# 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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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 NOcds!

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

The supposition that you can complete those infinite lists in the first place is impossible. You can't go on to step 2 without completing step 1. Therefore, claiming to be able to generate a unique number outside the scope of infinity is a logical fallacy. Cantor's "diagonalization proof" is proof of nothing. And, self-referencing breaks reality. With or without math. It's poetic nonsense and shouldn't be used to prove or disprove anything.

17:00 ish Prime factorization for developing permutation indices is so obvious and yet so weird to me, for some reason.

Fascinating. The micro-makro repetition the machine showed at the end - great. And it’s really well explained although this was hard at the border of my grasp tbh.👌

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“There is no proof for the statement with the Gödel number ℊ” is just an arbitrary statement, having nothing to do with mathematics. You take α sentence that does not have α mathematical solution, assign it α number, then throw it into α mathematical system, of course it doesn’t make sense. That doesn’t create α hole in the system, just α piece that isn’t supposed to fit. Like α pretzel in α bag of Cheez-Its. Am 𐊦 missing something?

It.would be pretty hard to prove considering the largest number continuously getting bigger and bigger. So would the twin primes keep going infinitely? Or would they stop at some number we don't even know yet? Maybe. Maybe not. Maybe you could prove something like that and I'm just wrong. But thats why I would say it's impossible to prove. But again I really don't know

10 years back i code this problem in C.. input were very small. max was 10X10 but rule were same.

Fatally brilliant! 👍👍😎

The flaw of Mathematics is aligned with the flaws of Language and Communication itself, which is: It is not singularly originated, but an attempt to understand and be understood by what is Infinite and Eternal.

This is actually a kind of silly problem, just make an exception set the rule. Kinda like how 2 is the only even prime number.

Lol plane tickets

Infinities are weirdest !!

@veritasium I would like to meet you personally ,please !!!

It's sad to here that greatest humans ever lived ended up suiciding 😕

Math has a fatal flaw - correct there's meant to be an s at the end!

I wish I understood this

joke's on you. I just watched this video again reflected through a mirror, which means I was not watching it on a device that stems from Hilbert's and Turing's contributions, but rather on a device that predates them both

I'm not sure I understand the godel card which causes a problem. Because in order for the card to exist, it's godel number must exists, which encodes the cards which prove it. So the proof can't exist, which means the card itself can't exist because it is referencing the non existing proof. So surely all it proves is that the question isn't valid.

existing

Lol that guy died of being obese not of covid.

But what if the barber is a woman?

I fail to see why the numerical proof system was needed to show the paradox in "This statement can't be proven by this system."

Goosebumps when you revealed the Game of life simulating itself 🤯🤯🤯

2:32 what is the background music here? Does anyone know?

As a computer science student who has gained some acquaintance with these concepts, I absolutely love this video! It's a great way to invoke interest in theoretical computer science!

Was tired so at first I thought it said “meth has a fatal flaw”

Yeah, its too expensive.

I can understand EVERYTHING here except ONE thing. Did the damn barber shave himself or not? pt2 required.

It's not a flaw. Nor is it fatal.

phfffff...XDD......you really want the answer? MATH IS FINITE, IT STARTS FORM 0 AND IT ENDS TO 9, EVERYTHING ELSE IT'S A LOOP ( think about when you draw a circle you start from point 0 and when the circle it's complete, it ends to point 9, after that it's repeated point 10 to point 19, etc.)

@Релёкс84 actually you should loop it around for 1001 times. brilliant isn't it?

Next time I owe you $1001 I'll give you $1 and tell you to loop it around.

So... is math what it is or is it what we found it to be?

if there are true statements that cannot be proven then the fact that God exists must be a true statement that cannot be proven.

I've watched your videos for quite a while now and this is by far the best work you've done so far! Great job!

“There will always be true statements that cannot be proven” for all my skeptic friends who say they don’t believe in God without prove.

he speaks fondly of those German and European "Scientists" and about their "extraordinary discoveries", But when a scientist with an Asian name comes up, he just calls him a "mathematician". It is disgusting to see such things, trying to make people believe that Europe's position of power and wealth was all because of their intelligence, hardwork and education, when we all know that it was by means of force, war and slavery. If you weren't aware of this, then please check for the scientists who actually were the first to make these discoveries, and if you already knew about all this, then I hope you burn in hell.

then no math pls deleted it from syllabas

Turing did not "invent the computer". Turing did not "lead the team".

I don't think math will ever reach any pure truth like physics or chemistry. All notation, theories and symbols will never be able to truly describe the intricacies of the world accurately.

6:00 maybe I'm stupid but if 9 goes back to 8 and goes back to 9 because you add 1 then eventually the real number wil just oscillate between 8 and 9 right? Sry for bad English

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I did not know math is philosophical, and I'm an engineer

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this was fantastic

ALAN TURING WAS GAY?

Mathematicians 🤝 philosophers Giving peeps existential crisis

David Hilbert: "We must know, we will know" Derek Muller: "The truth is, we don't know, and sometimes, we can't know"

I understood about 40% of this video, still loved it.

I donno I see a glaring issue with referring the self and then not being able to solve the problem because the system is stuck in a loop that is inevitable because a loop was the start of the system querying.

So… the pot thinking about its self and can’t decide if it is or not because it thinking about it self

everything you say depends upon how you define the words you use to say it. These definitions are by definition unstable, and therefore you can never fully know the exact meaning of what was said, unless you deny all instability of definition or interpretation or usage, which all change according to need, subject, experience, and intention, so you ultimately cannot fully know what journey meanings will take. I wish anyone good luck trying to know answers in advance of their calculation and expression. They are by nature out our control, as is the entire concept of stable, complete, pure and truthful Mathematics. But without mystery, there is surely only death, no colour and no sound.

17:50 Wouldn't it be easier to just have the Gödel number of a sentence be the concatenation of the symbols that make it up as digits in base + 1?

@Paul Mahoney That is blatantly false

@lord_ne Maybe, but the usage of prime numbers is the only thing in any system that can absolutely guarantee a unique set of numbers.

@Paul Mahoney But that's the point, we express them in base + 1. So there are no single-symbol cards in the double digits.

@lord_ne Because there are base cards in the double digits. If the last two digits of a number are 14, does that refer to cards one and three, or card 13?

@Paul Mahoney But how could they possibly concatenate to the same number if each card is a single digit in base + 1 ?

So 1 is equal to 2?

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Yeah, it's made up.

Bravo! What a nice presentation! Thank you, from the bottom of my heart, before someone discovers that there is a hole there!. You really know how to make understandable the unintelligible. Congrats!

21:00 this had me in a mental pretzel lmao 🤣

This video is Turing Complete; it halted at the end.

Is this why there are always gonna be bugs in the programs ?

Can you please do an episode on whether quantum computers have capabilities beyond a Turing machine? Perhaps it's an easy answer...

Countable infinites.....lol nah

it seems like the godel number g is only a paradox of "if it's true it's false, if it's false it's true" because it was given the godel number of g. but i guess eventually, if godel numbers are infnite, there would be a godel number just like it eventually. idk my brain is trying to understand lol

There is no such thing as something true you can't prove....

There is.

when i look up godel numbers, i unfortunately get some charts that show the same godel number meaning 2 different things.

Oh wow, nice to see the article by Tony Cubitt, David Perez Garcia and Michael Wolf here!

Im, sorry, I know this sounds a bit arrogent. But for the diagnalization proof. If you made your list in actual order. So 1 is 0.1, 2 0.2, 3 0.3, 10 0.0 then at 11 you get 0.01. Then you do that infinitly. Could you not just make this new number, follow the new index you would get, and then find that exact number having been made. Especially if 9 rolls back to 8. If you had a number of 0.999 then you could go to get 0.888 from the diagnal which from the previous organization you know 0.888 comes far before 0.999 comes, so it is already drawn. We know if the number has 9 anywhere in a number, then if the index is organized propperly, then we can know with 100% certinty that the number made has apeared.

The diagonal number you get from your list will be 0.211111... forever. The problem is that you're only listing numbers with finitely many digits, and so you're missing all real numbers with infinite digits, such as 1/3, pi/10 etc. and of course, 0.2111... as well. So everything's the same as in the video: the diagonal number is not on your list, nor are infinitely many numbers that are left out.

So "Because" doesn't count as an answer?