These are the people she listed birth dates for in her first number, .12171770060818601107186703... Johann Friedrich Schubert, Composer (12-17-1770) Alicia Boole Stott, Irish Mathematician (06-08-1860) Marie Curie, Physicist (11-07-1867) Don't know what the last 03 is for though, but these are all great people :)
Feeling so envious right now of people who can follow this stuff. It starts like "numbers!" and then arrives at "all life matters!" and I just jolt up like "wait, are we there already?! "
"... which means an infinite list of real numbers will quite litterally be missing ALL of them". Gives me shivers every time I hear it. I love Vi Hart!
No, the proof is correct. It's a proof that given any function from natural numbers to real numbers there are some real numbers that the function doesn't cover; but in fact the set of numbers that it doesn't cover has the same cardinality as real numbers themselves. (The proof is similar to a proof that Cantor's discontinuum is uncountable and has the same cardinality as reals.)
@@MikeRosoftJH I know this was 5 years ago, but to be fair you responded to a 2 year old comment. You are not disagreeing with them, they did not say anything contrary to your statement.
Well done. I just got back from babysitting my grandkids. I was talking with the one in third grade and infinity came up. He informed me there were countable and uncountable infinities and be knew the integers and the reals were two different examples. This is same kid asking good questions about time dilation after seeing Interstellar. I wanted to find a good example of Cantor's Argument that would make sense to him. You rock. Very accessible and wonderfully accurate.
@@countingfloats That's just a childish argument. "I am right because I say so, and if somebody disagrees with me, then la la la, I can't hear you." Infinite sets certainly can be compared, and they can be compared in *precisely* the same way as finite sets. Just like there's no way to put ten guests in nine rooms such that every guest has a room and no two share a room, so there's no way to map natural numbers one-to-one with real numbers. (And the proof that the latter can't be done is *precisely* the diagonal proof.)
@SpaceBoy Again, you certainly can compare infinite sets. Let ω be the set of all natural numbers. Then ω∪{ω} - a set which contains all natural numbers, and which also contains the set of all natural numbers itself as an additional element - is obviously a different set than ω (it contains the element ω, but ω doesn't contain ω). So the subset/superset relation is one way to compare infinite sets. There isn't a per se right or wrong way of comparing sets; it can just may or may not be useful. So is _this_ a useful comparison relation? Sure it is; otherwise mathematicians wouldn't use it. But there is one thing it can't do: it can't compare arbitrary sets. Is the set of all odd numbers bigger or smaller under this relation? It's neither - neither is a subset of the other. So is there some way to compare arbitrary sets? Cantor's insight was to use precisely the same way of comparing infinite sets, as is done with finite sets. Suppose you are sitting in a cinema, and see that all seats are occupied: nobody is standing, nobody is sitting on two seats, no seat is empty, and no seat has two people in it. Then you can say - without needing to actually count them - that there's the same number of people and seats. So the same relation is used with infinite sets: set A has the same cardinality (or the number of elements) as set B, if the two sets can be mapped one-to-one. Set A has no greater cardinality (no more elements) than set B, if A can be mapped one-to-one with a subset of B (in other words, A can be injected into B). (Note that a set is a subset of itself. And the following can be proven: if set A can be injected into B, and B can be injected into A, then the two sets can be mapped one-to-one.) Under this definition, natural numbers can be mapped one-to-one with even numbers; the mapping relation is n->2*n. Likewise, natural numbers can be mapped one-to-one with integers, because integers can be enumerated using the following infinite sequence: 0, -1, 1, -2, 2, -3, 3, ... - any integer n appears at a position no greater than 2*|n|+1. But real numbers can't be mapped one-to-one with natural numbers - given any function from natural numbers to real numbers, there is some real number which the function does not cover. (There are several proofs of this theorem, such as the diagonal proof.) So by definition, real numbers have strictly greater cardinality than natural numbers. (As I have said, this is just like you can't put ten guests in nine rooms such that no two guests share a room. You can only do this with a subset of the guests; in other words, there are more guests than rooms.)
Well done Vi, you've outdone yourself this time. All your other videos have been super fascinating and gave me some deeper insight into mathematical ideas but this one was the first to really open my eyes to an idea. I always thought I understood the different kinds of infinity but I don't think I actually did until now!
This is really an amazing video, such a seemingly debateable concept explained so simply. I loved it a lot, I wish I could learn to illustrate my lessons like you, and I'm not just talking about the figures, even the examples used and the casual axioms. I loved it so much and I feel so bad for the people who missed out on a great subject like Mathematics because we were conditioned to hate the subject since childhood (and didn't have teachers like you
I absolutely loved this metaphor in the Fault in Our Stars. Thanks for making this video :) It makes me want to go back and re-read the book... but maybe I'll just settle for seeing the film again.
It shows how beautiful math is when the already complicated concept of infinity is shown to be even more complicated. There are so many different types of infinity, but they don't actually contain every number. The concept of infinity seems relatively simple, but is much more complex, similar to many mathematical concepts. Even after watching this video, though, it's still a difficult thing to understand how a list of everything, doesn't actually have everything and that some lists of every number are larger than other lists of every number.
John Green himself quoted Vihart as the inspiration behind his spiel on universe, oblivion (and presumably infinities too) in TFiOS. johngreenbooks.com/questions-about-the-fault-in-our-stars-spoilers/#process
Saying all lives are equal because they all contain infinite moments is nice and all, but that's no more logical than saying an elephant is no bigger than a mouse since they both contain infinite points of space within their volume.
Vi, whyyyyyy did you give the "Whole Numbers" legitimacy? D: I'm pretty convinced that whoever deemed N = {1,2,3,...} the "natural numbers" and "W" = {0,1,2,3,...} the "whole numbers" did so just to make sure their Algebra class had enough arbitrary definitions to be tested on. Probably why we have teachers who say "you can remember that the whole numbers contain zero because 0 is a hole! Get it?" According to most mathematicians these days (especially if you're fond of ZFC!), N = {0,1,2,3,...} is the set of natural numbers, and if you want to exclude 0, just write N \ {0} or by some conventions N*. Bertrand Russell, the person who even coined the term "natural number", himself said that anyone with a good background in mathematics would see why 0 is the most logical place for N to begin. And Giuseppe Peano revised his axioms to start with 0 instead of 1. So what are the "whole numbers"? The integers. "Integer" as a word actually MEANS "whole" in Latin - that's where we get "integral" and "integrity" and even "entire" from. So when one of my students asks me what integers are, I tell them "whole numbers - the positive ones, the negative ones, and zero." It makes no sense to define "integers" as "kind of like the whole numbers but including the negatives this time"; instead we define them as "numbers that have no fractional part." I know that most textbooks in America stick with N = {1, 2, 3, ...} and "W" = {0, 1, 2, 3, ...} but come on - these are the same textbooks that say that a trapezoid has EXACTLY one pair of parallel sides (go tell that to calculus) or uses the [x] notation for the "greatest integer function" (instead of using the much more evocative ⌊x⌋ notation and calling it the "floor function"). The math books are behind the times. But you don't have to be, Vi. :D
Why are you bitching about terminology. Names have no significance; what counts is what they represent. The definition of the term is the idea/concept it represents, not the other way around.
PT Yamin You're right about the concepts being the important thing, but poor choices of terminology and notation for those concepts can confuse students. I deal with that every day.
I am yet to be convinced that zero is anything but an absence of number. It certainly is not an ordinal, and its only use in computation is to annul. It is also irrational (0:x::1:y) only 0! & n^0=1 have affect.
I define the integers to my students as "whole numbers and their opposites," but it is absolutely ridiculous that we have both whole and natural numbers in our curriculum.
I am eleven years old and I was getting bad grades then I saw your videos aaaaand I started watching them now my grades are up thank you for helping me to raise my grades and feel better about my homework
To summarize, if you have an infinite list of N decimal numbers, you can then make a new decimal following this rule: digit 1 is anything except the first digit of entry 1; digit 2 is anything except the second digit of entry 2; and in general, digit A is anything except the A'th digit of entry A. So that's all cool...but now your new number is different from EVERY number on your list in at least one place each (by definition). So you just made a number that's different from every number on an infinite list. Which shouldn't be possible--and therefore, the infinite number of real numbers is LARGER than the infinite number of countable numbers.
Oh yeah...it's all crystal clear now. X) My brain hurts! ;) I'm not sure what I might best be, other than myself, but it's definitely not a math being.
She does Cantor's diagonal argument in decimal. It's much easier to see/prove in binary. Every number has a unique binary representation. p1 = 01001010010101010101.... p2 = 11100011010101010101.... p3 = 00011010101010101010.... ... Let x = x1x2x3... such that x_n = 0 if the nth digit of p_n is 1 and x_n = 1 if the nth digit of p_n is 0. Therefore, x is not on the list, because it differs from every other number on the list in a least one place.
***** It's not intuitive at all. But much of infinity is not intuitive :) . It's not countable because by definition, the list was already countably infinite before the new number was created. What this means is there is a one-to-one correspondence with the natural numbers. Put another way, for each entry in the list, there is a unique natural number that can be assigned to it. And since the list is countably infinite, the reverse is also true: for each natural number, it can be assigned to an entry in the list. That's true from the start--but now we have a new entry. Since we've already assigned all the natural numbers to the original list, what do we assign to this newly created number? The answer: nothing. We can't assign a unique natural number to it because we've used them up, meaning the size of the real numbers is larger than the size of the natural numbers--it's uncountably infinite.
Dan Lewis The implication at the end is the main point. If time is infinitely divisible, all lives share the same number of infinite moments. Weirdly that would make a cat's 18 year life have the same number of moments as you have in your full life. Kind of mind blowing if true. There's no proof of this yet.
Vihart, I love you and your channel. You have a great way with words, and exceptional mind and voice. Thank you for sharing what goes on in your head. Sincerely, a loyal subscriber.
The next mindfuck: despite that the rational numbers are of "lesser" infinity than the real numbers, you will *always* be able to find a rational number between two arbitrary unequal real numbers.
I really like that you took a concept expressed in The Fault in Our Stars and made it into a concept related to math. I didn't really understand the majority of the video since I haven't learned about half of that stuff yet, but it was still extremely interesting. (My mom is now trying to explain basic concepts of calculus to me. Thanks) I think your video just convinced me that there truly is no such thing as an "infinity". It's a concept that's impossible to grasp... literally. Thank you for opening my mind to a subject I try to keep hidden in the back of my head. I think I'm going to keep watching your videos now because they're really cool.
Great explanation of all of this :D Also I really like how, in the end, you turned the great falsely taken metaphor in the book into a great mathematically correct one! (Arguably, though, it's not physically true, as you couldn't divide any given time span into smaller pieces than the planck time, since smaller time spans would loose all their physical meaning, so a moment really has to be a planck time long. Though I suppose that opens up a whole nother bag of worms)
Once when he was young, Ernst Zermelo cut down a cherry tree in his yard with an axiom of choice, thus ensuring that the set of firewood was non-empty.
My comment isn't about Vi Hart directly but about her lack of large viewership. It looks like this vid will get about a third of a million views in its first year. Isn't this vid about an important math subject that every math student (limited, of course, to the English speaking portion of the world) must understand when they reach a level after the basics of number theory? Aren't the math educators of the world familiar with her? I would assume if they were, they certainly would point their students in her direction. And for those people studding outside a formal school framework, assuming they have a internet connection, wouldn't they be automatically directed to her in one search or another? There should be millions of people interested-in or studying-formally this subject. Why aren't there more views on this vid? Idk, maybe RU-vid hasn't penetrated as far as I think it has. Maybe educators aren't ken on telling students about outside-the-course material. Whatever the problem, it's certainly not the quality of the vids.
Nehmo Sergheyev if you're studying abstract algebra or set theory or something similar where you're learning about groups and sets and their sizes you won't get much help from a video like this. it lacks rigor. it's a great way to open people's eyes for how cool math can be, but it doesn't really teach you how these things REALLY work, other than at a pretty superficial level, and nowhere near enough to actually work with. you're better off getting a decent professor, some book and spending tens and hundreds of hours actually doing work.
True enough, I'm sure... I've had a major mental block when it comes to math studies and Vi Hart opened my eyes and heart to the "shape" of math, which is what I was missing before. Now I feel confident to pursue avenues of math I was intimidated by... for that I am grateful.
I genuinely teared up at the end. I learned a lot of interesting things too, before that point, but yeah. Thank you for this whole thing. It's awesome.
Theres no such thing as infinity. thers an end. It uses energy to get your brain to think, and theirs finite energy, so finite numbers because numbers re a pigment of our capabilities and just metaphors. You coud probably calculate how many numbers you could have if all energy went into thinking up new numbers....
I read a book called "The Advent of the Algorithm" in which they discuss this exact topic - I thought I had grasped it then, but now realize I hadn't. All I can say is thank you. Beautiful, beautiful, beautiful...
Is this an example of an "incomplete" problem from Godel's incompleteness theorem? You said the size of the infinity cannot be proven within the standard axioms, which seems to imply and incompleteness-like problem.
The way I think of infinity is if you're running on this sphere but you don't know it's a sphere and if you do know then you don't know how big it is and everything looks the same so you can't tell. Infinity is everything you can't see you don't know if you've run around the sphere 3 times of 4 or 5 or maybe it's so big you haven't even run around it once yet. The horizon line is countable infinity you can kind of see it (Which is to say kind of conceptualization it) but if you try to catch it you'll never get any closer to it and all the bigger infinities can't be seen and you don't know how much you can't see so you don't know how big or far away it all is. If this makes any sense or is just completely babel please tell me.
A life cannot have infinite moments because the smallest possible amount of time is a Planck time so the maximum amount of moments in a life has to be equal to the length of the life divided by the Planck time..... Sorry philosophy but physics trumps on this one...
+TGslayer Hearthstone I heard, the number of times the Planck scale is missinterpreted by people on the internet every moment, contradicts the concept of Planck time.
whilst that may on the surface appear to be true the fact of the matter is that the Planck time is the shortest length of time that anything significant can happen in. So as far as i'm concerned an length of time less then the Planck time should not be considered a moment because nothing can take place within a length of time smaller then that. In addition saying that something is only a theory is a little misleading. Almost everything we know about science is simply theory however we have so much mathematical as well as physical proof for it that it is fair to count it as fact. This apply for Max Planck's quantum theory just as much as much as it applies for evolution or atomic theory. I strongly recommend that you go read some of the research around quantum theory particularly Max planks work it is very interesting.
TGslayer Hearthstone many things can happen with the time shorter than a Planck time, an exact copy of our planet could appear and disappear, something so large you cannot comprehend it could do the same, or something too small to see with our current technology, maybe even a wormhole, too small for anything to fit of course.
“You can divide infinity an infinite number of times, and the resulting pieces will still be infinitely large,” Uresh said in his odd Lenatti accent. “But if you divide a non-infinite number an infinite number of times the resulting pieces are non-infinitely small. Since they are non-infinitely small, but there are an infinite number of them, if you add them back together, their sum is infinite. This implies any number is, in fact, infinite.” “Wow,” Elodin said after a long pause. He leveled a serious finger at the Lenatti man. “Uresh. Your next assignment is to have sex. If you do not know how to do this, see me after class.”
I liked the drawings of people they were pretty fun and added a lot to the video. I was not confused by this video because it is such a straight forward and clear explanation. This inspired me to think more about infinites because where else can this apply like in infinities containing infinites.
Sam Auciello You wouldn't need a quantum computer in that case. The whole idea of quantum computers is being able to tackle NP problems. If P = NP, the usefulness of a quantum computer would be greatly diminished.
Wow. This one hurt my head. I think I'll have to watch it again later when I'm less tired. Fantastic video as always Vi; thanks! There seems to be more meaning than just maths in the last 10 seconds of the video too... :-)
Wait, but if you can add a number that isn't on the list to the complete list, than the list wasn't complete and the whole premise is invalid. (Also, how does 0.99...=1? I have been told this as far back as I can remember dealing with decimals, but no one cares to explain it. All I can assume is mathematicians are lazy, or someone a long time ago wrote something that became misunderstood and accepted as truth like with taste buds having sectors. How can n= not n without some kind of rounding or weird trickery?)
The premise isn't invalid. The premise is that the list is infinite, and is a one to one map of the positive integers onto some set of real numbers. The argument is that if that set of real numbers contains all the real numbers, then the size of the set of the real numbers is the same as the size of the set of integers. However, as was proven in the video, it is always possible to find a real number not in the set, regardless of what set you use. Therefore the set of real numbers must be bigger than the set of integers.
You are somewhat correct. Any proof that starts with "suppose we have an X such that Y" and concludes with "we cannot have an X such that Y" is flawed. However, I would formulate Cantor's diagonal proof as such: - Suppose we have a list of reals. - Then we can make a real number that is not on the list. - Thus this list is incomplete. This proves that any list of reals is incomplete, so there cannot be a complete list of reals, so the set of reals is not listable. As far as 0.999... = 1 is concerned, it's neither weird rounding nor trickery. It's just two different ways to write the same number. We could also write 1 as 1.000..., and it would still be the same number.
she has a video on her channel on the .999.....=1 thing. I wouldn't be able to explain it myself (I understand about half of each video.... sophmore in high school here who got a C- in algebra last year, but hey, I still find it interesting.)
The point is that the premise is invalid. Its called a proof by contradiction. You make a statement assuming something is true, but then you show that if it is true, it's not true, so it's definitely not true. If you look at more it starts to make sense. For example one way to prove that there are an infinite number of primes is to show that if you claim to have a complete finite list of all the primes, there has to be another one, so the list is never really complete.
I actually really like this channel. Mostly because her videos are kind of like overthinking things which I do ALOT! I overthink almost anything and everything.
This is anathema to me ! James Grimes also says infinity is not a number, infinity is an idea, a process ! So, if I let you play your game, and say: DEF: if I can't make a mapping then this idea is BIGGER than that idea. Then you may have invented a consistent game, which admittedly is what, for many mathematics is about, but it's trivial because it is CIRCULAR - that is it is 'simply' true by definition. (p.s. I qualify 'simply' because all math can be viewed true because of the definitions we choose. The point is we have ARBITRARILY changed from one conception of size to another such that we incorporate an abstract PROCESS) If I much more sensibly define a list to have size if it is FINITE, in lieu of Grimes's comment that infinity isn't even a number, then your proofs are broken and I have a sensible regime.
I'll go further. Intuitively SIZE is the most basic, first dealt with, MEASURE. You break from this conception WITHOUT justification. And as such, undoubtedly escape the constraints / fundamental properties of 'measures' MEASURE THEORY where B(S) is a Boolean algebra. m: B(S) --> [0, inf.) s.t. for A int.sect B = { } we have: m(A union B) = m(A) + m(B)
You'll also have a much less interesting system, with less scope for exciting new results. Mathematics doesn't really have to care at all what's going on in the real world.
Why are you scared of examining the concept of infinity? What is the point of math and how can it advance if we won't explore new territories? We consider all the natural numbers (for example). We can understand that there is an infinite amount of them because there is no biggest natural number. Any finite list can always be added to to make a greater list, no matter how many we list. Maybe this is breaking away from a concrete conception of size, but how else do we describe all the naturals without saying there is an infinite amount of them? That's simply the way it is. We then consider all the reals. We understand that there is an infinite amount of those as well. But we can't fit those in the constraints, a pattern, a linear list as we can the naturals. In a way we can't even consider all the reals because there are too many of them! Not only can we always add to any finite list we consider, but we can always add to any infinite list! The amount of reals is to the amount of naturals what the amount of naturals is to a finite set. Dare i ask if it isn't obvious that the amount of reals is far more numerous than the amount of naturals, even if they are both infinite? I don't know any way to describe this in words other than "a bigger infinity". This is not true by definition, this has been deduced. PS: Vi said herself in her previous video that infinity itself is not a number but rather a concept, but there are numbers that are infinitely big, such as the aleph-numbers.
Beauty is in the eye of the beholder, what exciting results ? What I object to is the 'mathematical WOO' factor. The is no excitement in the sense that is implied once you realise all normal notions have been alternatively defined to suit. I think I have as good as conceded that from a pure recreational / pure mathematical point of view, it may be fun / interesting (in that realm) - to compute consequences and properties. Just don't go telling me that I, in my regular context of size, need to be aware of these 'bigger' infinities, because they are meaningless outside the sensible, established, intuitive definitions.
Volbla / MadaxeMunkee - pls attend to the issues raised.
10 лет назад
Hola tengo la traducción de exactamente 3 de tus videos a español ¿Cómo te los puedo enviar para que estén de manera oficial en tu canal de youtube? ¡Saludos!
This is fantastic timing. I watched The Fault in Our Stars today. I think it did the book justice. I also take it very much to heart that Hazel Grace is not a mathematician.
.99999999999.... = 1? No, no, a thousand times no. No matter how many 9s you write, you still don't get all the way to 1.0. Why? Because you can always get closer to 1.0 by writing another 9 at the end of the number. If you could get all the way to 1.0 by writing a sequence of 9s after a decimal point, then you couldn't add another 9 to it and therefore it wouldn't be an infinite number of 9s.
I like the way this video was set up, with the quote from the fault in our stars and the aestetically pleasing colors and sketches. The idea of this video kind of blew my mind. I didn't know that any infinities could be larger than another infinite. I still don't really understand how that's possible, but I thought is was a really interesting video.
Apart from having a really annoying voice and way of speaking, this video is nonsense. The problem is the question of whether infinity exists is a philosophical one and not a mathematical one. This woman is just using word games to suggest that infinity exists and also that there are multiple versions. There is not and can never be such a thing as infinity. There can only be a concept of infinity. All science and philosophers believe that time had a beginning so there is no such thing as infinite past and no matter how much time stretches on there can never be an end to conceptual infinity as you could always conceive of more so infinity can never actually exist. It's just a useful concept for addressing issues.
"All science and philosophers believe that time had a beginning " is a logical fallacy. Also she clearly makes a distinction between the philosophy and the mathematics at the beginning of the video.
If you wanna try and expiriance Cantor's sad life just imagine that most of the mathematicians in the world are like this guy (the original commenter) and they all say that you are wrong while the only wrong thing is that the world was not ready to accept your work
Love infinities :D Also you reminded me of the song Finite Simple Group (of Order Two). I can only recommend it! So many math-jokes! (I need to learn a lot more to understand them all, but hey always good to have some motivation, right? ^^)
I enjoy your videos. You fall into the group of tubers like vsauce and vertiserum.... Vert... Vertsi... Vert something vert science guy.. And numberphile... Very informative and entertaining. Oh also minutephysics.
When I was much younger I attended Jethro Tull concerts... on acid. Now, in my dotage, I float along on the lyricism of V Hart. Similar but somehow more enriching.Once you've seen God in It's myriad forms, you cannot fear Death. Thanks V.
