Math Has a Fatal Flaw

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Prieš 2 mėnesius

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 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.


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. -

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

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

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

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. -

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

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

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

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. -

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

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

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

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 (
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


Joaquín Hinojosa
Joaquín Hinojosa Prieš 2 val
My brain hurts
XeL NaGa
XeL NaGa Prieš 2 val
well if you knew what they feed the cattle youd think youd get poissoned too lel worst monocrop used for cattle full of pesticide and gmo. basicly giving the cheapest worst most toxic chemical food possible for cattle. the steak gotta be cheap :D
Politics Prieš 2 val
Glad there is a mathematical answer to why I can't figure out the truth about Covid!
Adam McKinzie
Adam McKinzie Prieš 2 val
If the game of life can run the game of life, then it’s possible that the game of life could run the game of life running the game of life and continue to do so on an infinite scale
Sven Croon
Sven Croon Prieš 3 val
only a mathematical system that has no axioms, can ever be truly complete, consistent and decidable. Axioms are the evil that corrupts everything ;-)
Thomas Klugh
Thomas Klugh Prieš 3 val
Now my head hurts.
Marshall115 Prieš 3 val
give it a few decades-current math is wrong-it will change but still just be humans way of measurement.
neil unger
neil unger Prieš 4 val
And here I read this as Meth Has A Fatal Flaw.
EliteTrollingG Prieš 4 val
5:16. I just wanna ask, why don't mathematicians just agree on what number natural numbers and real numbers should stop at. Like the end of the number line is 10,000,000,000,000,000,000,000,000,000,000 starting today. And the end of real numbers is 0.999999999999999 until it goes to 1. I mean that would ruin all the work we have put into trying to find the end of pi, but my question is, WOULD it help solve problems mathematicians have been trying to solve for centuries, or would it just create more problems? I'm actually curious.
Piotrek Prieš 6 val
Hi, i ve got some idea, about part about cantor set theory. If u try to groupe natural numbers and numbers between 0 and 1 like this: for 1, theres 0.1. For 27, theres 0.27. For 29010123, theres 0.29010123. Then every Natural number, will be grouped with exactly one number between 0 and 1, and also if u try doing something like in the video, u just take another natural number matching this one. Does it work? Please lemme know.
Piotrek Prieš 5 val
@Brauggi the bold oh, u re right, thank u. I didnt considered irrational numbers. But still, i dont believe this😒, even if the proof showed by cantor is 100% true and logic, my mind cannot take this. :c.
Brauggi the bold
Brauggi the bold Prieš 5 val
It does not work. You will only ever hit rational numbers which have a decimal representation that ends in repeating zeros that way. Your mapping will for instance not include ANY irrational number between 0 and 1.
Sumit Juyal
Sumit Juyal Prieš 6 val
Wow, self-reference, what a beautiful link, this channel is the perfect blend between epistemological philosophy and mathematics, as someone who is a math graduate I would've loved to be introduced to subject histories like that before mindlessly trying to solve problems and pass the exam. People need to normalize and popularise the idea that mathematics applied philosophy which is just applied inherent logic. Your channel does wonders for us appreciators of the mind but are still rigorous in our understanding of the same.
Dylan Hase
Dylan Hase Prieš 8 val
I want to launch my own branch of math! How do you do that
H Prieš 8 val
Maybe I don't understand it but wouldn't the diagonal number be on the list since the list goes to infinity and because at one point you will have to roll back 9 to an 8 making it equal to that number in the diagonal the minute you add +1.
Abdullah Ahmed
Abdullah Ahmed Prieš 8 val
9:25 need explanation
Azio Prism
Azio Prism Prieš 10 val
Having 2 apples is impossible. Identical numbers are not possible. What makes numbers and apple differ is the environment around them :9
Croldfish Prieš 11 val
1:25 i forgor :skull:
Heaven&Hell Prieš 12 val
none of this can be proven to be true.. concurrent quantum states prevent this.
KLEIS Prieš 12 val
Math can prove a lot of things but it can't prove life, it is coz of math is itself incomplete by knowledge of human has created, completeness or creation of all things that has life is fundamental things that it is not related by math. So math is not absolute and it is just one of many knowledge to prove something by what creation has created
TheBeast Prieš 12 val
just learned set theory for my computer science degree really interesting stuff
Bijou Smith
Bijou Smith Prieš 13 val
@28:20 so, the undecidability of the spectral gap property amounts to the first proof quantum physics, as we conceive it presently, does not admit reductionism? Have I got that right? That's pretty signifcant for philosophy of science, which has previously generally operated under the paradigm that science (whatever it is) is reducible _in principle_ to base physics, through obviously not always in practice. So either that's a false paradigm or quantum physics is not base.
ayy lmao
ayy lmao Prieš 15 val
Russell's paradox is a violation of the law of excluded middle. Ergo the law of excluded middle is wrong. Ergo superposition.
Kim Tae Hwan
Kim Tae Hwan Prieš 16 val
The voice is good though I'd never understand the contents...
Afqwa Prieš 16 val
This really murders the idea that math is some kind of divine tongue bestowed upon us by the gods. Murders it in its crib by smothering it with a pillow. Why does math work . . . _uhhhh sometimes it has empirically useful results._
Giap Chin
Giap Chin Prieš 17 val
"This is the game of life, running on the game of life." Then proceeds to slow zoom out. Wow my mind literally was blown.
Andrew C. Mumm
Andrew C. Mumm Prieš 18 val
Enter quantum entanglement... a proof can be true and false at the same time until observed. And when observed, the universe splits into different realities... :D
Ascot Prieš 18 val
Mind Blown
Andrew C. Mumm
Andrew C. Mumm Prieš 19 val
The game of life animations at the start are awesome - does anyone know how they were made?
Gerard Ligonde
Gerard Ligonde Prieš 21 val
The erratic canada methodologically dare because oboe endosonographically sin aboard a tall afghanistan. curious, meek broccoli
enthusiasticGeek Prieš 21 val
25:57 and it vanishes in a puff of logic
hieu dang
hieu dang Prieš 22 val
somehow i heard "godel" as "good old" for almost the entire video
Babe Root
Babe Root Prieš dieną
The language of God...the Creator/Designer, and mind of ALL information. The language of God...Mathematics. Amazing...isn't it? ☀️
Andre Prieš 13 val
_"The language of God...the Creator/Designer,"_ There is no god and no creator.
ayy lmao
ayy lmao Prieš 15 val
More like word salad
flobbie Prieš dieną
Yeah, but it is neither a flaw, nor is it fatal.
Andre Prieš 7 val
@flobbie _"It just is that way. This does not make math any less useful to me."_ If there is even one contradiction it is. But we have never found any. It is just that there is no proof.
flobbie Prieš 8 val
@Andre, i don't see that this makes it flawed. It just is that way. This does not make math any less useful to me.
Andre Prieš 10 val
@flobbie _" i don't understand. There are no flaws in math."_ You can't prove that a formal system is free of contradictions inside that system itself (for those systems that fullfill all conditions of Gödels theorems). That is a problem, because if there is a contradiction (even a contradiction we do not know today) then you can prove anything. Math is useless then. You maybe can prove this in some other system, but then how do you know that this system is free of contradictions? _"Your axiomatic system may be flawed, as it is contradictory. "_ And for many, many of those systems you cannot prove that it's not. _". But there is no flaw in the general way of how somone is supposed to process language."_ A "language" that contains a proof for A and not A is not really useful. _"Do some math and you will see there is no flaw."_ You can't prove this by examples. Do you even know what math is?
flobbie Prieš 11 val
@Andre, i don't understand. There are no flaws in math. What is that even supposed to mean. Your axiomatic system may be flawed, as it is contradictory. Or your proof may be flawed as it contains errors. But there is no flaw in the general way of how somone is supposed to process language. Do some math and you will see there is no flaw.
Andre Prieš 13 val
Not being able to prove consistency is something I would call a "flaw".
Robin Collins
Robin Collins Prieš dieną
The actually ice cytologically add because brother noteworthily note an a tacit volleyball. fanatical, subdued pine
darth biernot
darth biernot Prieš dieną
My brain nearly exploded while watching this! People always say: "Math is logic" I personally am more comfortable with language. I speak German natively, English and French fluently but when it comes to a simple mathematic equation with a variable in it my brain goes: "system just crashed due to missing math.dll" 😆
MrBizaaro Prieš dieną
Excellent Video ! Feels like one of your best
Jeffrey Wiegley
Jeffrey Wiegley Prieš dieną
Holy crud!!!! I *finally* understand the diagonal proof after 30 years... Prof. Leonard Adelman (The 'A' in RSA) used in Gödel's Incompleteness theorem in Second Order Logic class and I blew that on the final exam. Oh... I can die happy now.
Matthew Means
Matthew Means Prieš dieną
Alan Turing a story is my favorite depiction of humanity. A single human was enough to determine the difference between a world with and without Nazi Germany thru his contributions which would have otherwise plummeted us into a darker world with less insigh, yet in response to his glorious contribution to humanity, he was treated as a plague for being gay and made the other on premise of his differences that hurt no one. Humanity will respond to it's very saving with its own doom.
why bread
why bread Prieš dieną
This video and his “how a infinite hotel ran out of rooms” video match up I just thought about it
Peter Shmain
Peter Shmain Prieš dieną
Cantor's diagnolization proof is incorrect because when you use that method to think of a new number, it must then also be assigned to a new index which is just 1 more than the previous number therefore disproving infinite inequality. However, that's not to say that his ideas are incorrect. It's just that this proof doesn't completely work in this way
Andre Prieš 13 val
_"Cantor's diagnolization proof is incorrect"_ No. _"when you use that method to think of a new number, it must then also be assigned to a new index"_ No. _"t 1 more than the previous number "_ What "previous number"? _"However, that's not to say that his ideas are incorrect."_ The idea and the proof is correct. _" It's just that this proof doesn't completely work in this way"_ It does.
Michael Borisow
Michael Borisow Prieš dieną
I don't know how many of all 8 million of yall are understanding this, but I'm gonna have to pause and look up stuff from this vid another ∞times before I understand anything
Aymane Sghiar
Aymane Sghiar Prieš dieną
Best video I've watched in a looooooooong time.
Anaya Barata
Anaya Barata Prieš dieną
So is the fact that not all true things can be proven also unprovable?
Andre Prieš 13 val
It is the opposite.
Robin Hodson
Robin Hodson Prieš dieną
But this depends upon the assumption that recursion and logic contradictions disprove systems. That's not necessarily universal, otherwise we wouldn't be capable of comprehending them. Saying "This is incomprehensible," an apparent paradox, is actually comprehensible, and thus not not invalid.
costaran Prieš dieną
James Corey
James Corey Prieš dieną
Thank you for this waste of time.
Michael MAnville
Michael MAnville Prieš dieną
More like insufficient computational ability to prove true, but can never be prove false.
Релёкс84 Prieš dieną
Absolutely nothing to do with "computational ability"
Boysfifa010 Prieš dieną
What does it say about me that i read the title as "Meth has a fatal flaw" ??
Furqan Siddiqui
Furqan Siddiqui Prieš dieną
That's the reason I love watching this channel. It forces me to "THINK"
James White
James White Prieš dieną
I don't know if there is truth to be found studying mathematics, but there is much beauty to behold.
Jacob Alexander
Jacob Alexander Prieš dieną
Godel hurts my damn head. How would you come up with that.
Prototype 81
Prototype 81 Prieš dieną
To say things like "always" or "we will never know" is a fallacy. This is similar to clickbait. Nobody can day for certain what can be or cant be possible in the future. That is a simple fundamental of life. So... why is this guy using terms that are incorrect?... What else is he invorrect about. Why trust this guy on anything when hes obviuosly romancing the structure.. Lame.
Mr Blue
Mr Blue Prieš dieną
1 + 2 = 4
János TÓTH
János TÓTH Prieš dieną
Lobachevsky and Bolyai, Gauss is at most the third.
kamahll goodarz
kamahll goodarz Prieš dieną
Perhaps maths really is the language of reality in so much as they are both paradoxical
Solar Plexus
Solar Plexus Prieš dieną
I knew it.. Remember "Computer Code Discovered In Superstring Equations" ?
Galina Zwerlein
Galina Zwerlein Prieš dieną
We don't know what we don't know.
Reilly 25
Reilly 25 Prieš dieną
Can you not? Math is hard already stop giving them ideas on making it harder I'm still studying just do it after I graduate ty
infinite define
infinite define Prieš dieną
I don't have to know any of this. I'm richer than yall could ever be. But I only know arithmetic. Chile imma count my money now. Also I don't actually have physical money anymore
Adam Keeley
Adam Keeley Prieš dieną
There is no preferable side of the equal sign.
Gleichtritt Prieš dieną
Well now you have it, I am suddenly interested in math....
Fat Al
Fat Al Prieš dieną
Hrhr he said "googleplex" like in google... SMART!
Pieter Prieš dieną
At 17:17, where do the prime numbers come from? 2, 3 and 5. Is it just because those are the first three prime numbers, and the equation holds three elements, or is it something else?
Spider-Jonah- Man
Spider-Jonah- Man Prieš dieną
It’s boring
Noriaki Kakyoin
Noriaki Kakyoin Prieš dieną
Somehow I understood what was going on and it facinated me even though I could never explain it myself
Jesse Thomas
Jesse Thomas Prieš dieną
Trying to use math to prove math is like trying to use light to prove light. As you would say, this is a self reference. The light needs the dark to be proven. Math is part ‘logos,’ the logical foundation of all that exists. It’s part of the source of all things.
Sam Re
Sam Re Prieš dieną
Godel, Escher, Bach
Frank Harcourt
Frank Harcourt Prieš dieną
This guy is a total moron and he has no idea why. Everything he says is crap. The universe does not work the way he says it does.
James Sonke
James Sonke Prieš 2 dienas
The Barber Paradox is not a paradox. Just because something is illegal doesn't mean it's impossible to do. Just because it's illegal for the Barber to shave someone who shaves themselves doesn't change the fact that he is probably going to shave himself It's just an imperfect law. And just because each set supposedly cannot contain itself doesn't change the fact that R (itself) was made up. It exists as an idea. And ideas are infinite
Zora Marslink
Zora Marslink Prieš 2 dienas
Infinites sort of have different sizes though, so it's not exactly obvious.
Grilled Flatbread
Grilled Flatbread Prieš 2 dienas
This is really similar to not knowing whether you are dreaming, alive, or a simulacrum
John Woodall
John Woodall Prieš 2 dienas
Math is God. -True, yet unprovable -Consistent as far as we know -Unable to be fully understood
Hedgehog3342 Prieš 2 dienas
Math is honestly quite confusing at times. Definitely not my strong area.
Adobe Reviews
Adobe Reviews Prieš 2 dienas
The sad salmon ipsilaterally untidy because yugoslavian genotypically switch underneath a untidy edward. boring, versed milkshake
Philip Berthiaume
Philip Berthiaume Prieš 2 dienas
I'll stick with 2 + 2 for now, thx....
Jesus Shuttlesworth
Jesus Shuttlesworth Prieš 2 dienas
Godel grief
Teflon  musk
Teflon musk Prieš 2 dienas
What a good ole number
KZisNBKosplay Prieš 2 dienas
Math is just a Riddle.
Seth Solomon [Student]
Seth Solomon [Student] Prieš 2 dienas
My life is a lie
Tea Drinker
Tea Drinker Prieš 2 dienas
The set that contains everything can't contain nothing
Alaa Ashraf
Alaa Ashraf Prieš 2 dienas
Colby Black
Colby Black Prieš 2 dienas
Little “g”. Gods mathematical devil in the equation of life.
San Ien Jao
San Ien Jao Prieš 2 dienas
Very interesting yet I understood less then 50%... 🤣
Daniel Methner
Daniel Methner Prieš 2 dienas
Georg Cantor... That was the name of my high school :D
Michael Fitzgerald
Michael Fitzgerald Prieš 2 dienas
Sounds like you need a second barber lol.
Fractalator _
Fractalator _ Prieš 2 dienas
Yes, but the machine h is flawed in this situation as there is no paradox output. Even if we add the rules to h+: > if h outputs paradox then halt, > if h outputs halt then go into a loop, > if h outputs loop then create a paradox, and pass h+ into h+ as both instructions and input, h+ will just halt. If it halts, h is correct, because, to get to the point of halting, h has to detect there is a paradox. The fact that h+ has stopped does not mean that h is wrong as a paradox occurred before halting. Reasons for this: > h would be wrong to output halt as a paradox occurred before halting. > h+ will not go into an infinite loop as I have already pointed out that h would be wrong to output halt. > h+ will not create a *second* paradox because, as stated above, there is no chance of h+ entering a loop. If you then pass that h+ into h, then the output will still be paradox. Now, there is complete decidability. Btw, I'm not entirely sure if this is correct and I would be highly interested in arguments against it: "If you think that something is true, you should try as hard as you can to disprove it." - Derek Muller.
Llama Vicky
Llama Vicky Prieš 2 dienas
3:20 me: you forgot about the youtube algorithm, why was this recommended to me and why am i watching it
an 17121981
an 17121981 Prieš 2 dienas
Stopped watching after narrator started bs claims touring was the brains behind cracking enigma when in reality he just improved the already working system designed by rejewski, rozycki and zygalski
Rakshath G. Poojary
Rakshath G. Poojary Prieš 2 dienas
30:05 goosebumps 🤯
BOB A Prieš 2 dienas
Since the "rules" are made up, you can't use these examples as proof of anything other than math needs rules to work (i.e. its all made up and not a naturally occurring truth).
Abderrahim Benmoussa
Abderrahim Benmoussa Prieš 2 dienas
There is something inherently limiting in the universe by itself and even more in our brains. We are not infinite, our body and brains have limits and that is probably our Turing limit. The universe limits and wether it is infinite or not (and if yes, can someone tell me how it is growing ?) Is the universe limit. I think the universe is finite but infinitely divisible like the distance between 0 and 1. But I am bad at doing maths so I can't prove it xD
Nigel Siya-d
Nigel Siya-d Prieš 2 dienas
Azizur Rahman siyam
Azizur Rahman siyam Prieš 2 dienas
the cards literally went above my head.
Kenichi Sasaki
Kenichi Sasaki Prieš 2 dienas
ma head HELP
Андрей Иванов
Андрей Иванов Prieš 2 dienas
The set of all sets is empty. This axiom resolves all contradictions of the set theory . Because the empty set containes themself. :-)
Андрей Иванов
Андрей Иванов Prieš 8 val
​@Дмитрий Косолобов >> This leads to a much much weaker theory that cannot express all mathematic What do you mean "all mathematics"? As far as I know there is impossible to build "all mathematics". Gödel is the witness. :-) I look at mathematical theory from practical point of view. My approach makes fractals very simple objects and allows to solve differential equation in integers. My next goal is clear theoretical turbulence theory without any empirical assumption. It is impossible with traditional approach. :-)
Дмитрий Косолобов
Дмитрий Косолобов Prieš 9 val
@Андрей Иванов Apparently, you try to define a set theory in which only a predicate "A is a subset of B" is defined (in the traditional set theory there is a predicate "A contains B"). This leads to a much much weaker theory that cannot express all mathematic (the traditional set theory can do this). So, such theory is of no use in the discussed context (the interest is to find a theory that is consistent and can express all or at least most of mathematics). Or maybe I misunderstood something in what you said.
Андрей Иванов
Андрей Иванов Prieš 10 val
@Дмитрий Косолобов >> empty set is denoted by {} Element can be denoted as {a}. What the difference between {} and {a}? You try to explain me traditional approach. Change definition and system of axioms and you eliminate all contradiction in set theory. :-)
Дмитрий Косолобов
Дмитрий Косолобов Prieš 12 val
@Андрей Иванов You didn't understand, it seems. What you say is wrong: the empty set is not an element of every set; what is true is that the empty set is a _subset_ of every set. This is what confuses you. If the empty set is denoted by {} and a set of elemens a,b,c is denoted by {a,b,c}, then {{}} denotes a set that contains the empty set, but the empty set {} does not contain {}.
Андрей Иванов
Андрей Иванов Prieš dieną
​@Дмитрий Косолобов According to definition empty set is an element of every set without exception. So empty net contains itself by definition. I don't see reason to argue about definition. :-)
Guillaume Ohz
Guillaume Ohz Prieš 2 dienas
I dont understand why you allow self reference as an axiom, in the first place. Why the barber is not a woman? Or a special citizen that has other laws.
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