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



    I dont get how Candor's diagonal proof works. If u have already listed out all real numbers between 0 and 1 how r u going to make one more that stands out from them? Imagine listing all the 3 digit numbers and using the diagonal to create a new one. 100 101 102 etc You would only need first 3 such numbers and their diagonal gives u 213 which would already be in the list. It doesnt work but one might say since its a list of infinites, theres always a new number but in that case how would u list out a pair of real numbers anr natural numbers into a box in the first place? B

  • Xavier Chua
    Xavier Chua

    Him: takes 3 cards to form 0 = 0 Me: Yes finally understood something Him: *flip cards over to show 6 5 6" Me: I'm out

  • Brian Chadwell
    Brian Chadwell

    Math will never complete. It's an honorable pursuit.

  • Brian Chadwell
    Brian Chadwell

    Natural numbers are about as real as the human concept of time.

  • Varrant green
    Varrant green

    Logic paradoxes are kinda humerous because logic has more to do with language processing of consecuted concepts and relations to those concepts that aren't nessasarily meaningful. Like the enemy of my enemy is my friend thing from the office is just a serries of sayings that don't really mean anything strung together. Yes it produces a paradox when strung together like that. But that dosent mean the sayings reflect reality or what they attempt to conceptualize of even applicable to the situation.

  • Micah Ball
    Micah Ball

    Not a fatal flaw! A glorious opportunity for discovery!

  • Honk Honkler
    Honk Honkler


  • Nancy Hey
    Nancy Hey

    “Not everything that is true can be proven “. This tells me that mathematicians make a lot of assumptions about the truthfulness of of certain statements based only on the probability that they are true.

    • Релёкс84

      Yea, except that's absolutely not how it goes.

  • Google Email Address
    Google Email Address

    Alright...How much did YoyTube pay you to throw in that subtle propaganda Covid 19 BS comment. You could have simply said, "A scientist who recently passed away" but no...Thankfully I have a choice to longer view this channel...

  • Max Telmanov
    Max Telmanov

    0:00 Yeah big brain time 3:45 So good so far 8:45 Yeah I kinda get it 14:45 Well it sounds overcomplicated but I guess they have more experience and background in understanding that abstract stuff 25:14 I pretty much lost the idea why we needed that in the first place but again it still makes some sense. I guess. 27:17 That's it, I give up, these people are insane and I can't let their demons infect me, I have a life to live. Thanks veritasium for showing me how much pop-sci is too much pop-sci.

  • Gummiodler

    I dont get why people always try to point out something in infinity... Those things are not good 4 a human brain

  • Rishabh Singh Thakur
    Rishabh Singh Thakur


  • Metodones

    For Cantor's diagonalization proof, wouldnt take the inverse of the index also work? i.e. for every natural number, their multiplicatie inverse would be between 0 and 1, but you'll always have infinite real numbers between 1/n and 1/(n+1). So, you have more real numbers between 0 and 1 than natural numbers

    • Релёкс84

      That would prove there are at least as many real numbers as there are natural numbers, but on its own it's sufficient to prove there are more.

  • Ed Gutz
    Ed Gutz

    If there's never an answer? Why try searching for it? Have fun and make love lol.

  • Pi Man
    Pi Man

    You've been absolutely KILLING it in the recent run of videos; each one a 'dinger' blasted well past the outfield wall. Keep'em comin'!

  • Raven Bona-Ide
    Raven Bona-Ide

    Soooooo basically it’s a complicated system of faith

  • Yacine ALG
    Yacine ALG

    I didn't understand everything but i sure learned something

  • Leif Maelstrom
    Leif Maelstrom

    "if you abandon a book later it will die" is pretty raw

  • Times Face in Energy
    Times Face in Energy

    Conscious love in wisdom, Powers3s our universe. 😅TESLAS 369 foundations in light. Cause& Mirror COS MOS "IAM" DI^VINE MinD1 ALL ConCeption 🌀1IOVE⚖ Quantum=0=÷🌀1+1=3 ♂️TIMESpacE♀️ +01 =O=-01 ♀️SPACEtim♂️ -01=O=+01 LikeMindIN ONE fatherTIME & MOther NATURE.. COcreating=

  • Bouthayna Nalouti
    Bouthayna Nalouti

    This is by far the best video I have ever seen, thank you so much for making it

  • Wally Cheladyn
    Wally Cheladyn

    I wonder what this means for the simulation theory.

  • Kamila G. B.
    Kamila G. B.

    I just feel a little sad for Russel for put so much effort in a dictionary book of logic sentences and a fewer payed attention. Amazing video!

  • Randy Diaz
    Randy Diaz

    That isnt a proof that math has a fatal flaw but that the analytical framework used to analyze math had a flaw

  • Kevin Mu
    Kevin Mu

    Now I feel so smart

  • billybobobenner

    Maths has a flaw eh?

  • rob strong
    rob strong

    I think an analogy would be: if it rains tomorrow I won't go outside; tomorrow comes and goes and there was no rain. The question would be, to those who wouldn't have knowledge of my movements, did I go outside even though it didn't rain? They would certainly know that I stayed inside had it rained; but could not prove that I went outside while there was no rain, unless I told them. I think that makes sense.

  • ChimneONE

    UZmilk: giving me free interesting and nicely animated maths video. My school: we don’t do that here.

  • Erik Smith
    Erik Smith

    That cantors diagonal proof looks like unabombers journal code … at least it reminded me of that

  • Truman Woodyard
    Truman Woodyard

    As Captain Kirk Said to Norman:" Everything Harry Mudd says is a lie ." Then Harry Mudd Says-"Norman, I am Lyiing." You figure it out.

  • Vivek Yadav
    Vivek Yadav

    Gödel's mathematical system which he used to prove things about all Mathematical systems, is also prone to this paradox. It is entirely possible that the _G_ paradox is exactly the problem which is unprovable in his meta-system, and honestly it seems plausible to me. This would mean we still don't know that ALL the mathematical systems are unprovable or just Gödel's (that statement felt pretty "paradoxical" anyways)

    • Gödel didn't prove that every mathematical system (axiomatic system) is incomplete. The mathematical systems for which he has proven incompleteness need to satisfy some conditions (for example they are not too "simple").

  • Randy Clark
    Randy Clark

    The lucky fired actually shrug because granddaughter critically hope of a stimulating yak. actually, cloistered wind

  • thevoidzzz _
    thevoidzzz _

    As an electrical engineer, this video caused me to have an existential crisis

  • Jimwell Bautista
    Jimwell Bautista

    the poor graphic artists man... imagine conceptualizing this

  • truthjester

    I knew it, math is horseshit.

  • Mauritius Dunfagel
    Mauritius Dunfagel

    Oh God, if you’re a Trumper, lease don’t watch this. You grasp on reality is far too tenuous!

  • Brett Hansen
    Brett Hansen

    I'm about halfway through the video (and loving it), but it strikes me as very odd to describe something math can't do as a "flaw". My vacuum cleaner can't bake me a cake, but calling that a "flaw" would seem to be a misunderstanding of vacuum cleaners.

  • NightUniverse

    Pewdiepie tried to sound as smart as this guy before lolololol

  • Final D
    Final D

    Sry. I got confused on the diagonalization proof. Couldn't the naturals be extended to the left and use the same logic to generate a new natural not contained in the list, by, instead of mapping 1 to .7438... mapping it to the natural ...8347, with the same "infinity" to the left? I am sure I have read something similar to this before but I don't recall why this was not correct.

    • Final D
      Final D

      @Релёкс84 Okay, thx. I think I got it, an infinite sequence to the left can't be an integer because it as to end somehow, but on the real spectrum it can be infinite so the diagonalization works only for the reals, and thus we can engineer a real that can't be mapped from the list but not an integer.

    • Релёкс84

      @Final D You can absolutely apply the diagonal method to a list of integers, but what you'll get is a string of characters ending in an infinite sequence of 1s, so while it is provably not on the list, it's also not an integer, so it's not missing from the list.

    • Final D
      Final D

      @Brauggi the bold but they would only need to end at the digit used for diagonalization, which is ever increasing but also a natural, or is this incorrect? Or, alternetively, if by imagining zeroes to the left couldn't a natural be obtained applying same procedure going from right to left? Sorry, if these questions are too impulsive, I will try to do the proper diligence meanwhile, don't want to plague the comments w/ uninteligent mumblings.

    • Brauggi the bold
      Brauggi the bold

      A natural number does not have infinitely many digits "to the left". They always end.

  • Syed Shahar
    Syed Shahar

    Hey @Veritasium , I'm wondering, how many scientists end up rich ? So many contributed to science that leads to everyday tech we use, but how many make money off it?

  • thomas aquinas
    thomas aquinas

    Every intellectual pursuit has unknowns. Heck, pre-history is just a pastiche of theories, conjectures, and fables. Astronomy is our best guess. Most things like luminosity, distance, etc. are based on stars that we know, as if they're all made in the same factory. So math has many avenues that are unknown, the scope beyond even our computers. That's the province of calculus, and that's why most of us don't even try to master it...

  • charles sanson
    charles sanson

    How could you say is is true if it ain't, a paradox I guess.

  • Nanda Desu
    Nanda Desu

    Poor barberman, he just wanted a peaceful job at the village

  • Argonian Merchant
    Argonian Merchant

    Just get a second barber nerds.

  • Ar Back
    Ar Back

    maybethat is a stupid question, but why can't we just use diagonalization Proof on natural number

    • Релёкс84

      That's not at all a stupid question. In fact, it's an extremely relevant question. Go ahead and try it out, see what happens!

  • Jami StarBrite
    Jami StarBrite

    this is over my head

  • Watchful Wendigo
    Watchful Wendigo

    The proper way to go mad.

  • Joker Smith
    Joker Smith

    It's not a flaw but a feature.

  • Omar Othman
    Omar Othman

    My man been watching crash course philosophy.

  • tom bash
    tom bash

    Just goes to show that too much thinking (especially overthinking) can lead to disaster. You can end up paranoid and suicidal. Not worth proving a point that turns out, so far, to be undecidable. I might conclude from this that even mathematics is not a perfect science and has its own flaws, or maybe the flaw lay in the logic applied to arrive at an answer which is unattainable with current tools. This video did spark my interest and showed that if you're creative enough you can devise your own system to test and explore mathematical probabilities. So much to explore; so little time.

  • The Plasmic Alchemist
    The Plasmic Alchemist

    Does this mean I can use this video as an argument against my math professors as to why I don't wanna use Pearson for their assignments?

  • Abdul Alhaz
    Abdul Alhaz

    Half way through i was going to make a comment about math that drives you insane, only it turned out to be true and now I kind of feel bad.

  • Paul Henze
    Paul Henze

    That last sentence tho...

  • Mohand Arezki ACHERIR
    Mohand Arezki ACHERIR

    At the bottom there's : Faith.

  • OmnissiahZelos

    as a mathematician, I hate you for using subset symbol for if then, it is => not what you use.

    • Релёкс84

      "as a mathematician" is not a phrase you're allowed to use as an early teen with an furry dragon profile pic. Besides, a mathematician would know that this notation is valid.

  • Ned Simlish
    Ned Simlish

    You lost me at "R is the set of all sets that don't contain themselves, which by definition means it must contain itself" huh?!

  • Americus Maximus
    Americus Maximus

    Why I majored in history.

  • Rares Popa
    Rares Popa

    I don't follow your logic when you demonstrate that a math system cannot be decidable. For me, the Turing example is a prove for a system not being able to be complete, rather than not being decidable. If I already know I'm not complete, I am allowed to say I have no answer for a particular statement, (like for love and other things), especially when that statement contains a bold claim about being able to say something about everything, (which I already know It can't know, since I know I'm not complete). For me, if a system has no claim for being complete, it can be both consistent and decidable with no problem. Probably the same goes for a system complete and consistent but not decidable, or a system complete and decidable but not consistent. I don't know if yours categories (complete, consistent, decidable) are entirely disjointed notions, but seems to me you are playing at the border of the notions and you create confusion. It's enough to say if a statement is derived from the axioms to have that system decidable. (I don't care for the statements with different answers like "no", "I don't know", "ambiguous", etc.)

  • Quarail Hale
    Quarail Hale

    In real analysis, you can prove a statement is true by contradiction. For example, we can can prove that there are an infinite amount of numbers between 0 and 1 by assuming there is a finite list of numbers and then disproving that. Why can’t you do that here with the set of twin prime numbers? I think the Godel “g” is just a false statement. You can find a proof for that statement, and you proved that instance. It just so happens that that instance says “there’s no proof”

  • Snuhith R
    Snuhith R

    Does anyone know that this video has a post credit scene?

  • XTheMan

    This is a reupload ?

  • Timbo dong
    Timbo dong

    MATHS !!!!!


    Game of life is bs. See you can build a program which can run on entire space and matter in the universe with all of the energy in universe, so will you call it a great program, NO. See universe is an thermodynamical system, every point in it has we don't how many possibilities of moving forward (we don't know how many = infinity). So unless you find out how many dimensions are in our universe, and how many tangents we can make on a point which is at the end of a function. Then we don't know the actual meaning of infinity. Again this game is B.S.

  • SocioStache

    All this video proves is that philosophers and mathematicians have way too much time on their hands.

  • Connor Kearley
    Connor Kearley


  • Connor Kearley
    Connor Kearley


  • Connor Kearley
    Connor Kearley


  • Connor Kearley
    Connor Kearley


    • Connor Kearley
      Connor Kearley

      us 😥😰

    • Connor Kearley
      Connor Kearley


    • Connor Kearley
      Connor Kearley

      cannot top man see

  • J L
    J L

    Math is just one big minesweeper game.

  • Allen Rubin
    Allen Rubin

    What do mathematicians smoke

  • Nguyễn Tiến Sơn
    Nguyễn Tiến Sơn

    2:07 it looks like a flying machine in Minecraft lol

  • Andosan

    3:58 makes me wonder when does a pair a shoes become a pair? Is it when they are individually made and destined to come togheter as a unit? Is it when they are placed in the same box and waiting to be delivered? Is it when they're bought and seen as the owner as a pair. Say you lose a shoe but you keep other, are the two still a pair or did they just become a shoe? What if I own two sets of identical shoes, do they only become pairs when I pick out a shoe for each foot? What if I mix two shoes from two completely different pairs is that now a "true pair" of shoes and are the leftover shoes a pair or not?

  • Antoine Bilbord
    Antoine Bilbord

    We all are the whole universe in itself through an unpredictable infinity. Our decisions matter. YOU matter

  • Call me Ishmael K
    Call me Ishmael K

    These paradoxes are actually a proof of God.

  • Call me Ishmael K
    Call me Ishmael K

    Amazing what people could accomplish before TEEVEE

  • VanLaw Prime
    VanLaw Prime

    This just goes to show that the smarter you are the more ridiculous the things you argue over. "My infinity is more infinite than your infinity." Blah blah blah

  • Mikey Tha Fsh
    Mikey Tha Fsh

    Remind me to circle back and check if my other comment got scrubbed ;)

  • Christian Lorenzo
    Christian Lorenzo

    4:23 E

  • Asante Amutse
    Asante Amutse

    The beginning of your video literally contradicts its self

  • ✗ThatThieff✗

    Wait so R=Schrodinger cat?

  • Jorge Molina
    Jorge Molina

    Many random crap XD

  • Tyler William Hutton
    Tyler William Hutton

    its called infinite forever

  • IbleedINC

    Well now I feel really stupid…….or DO i?…. lol

  • The Christian Gamer
    The Christian Gamer

    Interesting video

  • The Christian Gamer
    The Christian Gamer

    All I’m gonna say is Jesus loves you all and wants you to be saved and go to Heaven and wants the best for you and just loves you and wants a relationship with you. If you want to know more then please message me. Regardless wish you all the best everyone and have a great blessed day and life! GOD bless

  • Metal Steel
    Metal Steel

    Turing gets flogged by britans for being gay. yet america is the evil one.

  • Seek Truth
    Seek Truth

    Is this anything like the theory known as "Life sucks"?

  • Jose Garcia
    Jose Garcia

    @12:27. Can someone please break diem the mathematical expression for "No human is immortal".

  • Udayan C
    Udayan C

    Now answer a similar question, ehm... 👇 "What happens if a snake eats itself.

  • mikioPL the gamer
    mikioPL the gamer


  • Emily Dunlap
    Emily Dunlap

    Check out Godel, Escher, Bach: The Eternal Golden Braid. It is a goddamn trip

  • Transcendence

    Which means the only purpose of Humans is exploration or eternal exploration too,if that is possible. Seems like the system doesn't want us bored 😘,thank you universe ❤️.

  • Mr. Khan
    Mr. Khan

    Our ancestors f**ed us badly

  • ADAMEC studio
    ADAMEC studio

    Visuals lured me in, but happy for the content instead in the end.🙏🏽🥰

  • Erik Brakkee
    Erik Brakkee

    Don't you like the completeness of mathematics that allows it to prove its own incompleteness?

  • Darth Mage Productionz
    Darth Mage Productionz

    R.i.p. Conway

  • Perry Quinton
    Perry Quinton

    The troubled bicycle conjecturally interfere because mine phytochemically care following a flashy latency. knotty, taboo stream

  • Liu Abram
    Liu Abram

    I have been thinking this kind of question for so long. Somehow I know the answer pointing to God, the different perspective of God Himself. I would say that is why God first introduced Himself, "I am who I am." This answer not only answered Moses' question of His identity, but also His other identities. He is a "I am" God that is an enough answer for us, the creatures. And of course we are not able and shouldn't be able to define Him who is in the center of that existing but undefinable hole. Like Paul wrote in Rome letter: 33 Oh, the depth of the riches of the wisdom and[i] knowledge of God! How unsearchable his judgments, and his paths beyond tracing out! 34 “Who has known the mind of the Lord? Or who has been his counselor?”[j] 35 “Who has ever given to God, that God should repay them?”[k] 36 For from him and through him and for him are all things. To him be the glory forever! Amen. God is the answer if you were trying to find one, however, He beyonds Q&A and is the fullness who fills everything in every way. Here is another issue or question for you: What is the "Truth" or what does it mean? God bless you.

  • Kevin Lewis
    Kevin Lewis

    OK I don't like this but you see 2 people walking towards you. You think they are 2 people, but every person is made of all his past experiences and the people that have affected that person in some way, or the traits passed down from family so each of those people is actually an infinite number of people and souls. This also leaves the door open for evil or good to have affected these people. They are no longer 2 people. Though that is possibly all you see. So gravity is actually antigravity and light as a feather stiff as a board is a frame of mind and can be done by any 2 or 3 people on any person no matter the person's size

  • I'm just a guy
    I'm just a guy

    Damn, ig if theres an after life in something this is what i can do for the rest of time

  • James Barralet
    James Barralet

    If there are truths that cannot be proven, then why is all scientific research founded on establishing proof? Yogis are more comfortable with knowing things that cannot be proven. That knowledge comes from an open mind and experience. Not books.