One night I was extremely bored at work and had nothing to do but look at the different kinds of infinity on Wikipedia. Despite quickly running out of articles I could even slightly comprehend, there was still a kind of shock-and-awe spectacle to the whole mess that kept me trudging ever onward into bigger and less comprehensible cardinalities. Later entries on the list often seemed to swallow the previous ones whole, dwarfing the sense of scale I'd just grown accustomed to like godzilla eating a whale. Eventually, the size of the infinities became unnerving. I could no longer read their definitions. It seemed as though there should have been a faint rumbling in the background- that the lights should have flickered, or that spasming artifacts should have started flickering across the pixels of my monitor. The nomenclature didn't help. A lot of the names these things were given suggested children on a playground trying to think of the strongest super-power. Children who were really into Lovecraft. Further still, the infinities' bulk began inflicting harm on the structure of mathematics. They required specialized axioms to define, or to prevent their existence from proving that 1 = 2. They created paradoxes in their wake like black holes warping spacetime. I don't know if I would have found any of it half as interesting if I wasn't so incredibly bored, but at the time it was a pretty good way to kill two hours.
You are on the right track, you came up with a short essay which makes no sense whatsover. don't quit, go to the beginning of it and copy it over and over. Than you will have touched the face of inifinity, like the author of this RUclips video. Peace !!!
+Ekaterina Nosenko It will depend on who thought you and where you learned about it. A pal of mine did a PhD in this stuff about 10 years ago, he referred to it as Aleph Null.
Philosophical Owl To begin understanding infinity, take amount of the second that passed by. Now what happened in the last 10 minutes. Then the last hour. What happened yesterday. A week. A month. 6 months. A year. Two years. Three years. Five years. 10 years. Your entire life. Childhood. Your birth. Now imagine all of human history, the creation of earth, the creation of the sun, and even more. I call this “The Infinity Meditation” if you want to realize how small your problems are in spacetime continuum.
This was the first among your videos that I wasn't able to follow, but it is also officially my favorite. I have been both fascinated and frustrated by infinity, and this just opened my view of the subject much wider. Now I have a direction to begin learning...
You state that you were frustrated by infinity, did you take infinity to small claims court and asked for refunds ? What a hoot !. What you were witnessing was not infinity but too much lasagna...or beer or watching Tom Brady blowing his chance to win the SuperBowl.
A box containing all boxes is easy to make, in theory. Just make a box (for simplicity's sake, mark the inside with an "I" and the outside with an "O") with no boxes in it, and turn space inside out. All boxes are contained in the box. The outside of the box will then be inside the inside of the box, which you know is the inside since it has an "I."
so what no it not rabbits are good I loòøöôvveee rabbits want to think what I just wrote you love rabbits will have as pets you protect them willingly participate in promoting there well and listening to vihart is where rabbits. another thing you will repeatedly saying to your self I love rabbits there my pets!
theScholarlyWalrus It's a quote from The Fault In Our Stars. I don't think it's entirely random that ViHart made a video on this the day after the premiere of the movie...
***** Not really, a lot of ViHart's followers are also Nerdfighter, and I think she is a Nerdfighter as well. It could be a coincidence, but I don't think so.
You are carrying the torch for Lillian R. Lieber beautifully! Her book called Infinity exploded my middle school mind (in a good way), and your video brings both nostalgia and fresh insights. Thanks!
Yes, when people in the community talk about the FGH (fast-growing hierarchy) to quantify the growth rates of certain functions, ordinals are used I believe
Pretty sure it's countable considering it's just the limit / fixed point of Omega0 / Aleph0, which means it's a (countably) infinite power tower of infinities
***** many people have said this many times. because it is a fact its like saying 2 is bigger then 1 in reality it should not be a reference rather way. but in the back of the book he says directly that he was refraining vihart. so thats who he was refrancisng
***** I think I missed that part... Because I have the book, and nowhere in my copy did it say that specific part was referencing ViHart... She is in the acknowledgements, but it's not specific.
***** How does that make it from her? Maybe it was some other math stuff that he was acknowledging her for. You're making an assumption, which, though somewhat likely, is not definite.
From what I was actually able to follow of this video (I'm not mathematically minded), it was fascinating. I had no idea there were different types of infinity. Thank you!
Great video! You covered some things even I didn't know about! I should say, there's also a whole class of infinities greater than uncountables, but smaller than absolute infinity called inaccessible cardinals, where you essentially have to add a second (third, fourth etc.) axiom of infinity to describe these numbers. There's an interesting graph (I think it's in VSauce's video on infinity) on the relative sizes of all these inaccessible cardinals
You state and I quote " Great video! You covered some things even I didn't know about! I My response to you is this . She also covered things which do not exist, and nobody knows about. Her covering it did not add to your understanding of infinity, just confused you infinitely more hopelessly. Peace bro !
@@aligator7181 All of these things are established concepts, and I have at least a surface knowledge of all of them. So I'm not sure what you're talking about.
I just really enjoyed this video. I love thinking about these kinds of things. Now I am just sitting here feeling extremely excited about all the infinities and I feel so happy.
That was absolutely excellent. I'm a grad student in maths myself, and I've kinda been thinking of making something like this for a long time, but this is done waaay better than I could have managed. Wow.
I love how she explains ordinals :). By the way, there are uncountably many countable infinities. In Googology Wiki, we would commonly use big Omega or w_1 for the first uncountable infinity.
I love this! I was looking for a some sort of informative medium that would totally blow my mind with incredibly complex mathematics. I have only seen this one video and I can confidently say I love you Vihart!
It's been many years since I watched Vi with any sense beyond "Oh hey, here's some nostalgia from a simpler time" (for reference, I first started watching the "doodling in math class" series in about third grade, and I'm now a senior in high school). Since then, my fascination with anything math has been... well, thoroughly crushed. In school, I only ever learned a very specific version of the "how" of mathematics--how to do this or that equation, how to find this or that solution, etc etc, but I never figured out the "why" (why are we doing this, why does this matter) or the "what" (what are we doing, what is this supposed to be, what of any other parts of math does this even relate to), or other versions of the "how" (how do i figure out the "why" and the "what" myself, how do people even discover this stuff, how do people enjoy ANY of this)... The methodology of completing what we were given always clicked (even through dyscalculia, somehow, although numbers can still evade me--don't ask me about the time in freshman year that I added 49 and 18 several times over and continuously got 57 until I actually, finally plugged it into a calculator upon my baffled teacher's request), but never anything else. I couldn't see for a long time how I thought any of this could be fun (although, when we got to the Fibonacci sequence in my math class last year, I remembered it from the mini-series about Fibonacci and spirals and plants and got excited because that was something I understood. And earlier this year, in my AP literature class, my teacher (the coolest guy, possibly) mentioned how he would "teach [us] how to write the perfect essay using the Fibonacci sequence", which he hasn't done yet, but he warned me not to pop a blood vessel when he saw my enthusiasm for the Fibonacci sequence). But today, for the past few hours or so, I've been binge-watching Vi's stuff again. Starting with the doodling in math class series, because that was got me started in everything (I've yet to go back to the hexaflexagons, but I'll get there eventually, I'm sure). And I think it's helped me come up with some sort of answer, or at least part of an answer, that gets me curious and excited about math for the first time in so long--it's a way to conceptualize patterns (and/or the lackthereof, in certain cases). It's a game of logic, it's a puzzle, it's a knot to untangle that stretches out to as many infinities as there could possibly be, and those are things I can look at and hold, things that feel more complete than "Well, you won't always have a calculator in your pocket!" or "Well, this is how mortgage rates are calculated, which will be useful when you try to buy a house some day" ("and why exactly do we need mortgage rates when, with all of the mansions and hotels and empty condos, we could house so many more people and get rid of so much more suffering? what part of mortgage rates is necessary, why should this be a thing that would be useful for me to know?" is what I did not ask my teacher, because I'm pretty sure he didn't want an hour-long dissertation on why housing is a basic human right that should be met whether or not someone can pay for it, because basic human rights shouldn't be pay-to-win-or-else-you-die-and-or-become-homeless-thus-making-you-more-likely-to-die-anyway--but I digress). In all, Vi, if you see this... thank you for reigniting my passion for numbers and patterns, for spirals and curves, for infinities of every different size. I think I might reconsider my feelings on math and try looking at it at another angle, now that I have the time (no way in hell I took another math class when I got all my math credits two years ago and my class last year just counted for an elective) to look at it and digest it in a way that makes sense and is fun for me :)
Small quibble in the projective geometry section. You didn't say this, but you drew it on the page, and that is the parallel lines meeting at infinity. Any two parallel lines will meet at one point, not at two points as is shown in the drawing. (I realize you're aware that it's the real projective line at issue, and that it doesn't operate like the extended real number line in that the real projective line has one, and only one, point at infinity added to it, rather than two. But for the purposes of giving flavor to the video, I doubt axiomatic precision would be of much use.)
The two symbols she drew of the lines "meeting" on paper were at the same point though. Projective infinity (when projected into a Euclidean R^2) basically just looks like a halo beyond all finite regions, even though that "halo" represents a single point, and the lines would *appear* to meet there at both extremes. :3
Happ MacDonald indeed, it does. I was gaming when I was watching this so I missed her having designated it as such (and, indeed, she even specified that -inf = inf. I glanced at the line and saw two different notations for what should have been the same point. Clearly, I missed that.
"Meet at infinity" is another way of saying "don't meet" There is neither one point nor two where parallel lines meet, since a point can not be placed at infinity.
We learned about this in math class around a week ago. I got super excited and squealed when the teacher announced it because it reminded me of you and this video. Everyone stared at me....
That last segment when you were giving examples from what fields these infinities come from gave me a rush. I love that type of stuff. I'm not a mathematician, I'm studying neuroscience. I've always known math was everything in science, that science finds it origins in math and to do most science a good understanding of math is required, but showing just how pliable it is for all different uses reminds me how awesome it is. I'm still in my undergrad for neuro, but I actually can't wait to learn one of the many fields of math that will most help with my research/projects in the future. Good job, Vi! Another brilliant video!
An interesting corollary of this video, if you pick a number at random, the probability of you picking any specific finite number is zero. Now, pick a number at random. No: this one is not random, try again. Euh... see you after eternity.
is it that easy? i mean i understand, the idea: you have a set with infinite elements and pic one out of it, BUT is picking thins out of an infinite number possible? you can not pic any number out of the natural ones, you are limited by for instance the (life)time you have to put down that number. hm, i wonder (if or actually) how a theoretical model that does what you say, would work of course you could imagine this to be a geomatricaly distributed experiment. but than you would not pic out "one element", it is more like pointing at one element, and if you are "close" to the specific number (which means you are in a defined intervall around that specific number) you succeeded.
It is impossible to pick a random number out of the set of all real numbers, as the number of real numbers is uncountably infinite. If you limit yourself to algebraic numbers, which are countably infinite, you can pick a number at random and not have the probability of any specific number be zero. However, if you impose the limitation that the probability of picking each number must be equal, then the probability of any specific number is zero.
+Tahititoutou "Randomness" just means that every element has probability. The "finite numbers" are uncountable, but you can say something like "1 has 1/2 probability, 2 has 1/4 probability, 3 has 1/8 probability, and so on."
Sort of, but there are practicalities. A human only lives for a finite time so only has a finite set of numbers to choose from in the first place. So in reality the probability of a person choosing a specific number will be nonzero for some numbers.
Incredible. I really enjoyed this, so thank you. I'm impressed by the breadth even though I know almost nothing about the subject. (I self-identify as finite, but try not to be closed minded :-). At any rate, seeing a survey like this then a summary on one sheet, at least gives me some framework to plug things into...and to better see how much I don't know.
What a hoot! I think this was the most challenging for me (and one of the most fun) videos ViHart has ever done! I'm somewhat familiar with multiple infinities, but this introduced a bunch of mathematical concepts that I'd only heard about and some I've never even heard of.
Yay, one of the many Vi Hart videos that were super informative and went almost totally over my head! (except for the topology part, I'm a fan of geometrics and whatnot, so that was easier to digest).
Big Omega deserves a bit more than what you give it here. if mathematics (ie set theory) is consistent, which it almost certainly is, then Big Omega exists - it just isn't a set. it exists both outside of the system as an intellectual concept, and virtually inside the system, because set theory is self-descriptive. we can emulate Big Omega in a sort of virtual set theory and talk about it just like any other set, even though the "real" one isn't a "set". see Skolem's paradox for more info. very deep metamathematics. the term "class" is sometimes used for a collection which cannot be a set within the scope of the system. Cantor equated the class Big Omega with God. and I totally agree.
Cantor's formulation of the transfinite ordinals and cardinals was revolutionary, but infinitesimals did not fit easily into it. Mathematicians were skeptical of the infinitesimal for centuries because no one was able to create a definition consistent with the laws of arithmetic, until years after Cantor's death. Since the limit had already been formalized, infinitesimals were no longer necessary for doing calculus, and many discarded them as obsolete and inconsistent. The fact is that infinitesimals actually represent a completely different concept from ordinals and cardinals. Ordinals represent well-ordering, cardinals represent counting, and infinitesimals represent measure. You don't need to consider measure to understand the Absolute Infinite. How does measure contribute to the concept? Perhaps my idea is limited as well!
+gingergamergirl98 I feel slightly smarter every time I watch this video. My understanding gets a tiny bit more nuanced. It's like throwing grains of sand into a bowl the size of the Sun, but it's still progress :)
Great video, thanks for making it! I always love watching your stuff, even though math isn't really my forte, it's always neat to see these concepts, like, the infinities of the decimals between real numbers like you showed. I really hope you didn't get sharpie on your sleeves though, I couldn't stop thinking of that toward the end!
Nope, every time you watch it, no matter how little you know, you're gonna learn something new. You can watch it again and again to learn new stuff each time, until you understand everything. So it is appreciable and need not be completely understood in one go for it to be appreciable.
Pretty sure I didn't miss it, but what about 1+2+3+4...= -1/12? There's some wacky proof involving it, but I can't wrap my mind around how it makes any sense.
Taiga Aisaka Noy sure about what to think, but it's a result used and it works and describes reality so if it's incorrect then I think a big part of the quantum theory might be wrong.
No, you have to stop there! That equal sign does not represent what you think it does. It is not the same thing as 1=2/2. Rather, it assigns a "meaningful value" to the series. This value describes the series in a mathematical way, but DOES NOT ACTUALLY DEFINE THE INFINITE SUM.
I love Vihart. I have absolutely no idea what she is saying most of the time. However she says it well and I feel smarter after I hear her out...just like my wife. Keep the videos coming.
It's serious when the transcript says "unintelligible." 14:30 says "Primitive Recursive Arithmetic" I remember when recreational math videos like Vihart motivated me to dive deeper into number theory... now starting college, this is tempting me to major in math :P (either way it's cool tho)
I used to watch Vihart all the time, then moved on to Numberphile and 3Blue1Brown and the like. Just came back here because of a video on Gondel’s Incompleteness Theorem.
I kept thinking when is she going to say that and whether I should post this quote, but then I thought everyone else is going to post it anyway so I should focus on what she is saying.
KS Ng colethewalrus Yup. It was made. A lot. So many times in the comments that the comments have a larger percentage of this joke (or references to it) than comments regarding the video itself, it seems.
I thought this video was very fascinating! What I like about the video was how the concepts and thought processes were drawn out, not only did that help auditory learners, but it also allowed for visual learners to take part in learning. Honestly, I was so confused the first time I watched, and I had to replay it about two or so more times. After grasping some if not most of the ideas being displayed, it inspired me to realize how large - for a lack of a better word - the world is. Everyone has their own way of perceiving things, whether it be looking at the micro or macro. While watching, I noted that infinity is an abstract concept, - key word being concept - meaning that someone's infinity is not the same as another's. What stuck to me the most the list of the infinite "all real numbers." That inspired me to do a little research! The real numbers include all rational numbers, fractions, and irrational numbers. Although it might be infinite, it does not include imaginary numbers. However, an imaginary number can be written as a real number! Take for example, the square root of negative one would be "i," an imaginary number. When squaring the "i," it will be negative one (-1), which is a real number. There is also the line at infinity. Ever since about first grade (give or take a year), students were taught that parallel lines will never meet, and that they will go on for infiinty never meeting. In geometry, the line of infinity, also called the ideal line, is added to a real plane to essentially stop the result of a projective plane. This raises my question of is this an infinity ending? In projective geometry, any pair of lines always intersects at some point - at infinity! Which brings me to a final thought; is infinity the endpoint or starting point of an infinity?
I wish I could sit down with you and talk for hours about this. It's soo amazing. I study computer science, so I am familiar with aleph 0 and the non-countable infinities. All the other infinities are new to me. I'd really like to get to understand the formal definition of everyone of them. A question for you ¿Is 0 a member of the natural numbers? PS: I think the Real numbers are called like that because they are the better at describing reality
GenaTrius I love the word 'agglutinating'. It's new to me and I always love learning a new word. I've looked it up but to help me understand it, what names would work better for you?
BobSkiz1 I'm not completely sure if I understand this correctly, but I think it is just a language where you take a root and keep adding suffixes/prefixes to get new words. Newspeak from 1984 comes to mind, where you say "good", but then a more intense good is "plusGood," and an even more intense version is "doublePlusGood." And then if you wanted a really bad word, you would say "doublePlusUnGood"
MrAmazingAwesomeness You are right :)) My mother language is agglutinating and it works like that. Except you can't keep adding pre/suffixes foever :)) Some examples: "jó"="good", "jobb"="better" (="plusGood" :D), "legjobb"="best" (="doublePlusGood" :D) and "doublePlusUnGood" would be "legkevésbé jó". See, that already split into 2 words :)) Another thing I wanted to add were the names for infinity: Countable is "megszmálálható" and "uncountable' is "megszámlálhatatlan". The others greatly resemble the English equivalent ("cardinal" is "kardinális")
I gave up at 10:40. You lost me. I feel dumb. (I have a bachelors in Electrical Engineering and a Masters in Secondary Math Ed, and I studied Hyper-real numbers in grad school.) Vi, I bow to your awesomeness.
Great video. Most videos stop at the fact that real numbers are a higher level of infinity, at best only acknowledging that there are infinite infinities without giving an explanation how that comes to be.
I've been contemplating a shape I may have made up that I'm presently calling a meta-helix. It's basically an infinitely scaling helix made of helixes. As if you were to take a phone cord and wind it into a helix, and then take that larger helix and bend it into a helix, etc... and of course if you looked closely at what you thought was a phone cord you would see that it is actually made of a smaller helix, and that helix a smaller one, etc... I really need to make a model, I feel like it still sounds more confusing then it is. Well, anyway if you can visualize the infinite scaling helix I'm talking about, snip a little segment out of it. You can identify the locations of the ends of your segment and define the space it occupies, but if you pull on the ends to measure it how long it is, you will never stop pulling because a segment of this shape is still infinitely long! The length cannot be measured even though you can locate both ends of the segment!
I am the only one who thinks one of the reasons why Vi made this video is TFIOS? :) "I cannot tell you how thankful I am for our little infinity" Or I'm just crazy.
She was actually mentioned in the credits thingy in the back of the book ( My mind had a blank so I have no idea what it's called atm) and when I saw that I fangirled a bit
Okay so I'm not trying to stir anything up, so please, nobody criticize me for my question. The whole idea that an uncountable infinity is "bigger" than aleph null doesn't seem right to me. I'm bad at explaining but I'll try my best. So, there are essentially an infinite number of real numbers between, say 1 and two.The amount of numbers between these two is, at least how I see it, equal to aleph null. If someone would like to argue otherwise, go ahead, this question is for the pursuit of knowledge. Anyways, that means that for every counting number, there is an infinite amount of counting numbers between them, in a sense. So there are aleph null counting numbers and aleph null numbers between each of those counting numbers, which would mean that, hypothetically, there are aleph null to the power of aleph null real numbers. Again, I could be doing something very wrong here, its late and my mind isn't too sharp right now. Vi mentioned that aleph null to the power of any number would still be aleph null, which makes perfect sense to me. However, this would mean that aleph null to the power of aleph null is stillaleph null. Again, this is fine, just like how 1^1 is 1, no rules are being broken (as far as I know). So by this idea, aleph null would be an uncountable infinity in itself, even though it is not. And this is where I do not fully understand. My assumption is that I made a false assumption (probably several) along the way but I'm not positive since theoretical mathematics isn't really my thing. If anyone could clear this up, I'd really like to know please. TL;DR: If all real numbers (an uncountable infinity) is aleph null to the power of aleph null, and aleph null to the aleph null is also aleph null, then how is an uncountable infinity "larger", so to speak, than aleph null?
Short answer: a (finite) number to the power of aleph null is bigger than aleph null - 2 to the aleph null is at least aleph one Long answer: There are as many real numbers between 0 and 1 as there are on the whole ("countably") infinite number line - at least aleph one. If there were only countably many real numbers between n and n+1 then you'd have aleph null disjoint intervals with aleph null numbers in each for aleph null times (NOT to the power of) aleph null numbers, which would be aleph null still. Instead, you have, call it C (for "continuum") real numbers in any finite interval, and a countable number of such intervals, so you have aleph null times C (or C times aleph null) which is C real numbers in total. Proving that C is actually bigger than aleph null is tricky - the standard approach is to use something called the Diagonalisation Argument, which says that if you listed every real number between 0 and 1 then you could use that list to write a string of digits that you'll never find on the list, but represents a real number between 0 and 1 with only one standard representation, so a real number which isn't at any finite position on the list. Since the defining feature of countable sets is that you can make a list of them which includes each member at some finite position, that proves that the real numbers between 0 and 1 aren't countable. The question of whether C is aleph one or not is generally regarded as undecidable - it's been shown that it's independent of the standard "rules" for (infinite) sets ("ZFC") - meaning you can chose whether you want it to be true or false and continue doing sensible maths from there, and people have done both...
Vihart, you are the infinite teacher. I know that doesn't make sense, yet it does make sense. And there aren't enough words to describe that. Still, it exists. It is. Thank you
I really like how you incorporated irrational numbers into your explanation. I'm confused because you're making it sound like you can quantify infinity. Your video inspired me to look for more types of infinity. What an interesting perspective on infinity!
No! You're wrong! They're adopting a set of axioms and then then they're taking shit. ps. This comment is still provable mathematically if you meant "talking" instead of "taking."
They're taking the shit to a whole new level. Like if you were completely lifetime dedicated to whatever nonsense what-if you discussed with your mates and worked out all the rules and consequences...
Hi -- my 8 year old woke up this morning telling my 4 year old about your videos! "It's mostly math stuff, but drawn funny. There's one where she says "hi I'm a happy square!" It's hilarious!" Thank you for being you.
I love the fact that even if most of your viewers don't really understand your videos (me included) you just DON'T CARE and do them anyway, doodling happily, as if we understood. Amazing! :D
People disliked this video? Please return to watching The Kardashians, and stop down thumbing things because you can't understand it. I mean, really, why? Math haters or haters of Sharpies? Or wearing finger-less gloves while using Sharpies? Or people so math smart they disagree with everything she said? I guess there could be 200 of them. It's possible.
+SafetySkull If you accept the Axiom of Choice, then nope! You can prove that aleph null is the smallest transfinite cardinal number. This means that there is no cardinal number which is infinite in value that is smaller than aleph null. How this is done: You can show that every infinite subset of the natural numbers can be placed in one-to-one correspondence with the natural numbers. Thus, every infinite subset of the natural numbers has cardinality aleph null. Then if there were a smaller "infinity" than aleph null, there would have to be an infinite set which is in one-to-one correspondence with a subset of the natural numbers (which would have to be infinite). But by the above fact, all infinite subsets of the natural numbers have cardinality aleph null, which would make this "smaller" infinite set also have cardinality aleph null, which is a contradiction. The proofs of these facts are not trivial, and your question is very good! Here, you don't technically need the full strength of the Axiom of Choice. You just need the Axiom of Countable Choice (which is a weaker version). But if you reject the Axiom of Countable Choice, it is possible to find a smaller transfinite cardinal number than aleph null.
+SafetySkull Make sure you have your types correct. Can you compare a real number to a cardinal number? Cardinal numbers should be thought of as sizes of sets. Does pi number of elements make sense? Do you mean "Is that a cardinal number that exists between the cardinal of the size of the set of real numbers and aleph naught, which is the size of the set of the natural numbers?" This is known as the continuum hypothesis. It can be shown to be independent of ZFC (Zermelo-Fraenkel plus Choice). So take it or leave it. If you mean "Is there any cardinal number larger than all finite cardinals yet smaller than aleph naught?" No, aleph naught is the first infinite cardinal. I don't think we need the axiom of choice. Of course, you may feel free to use it.
I am almost certain there isn't a number that fits your description, though your conjecture could be undecidable. Sadly, I'm not a mathematician, so I don't know the definitive answer...
The problem with your question is that the order of the infinities, as presented here, does NOT coincide with the natural order on any number system bigger than the positive integers... In fact, these infinite numbers are rather to be perceived as sets than as numbers. And aleph 0 is simply the set of all natural numbers, which can be perceived as a number itself (in set theory, one usually perceives the number n as being the set {0,1,...,n-1}). So, while one could repair your question in one way or the other, what you ask here is simply an ill-posed question.
I just like to say that omega + 1 is not bigger than omega, it just goes after omega. Ordinals represent order, and cardinals represent a specified amount of stuff. It's important not to confuse the two.
Bella Hatch You don't- you just think you do because you don't know the true nature of maths. You might hate the school subject, but you don't hate all of maths.
i thought this was a great, very informative video and the way you write it all down in colors is cool. I've always wondered about the whole infinity concept and this answered a lot of my questions you explained it and gave good examples that helped.
watching this makes me feel like i've just taken the red pill and the blue pill, crushed them into a single, purple powder and snorted it
+Cursed Phoenix
Not only that but the feeling should last for a couple of days at least
Blue Spirit months....
Isn't that an xkcd comic?
+Luca Scharrer Yes it is.
Alex Robinson Years
One night I was extremely bored at work and had nothing to do but look at the different kinds of infinity on Wikipedia. Despite quickly running out of articles I could even slightly comprehend, there was still a kind of shock-and-awe spectacle to the whole mess that kept me trudging ever onward into bigger and less comprehensible cardinalities. Later entries on the list often seemed to swallow the previous ones whole, dwarfing the sense of scale I'd just grown accustomed to like godzilla eating a whale. Eventually, the size of the infinities became unnerving. I could no longer read their definitions. It seemed as though there should have been a faint rumbling in the background- that the lights should have flickered, or that spasming artifacts should have started flickering across the pixels of my monitor. The nomenclature didn't help. A lot of the names these things were given suggested children on a playground trying to think of the strongest super-power. Children who were really into Lovecraft. Further still, the infinities' bulk began inflicting harm on the structure of mathematics. They required specialized axioms to define, or to prevent their existence from proving that 1 = 2. They created paradoxes in their wake like black holes warping spacetime. I don't know if I would have found any of it half as interesting if I wasn't so incredibly bored, but at the time it was a pretty good way to kill two hours.
You are on the right track, you came up with a short essay which makes no sense whatsover. don't quit, go to the beginning of it and copy it over and over. Than you will have touched the face of inifinity, like the author of this RUclips video. Peace !!!
When you make a essay for a RUclips comment but can’t right a school essay.
Interesting. You've really kicked me off my mathematical high horse with this one.Anyway, Aleph Null sounds like a badass spy name.
It's called aleph zero
+Ekaterina Nosenko "Null" is German for "Zero", so this might be a valid notation
+Ekaterina Nosenko It will depend on who thought you and where you learned about it. A pal of mine did a PhD in this stuff about 10 years ago, he referred to it as Aleph Null.
Yeah kinda like Artemis Fowl
@@katnos4609 I've also heard aleph nill, but aleph null is definently the coolest
I don't understand anything she's saying yet I'm still watching
Philosophical Owl To begin understanding infinity, take amount of the second that passed by. Now what happened in the last 10 minutes. Then the last hour. What happened yesterday. A week. A month. 6 months. A year. Two years. Three years. Five years. 10 years. Your entire life. Childhood. Your birth. Now imagine all of human history, the creation of earth, the creation of the sun, and even more. I call this “The Infinity Meditation” if you want to realize how small your problems are in spacetime continuum.
Tis the magic of Vihart
exactly
Imagine if vi and vsauce collabed
I WANT VISAUCE
PLEASE
bUT VSAUCE + VI HART + TED ED
VTarded?
VISauce
Lol I came here From vsauce’s video!
heart sauce
biggest infinity? windows start up time
Knighty02 Especially windows 10
😂😂
Biggest infinity? Trump's ego.
🤣
IE9 loading time > Big Omega
This was the first among your videos that I wasn't able to follow, but it is also officially my favorite. I have been both fascinated and frustrated by infinity, and this just opened my view of the subject much wider. Now I have a direction to begin learning...
You state that you were frustrated by infinity, did you take infinity to small claims court and asked for refunds ? What a hoot !. What you were witnessing was not infinity but too much lasagna...or beer or watching Tom Brady blowing his chance to win the SuperBowl.
8 years later and I actually COULD follow the majority of this video...
8:24
Go to your corner, Zero. Stop being weird and play nice.
Just kidding buddy. C'mere and gimme a hug.
(sound of explosion)
A box containing all boxes is easy to make, in theory. Just make a box (for simplicity's sake, mark the inside with an "I" and the outside with an "O") with no boxes in it, and turn space inside out. All boxes are contained in the box. The outside of the box will then be inside the inside of the box, which you know is the inside since it has an "I."
how do you turn such a box inside out? is this box abstract or physical?
both and neither. once you wrap that around your head, turn space inside out again and you've now wrapped your head around it.
That doesn't sound easy.
Ty Prince That's what she said.
but it wouldnt contain itself, hence it wouldnt contain all boxes
More Vi brilliance! Watching this is like taking the red pill with no end to the rabbit hole.
so what no it not rabbits are good I loòøöôvveee rabbits want to think what I just wrote you love rabbits will have as pets you protect them willingly participate in promoting there well and listening to vihart is where rabbits.
another thing you will repeatedly saying to your self I love rabbits there my pets!
The infinite beauty of mathematics
I agree
"Some infinities are bigger than other infinities"
Yes, we know. What did you think you would accomplish by posting this?
theScholarlyWalrus It's a quote from The Fault In Our Stars. I don't think it's entirely random that ViHart made a video on this the day after the premiere of the movie...
Ze Rubenator well that phrase is far older then that book so its a bit of a streatch
***** Not really, a lot of ViHart's followers are also Nerdfighter, and I think she is a Nerdfighter as well. It could be a coincidence, but I don't think so.
in the TFIOS acknowledgements there say Vihart:,)
You are carrying the torch for Lillian R. Lieber beautifully! Her book called Infinity exploded my middle school mind (in a good way), and your video brings both nostalgia and fresh insights. Thanks!
I am on a quest of finding a Vi Hart video I can actually understand. Still looking...
Have you found one
the logarithm one is pretty understandable.
Doodle music?
how to draw a perfect circle
that was literally the only one I understood
*Googology intensifies*
I thought Googology specifically referred to very large finite numbers and talks about things like computability.
Infinity is part of googology
Yes, when people in the community talk about the FGH (fast-growing hierarchy) to quantify the growth rates of certain functions, ordinals are used I believe
+Gingeas is epsilon0 countable or uncountable
Pretty sure it's countable considering it's just the limit / fixed point of Omega0 / Aleph0, which means it's a (countably) infinite power tower of infinities
remember back when "math" was just memorizing multiplication tables in 3rd grade?
No.
Yes.
I got bored with that and skipped to algebra.
Hahahaha..ha... No. :')
This was an infinitely long time ago.
So... Some infinities *are* bigger than others... Also... There's a lot of infinity...
he was referencing vihart when he wrote that. she has said it befor
***** many people have said this many times. because it is a fact its like saying 2 is bigger then 1 in reality it should not be a reference rather way.
but in the back of the book he says directly that he was refraining vihart.
so thats who he was refrancisng
***** I think I missed that part... Because I have the book, and nowhere in my copy did it say that specific part was referencing ViHart... She is in the acknowledgements, but it's not specific.
Skystarry75 he put her in his acknowledgements. and she herself has said this before.
how is it not from her?
***** How does that make it from her? Maybe it was some other math stuff that he was acknowledging her for. You're making an assumption, which, though somewhat likely, is not definite.
From what I was actually able to follow of this video (I'm not mathematically minded), it was fascinating. I had no idea there were different types of infinity. Thank you!
Why is math like... Greek Greek Greek Greek... HEBREW
Because English.
+LibertyLikes Because some of it the greeks discovered, and some of it the hebrews
Because the symbols looked cool.
+Ryan Lintott
hahaha
"the hebrews"
is that racist?
Great video! You covered some things even I didn't know about! I should say, there's also a whole class of infinities greater than uncountables, but smaller than absolute infinity called inaccessible cardinals, where you essentially have to add a second (third, fourth etc.) axiom of infinity to describe these numbers. There's an interesting graph (I think it's in VSauce's video on infinity) on the relative sizes of all these inaccessible cardinals
You state and I quote " Great video! You covered some things even I didn't know about! I
My response to you is this . She also covered things which do not exist, and nobody knows about. Her covering it did not add to your understanding of infinity, just confused you infinitely more hopelessly. Peace bro !
@@aligator7181 All of these things are established concepts, and I have at least a surface knowledge of all of them. So I'm not sure what you're talking about.
I just really enjoyed this video. I love thinking about these kinds of things. Now I am just sitting here feeling extremely excited about all the infinities and I feel so happy.
That was absolutely excellent. I'm a grad student in maths myself, and I've kinda been thinking of making something like this for a long time, but this is done waaay better than I could have managed. Wow.
I love how she explains ordinals :). By the way, there are uncountably many countable infinities. In Googology Wiki, we would commonly use big Omega or w_1 for the first uncountable infinity.
You mean ѡ+1?
I love this! I was looking for a some sort of informative medium that would totally blow my mind with incredibly complex mathematics. I have only seen this one video and I can confidently say I love you Vihart!
It's been many years since I watched Vi with any sense beyond "Oh hey, here's some nostalgia from a simpler time" (for reference, I first started watching the "doodling in math class" series in about third grade, and I'm now a senior in high school). Since then, my fascination with anything math has been... well, thoroughly crushed. In school, I only ever learned a very specific version of the "how" of mathematics--how to do this or that equation, how to find this or that solution, etc etc, but I never figured out the "why" (why are we doing this, why does this matter) or the "what" (what are we doing, what is this supposed to be, what of any other parts of math does this even relate to), or other versions of the "how" (how do i figure out the "why" and the "what" myself, how do people even discover this stuff, how do people enjoy ANY of this)...
The methodology of completing what we were given always clicked (even through dyscalculia, somehow, although numbers can still evade me--don't ask me about the time in freshman year that I added 49 and 18 several times over and continuously got 57 until I actually, finally plugged it into a calculator upon my baffled teacher's request), but never anything else. I couldn't see for a long time how I thought any of this could be fun (although, when we got to the Fibonacci sequence in my math class last year, I remembered it from the mini-series about Fibonacci and spirals and plants and got excited because that was something I understood. And earlier this year, in my AP literature class, my teacher (the coolest guy, possibly) mentioned how he would "teach [us] how to write the perfect essay using the Fibonacci sequence", which he hasn't done yet, but he warned me not to pop a blood vessel when he saw my enthusiasm for the Fibonacci sequence).
But today, for the past few hours or so, I've been binge-watching Vi's stuff again. Starting with the doodling in math class series, because that was got me started in everything (I've yet to go back to the hexaflexagons, but I'll get there eventually, I'm sure). And I think it's helped me come up with some sort of answer, or at least part of an answer, that gets me curious and excited about math for the first time in so long--it's a way to conceptualize patterns (and/or the lackthereof, in certain cases). It's a game of logic, it's a puzzle, it's a knot to untangle that stretches out to as many infinities as there could possibly be, and those are things I can look at and hold, things that feel more complete than "Well, you won't always have a calculator in your pocket!" or "Well, this is how mortgage rates are calculated, which will be useful when you try to buy a house some day" ("and why exactly do we need mortgage rates when, with all of the mansions and hotels and empty condos, we could house so many more people and get rid of so much more suffering? what part of mortgage rates is necessary, why should this be a thing that would be useful for me to know?" is what I did not ask my teacher, because I'm pretty sure he didn't want an hour-long dissertation on why housing is a basic human right that should be met whether or not someone can pay for it, because basic human rights shouldn't be pay-to-win-or-else-you-die-and-or-become-homeless-thus-making-you-more-likely-to-die-anyway--but I digress).
In all, Vi, if you see this... thank you for reigniting my passion for numbers and patterns, for spirals and curves, for infinities of every different size. I think I might reconsider my feelings on math and try looking at it at another angle, now that I have the time (no way in hell I took another math class when I got all my math credits two years ago and my class last year just counted for an elective) to look at it and digest it in a way that makes sense and is fun for me :)
"Its the most numbery number"
you are my favorite mathematician
The most mathematiciany mathematician.
Small quibble in the projective geometry section. You didn't say this, but you drew it on the page, and that is the parallel lines meeting at infinity. Any two parallel lines will meet at one point, not at two points as is shown in the drawing. (I realize you're aware that it's the real projective line at issue, and that it doesn't operate like the extended real number line in that the real projective line has one, and only one, point at infinity added to it, rather than two. But for the purposes of giving flavor to the video, I doubt axiomatic precision would be of much use.)
The two symbols she drew of the lines "meeting" on paper were at the same point though. Projective infinity (when projected into a Euclidean R^2) basically just looks like a halo beyond all finite regions, even though that "halo" represents a single point, and the lines would *appear* to meet there at both extremes. :3
She actually did qualify it with "same point" on the drawing at 11:25
Happ MacDonald indeed, it does. I was gaming when I was watching this so I missed her having designated it as such (and, indeed, she even specified that -inf = inf. I glanced at the line and saw two different notations for what should have been the same point.
Clearly, I missed that.
"Meet at infinity" is another way of saying "don't meet" There is neither one point nor two where parallel lines meet, since a point can not be placed at infinity.
Hmm, never seen the inside of a maths classroom, that's manifest.
I wonder if Vsauce got his ideas from this video. Great job!
No
Anau Naga great argumentation
CrashTestFoetus I doubt it.
CrashTestFoetus He worked with mathematicians at some university.
Stéphane Dubedat
I know, i just blew the audience away with my argument
That's a vast amount of perspective across a lot of different fields for a 14 minute and 55 second video. Beautifully humbling video. Thank you.
What a wonderful 4th of an hour this was. Infinitely grateful to such a stellar contribution to the inter-webs.
"accidentally make one equal two" ...*chuckles* I have no idea what to say. ==the end==
This is the first time I've understood absolutely fuck all on a Vihart video.
+Ω
All hail the mighty Big Omega
Zogg from Betelgeuse Please make more videos!?
Zogg!!! Where have you been?
Ω^Ω
^^
aleph omega omega omega omega omega omega omega omega omega omega omega omega omega
We learned about this in math class around a week ago. I got super excited and squealed when the teacher announced it because it reminded me of you and this video. Everyone stared at me....
That last segment when you were giving examples from what fields these infinities come from gave me a rush. I love that type of stuff. I'm not a mathematician, I'm studying neuroscience. I've always known math was everything in science, that science finds it origins in math and to do most science a good understanding of math is required, but showing just how pliable it is for all different uses reminds me how awesome it is. I'm still in my undergrad for neuro, but I actually can't wait to learn one of the many fields of math that will most help with my research/projects in the future.
Good job, Vi! Another brilliant video!
Maybe I shouldn't watch this the night before a ski race for which I have to wake up at 6 am...
An interesting corollary of this video, if you pick a number at random, the probability of you picking any specific finite number is zero.
Now, pick a number at random. No: this one is not random, try again. Euh... see you after eternity.
is it that easy? i mean i understand, the idea: you have a set with infinite elements and pic one out of it, BUT is picking thins out of an infinite number possible? you can not pic any number out of the natural ones, you are limited by for instance the (life)time you have to put down that number. hm, i wonder (if or actually) how a theoretical model that does what you say, would work
of course you could imagine this to be a geomatricaly distributed experiment. but than you would not pic out "one element", it is more like pointing at one element, and if you are "close" to the specific number (which means you are in a defined intervall around that specific number) you succeeded.
It is impossible to pick a random number out of the set of all real numbers, as the number of real numbers is uncountably infinite. If you limit yourself to algebraic numbers, which are countably infinite, you can pick a number at random and not have the probability of any specific number be zero. However, if you impose the limitation that the probability of picking each number must be equal, then the probability of any specific number is zero.
Tahititoutou damn, dats deep
+Tahititoutou "Randomness" just means that every element has probability. The "finite numbers" are uncountable, but you can say something like "1 has 1/2 probability, 2 has 1/4 probability, 3 has 1/8 probability, and so on."
Sort of, but there are practicalities.
A human only lives for a finite time so only has a finite set of numbers to choose from in the first place. So in reality the probability of a person choosing a specific number will be nonzero for some numbers.
this hurt my head
oldcowbb same
your headache is medical issue solves nothing for anyone so why you give me a solution a solutions in theory descraction take to see if it works.
You forgot one: "Our love of Vi Hart's videos" Infinity. Clearly greater than even Big Omega.
I'm already lost at the 3 minute mark, but the delivery and presentation is so mesmerizing that I can't stop watching
Thank you Vi for being true to yourself. This is amazing!
Wow... DEEP thinker:-B :-/ (+comment)
She probobly has to spend so much money on notebooks
She won't spend too much on notebooks. Sharpies are the real money waster (Sharpies are fabulous though)
QTHERESSERECTION I understand where your username comes from and I love it. DFTBA :D
***** Thanks! Nerdfighters unite! DFTBA!
worth
Incredible. I really enjoyed this, so thank you. I'm impressed by the breadth even though I know almost nothing about the subject. (I self-identify as finite, but try not to be closed minded :-). At any rate, seeing a survey like this then a summary on one sheet, at least gives me some framework to plug things into...and to better see how much I don't know.
What a hoot! I think this was the most challenging for me (and one of the most fun) videos ViHart has ever done! I'm somewhat familiar with multiple infinities, but this introduced a bunch of mathematical concepts that I'd only heard about and some I've never even heard of.
Yay, one of the many Vi Hart videos that were super informative and went almost totally over my head! (except for the topology part, I'm a fan of geometrics and whatnot, so that was easier to digest).
Big Omega deserves a bit more than what you give it here. if mathematics (ie set theory) is consistent, which it almost certainly is, then Big Omega exists - it just isn't a set. it exists both outside of the system as an intellectual concept, and virtually inside the system, because set theory is self-descriptive. we can emulate Big Omega in a sort of virtual set theory and talk about it just like any other set, even though the "real" one isn't a "set". see Skolem's paradox for more info.
very deep metamathematics. the term "class" is sometimes used for a collection which cannot be a set within the scope of the system.
Cantor equated the class Big Omega with God. and I totally agree.
Funny thing is that Cantor thought infinitesmals were absurd. So his idea of Absolute Infinite itself would be limited.
Cantor's formulation of the transfinite ordinals and cardinals was revolutionary, but infinitesimals did not fit easily into it. Mathematicians were skeptical of the infinitesimal for centuries because no one was able to create a definition consistent with the laws of arithmetic, until years after Cantor's death. Since the limit had already been formalized, infinitesimals were no longer necessary for doing calculus, and many discarded them as obsolete and inconsistent.
The fact is that infinitesimals actually represent a completely different concept from ordinals and cardinals. Ordinals represent well-ordering, cardinals represent counting, and infinitesimals represent measure. You don't need to consider measure to understand the Absolute Infinite. How does measure contribute to the concept? Perhaps my idea is limited as well!
for some reason you remind me of snape when he wass waffling about what potions can do
That's a weirdly accurate way describe it...
This video makes me feel _incredibly_ moronic.
Sunera Perera When is feeling like a moron a good thing? Haha
+gingergamergirl98 I feel slightly smarter every time I watch this video. My understanding gets a tiny bit more nuanced. It's like throwing grains of sand into a bowl the size of the Sun, but it's still progress :)
like infinitely?
Great video, thanks for making it! I always love watching your stuff, even though math isn't really my forte, it's always neat to see these concepts, like, the infinities of the decimals between real numbers like you showed. I really hope you didn't get sharpie on your sleeves though, I couldn't stop thinking of that toward the end!
Whether or not you've ever considered teaching as a profession you are a wonderful math teacher
I feel like you already need to know what's going on here to truly appreciate it ...
+LittleOxfordSt Yeah, I wish instead of one crazy fast explanation, a series of videos explaining each would have been less dizzying.
+MathProofsable
What I think is really fun is rewatching the videos after studying (at least) one of the subjects mentioned.
Nope, every time you watch it, no matter how little you know, you're gonna learn something new. You can watch it again and again to learn new stuff each time, until you understand everything.
So it is appreciable and need not be completely understood in one go for it to be appreciable.
You are really weird, but in a good way. The world needs more people like you.
Pretty sure I didn't miss it, but what about 1+2+3+4...= -1/12? There's some wacky proof involving it, but I can't wrap my mind around how it makes any sense.
Quantum Physics...
Numberphile did a video about that.
Thinnestmeteor
I'm aware, but I was wondering what Vi Hart's view on it was. It seems pretty controversial.
Taiga Aisaka Noy sure about what to think, but it's a result used and it works and describes reality so if it's incorrect then I think a big part of the quantum theory might be wrong.
The proof is actually pretty fundamental, and simple compared to the other ones.
No, you have to stop there! That equal sign does not represent what you think it does. It is not the same thing as 1=2/2. Rather, it assigns a "meaningful value" to the series. This value describes the series in a mathematical way, but DOES NOT ACTUALLY DEFINE THE INFINITE SUM.
I love Vihart. I have absolutely no idea what she is saying most of the time. However she says it well and I feel smarter after I hear her out...just like my wife. Keep the videos coming.
It's serious when the transcript says "unintelligible." 14:30 says "Primitive Recursive Arithmetic"
I remember when recreational math videos like Vihart motivated me to dive deeper into number theory... now starting college, this is tempting me to major in math :P (either way it's cool tho)
I used to watch Vihart all the time, then moved on to Numberphile and 3Blue1Brown and the like. Just came back here because of a video on Gondel’s Incompleteness Theorem.
You are a math poet. respect.
I would have understood more if TFIOS wasn't in my head. Great excuse to rewatch this video and get more mind blown.
"Some infinities are bigger than others..." Surely this joke has already been made, but I couldn't help but think that the whole time.
I kept thinking when is she going to say that and whether I should post this quote, but then I thought everyone else is going to post it anyway so I should focus on what she is saying.
KS Ng colethewalrus Yup. It was made. A lot. So many times in the comments that the comments have a larger percentage of this joke (or references to it) than comments regarding the video itself, it seems.
colethewalrus yeah I just got back from the fault in our stars like 5 min ago and freaked out when I saw this video
Sometimes I mostly understand the math in these, today I do not. *woosh* over my head.
Not so much. I can't fathom any of it. I think I'll stick to algebra.
I thought this video was very fascinating! What I like about the video was how the concepts and thought processes were drawn out, not only did that help auditory learners, but it also allowed for visual learners to take part in learning. Honestly, I was so confused the first time I watched, and I had to replay it about two or so more times. After grasping some if not most of the ideas being displayed, it inspired me to realize how large - for a lack of a better word - the world is. Everyone has their own way of perceiving things, whether it be looking at the micro or macro. While watching, I noted that infinity is an abstract concept, - key word being concept - meaning that someone's infinity is not the same as another's. What stuck to me the most the list of the infinite "all real numbers." That inspired me to do a little research! The real numbers include all rational numbers, fractions, and irrational numbers. Although it might be infinite, it does not include imaginary numbers. However, an imaginary number can be written as a real number! Take for example, the square root of negative one would be "i," an imaginary number. When squaring the "i," it will be negative one (-1), which is a real number. There is also the line at infinity. Ever since about first grade (give or take a year), students were taught that parallel lines will never meet, and that they will go on for infiinty never meeting. In geometry, the line of infinity, also called the ideal line, is added to a real plane to essentially stop the result of a projective plane. This raises my question of is this an infinity ending? In projective geometry, any pair of lines always intersects at some point - at infinity! Which brings me to a final thought; is infinity the endpoint or starting point of an infinity?
I love your videos, but my brain stops working while I watch them.
I wish I could sit down with you and talk for hours about this. It's soo amazing. I study computer science, so I am familiar with aleph 0 and the non-countable infinities. All the other infinities are new to me. I'd really like to get to understand the formal definition of everyone of them.
A question for you ¿Is 0 a member of the natural numbers?
PS: I think the Real numbers are called like that because they are the better at describing reality
The omega plus one basically solved my lifelong question of what happens if you add one to infinity
I know right. Thank you Vi Hart!
Yeah, but it doesn't work with all infinities so that question doesn't have a single answer.
I think the biggest barrier to understanding all this is the names. I get the feeling that this would work a lot better in an agglutinating language.
Only if you like dealing with infinitely long words.
GenaTrius I love the word 'agglutinating'. It's new to me and I always love learning a new word. I've looked it up but to help me understand it, what names would work better for you?
BobSkiz1 I'm not completely sure if I understand this correctly, but I think it is just a language where you take a root and keep adding suffixes/prefixes to get new words. Newspeak from 1984 comes to mind, where you say "good", but then a more intense good is "plusGood," and an even more intense version is "doublePlusGood." And then if you wanted a really bad word, you would say "doublePlusUnGood"
MrAmazingAwesomeness You are right :)) My mother language is agglutinating and it works like that. Except you can't keep adding pre/suffixes foever :))
Some examples: "jó"="good", "jobb"="better" (="plusGood" :D), "legjobb"="best" (="doublePlusGood" :D) and "doublePlusUnGood" would be "legkevésbé jó". See, that already split into 2 words :))
Another thing I wanted to add were the names for infinity: Countable is "megszmálálható" and "uncountable' is "megszámlálhatatlan". The others greatly resemble the English equivalent ("cardinal" is "kardinális")
tirocska Cool :) is that Hungarian?
I gave up at 10:40. You lost me. I feel dumb. (I have a bachelors in Electrical Engineering and a Masters in Secondary Math Ed, and I studied Hyper-real numbers in grad school.) Vi, I bow to your awesomeness.
I love the way you speak like it's a poem with metaphors and yerwith real clear info
"some infinites are bigger than other infinities" -TFIOS
This is why am a Philosopher. Math just give me headache, although it's an area I'm really passionate about.
So there's infinity types of infinity?
No, there are more than infinity types of infinity
Mad Drill , overthehedgeisamovie - there are Big Omega types of infinity :D
thebarbes - RUclips Now we're going way too far! Infinite but not big omega!
thebarbes - RUclips
plus Big Omega?
Mad Drill Infinite types of infinite types of infinite types of...
Great video. Most videos stop at the fact that real numbers are a higher level of infinity, at best only acknowledging that there are infinite infinities without giving an explanation how that comes to be.
It's incredible to me to think about the actual concept of infinity. It makes everything so minuscule even being intangible itself.
I've been contemplating a shape I may have made up that I'm presently calling a meta-helix. It's basically an infinitely scaling helix made of helixes. As if you were to take a phone cord and wind it into a helix, and then take that larger helix and bend it into a helix, etc... and of course if you looked closely at what you thought was a phone cord you would see that it is actually made of a smaller helix, and that helix a smaller one, etc... I really need to make a model, I feel like it still sounds more confusing then it is. Well, anyway if you can visualize the infinite scaling helix I'm talking about, snip a little segment out of it. You can identify the locations of the ends of your segment and define the space it occupies, but if you pull on the ends to measure it how long it is, you will never stop pulling because a segment of this shape is still infinitely long! The length cannot be measured even though you can locate both ends of the segment!
Tungsten filament for light bulbs is actually made into that shape, but not infinite of course :D
that's called fractal geometry.
oh dear I need to lie down in a dark room...
I am the only one who thinks one of the reasons why Vi made this video is TFIOS? :)
"I cannot tell you how thankful I am for our little infinity"
Or I'm just crazy.
You're crazy ye.
She was actually mentioned in the credits thingy in the back of the book ( My mind had a blank so I have no idea what it's called atm) and when I saw that I fangirled a bit
I like watching these videos in the same way I like watching someone do a sculpture in a grain of rice. Just amazing stuff.
OMG I could listen to you talk about math forever!!
Is that weird? It feels weird.
I feel weird about how much I love this video.
Spare my brain please! Have mercy
you should try being me its just that this is to advanced for me but the scary part is that i understand it all and i am only 12
Yeah I used to be like you. Then I took a puberty in the knee :P
goeiecool9999 lol i hope i keep understanding everything when i grow up and puberty has to much power
Ikr. Are you a nerdfighter?
This wouldn't have anything to do with The Fault In Our Stars release today...
Okay so I'm not trying to stir anything up, so please, nobody criticize me for my question.
The whole idea that an uncountable infinity is "bigger" than aleph null doesn't seem right to me. I'm bad at explaining but I'll try my best.
So, there are essentially an infinite number of real numbers between, say 1 and two.The amount of numbers between these two is, at least how I see it, equal to aleph null. If someone would like to argue otherwise, go ahead, this question is for the pursuit of knowledge.
Anyways, that means that for every counting number, there is an infinite amount of counting numbers between them, in a sense. So there are aleph null counting numbers and aleph null numbers between each of those counting numbers, which would mean that, hypothetically, there are aleph null to the power of aleph null real numbers. Again, I could be doing something very wrong here, its late and my mind isn't too sharp right now.
Vi mentioned that aleph null to the power of any number would still be aleph null, which makes perfect sense to me. However, this would mean that aleph null to the power of aleph null is stillaleph null. Again, this is fine, just like how 1^1 is 1, no rules are being broken (as far as I know).
So by this idea, aleph null would be an uncountable infinity in itself, even though it is not. And this is where I do not fully understand.
My assumption is that I made a false assumption (probably several) along the way but I'm not positive since theoretical mathematics isn't really my thing.
If anyone could clear this up, I'd really like to know please.
TL;DR: If all real numbers (an uncountable infinity) is aleph null to the power of aleph null, and aleph null to the aleph null is also aleph null, then how is an uncountable infinity "larger", so to speak, than aleph null?
Short answer: a (finite) number to the power of aleph null is bigger than aleph null - 2 to the aleph null is at least aleph one
Long answer:
There are as many real numbers between 0 and 1 as there are on the whole ("countably") infinite number line - at least aleph one. If there were only countably many real numbers between n and n+1 then you'd have aleph null disjoint intervals with aleph null numbers in each for aleph null times (NOT to the power of) aleph null numbers, which would be aleph null still. Instead, you have, call it C (for "continuum") real numbers in any finite interval, and a countable number of such intervals, so you have aleph null times C (or C times aleph null) which is C real numbers in total.
Proving that C is actually bigger than aleph null is tricky - the standard approach is to use something called the Diagonalisation Argument, which says that if you listed every real number between 0 and 1 then you could use that list to write a string of digits that you'll never find on the list, but represents a real number between 0 and 1 with only one standard representation, so a real number which isn't at any finite position on the list. Since the defining feature of countable sets is that you can make a list of them which includes each member at some finite position, that proves that the real numbers between 0 and 1 aren't countable.
The question of whether C is aleph one or not is generally regarded as undecidable - it's been shown that it's independent of the standard "rules" for (infinite) sets ("ZFC") - meaning you can chose whether you want it to be true or false and continue doing sensible maths from there, and people have done both...
rmsgrey Never thought of it that way, thanks for clearing it up. Much appreciated man
Vihart, you are the infinite teacher. I know that doesn't make sense, yet it does make sense. And there aren't enough words to describe that. Still, it exists. It is. Thank you
sooooo..... Basically the best thing I have ever seen! thank you so much for making this.
never before have i felt so STUPID...
My head hurts now... :-)
I had to take about 2 aspirins after watching this :( .
Kaelyn Willingham about? :D
Stefan Dobre Sure, let's call it an even 2. ;)
In the best way possible. :D
You u no make fault in our stars reference? ?!??!?!?!?!??!!?!?!??!?!?!?
mediocre
I think the video as a whole was in reference to tfios
+thepeteris OMG I THOUGHT I WAS THE ONLY ONE THAT SAW VIHART AT THE ACKNOWLEDGEMENTS
+thepeteris OMG I THOUGHT I WAS THE ONLY ONE THAT SAW VIHART AT THE ACKNOWLEDGEMENTS!!!
Vi, this brings peace and happiness to my soul ^_^ I love being a mathematician.
I really like how you incorporated irrational numbers into your explanation. I'm confused because you're making it sound like you can quantify infinity. Your video inspired me to look for more types of infinity. What an interesting perspective on infinity!
This video made me realize something: mathematicians are just taking the shit.
No! You're wrong! They're adopting a set of axioms and then then they're taking shit. ps. This comment is still provable mathematically if you meant "talking" instead of "taking."
They're taking the shit to a whole new level. Like if you were completely lifetime dedicated to whatever nonsense what-if you discussed with your mates and worked out all the rules and consequences...
The infinite question is: is Vi Hart really a triangle?
Hey, Vi, do you have a PO Box or something?? I am a knitter and I want to send you cute math arm warmers!!!
Hi -- my 8 year old woke up this morning telling my 4 year old about your videos! "It's mostly math stuff, but drawn funny. There's one where she says "hi I'm a happy square!" It's hilarious!" Thank you for being you.
I love the fact that even if most of your viewers don't really understand your videos (me included) you just DON'T CARE and do them anyway, doodling happily, as if we understood. Amazing! :D
At least we DO get a "sense" of how many things we don't understand, how they look like, and how cool it would be to know them.
People disliked this video? Please return to watching The Kardashians, and stop down thumbing things because you can't understand it. I mean, really, why? Math haters or haters of Sharpies? Or wearing finger-less gloves while using Sharpies? Or people so math smart they disagree with everything she said? I guess there could be 200 of them. It's possible.
Now I'd like to know what these words are called in german...
Is there a number larger than any real number but smaller than aleph null?
+SafetySkull If you accept the Axiom of Choice, then nope! You can prove that aleph null is the smallest transfinite cardinal number. This means that there is no cardinal number which is infinite in value that is smaller than aleph null.
How this is done: You can show that every infinite subset of the natural numbers can be placed in one-to-one correspondence with the natural numbers. Thus, every infinite subset of the natural numbers has cardinality aleph null. Then if there were a smaller "infinity" than aleph null, there would have to be an infinite set which is in one-to-one correspondence with a subset of the natural numbers (which would have to be infinite). But by the above fact, all infinite subsets of the natural numbers have cardinality aleph null, which would make this "smaller" infinite set also have cardinality aleph null, which is a contradiction.
The proofs of these facts are not trivial, and your question is very good!
Here, you don't technically need the full strength of the Axiom of Choice. You just need the Axiom of Countable Choice (which is a weaker version). But if you reject the Axiom of Countable Choice, it is possible to find a smaller transfinite cardinal number than aleph null.
+SafetySkull Make sure you have your types correct. Can you compare a real number to a cardinal number? Cardinal numbers should be thought of as sizes of sets. Does pi number of elements make sense? Do you mean "Is that a cardinal number that exists between the cardinal of the size of the set of real numbers and aleph naught, which is the size of the set of the natural numbers?" This is known as the continuum hypothesis. It can be shown to be independent of ZFC (Zermelo-Fraenkel plus Choice). So take it or leave it. If you mean "Is there any cardinal number larger than all finite cardinals yet smaller than aleph naught?" No, aleph naught is the first infinite cardinal. I don't think we need the axiom of choice. Of course, you may feel free to use it.
I am almost certain there isn't a number that fits your description, though your conjecture could be undecidable. Sadly, I'm not a mathematician, so I don't know the definitive answer...
The problem with your question is that the order of the infinities, as presented here, does NOT coincide with the natural order on any number system bigger than the positive integers...
In fact, these infinite numbers are rather to be perceived as sets than as numbers. And aleph 0 is simply the set of all natural numbers, which can be perceived as a number itself (in set theory, one usually perceives the number n as being the set {0,1,...,n-1}). So, while one could repair your question in one way or the other, what you ask here is simply an ill-posed question.
Thanks for taking time out of your busy schedule of watching microwave ovens to make a very interesting and complex analysis of infinity. Loved it.
I just like to say that omega + 1 is not bigger than omega, it just goes after omega. Ordinals represent order, and cardinals represent a specified amount of stuff. It's important not to confuse the two.
Long line is long.
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Gal's smart.
I have no clue what's going on. I hate Maths, it's the bane of my life. I don't understand jack shit in this video. Yet I love it so much????
Bella Hatch so many homestuck icons on VI's videos for some reason
Homestuck fandom and math-junkies contain the same kinds of people.
I hate maths
Bella Hatch You don't- you just think you do because you don't know the true nature of maths. You might hate the school subject, but you don't hate all of maths.
If pi had a face, I would kick it in the face
I honestly love your voice and the way you tell us about math. It's amazing
i thought this was a great, very informative video and the way you write it all down in colors is cool. I've always wondered about the whole infinity concept and this answered a lot of my questions you explained it and gave good examples that helped.