infinity, so there is no "the limit of 1/x", such a thing is not defined.
@@xXJ4FARGAMERXx f(x) ≠ 1/x, read again
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
The problem of all these different contortions and trying to figure out sets of numbers is that there is no such thing as infinity . When you realize that, then all these problems go away. Numbers originate from counting objects, real objects. Even in your imagination you are counting objects. You have stored pictures of what one is , two, three, up to about 12 and after that it gets kind of blurry. In order to prove a number exists , you really have to count up to that number and count one at a time. No short hand, as in multiplying or exponentiation can be allowed. The problem is induction. With all finite numbers you assume you can always add "one" to a number and get the next higher up number. You assume the next higher up number exists. If you think about it, there are only 10^80 particles in the universe. So you could not count higher than this. Before you reach 10^80 , you will run into other problems. If you are using a computer to keep track of your count, you will have to keep adding hard drives or memory to this computer to count higher. When you reach a certain point , the computer will be so massive , it will collapse into a black hole. There exists a number such that if you try to add "one" to it , you will find out that it will be impossible. That number is the largest number. It will be nonsense to speak of bigger numbers.
This means infinity is just nonsense and so is zero. You can't have zero of anything. The empty set ( ) is supposed to have nothing between the brackets. But in the real world you can't do this. There are things tunneling in and out of the brackets. Things can appear between the brackets for short times and disappear. Empty space is teaming with activity.
There are not infinitely many numbers between zero and 1 or between 1 and 2 etc. They are quantized. There are probably numbers that are just impossible. This is the real world. July like an electron in an atom. It can jump from one state to the next higher state without passing the intermediate energies. It does a quantum leap. So to figure out what one plus one really is, you have to add up all the possible ways , combinations and permutations there are to count things from one to two. Then divide. You will probably find out that it doesn't equal the classical two. It will probably be slightly less than two due to tunneling. The way Richard Feynman added up the electron paths was to take every possible path in the universe an electron could travel and then divide to get a distribution. This is how numbers should also be treated. Numbers represent collections of objects. Most mathematicians are stuck in the classical world. Newtons world does not exist.
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)
I love your way of explaining the concept!
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
So if I understand correctly, we should use ∞ when we talk about a limit (like "what happened if this sum "goes" to infinity?) and we should use the hebrew cardinality symbols when we talk about infinity "as a number"?
I think this should be taught in a lecture on infinite sets, it will clear the confusion of "an infinite set can be bigger than another infinite set" quite easily since this confusion comes from the use of the same symbol for the cardinality of those sets. If you just introduce the hebrew cardinality symbols, you have different 'numbers' for the cardinality of the set and there is no confusion anymore!
Plus it does not take that much time to get a basic understanding as you've shown with this video, maybe 10-15 minutes of the lecture to introduce this new number systems and maybe even the associated operations. Maybe it should not even be in the final test associated with that lecture, but just here to clarify a part of the lecture that can be confusing when you meet it the first time.
Anyway great video as always!!!
Infinity as a set. Mathematicians sidestep the question of what is a number by creating a new mathematical object called a set and then defining all numbers as a set. All numbers are sets but not all sets are numbers. For example the set {1, 3, 63876} is not a number it is a set of finite collection of numbers but not a number. Another example is {g, z, k, w} is a set of finite collection of the english alphabet and is not a number. Infinity is a set it is an infinite collection of numbers but not a number.
@@kazedcat Ok thanks for the explanations, all is clear now :)
It basically depends on the context. It wouldn't make sense to write the limit as x goes to bet 0 because that is not what the limit is trying to convey. In calculus, there is only 1 infinity, so writing bet 0 or bet 1 is absolutely the same. So yes, it is very dependent on the context and the framework you are working under. They are refering to different ideas about infinity
I cant stand the fact that mathematicians have tried to rationalize infinity. The fact that it is commonly believed that there can be multiple infinities is astounding, and it is a critical misunderstanding of what infinity entails. An infinite series of ones is equal to an infinite series of twos, simply because infinity is not a number. In any other context, those two series of numbers would certainly create two different outcomes, but infinity doesnt work that way.
Mathematicians don't talk about 'multiple infinities'. They talk about infinite sets with different cardinality. You can't just talk about 'an infinite series of ones'. If you mean an infinite sequence, then the size of your (multi)set is aleph0. But there are other infinite sets with larger cardinalities
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.
GREAT VIDEO! Liked and subscribed ❤❤❤❤❤
I mean, for counting the real numbers, you don't have to count them in order, right? Like, you started at 0 when including negative numbers because you can't start at the largest negative number, since it doesn't exist.
So, why do we have to start at the rational number directly after 0? What if we start at 0, but then skip to 1, and then go *back* to fill in the gap? So we start with 0, then 1, then go back to .1, .2, .3, .4, .5, .6, .7, .8, .9... then .01! We're counting all the numbers between 0 and one in a weird way, where it's like normal, but instead of going from .9 to 1, we put the 0 in *front* of the 1. Normally you go from 9 to 10, we go from 9 to 01, and we keep the decimal place in front of the number every time, .9 to .01. it's like we've treated the decimal place as a mirror.
We can do this between every number, since again, we aren't constrained by where we have to start or go. 0, infinite numbers, 1, infinite numbers, 2, and we go on forever. Rational numbers have been made countable!
Canters diagonalization assumes that the decimal place is the same for every number, but in order to ignore that you'd have to assume that there aren't any numbers prior to the decimal place. If there are, you now have to do the diagonal in both directions at the same time. Otherwise, doing so points out a small issue. Lets say your table looks like this
2.5
1.6
Generally, you assume that the decimal places have an infinite number of zeros after them, because otherwise you'd run into an issue where you try to do this:
.1
.2
Diagonal
.X
.2_
Instead, we do this
.X000000
.2X00000
So now that we're doing this in both directions, it looks like this
000002.500000
000001.600000
And we read it diagonally
00000X.X00000
0000X1.6X0000
Ok. Lets actually just not include the decimals, then! And for fun, lets do the integers in order
000000
000001
000002
000003
Lets also have it so we add 1 to whatever number we read
00000X
0000X1
000X02
00X003
If we continue this forever, we know for certain that the resulting number will be
...111111111111111111111111
Allegedly, a brand new number! But it's just because the rate that our new number grows, and the rate that the integers grow, is different. Instead of numbers, lets say uts in binary.
00000
00001
00010
00011
00100
Each progressive number we count takes longer to change than the previous one, but if we let it go forever, we can assume it would reach ...111111111111111, right? But it would do so after the diagonal test does so because every digit takes the same time to change.
In short, we can prove that decimals are countable, and the proof against it can be applied to real numbers, which we know for a fact is countable. It's proving something incorrectly
That’s creative, but unfortunately it doesn’t work. It’s actually a common thing people try.
Sure, you can treat the decimal as a mirror. But then, which natural number is assigned to 0.333…? Or to pi?
This was wonderful
I find it interesting that reality is filled with immensely complex situations. Assortments of abstractly connected thing's. Yet math can have a very rigid (watered down perspective) There's no way we can actually define every single aspect to a certain system. Math can definitely help find "approximate circumstances to a situation" but it's to linear to tell the entire story. It feels like math sketches a outline but isn't able to fill in all the blanks. Due to this paradox that this video covers.
Your vlog looks good. Perhaps you can talk about p-adic numbers or perplex numbers in your next vlog.
our teacher used to say that if ∞ was a number we would write set R as [-∞, ∞] and not (-∞, ∞)
But then the ♾️ symbol is no-longer a number, so should you try to divide by zero, then you will get an error message.
I love your videos, but I feel like this video… kinda loses the plot on itself. It feels like a CGP Grey and Veritasium video, with the wrong ingredients. I suspect this is due to you getting too close to the “What is True?” dimension that Grey warns a LOT about. Even though I’ve seen other videos on this topic, I still finished the video confused.
I still do like the video, Im sorry if this comes off negative. I feel its a hollow compliment but the visuals and audio are really good! Please keep making more videos 😊
6:44
What's stopping us from doing a similar proof for integer numbers?
1. Assume you have a list of integers.
2. Make a new integer that's one more than the "last" integer. (if "... and so on" is alllowed in the original proof, doesnt that imply that there is an ending to the list?)
3. This number is not in the list.
You're thinking of a list as a finite object. In the video, "list" is used as language for "a way to assign something to each of the infinite number of integers." So, we are "listing" infinite real numbers when I say to make a list of them. A real number can have an infinite number of digits, so it's reasonable to define a real number that has a digits for each of the infinite quantity of numbers in the list. The idea is that if you only have "as many" real numbers as integers, you haven't listed all the real numbers. In contrast, you can easily put all the integers in one of these infinite lists by just assigning each integer to itself.
The more accurate way of putting all that that doesn't use language like "as many" to gloss over the details is that there's no way to match up integers to real numbers such that every possible real number gets a unique integer. What the proof is saying is that if we have made a way to assign integers to real numbers, there is a real number that is not matched up with any integer at all. This goes against the intuition that because integers are infinite, we must be able to pair them up one-to-one with other things we have infinitely many of. Because of this, we say there's "more" real numbers than integers.
There is one obvious infinity that exists and that’s space. Space cannot be finite because that would be something existing within a greater context of nonexistence. You can’t have something existing within something that doesn’t exist, that would be a structural paradox. Because space is a tangible construct, it’s infinite value can be given a numeric number. What would that number be? Well, since space has three tangible spatial properties being its height, width, and length, each of those (properties) value can be assigned “one,” or in other words, 1 cubed; and since it’s an infinity, it would have to be written as “one cubed times infinity. [That’s one height, times one width, times one length, times infinity.] One is the working number, and infinity is the constant. How can one be a working number you might ask; well, one can be divided to finity within the greater context of infinity…
Integers are not finite in length. They have an infinite amount of zeros preceding and following them. However, it would be overly burdensome to list all of the infinitely many zeros, so those zeroes are truncated, but they still exist.
The idea of different sized infinites should therefore suggest that there can be different sized zeros.
Instead think of the decimals as representing some full integer number of infinitesimals, and the whole number as representing how many times you have listed all of the infinitely abundant infinitesimals.
5:25
Should we consider set's density in place of its cardinality in that case ?
Imagine that surface area is Aleph_3 and volume is Bett_3; therefore, they are not the same size, but the cardinality of the categories about them is. So it is both, but only past Aleph_2.
Hey wait a minute, this is really good!
14:48 It's not completely true that you need a more powerful system to proof that peano arithmetic is consistent. Gerhard Gentzen showed what you can proof Peano arithmetic is consistent, as long as a certain other system used in the proof does not contain any contradictions and the other system used in his proof is neither weaker nor stronger than the system of Peano axioms.
The segment on rational numbers (starting at 4:15) doesn't have me convinced that its cardinality is identical to integers. What about the infinite number of rational numbers between 0 and 1 plus the infinite number of rational numbers between 1 and 2 plus...the infinite number of rational numbers between any two integers x and x+1? Where do they fit in the unique pairings?
You have numbers like 5/4 which were included. They are larger than 1 and smaller than 2. You can find any rational number in the listed ones.
There are countless ways of showing the cardinality of rational numbers is the same as the cardinality of integers, you can relatively easily look them up
The union of a countably infinite collection of sets, each of which is countable, is always a countable set.
@@rsm3t not true. Union of finitely many countable sets is countable. Union of infinitely many countable sets is uncountable
@@methatis3013 False. Pick up a book on set theory ffs
You maths dudes might be able to help out the quantum nerds by working out what number fits between zero and the next number up, this being the mass of an atom or the charge on a quark etc, and find which number when fitted into a set makes an illuminating change. This is just an experiment and may not yield results, but may be a way to break our present understanding and make a discovery. The presumption I have made is that the smallest possible physical quantity that exists in the physical world is the key to a puzzle. The overarching view is that numbers are a quantity, no matter what else we can do with them.
Mathematics can be related to & is often motivated by real-world problems, but it should not be defined by them. For example, you want the smallest positive mass that can exist, but it's not useful to make that the smallest positive number. You didn't specify the units, you might have to compute the gravitational attraction between two such masses, and numbers don't exist just for physics calculations.
I didn't specify anything. I've lost track of the video but essentially knowing myself I have probably tried to prompt a solution to a puzzle. I think I have suggested that which you say does not apply, that numbersMUST apply to the physical world.In the circumstance you suggest for example other calculations such as gravitational forces might really need to be faced up to as aspects of the original rather than numerical values of their own. Thanks for your reply. I am sure I am looking for what may be a new way to calculate,in order to help out the scientific community who are looking for a unifying principle.@@jorgenharmse4752
Arguably 0 isn't a number either for many of the same reasons, since a lot of exceptions have to be made for 0
0^0 is undefined, and so is n/0, among other examples.
There's also no consensus on whether or not the natural numbers include 0.
In many ways infinity and 0 are each other's inverses. 0 is the limit for endlessly shrinking values, while infinity is the limit for endlessly growing values
0 is pretty well defined though. It is as equally well defined as 1. 1 is the limit of endlessly shrinking values around 1, just like 0 is the limit of endlessly shrinking numbers around 0. There are also many operations that make exceptions for 1 as well, for example the logarithm with base 1. 0 is as much of a number as any other
0 is, in fact, well defined by a Dedekind cut which is currently a way by which we define all real numbers. Feel free to look it up
all of your videos are amazing.
It may not be a number, but in the world of the affinely extended real number system, and the real projective line (those are things go check them out on Wikipedia), it is.
I am not a mathematician or even a student of it. I do enjoy good explanations of it though. So thanks.
One of the things that bugs me about infinity is that nothing we ever deal with has infinity. Whether infinite time or infinite numbers of things. I looked at ZFC axioms and in there is an axiom that basically magics infinity into existence. The axioms being things you take as truth without proof. In other words , they just magicked it into existence and all the "infinity" talk I suddenly justified. So I've always wondered if you take that little toy away from mathematicians or physicists,, do lots of the paradoxes disappear. Suddenly you can't infinitely divide up space. Or the infinities that plague the joining of QM and GR vanish.
I don't know, but to me a lot to the worst nonsensical stuff disappears when you take away the infinity toys from maths/science and deal with reality.
I'm a little confused on counting rationals and reals. The cardinality of reals should be the same as irrationals. But between any 2 irrationals is a rational. How then can there be more irrationals than rationals. Pick sqrt(2) as a starting point. To get to the next irrational, either on the left or the right of it on the number line you necessary pass a rational. It seems impossible to start at an irrational number, travel along the number line any amount and count more irrationals than rationals?
Is it maybe the case that, in the same way size is ill-posed when it comes to infinity, so to is counting? Or is there something else i'm missing here?
edit: also fantastic video, I sincerely hope you make many more in the future.
The issue here is not necessarily counting, but that there is not a "next" irrational after some particular irrational. If you pick any two non-equal irrationals, they have an infinite quantity of rational and irrational numbers between them. It also happens that the cardinality of the set of rationals you skipped over is beth zero, while the cardinality of the set of irrationals you skipped over is beth one.
@@JoshsHandle I think I get it but will definitely have to digest this more. I'm trying to construct a way where I don't skip any numbers but I guess that is impossible?
The statement that there exists a rational between any 2 irrationals, whilst true(?) says nothing about how many rationals and irrationals there are in this interval. And there is no method of zooming into the number line to notice a string of numbers i1,r,i2 with no other numbers between them, where i's are irrational and r is rational. Even at infinite magnification if that even makes sense.
Infinity is weird! But I think I'm slowly getting there. Thanks!
"The cardinality of reals should be the same as irrationals." That happens to be true, although proving it without the axiom of choice requires a bit of work. (Given the axiom of choice, the cardinality of the irrationals is aleph_alpha for some alpha, so beth_1 = aleph_0 + aleph_alpha = aleph_alpha.)
Great maths content aside, your wolf animations are too damn cute
Great video ! I'd be a bit more careful about terminology though ; integers != natural numbers (for example)
awesome video, love to see these
9:44
You got it Wrong, sqrt of 4 is 2 not -2. Sqrt of any possitive number will give a possitive number
I don't agree that, sqrt 4 =2 or -2 because you can try (-2)^2 then you got 4 .Sqrt just reverse
A wild interrobang appeared!
18:12 I think it's misleading to say 'get closer to infinity'. Infinity means endlessness, which is why it's not a number which is after all a finite quantity - saying you're 'getting closer to infinity' is a contradiction in terms, because you can't reach infinity. If you did reach infinity, by counting long enough then the really big number you counted to would just be a big number with an endpoint on the counting line, not endlessness.
I think it would be more accurate to say if you continue further and further or something.
But that's a minor nitpick, I'm sure everybody knows what is meant, maybe just it encourages people to thinking about the concept the wrong way.
You could also use the formal definition of a limit at infinity, but that is too pedantic and needless for what this video is trying to achieve
Great content! A new 3b1b is rising to bless us
Excellent video! I just have a bit of constructive criticism regarding the thumbnail of the video, that says “This is not a number”. I think it is a bit misleading to say that infinity is or isn’t a number, since the word “number” is ambiguous to begin with. It is not a natural, real, or complex number. But it is a Hyperreal, transfinite cardinal, tranfinite ordinal and surreal number. So it depends on the context.
I actually thought about this when making the thumbnail and introduction. I very specifically chose the infinity symbol because that does not represent a number, that symbol is only used in calculus. I do agree that it is reasonable Aleph/Beth numbers numbers.
@@JoshsHandle perfect! Them I take back my criticism, I agree that ∞ is not a number (in the common way it’s used). I congratulate you again for the amazing video and animations!
Well when it comes to hyperreal numbers, we aren't using infinity per se. We are using ω and ε which can be thought of as infinitelly large and infinitely small, but they aren't exactly infinity. They are a simplification of the concept
@@methatis3013 really? I hadn’t heard that, what’s the difference?
@@AndresFirte well the difference is that infinity is just purely a concept lol. We use hyperreals to tame that concept. Hyperreals work with addition and all other operators as regular numbers. For example
ω +1 > ω
even though ω is thought of as infinitely large.
Out of curiosity what software do you use ? Looks like a 3D Manim to me
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?