## Math Has a Fatal Flaw

Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer. This video is sponsored by Brilliant. The first 200 people to sign up via brilliant.org/veritasium get 20% off a yearly subscription.

Special thanks to Prof. Asaf Karagila for consultation on set theory and specific rewrites, to Prof. Alex Kontorovich for reviews of earlier drafts, Prof. Toby ‘Qubit’ Cubitt for the help with the spectral gap, to Henry Reich for the helpful feedback and comments on the video.

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References:

Dunham, W. (2013, July). A Note on the Origin of the Twin Prime Conjecture. In Notices of the International Congress of Chinese Mathematicians (Vol. 1, No. 1, pp. 63-65). International Press of Boston. - ve42.co/Dunham2013

Conway, J. (1970). The game of life. Scientific American, 223(4), 4. - ve42.co/Conway1970

Churchill, A., Biderman, S., Herrick, A. (2019). Magic: The Gathering is Turing Complete. ArXiv. - ve42.co/Churchill2019

Gaifman, H. (2006). Naming and Diagonalization, from Cantor to Godel to Kleene. Logic Journal of the IGPL, 14(5), 709-728. - ve42.co/Gaifman2006

Lénárt, I. (2010). Gauss, Bolyai, Lobachevsky-in General Education?(Hyperbolic Geometry as Part of the Mathematics Curriculum). In Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (pp. 223-230). Tessellations Publishing. - ve42.co/Lnrt2010

Attribution of Poincare’s quote, The Mathematical Intelligencer, vol. 13, no. 1, Winter 1991. - ve42.co/Poincare

Irvine, A. D., & Deutsch, H. (1995). Russell’s paradox. - ve42.co/Irvine1995

Gödel, K. (1992). On formally undecidable propositions of Principia Mathematica and related systems. Courier Corporation. - ve42.co/Godel1931

Russell, B., & Whitehead, A. (1973). Principia Mathematica [PM], vol I, 1910, vol. II, 1912, vol III, 1913, vol. I, 1925, vol II & III, 1927, Paperback Edition to* 56. Cambridge UP. - ve42.co/Russel1910

Gödel, K. (1986). Kurt Gödel: Collected Works: Volume I: Publications 1929-1936 (Vol. 1). Oxford University Press, USA. - ve42.co/Godel1986

Cubitt, T. S., Perez-Garcia, D., & Wolf, M. M. (2015). Undecidability of the spectral gap. Nature, 528(7581), 207-211. - ve42.co/Cubitt2015

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Special thanks to Patreon supporters: Paul Peijzel, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Juan Benet, Ruslan Khroma, Robert Blum, Richard Sundvall, Lee Redden, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson, Sam Lutfi, Ron Neal

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Written by Derek Muller, Adam Becker and Jonny Hyman

Animation by Fabio Albertelli, Jakub Misiek, Iván Tello and Jonny Hyman

Math City Animation by Another Angle 3D Visuals (www.anotherangle.ee)

Filmed by Derek Muller and Raquel Nuno

Edited by Derek Muller

Music and SFX by Jonny Hyman Additional Music from Epidemic Sound

Additional video supplied by Getty Images

Thumbnail by Geoff Barrett

Associate Producers: Petr Lebedev and Emily Zhang

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გამოქვეყნდა 2 თვის წინ

1:25 the cell forgor

Why can't you just use the diagonalization thing in mirror for the index numbers, where you add 10 ^ index ?

This video literally gave me chills!

I am learning more math here in few hours than in 14 years of school.

"This sentence no verb" "This statement no proof" "It goes without saying that I trust you implicitly" "Yields falsehood when preceded by its quotation" x 2

Passed away in 2020 from COVID 19… I’m sure being nearly a century old didn’t help either.

If you list ALL real numbers between 0 and 1, how can you afterwards add another number that is different to all of those? It's impossible.

@timsen The point is the following. We start by listing all the natural numbers: 1, 2, 3 .... That is the way we count (naturally and also mathematically). Now we show that this is not the same a listing all the numbers between 0 to 1.

@Moyprod Than i don't understand the point. First, if we don't list an infite number of real numbers between 0 and 1 how can adding one extra number expand the amount to "infinite +1"? Second, doesn't mean adding an extra real number that we also add an extra natural number (or index)?

That is the point: *you don't list all numbers between 0 and 1* . You list the number 1, 2, 3....

Something that is thought, or proven, to be true, obtains a true-state modus operandi until proven false under one or more conditions. Something that is thought, or proven, to be false, obtains a false-state modus operandi until proven true under one or more conditions.

That doesn't apply to mathematics. If you prove something in math (which is a closed logical system), you can not "unprove" it. Unless you prove that mathematics is wrong.

>he passed away from COVID-19 Disappointing propaganda coming from a channel dedicated to truth seeking

I was expecting to learn about meth.

Love this channel.

Did I just take a hit of acid? Did I just understand all this or do I just understand that I can’t understand this? “All I know is I know nothing.”

What about quantum computers

Hi Veritasium! I would really like to know what is your source for what you say at 0m52s (it was proven that there are stuff in arithmetics that you can't prove). Or if anybody sees this comment, please tell me.

How do i write a sentence of words like (legends never die) in hilbert-style proof?????? i searched Google for an hour had no luck

clickbait, I solved all the puzzles

Thank you. My wife was almost in tears of beauty at times.

This filled my head like a helium balloon and like that balloon over time my head will slowly contract as my understanding starts to fade. I'm a sad, old helium balloon.

I was watching this but it must have not been stimulating enough because I fell asleep

Discrete math is coming back.

I've played Life, and this is not it. Cool, though.

Turing was gay, that is a explain a lot , i think the society played bad joke on him.

I don’t know why but 64 not being a square and being represented as an rectangle bugs me

teachers: you cant get smarter by watching youtube ...youtube:

RIP my brain

Is that a proof of other da-mention ?

6:25 I got lost here. Can someone explain why the Reals (0;1) are bigger than Naturals? Just because u know u can generate always new number doesnt make them bigger? Huh. Ty

_"Can someone explain why the Reals (0;1) are bigger than Naturals?"_ Assume they are not. Then there is a bijective function from the natural numbers to [0,1] by definition (both are infinite and both are countable by assumption). Such a function is an infinite list of numbers. Let us construct a number like shown in the video, so that each digit differs for any line n of this list at position n. (add 1 to the digit and subtract 1 in case of a 9). This number is clearly a real number in [0,1] but it is clearly not on this list because it differs from each line n at its n-th position. This means the function was not surjective (one number is missing) and that means it was not bijective. Contradiction. So the assumption was wrong and there is no such function. Because both sets are infinite, this means [0,1] is not countable infinite and there is no 1-to-1-relation between both sets. [a 1-to-1 relation is a bijection function between both sets] Because you can easily prove that all rational numbers in [0,1] are countable this means that there are indeed more real numbers in [0,1] than natural numbers. q.e.d. _"Just because u know u can generate always new number doesnt make them bigger?"_ No. But showing that it is not countable does the trick.

So confused but interesting lol the godel system lost me hahaha

Amazing Video.. actually gave me chills. Really exciting how you illustrated the development of these principles.

Ahh this is bringing back fond memories of my Finite Automata class in college... If only I knew of these videos then. Turing machines, mapping reductions, NFA and DFA's, turing complete, set theory, proofs by induction... Still don't know how I made it through that class

9:20 Me:💥👄💥

I understood the Cantors thingy about the diff infinite sizes than the whole infinite hotel paraodx vid

God made Hilbert, then God made Godel. Godel... 1-0 2-0

@32:18 "...you might think this realization would have driven mathematicians mad..." Kurt Goedel literally suffered bouts of insanity and quite a number of mathematicians also went crazy Cantor, Nash from the top of my head.

I'm undecided about this ( And that's the truth. Or is it)?

Okay so maybe I’m an idiot and am missing something obvious, but I feel like the proof with the cards and Gödel numbers isn’t valid and here’s my reasoning: The number ‘g’ is just an abstraction for an actual number so we have to assume that number can be reached by the same method by raising the components to create the card above the prime numbers. I thought about what the Gödel numbers of all the natural numbers are and since the Gödel numbers for 0 and the successor function are positive, as the natural numbers grow linearly, their Gödel numbers grow exponentially (or more quickly. I didn’t actually do any calculations, just reasoned that out in my head). This would mean that no card can have its own Gödel number on the front. And pair this with the fact that you still have to account for the rest of the proof on the card it doesn’t seem to me that this proof should be valid. Having said all of this, my intentions with this are purely to help me understand this better. I’m not trying to call out Veritasium for being wrong or anything like that. I admire his work and his videos are always fascinating. I just find this topic in particular very interesting and after thinking about it for a long time I figure I must be going wrong in my thought process somewhere and I really want to understand.

It has it's own number on the front because it uses self reference leading to the problems explained in the video

I am absolutely amazed by this video. I am a software developer with a solid amount of experience under my belt and therefor love diving deep into the inner workings. Despite that, math is a bit of a sore spot in my curriculum, so seeing something with this level of complexity explained so elegantly and simple (which is what I obviously strive for in my code) is absolutely fantastic to me. You made my day and I look forward to seeing more of your videos, respect!

Does G stand for God

I thought it was Frege rather than Cantor whose work was most impacted by Russell's insight.

Gödel just stomping on someone’s dreams one paper at a time

I flunked computational theory during my engineering degree. It took me 3 attempts to clear it. I hated Alan Turing’s work because I didn’t understand it, and why it was important to understand it. I swear I have never had Turing machines and the concepts of completeness, consistency and decidability quite so well. I strongly suspect if I had tutors like this in college, I would have learned with more enthusiasm. Thank you for this spectacular video

At "set theory"... I checked out.

molten brain

Basically there is a missing number that was lost

There are things we simply cannot measure, observe, yet they exist and have some atributes. So... yea, dont mess with infinity :D

....

When did math start in the human evaluation?

Time to create.... Math *2*

"If R contains itself if and only if it doesn't contain itself"... Had to listen to this section like 20 times over and over in order to understand this.

NO WONDER GAY PEOPLE ARE ALWAYS ONLINE!🤣🤧

What about Z1?

I thought I was finally gonna understand the whole video..... Dam u Godel..

How does one make a rule of randomization using math. Remarkable

Even if it didn't exist, the possibility of a reality where Godel does get poisoned always exists. Godel was terrified he was living in that reality and could never prove that he wasn't.

The sneaky c-clamp postprandially provide because sailboat early moor beside a delicious smash. addicted, tearful share

Why equations does not contain time and coordinates? E=mc2 but it's false at 5:52:43 am inside let's say SagitariusA? Then what? People create explanations based on their understanding of everything, but what if source is wrong? What if e=mc2 false? Where and when? And most importantly WHY? If to include that new data, mathematicians will have fun for a whaile :) while

almost 9M views lol

Here is another puzzle: how to dress and stay naked at the same time? ( half dressed thing won't work, because it's partially, but dressed, so...).

And who said " black holes" are black? Black is a color too, actually a mix of colors - take one out, and it won't be black anymore. Why absence of light is black? Universe is full of colors( clouds of elements, stars, planets, etc) and only mix of them seems like absence of anything - BLACK. Quadrillions of cosmic objets covering our sky completely, just distance difference mixes everything( all colors) up into black sky with lonely stars that closer then the rest of them and because of that we can see them. The rest of universe is a cocktail of colors that become...black to our 👀 eyes. If our eye would be like Hubble telescope, we would see btight, blinding light instead of blackness of COSMOS. Agree - that was well said:" blackness of Cosmos..." wo-o-ow... Ha-ha😄. And last point: isn't it paradox itself - many many stars, planets, holes😄, clouds of elements, comets, super novas and oldas :), EVERYTHING - gives us... Black sky at 🌃 night. People create unsolvable problems for themselves and then suffering from trying to solve it: what if... I add 1 to every second number in triangular square??? WTF do u need stuff like that? Purpose? Reason? Necessity? Why? What for? Prediction? People failed even in explanation of light behavior: when u look - its wave, when u dont - particle. How dumb that is? How the hack light knows if we have eyes open or not? Stupidest explanation, based on humans tinny knowledge of surrounding world. Here is something to think about: if we are made of stars(same elements) and were before part of universe, of everything, where then our quantum entanglement with elements that were next to us, where it go then? Why we don't feel it? We should, but not. The atoms that used to bear our electrons and vice versa, are far away, part of stars, but we don't feel no connection-not entangled😄, why???...

black is just an easy way to define void/lack of colors

Barber, huh? Has to shave himself only if he doesn't shave himself... What, if..... barber shave only half of his beard? He didn't shave himself and shave himself at the same time...? ΘΩΜ0¬Μ[0￦Г¬¡¿{ГΩ¤¤¡}ฯГฯΘΩ°¥÷0567%=1234567890.....=0({[ · )}]1×÷-+0-0.000000...1. Ha-ha. Just joking with " formula"😉. Paradoxes exist and can explained only by paradox, just like quantum paradox - wave and particle at the same time and coordinates... That's how close we can get in explanation of theory of everything(becase that what we trying to do - a unique formula that explain everything, even so called " black holes"). It's a new chapter in physics and new language will need to b developed. But humans are NOT capable of creating something complitely NEW that never existed in reality and nature. Humans only capable of mimicking existing(in nature) things by observation. Breakthrough will be done by a "mistake" in a system. A completely isolated and developed mind that will never have anything to compare to... NOTHING at all even physical abilities(muscle move, joints, tissue, etc.). That's how it looks like from little heights of my mind. Welcome to destroy this "THEORY" or explanation or vision of mine... .

It's all chaos. Almost infinite seemingly random combinations from a simple input.

Fact = ying yang principle ... funny

You have a really attractive and sweet sounding voice

Waitaminute: For any natural number n>0 i finde a real number e=1/2x that is 0, the first being 1/1*2=0.5 and every following being smaller than that. since there is also 0.6, 0.75, 0.987654321 there is an amount of real numbers more than natural numbers. q.e.d. no?

@Релёкс84 _"Just because you found a mapping that doesn't include all numbers, doesn't mean it is impossible to find one"_ Exactly.

@reckless glitch No, that's not a sufficient proof. Just because you found a mapping that doesn't include all numbers, doesn't mean it is impossible to find one - that would be true for finite sets, but not for infinite sets. For example, you can map all natural numbers to just even numbers, by multiplying them by 2 (0>0, 1>2, 2>4, 3>6 etc.) This mapping clearly misses numbers, but does that imply the set of natural number is smaller than itself? Well, technically you could say it is not bigger than itself. In fact, if you find such 'icomplete' mapping going both ways for two given infinite sets, then you can use Cantor-Bernstein to conclude they must have the same cardinality, but that's another story.

And another, for any given natural number you can find a real with the same amount of number 9 digits after 0, like 1->0.9 2->0.99 10->0.9999999999 and there are still indefinet numbers left.

read the title and saw "Meth has a fatal flaw" and immediately clicked, good video but kind of a let down after those expectations.

“ Math has a flaw” “Wait u mean A flaw like only one??”

The simplistic office regrettably grab because particle ethnically promise like a abandoned rise. ethereal, descriptive david

I'm more mind-blown that airline ticket prices are turing complete. Like.....how?? EDIT: I take it back, the game of game of life got me

I'm usually good at comprehending these topics. However, I am tired and confused. Good job though, it was enjoyable to watch.

6:35 can’t you just get rid of the decimal from the real numbers between 0 and 1 to get the same amount of matching numbers?

No this is a simulation....

The inventor if LIFE died of COVID. Let this sink in for a moment.

Wasn't expecting to see mtg arena in a math video

How do you even have perfect ads that make me want to learn more math

Hence why i still believes in the existence of God

😱😱😱😱😱😱😱 Self simulation was mind blowing

Love this video, and your butterfly collection

I felt like i was keeping up pretty well until the Godel numbers. Then my brain went smooth.

Dereck is transitioning to magic

Ok so, i know this is a VERY late response but i have a genuine question about the h machine, what exactly says that it cant print out both outcomes in the event of a paradox, or even just staggered outcomes, and it did do that, then its first outcome would be true

Therefore making it so that the machine is still correct

cant the barber hire another barber and that barber shave the barber that hired him and the barber that was just hired could get shave by the barber that hired him

That statement at the end was fantastic!

This video really is mathematicians find problems with math and try solve them with math

i forgor 💀

Though it is simply a footnote in the video, it disgusts me that even such a brilliant mind as Allen Turing was not enough for the homophobic UK. In the end, all that mattered to that society was who he felt love for in his own privacy and was driven to suicide. Today, the internet, accessed via machines, provides many safe spaces to people who identify as LGBT. Spaces where people feel safe to come out and explore themselves. And the world becomes progressively more and more welcoming to us. Turing's legacy is the incredible technology we use every single day, so normalized that it doesn't even register. But among that, his legacy is also creating the tools so people like him can be themselves with hopes they will not suffer the same fate as him.

Solve: 3x+1=? 0÷0=? If johnny has 3 melons but the train didn't arrive until 7AM what is the key answer to finding the meaning of life?

The last question depends entirely on the ripeness of the melons.

Well the only way I see it all is that we will not know what we're not supposed to... And that the human mind can only handle certain rigid sets of comprehensions and not everything... Einstein endeavoured to find the equation to well... Everything... And died in vain... Newton suffered a term of mental breakdown... But Hilbert's was worser for I think the drive to want to know it all and knowing it all is a must that disintegrated him at the end🤷... And for some weird reason the card findings ended with the letter G ... Well it's a good time to reflect... On what and who we really are... Take a look at the zygote... You'd see that Hilbert was wrong in we will know and we must know... It's a kind of tech beyond the human mind... As I say human thoughts can't go beyond the egg... Just the egg apart from the infinite amount of creations out there that are incomprehensible thus we humans do not have the answers to in the first place... And in closure I say that the letter G could also spell God 🙏✌️🇲🇾

That was a great video

If you think in the 4th dimension paradox don’t really exist because it’s true and untrue all at the same time and since math is a universal representation of reality it has to work in all dimensions even if it means we won’t get a “correct” explanation in our 3 dimensions, the thing is if we spend all day taking into account a 4th dimension most people can’t even comprehend let alone experience we’ll spend our life thinking about we won’t get anywhere

I wish I was smart enough to understand this stuff

👍🏻 💯

Best video I have seen on GEproms!

F

I've just been nodding through this for the past 30 mins without really understanding a thing. All I got is that we got computers out of it, so I'm cool.

I can tell where my life is leading right now. I watched the whole video not understanding one thing he was trying to make me understand smh fml ....

12:55 :O