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 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


  • Adam Keeley
    Adam Keeley

    There is no preferable side of the equal sign.

  • Gleichtritt

    Well now you have it, I am suddenly interested in math....

  • Fat Al
    Fat Al

    Hrhr he said "googleplex" like in google... SMART!

  • Pieter

    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

    It’s boring

  • Noriaki Kakyoin
    Noriaki Kakyoin

    Somehow I understood what was going on and it facinated me even though I could never explain it myself

  • Jesse Thomas
    Jesse Thomas

    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

    Godel, Escher, Bach

  • Frank Harcourt
    Frank Harcourt

    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

    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

    Infinites sort of have different sizes though, so it's not exactly obvious.

  • Grilled Flatbread
    Grilled Flatbread

    This is really similar to not knowing whether you are dreaming, alive, or a simulacrum

  • John Woodall
    John Woodall

    Math is God. -True, yet unprovable -Consistent as far as we know -Unable to be fully understood

  • Hedgehog3342

    Math is honestly quite confusing at times. Definitely not my strong area.

  • Adobe Reviews
    Adobe Reviews

    The sad salmon ipsilaterally untidy because yugoslavian genotypically switch underneath a untidy edward. boring, versed milkshake

  • Philip Berthiaume
    Philip Berthiaume

    I'll stick with 2 + 2 for now, thx....

  • Jesus Shuttlesworth
    Jesus Shuttlesworth

    Godel grief

  • Teflon  musk
    Teflon musk

    What a good ole number

  • KZisNBKosplay

    Math is just a Riddle.

  • Seth Solomon [Student]
    Seth Solomon [Student]

    My life is a lie

  • Tea Drinker
    Tea Drinker

    The set that contains everything can't contain nothing

  • Alaa Ashraf
    Alaa Ashraf


  • Colby Black
    Colby Black

    Little “g”. Gods mathematical devil in the equation of life.

  • San Ien Jao
    San Ien Jao

    Very interesting yet I understood less then 50%... 🤣

  • Daniel Methner
    Daniel Methner

    Georg Cantor... That was the name of my high school :D

  • Michael Fitzgerald
    Michael Fitzgerald

    Sounds like you need a second barber lol.

  • Fractalator _
    Fractalator _

    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

    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

    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

    30:05 goosebumps 🤯

  • BOB A
    BOB A

    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

    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


  • Azizur Rahman siyam
    Azizur Rahman siyam

    the cards literally went above my head.

  • Kenichi Sasaki
    Kenichi Sasaki

    ma head HELP

  • Андрей Иванов
    Андрей Иванов

    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

      @Дмитрий Косолобов >> The set of all sets should then contain itself. Accept that "set of all sets is empty" like axiom and all contradictions will be eliminated. Empty set is the only ordinary set. Turn on imagination. Do you remember "Сказка о рыбаке и рыбке"? :-)

    • Дмитрий Косолобов
      Дмитрий Косолобов

      The set of all sets should then contain itself. So, it can't be empty. A resolution could be that there are no sets at all. But we always assume that at least one set, the empty set, exists. Thus, the problem remains here.

  • Guillaume Ohz
    Guillaume Ohz

    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.

  • TheDeadPoet

    Most of us hit "Like" just for the title.

  • John Cantu
    John Cantu

    “We will never know everything with certainty”…… There you go, a true statement we can’t prove. But even that is without certainty.

  • can_of_woopwoop

    Rephrase... 'science has a fatal flaw'

  • Asma Angel
    Asma Angel

    The lean discovery dfly spell because cauliflower reassembly measure mid a grubby gruesome target. silent, festive turkey

  • Sure, Indubitably!
    Sure, Indubitably!

    Can't believe I see "math" and "fatal flaw" together except if the sentence is "your answer in this math equation has a fatal flaw" lol

  • zmunk

    Love the MTG shoutout!

  • El Mino
    El Mino

    Who are we? Where are we? Why are we still here?

  • Felipe El Guapo
    Felipe El Guapo

    2nd time thru... So has it been decided? Can set theory be on an infinite loop?

  • Araceli Perez
    Araceli Perez

    Bitches be like "bitches be like~" like they ain't like the bitch that be like, like ... bitch?

  • Peace22

    It seems like common sense to me, I haven't even finished a minute of the video. I however understand that just in the most basic view, if math is to have an infinity (as it does) than there can never be an end answer to many questions. If you have box, and in that box you have 3 problems and 3 solutions that can all intertwine, you will always have 9 possibilities of a problem/solution scenario. With math, and life, there are infinite numbers and infinite possibilities. You can't put infinity into a box. That's just one example, and I'm sure as I watch more I'll understand in greater detail and more about this topic.

  • dodiswatchbobobo

    There’s a popular saying that math is the universal language. It is false. Math is the language we are attempting to translate the universe into. It is man made. We do not have words for many many things.

  • Megananium

    Surprisingly i actually understood most of this

  • Daniel DaSilva
    Daniel DaSilva

    Using math and theoretical equations to prove scientific law 🙄 no one is even seeing what’s really going on?

  • Mordechai Vanunu
    Mordechai Vanunu

    Maybe everything has its time and cant be solved before we get there.The questions from 500 years ago are in some ways solved now and maybe 100 years from now questions we have now will be solved by the generation at that time.And i do believe there's a limit to the human capability.Just the fact we cant even tell by 100% what happens after we decease.

  • LiT TV
    LiT TV

    My brain hurts

  • Iuri MC
    Iuri MC

    5:53 You said "by the end of this process" but it is clearly an infinite process...

  • Diggensagg

    Why must the set of sets with more than 5 elements contain itself? Iam confused.

    • Moyprod

      I doesn't have to, but you can define one.

  • Jacobus Terhorst
    Jacobus Terhorst

    Even God does not know everything. Because humans have free will they are inherent unpredictable...

  • Steve

    0:16 hold up... if a turth cant be prrovien then its just a threory

  • Murat İlsever
    Murat İlsever

    I am ok with Cantor's proof. But what does it exactly tell us? Is there at some point a real number between 0 and 1 which cannot be mapped to a natural number?

    • Релёкс84

      It means that it is impossible to simultaneously map *all* real numbers between 0 an 1 to natural numbers.

  • Nethender

    Its moments like these where im glad other people did the hard thinking for me, because there's no way id think of any of this

  • Lamamri Aissa
    Lamamri Aissa

    i have determinate the formula of the tangent

  • Lamamri Aissa
    Lamamri Aissa

    do hear about fractale music

  • Lamamri Aissa
    Lamamri Aissa

    is there an undefined function in an interval but it is derivable and continuous in this same interval

  • Jeremy Lanier
    Jeremy Lanier

    If you could solve this then you could have the most balanced rpg in the world.

  • Ovid Bambusa
    Ovid Bambusa

    What´s undecidable about airline ticket systems?

  • Jim Frost
    Jim Frost

    The slit experiment, the work of the Japanese professor with water and how it's very structure is altered by our words/emotions, the effect of speech and love on plants, all these things point to the the concept that the world is physical, provable, and yet meta-physical, un-provable, malleable even. Understanding this, the answers to the the 3 Cs is obvious, as math is a representation of the universe which is both physical and meta-physical.

  • Bruce Wayne
    Bruce Wayne

    Godel Escher Bach


    I have learned more on ISnets from these sience/math people then school...

  • Nial Red
    Nial Red

    This also proves why robots and AI can never really be human

  • Nial Red
    Nial Red

    Can we call the hole as consciousness?

  • slkjvlkfsvnls dfhgdght
    slkjvlkfsvnls dfhgdght

    I don't really understand the difference between "complete" and "decidable"

  • Diego Savic
    Diego Savic

    Maybe start by reading ancient text... : In the beggining everything was chaos : ?

  • timthegameryeet


  • Dila Afdhol Santosa
    Dila Afdhol Santosa

    That's why i love math. Never ending Journey. Even create a new whole universe in it (ex : a computer)

  • Dim Willow
    Dim Willow

    stupid video .. you have no idea how powerful our computers and knowledge will be in the future..

    • dsBlocks

      you didn't watch the video, right?

  • Spoiler Alert
    Spoiler Alert

    How can you call theories true if the cannot be proven? Why try to understand why your Creator made things that he He did. All that you need to know is that's how He made it and it it works in mysterious ways. Ancient saying: Asking why, only leads to more questions.

  • Random stuff
    Random stuff

    I know you have worked hard on video and it's amazing ....... But the problem is I can't understand it

  • T-tus

    QUESTION, More real numbers then natural? Why did "we" not just index the new made up number and say. Eey look, thay are equal many again. Real number apair. You index it. Real number appear and you index it. We could index the natural numbers with real numbers. Whish should turn the table. Just add one to the natural numbers. And then use diagonalization to index it? What funny made up rule am I missing? =D If you are able to get a diagonalization number out of infinity amount of real numbers. Then you can also add one to an infinite amount of natural numbers. What am I missing? =D

    • T-tus

      @Mr_Magnus @Mr_Magnus The new number you get after using diagonalization. It feels like schrodinger's cat. Ether you have used up all of the natural numbers or you don't. As soon as you can start diagonalization you only prove that you had not used all the natural numbers to reach infinity. But maybe I am not listen good enough. The rules are set and I need to obey them better. I will have to think about some more. Thanks for the feed back =)

    • Mr_Magnus

      We can't index the new number because we've ran out of integers to index it with. You start with a list of every natural number, and map all of them to a real number, and then you point out that you've left some real numbers behind.

  • Jhake Pugosa
    Jhake Pugosa

    This video can either cure insomnia, or spread insomnia.

  • Jhake Pugosa
    Jhake Pugosa

    Mathematicians. The people with the most free time in their life.

  • Matthew Salvatar
    Matthew Salvatar

    So a computer cannot exist illogically, and yet would not exist without illogic. That quite interesting.

  • fleetstreet11

    The intuitionists were Kantians.

  • fleetstreet11

    Uncountable infinities (Real numbers): infinite infinities. Countable infinities (Natural numbers): finite infinities. Monadologies in monadologies, unity of unities.

  • ActionCamSports

    I'm only half-way through this video, but just wanted to stop and say I'm sorry I ever asked anyone, "How do you know that?"

  • Anubhav Saha
    Anubhav Saha

    Loved this video

  • Brian

    This is a really long MTG ad

  • Brian

    This is poetic 😢

  • Robbie8

    Thirty four thousand comments huh! Hard to get noticed on this channel. I am an Australian POSSUM! I hurt my ear playing with Stumbles at his tree. My Mumma said I stand out in a crowd! I'M HAPPY!!! 🤗🤗🤗

  • Patrick Gueriguian
    Patrick Gueriguian

    the diagonal proof doesnt apply to an infinite list imo. to me thats not a proof at all but trying to apply a concept that might work in the finite realm to the infinite realm

  • KÏd BÚddŸ
    KÏd BÚddŸ

    This was the best add for game of life I’ve ever seen

  • Marcin

    Absolutely best science show! Thank you for inspiring us!

  • Andrea Croci
    Andrea Croci

    One of the most interesting videos I have watched in a long time, although I have to admit I got lost in a couple of passages.

  • Johanna Richardson
    Johanna Richardson

    The past waiter unexplainably fool because bracket immediately twist before a decorous luttuce. noiseless, holistic ghana

  • wavingdragon

    This video shows me why there are only a few Turings in this world while the plentiful of mostly hairless apes on Earth have ignorance and violence instead.

  • MotorsportsX

    0 doesnt exist in the real world. only in math. there is never nothing. only equilibrium.

  • Anthony Gonzalez
    Anthony Gonzalez

    This video is really going to make me switch my major

  • vinit sharma
    vinit sharma

    Pls explain tht how godel proved true statements can't be proven

  • Super Pokemon
    Super Pokemon

    I'm fan of conways game of life

  • Luka Chanadiri
    Luka Chanadiri

    I think that's where god enters the chat.

  • Michael Lazarus
    Michael Lazarus

    Pretty sure Alonzo Church published his proof prior to Turing.