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., \u0026 Deutsch, H. (1995). Russell’s paradox. -

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

Russell, B., \u0026 Whitehead, A. (1973). Principia Mathematica [PM], vol I, 1910, vol. II, 1912, vol III, 1913, vol. I, 1925, vol II \u0026 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., \u0026 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


  • Oskar Tegby
    Oskar Tegby

    “Let’s make the title more clickbaity”

  • Shlibber Macshlibber
    Shlibber Macshlibber

    Is the flaw that its spelled MathS?

  • AliasUndercover

    I get it!

  • Emir Cenk
    Emir Cenk

    One of ur best !! TY

  • David Do
    David Do

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  • Parker Essential
    Parker Essential

    The nippy fight rationally pray because vault hypothetically develop over a quack ravioli. pastoral, ad italian

  • Samuel J. Rivard
    Samuel J. Rivard

    so you got a consistent inconsistency of paradox

  • Zeb 1
    Zeb 1

    So, if truth can exist without a proof, is it logical to say that God doesn't exist as there is no way of proving His existence?

    • People Legend
      People Legend

      Yes, but there is also no way of proving he doesn't exist at the same time

  • b bb
    b bb

    The useful cymbal periodically bore because twilight aetiologically box afore a absurd dessert. plausible, befitting parsnip

  • Bob Aldo
    Bob Aldo

    Do you have to have severe OCD to do this sort of math, or do you get OCD from doing it?

  • Alex Finiks
    Alex Finiks

    *Political correctness downplays the severity of an issue* : "forced to take hormones" doesn't convey WHAT was done to A.Turing... especially for those who haven't heard of him before~ Ppl nowadays cannot even express their thoughts straightforwardly without being accused of all sins of the world by those striving for political correctness... democracy & free speech in all its doubtful glory at play~ 😕

  • john almers
    john almers

    The adhesive chest subsequently hop because canadian posteriorly own of a drab sphere. selfish, puffy maria

  • Moadot720

    12:22 Tha went dark, fast... ...😂😂😂😂

  • Tom Van Waes
    Tom Van Waes

    He passed away from covid at 83, how about he just passed away? How old or how sick do you need to be to just pass away????

  • Jeff Navarro
    Jeff Navarro

    Great explanation!! My favorite book of all time is Godel, Escher, Bach which talks about all these ideas, and how they are related to music, art, and many other things through the idea of self-reference. I also have a copy of Godel's actual proof in book form...and I am unashamed to admit I can only understand the first paragraph...all else is complete gibberish to me.


    the game of life is maybe not decidable in every arbitrary case because the loop would be too big but many can be tested for stability. after all the deterministic nature means that equal states produce equal outcome. even stuff like gliders and such can be tested simply by doing the same equality check and shifting the reference frame. by splitting it up into regions that are noticably distinct, say the growing diamond, that all also applies to these because the individual regions all have a testable loop size and are stable within their reference frames while moving away from each other. if they move away from each other while leaving a trail of immobile or otherwise stable constructs, that means its growing.

  • James D
    James D

    3:30 *Listening while playing MTG arena* Okay now you really have my attention

  • Wendy Yang
    Wendy Yang

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  • Jonathan San Antonio
    Jonathan San Antonio

    Love this format. Voice & choice of music makes a great experience.

  • Donald Johnson
    Donald Johnson

    Time and chance happens to us all. It takes an intelligent man to realize that he is not that smart. We are ants trying to do calculus.

  • Donald Johnson
    Donald Johnson

    Kindergarten stuff.

  • Donald Johnson
    Donald Johnson

    Math does have a problem. Men have a problem understanding math. Somewhere God is laughing because man may one day be able to add 1 plus 1.

  • tarek al
    tarek al

    Why are mathematicians like this? What''s wrong with these people? and why do people like this exist... just to make us suffer

  • Spin-chan Art
    Spin-chan Art

    wait there is an order in which you define things... you either write a statement then calculate its Godel number or you pick a godel number, decompose it to find which symbols could be written on the card you can't define a statement and give it a Godel number simultaneously and expect it to match

  • Roger Dinhelm (Polyglot)
    Roger Dinhelm (Polyglot)

    God, I like those math proves: "Since they are very large numbers we can call them by letters just as well" "Hmm, ok, no big deal" "AAAND now it is a centerpiece of our proof" "Wait, what?"

  • phaseshifter3D

    Gödel broke me....

  • Mic_Glow

    Is the equation written on card G unsolvable or impossible to prove? Because if not it doesn't matter what else is written on the card... Like in one of those jokes where you write "turn this card around" on both sides of a paper. It doesn't have infinite pages. This whole setup (assigning numbers to symbols, than multiplying by primes, assigning new numbers etc.) is fishy.. seems to me a little bit like intentionally cornering oneself. Pumping water out of a bathtub, then putting a cup into it and upon finding no water in the cup concluding that water doesn't exist.

  • Yojimbo Aeon
    Yojimbo Aeon

    well thats because prime numbers are either THE problem or show the problem,( prime numbers shouldnt exist and only do because of our way of using numbers for math) or (we have not figured out a math to utilize prime numbers so)

  • I.O

    That first point makes no sense. All infinity are the same size. You can't prove that there won't be a number in the set of real numbers that will eventually reach the diagonal + 1 number. Just because you can't conceptualize it doesn't make it impossible. Nothing is impossible when it comes to infinity.

    • mar98co1

      Including there being different kinds of infinity? Lol Why write up a comment like this when you don't know the math behind the context? You can watch his previous video, or the one from vsauce for an into to the idea of different infinity. Or i can give a quick sketch up

  • Oliver Ryan
    Oliver Ryan

    Thanks for explaining this so well, really interesting

  • Absent Minded the SubCreator
    Absent Minded the SubCreator

    Thinkers during the Scientific Revolution were highly motivated, because by categorizing things like mathematics and scientific laws, they believed that they were discovering the language and mind of God, at least inasmuch as God used language and coding to “program” the universe. There is something wondrous about reaching what appears to be limits to our ability to comprehend those languages. It’s the reason why so many brilliant people were also very humble- the more you know, the more you realize just how much you DON’T know.

  • Icy_Fire369

    maths not math

  • Esconis

    I know the fatal flaw in Math... ...that it should be pronounced as MATHS!

  • Landon Gray
    Landon Gray

    “Not all infinities are the same size”

  • Evergreen641

    I would go as far as to say... We can't know. But we MUST try. For only then will we progress.

  • beeorganic

    Pfffft... With liberal outcome-based educated ALL answers are correct.

  • patrick mullot
    patrick mullot

    Absolutely brilliant! (no pun intended :-p)

  • zfierocious311

    from @6:00 to @6:30 the "diagonalization proof".... how does that prove that there are more real than natural numbers? For each real number that gets generated using the diagonal method... can't you just add 1 to the last natural number to match?

    • zfierocious311

      for every Real number, you could just remove the decimal point, and any 0's that were after the decimal, move them to the back instead. Boom, new Natural number, 1 to 1 between real and natural.

  • algaeikpond

    we're human

  • whatcan welearn
    whatcan welearn

    What a gem ❤️🇮🇳

  • CosyStudios

    Nailed it. Great video

  • mj 73
    mj 73

    Math Has a Fatal Flaw: The Mathematicians.

  • Samantha Atkins
    Samantha Atkins

    I have done some fooling around with Twin Primes and I am convinced it is true by pretty simple observations and arguments. Further I am convinced that there are an infinite number of prime pairs 2n apart for all positive integers > 1. No, I don't have an airtight proof yet.

  • Ert Hiles
    Ert Hiles

    You had me then you lost me bruh

  • solo lobo
    solo lobo

    Crazy to think that there are still an infinite amount of conjectures that are now considered unsolvable that actually are.

  • Robert Camacho
    Robert Camacho

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  • TheBlinkyImp

    The problem isn't self-reference, the problem is choosing to include the Axiom of Infinity in your construction of set theory.

  • RY0QHAirkwwI0sXTUXmPued9l0u5GxInjy1

    how about a light headed video about dophins?

  • Matej Mazur
    Matej Mazur

    "Not all infinities are all the same size" -Monkey Brian

  • Jimmy


  • Carrot Chaser
    Carrot Chaser

    This is the definition of overthinking something, this is like trying to measure something that CANT be measured, then acting surprised when it cant be measured. For something to be infinite it was to be infinite in all directions thus you cant have a staring point. just like the universe looks to be infinitely big it must also be infinitely small, and all of the evidence we have up to this point agrees with this statement. the same holds true for time, it spread infinitely into the future and spreads infinitely into the past, thus no starting or middle or end point. we like to think the universe has a starting point but that's just called anthropomorphize something, which normally is just to make us feel better (dopamine shot) not real fact, just like Disney movies. And as for the game of life, the game repeats its self if you put the same starting pattern, thus there is no mystery there. why try to make it seem like there is? You just got smart served overthinking mathematicians, boom... mike drop.

    • mar98co1

      Uhm, take a semi-line in geometry. It is infinite. It has a starting point. Take the game of life. Make a single glider. Play it. It has a beginning, but no end state Boom, mic drop?

  • Dilwhite4president

    Hilbert wanted to take an objective topic and subjectively express it. Math isn't incomplete because if it was then our universe would be incomplete aka chaotic, and life wouldnt exist. Math is the foundation for our universe because its actually real. From 0 and infinity, to negative numbers, to imaginary numbers, and real numbers they all have a role in our day to day reality. What Godel proved was that we as humans can never program a robot to live and experience consciousness the way humans do. Have great day everybody. This was a thumbs down from me.

  • Franciscop98

    I can't believe Covid took John Conway from us. I can't believe I didn't know sooner. That is a tragedy...

  • Guy Mahoney
    Guy Mahoney

    Just proof that using human numbers causes stupid problems, nothing wrong with how the concept of maths works. People making proofs that rely on human representations being broken are the problem; they're not doing maths, they're doing semantics.

  • Usman Shahid
    Usman Shahid

    Can someone explain 20:53 - doesnt that just mean that Godel's system is flawed, why does it mean that any mathematical system will never be complete? Doesnt that just mean that godel's system is not incomplete, and most logic models are incomplete?

  • kip LaRoy
    kip LaRoy

    Sad that Turing did so much to help humanity/ the allied war efforts just to be condemned for being himself. THAT WAS THE FAR RIGHT.. NOW WE HAVE TO BATTLE THE FAR LEFT.. WE CAN NOT LET EITHER GAIN ABSOLUTE CONTROL. IF WE DO WE WILL ALL SUFFER THE CONSEQUENCES!!!

  • Sofa King Hung
    Sofa King Hung

    If we know a pattern will continue infinitely that is proof not theory. I’m genuinely confused.

  • Sofa King Hung
    Sofa King Hung

    If we know a pattern will continue infinitely that is proof not theory. I’m genuinely confused.

  • Li Hua
    Li Hua

    4:23 bruh. Q is not happy.

  • Power Bastion
    Power Bastion

    Turing should be in everyone’s top 5 heroes.


    "Of" or "with" - 83 is above average, but he would never have died, if not for "it".....

  • M7 Rengy
    M7 Rengy

    I have a ton of maths homework to do and I'm watching this.

  • Francisco Ramirez-Mora (Zyshko)
    Francisco Ramirez-Mora (Zyshko)


  • John Walker
    John Walker

    This entire video is proof that humans are masochists who create their own problems

  • J.D. Shah
    J.D. Shah

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  • Soteriology101

    If math is incomplete is there anything better that is complete to replace it? Existics?

  • YellowWorld


  • Lee Ngugi
    Lee Ngugi

    Look at us. Learning mathematics willingly. Who would have thought?

  • Hairetikos

    I am an idiot, but is this basically the Achilles and the tortoise problem?

  • Commander DusK
    Commander DusK

    How can we be certain that something is uncertain...

  • Luciano Gomes
    Luciano Gomes

    i got lost in R set. The game of Life running the game of Life seems to me fractal

  • CamperVan-K

    Why do you say 'math' in America? When you're talking about plural, rather than singular? Maths is a subject, rather than a single entity. What's the reason for the missing 's'?!?!?!

    • kalidasmusic1008

      probably for the same reason we don't say musics.

  • alexb1886

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  • Kal2019

    Im too stupid to understand anything in this video :( Gonna watch Joe Rogan podcast instead.

  • Free Hat
    Free Hat

    Doesn't know what infinity means.

  • RayHeartsYou

    I really want to make a Spiders Georg joke right now, but my brain is broken lol

  • Andrew Eppink
    Andrew Eppink

    Man. Heard if Gödel's Incompleteness Thm. Not that l understood much of it. Disconcerting. Deflating. Like 2nd Thermodynamic Law, "You can't win. Only possibly break even or lose."

  • destro

    war conributed to creation a compuuter which leads to finish the war

  • Fırat KARA
    Fırat KARA

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    I thought it said meth and clicked only to see it was math

  • Logic Spike
    Logic Spike

    oh listen to the brain talk; blah, blah, blah, blah, blah. .... lol. Great Video! I'm going to take this knowledge, go over to my Father in-laws house, and mentally beat the crap out of him. :o) jk.

  • alexb1886

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  • opposed2logic

    reminder that a single swedish professor cracked the nazi codes on his own for fun, years before the british did it with a full team.

  • Newman Ortega
    Newman Ortega

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  • Felix Nielsen
    Felix Nielsen

    Next steps: 1. Prove that every unprovable statement can be proven to be unprovable. 2. Prove that every unprovable statement can be proven by proxy. fx there are many cases of "if true, then this can be proven", fx by brute force, given infinite time, in which case, if proven to be unprovable, we know it to be false. I'm sure there are reasons that these two point are not worth considering, but still, I would like to know. ;)

  • Bruh

    Thought it said meth had a fatal flaw…

  • Mike S
    Mike S

    Great, if mind-bending, video. So I guess when kids in high school asked why they should study math, the teacher should have said, "So one day you can understand whey math is incomplete, inconsistent, and undecidable." LOL Have they played life in 3 or more dimensions? How cool would it be if you helped start up a new Numb3rs TV series?

  • Aldon Dekock
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  • Joel Rey
    Joel Rey

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  • SuperSaiyan Stallone
    SuperSaiyan Stallone

    yea i had to stop at the real vs natural. new real number would exist. and would be slotted.