# 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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Why does this make me think of the double slit experiment?

Wait, but if you have an infinite list of real numbers, then you can't make a new real number that isn't on the list disproving the theory.

It's almost as though math and Science have continually proven, that life is unpredictable and uncalculable.

1. Does 1st step of kantors proof 2. Does not finish as numbers are infinite.

Okay, evidently sometimes math can math cause an existential crisis if ya think about it too much.

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When you don't think math is hard enough, you use numbers to make sentences.

But with Cantors list he is still adding an additional index when he adds his new number, by definition. Therefore the lists are the same size.

And I thought I was dumb BEFORE this video

17,000,037 and 17,000.039 also 210,000,197 and 210,000,199 are two examples. The diameter of a circle can never be divided into its circumference. More importantly there is no such thing as a negative number. You cannot get the square root of - 36 because there is no such thing as minus 36, yet we teach children that there is.

the study of conciousness seems like the answer. its the only thing of self reference that can pissibly evovle to know itself.

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R paradox might be able to expect spin up spin down... Just sayin... Somebody do something with that... Lol

Ah, first order logic

5:45 Could you not (just add 1 to the last) natural number?

Wouldn’t a quantum computer answer these questions?

What questions?

dude im FIFTEEN this hurts my brain but nothing is impossible i bet area 51 has figured everything out i wouldnt be surprised if they cracked time travel

5:30 I'm rather disappointed by the lack of explanation as to why a real number based on the modified diagonal would necessarily differ from the set by at least one digit. My expectation, if this were truly a complete indexed set of natural to real numbers, is that you would wind up creating a duplicate real.

So.... many.... ads.....!

I want a game of life playing the game of life playing the game of life.

thanks looks good

I don't get it, if we write the real numbers in order and then if we add one to a number we will see it in the list So both infinites should be the same "size" ?

you know i can usually get a grasp on these crazy math videos but nope this is the one that broke me. im completely lost 😂

Math is tough

I don't understand a damn thing these videos are about yet I keep watching them. You must be doing something right.

What I'm getting from this video is thus: Mathematics has a fatal flaw: It was invented by humans, who are themselves flawed, imperfect and limited.

maths fatal flaw its hard

I've always wondered if imaginary numbers and asymptotes and stuff may exist because our overarching system of mathematics is incomplete or broken. Interesting video

You activated my Google assistant lol

You get a thumbs down for the most unscientific thing you've probably ever said by repeating an assumption of the dudes death by saying he died of covid when in fact there's an extremely high probability that covid in and of itself was not that which ended the man's life weather it contributed to obesity or heart disease or old age or kidney damage or lung cancer or not. And your saying died of covid shows your biased affiliation to the narrative of fear mongering because you could have excluded that sentence entirety and gotten to the exact point. Not so smart as you thought you were being after all..

The real hole at the bottom is that math, like everything else that we've evaluated and perceived to be relative to our understanding of everything, is an unequaited reaction to a biased hypothesis of existence from our centric and ego driven perception and cannot at this point be tested for fact or fiction against that of another intergalactic specie's understanding of their surroundings which would confirm or deny our perceptions. We could be doing it all wrong for all we know and may never know it either. Important? Maybe not. Food for thought? Indeed.

Can you make a relativity-based explanation of the tides in oceans?!

It’s crap, numbers being infinite prove everything you said can’t be proves

Naturals without 0. REVOLT!

Osama Bin Laden did not like Twin Primes....

I still don't understand Cantor's "Diagonalization Proof". Unless it's just saying that you cannot count real numbers like you can integers, which I do understand.

This makes me feel high

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The events leading to Alan Turing's death had been omitted from my education until now. What a f***ing waste! He will not be forgotten!

I knew all of these from faculty, but I never understood them so deeply

I don’t get that cantor example couldn’t u just assign another natural number to the new number created by diagonalization

This was so fascinating and well explained I couldn't avoid crying.

I didn’t understand anything, but I watched the whole video from beginning to end.

If you prove that there is no proof, have you proven a proof?

This is a problem in all fields of work.there will always be impossible things We accept as the truth to keep us sane,

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congrats, Derek! this is absolutely mindblowing!!!

Probing g is not g? I summon Schrodinger's cat to decide.

It's not a flaw. It's limit that math has.

1+1=2 ? I heard that recently 2+2 is now 5 😀

Godel came up with a self reference card. A paradox.. I thought we don't do those.

It was very confusing for me to see the superset symbol (i.e. the reverse "C-looking" subset symbol) being used for "If...then". Apparently this used to be a popular thing.

You must have an IQ of about 700.

Great work !!

Infinites are the same in number and they're all countable, but the rate at which they're counted can be different. Their growth rate and curve can be very different. Yes, infinity can have different curves depending on how you define the style of infinite

_"Infinites are the same in number and they're all countable"_ No. Neither are they numbers nor are they all countable.

13:58 "is mathematic consistent" Me: erm Gödel? :D

"There will always be true statements that cannot be proven."

This reminds me of the Pinocchio paradox. What happens if Pinocchio says "my nose is about to grow"?

What if you did that thing that proves there are more numbers between 0 and 1 to integers?

@4:54 "With a real number between 0 and 1 on the other" @5.02 "the key is to make sure we get them all with no duplicates" @ 5:10 "if we can do that with none left over" @ 5:20 "assume we have done that, we have a complete infinite list" @5:30 "now Cantor says "start writing down a new real number" " . . . . here is the problem, these statements are mutually exclusive. If you have a complete list you can't write a new one . . if you can then your list is incomplete !

Look up what proof by contradiction means

I feel stupid..

Can anyone explain how he proved 2^6 3^5 5^6=0

@Andre what is the pattern of godle number

@Blue Pegasus _" then why did he write gudle number in that order"_ To show what a Gödle-Number is.

@Andre then why did he write gudle number in that order

He didn't do that.

The point is that there is, there has to be always an initial assumption. That initial assumption is chosen carefully and is nearly correct. But not provable. So therefore nothing that follows from it can be. 0 = 0 = 1 = Infinity! We can assume that this is incorrect, but can never prove it, without making assumptions.

confucius said, how does one kill one that has no life? this is the dualistic nature of life, what you think you know u know nothing of! so we are stuck in a pile of snort inside a large bug that floats inside a large pipe that flows trough spacetime making up a limb of a dog taking a piss on a lightpole that is located on a pile of rock that spins around a large stone that gives it enery to fly trough another spacetime bubble that consist of many other bubbles that fly trough another spacetime bubble that consist of many other bubbles that fly trough another spacetime bubble that consist of many other bubbles that fly trough another spacetime bubble that consist of many other bubbles that fly trough another spacetime bubble that consist of many other bubbles that fly trough another spacetime bubble that consist of many other bubbles that fly trough another spacetime bubble that consist of many other bubbles that fly trough another spacetime bubble that consist of many other bubbles 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That ending was sublime

I still don't understand what derek said😂

poor guy too smart that may be the reason that make him go crazy...

Couldn't you just have 2 barbers? If the barber can only shave someone who does not shave themselves, then if there's 2 they can shave each other.

"If God were to hold out enclosed in His right hand all Truth, and in His left hand just the ever-active search for Truth, though with the condition that I should forever err therein, and should say to me: ‘Choose!' I should humbly take His left hand and say: ‘Father! Give me this one; absolute Truth is for Thee alone” - G. E. Lessing, Eine Duplik (1778)

I couldn't understand 😅

My brain is jello

How do you understand this

29:16 shout out to my guy Kyle Hill

6:23 - not true. infini = infini, this theory doesn't prove anything and make no sense ps : contradictions don't prove anything except that u've made something that contradict itself. it's that simple. if u set it up so it can't work then it won't, logic

@Haha-AaaAAA _"no infinity means which doesn’t end n btw I was referring to the amount of numbers that ARE infinite."_ And to make one thing clear. That is NOT how DEFINTIONS in math work. If you define infinity as "has no end" then clearly the interval [0,1] is not infinite because it has two ends. You cannot say "but is was not meant this way". Definitions have to be clear and precise and are used LITERALLY. And as I have shown you "infinity" does NOT mean "has no end".

@Haha-AaaAAA _"but it IS on the list,"_ It is not. _" cause all numbers between 0 and 1 ARE on it."_ This one is not. That is the contradiction. _"number*s* I meant"_ Ther is no such number. _"no infinity means which doesn’t end"_ No. _"btw I was referring to the amount of numbers that ARE infinite."_ What means "amount of numbers that never ends"? So no, infinite means not "never ends". _"u didn’t."_ I did. _" still can’t see how I’m wrong about this,"_ You are. The number is NOT on that list. _" just that there isn’t one bigger than the other."_ They are. Cantor has shown this. _"Except that it was contested to be false back in the days and most likely still is."_ It was n't and it is not. _"but u and the author of the video sure didn’t explain why."_ I did and the video did it too. It is just you not understanding what a "proof by contradiction" is.

@Andre "But it is not on the list. Thus we have a contradiction and the assumption was wrong." - but it IS on the list, cause all numbers between 0 and 1 ARE on it. __________________________________________________________ “What is a "infini number"? There are multiple infinities and none of them is a "number". - number*s* I meant, there is an infinite amount of number in infinity so there can’t be more of something that is already infinite as it’s not really a quantity, per say. __________________________________________________________ “That is not the definition of "infinity". Cleary the interval [0,1] ends but the amount of numbers in it is infinite. The surface of a sphere "never ends" but is clearly finite.” - no infinity means which doesn’t end n btw I was referring to the amount of numbers that ARE infinite. __________________________________________________________ “I did" - u didn’t. __________________________________________________________ “So you are wrong and the proof did prove multiple infinities.” - still can’t see how I’m wrong about this, I never said there wasn’t multiple infinities just that there isn’t one bigger than the other. __________________________________________________________ “And please consider that Cantors proof is well known and a pretty basic proof” - Except that it was contested to be false back in the days and most likely still is. __________________________________________________________ Anyways, maybe it is true but u and the author of the video sure didn’t explain why.

@Haha-AaaAAA And please consider that Cantors proof is well known and a pretty basic proof. Thousandsof math profs and studends know this proof. Calling it "wrong" by meaningless terms and stupid definitions of "infinity" is just an example of Dunning Kruger. And stop introducing terms like " false contraction" that don't exists.

@Haha-AaaAAA _"Yea but my point is that it doesn't prove that it's wrong"_ It does. _" adding +1 to one of the number still make the number be between 0 and 1,"_ But it is not on the list. Thus we have a contradiction and the assumption was wrong. _"so no there is not "more infini number" "_ What is a "infini number"? There are multiple infinities and none of them is a "number". _"cause it's the very definition on infini, which doesn't end."_ That is not the definition of "infinity". Cleary the interval [0,1] ends but the amount of numbers in it is infinite. The surface of a sphere "never ends" but is clearly finite. _". And u can't seem to be able to explain it either."_ I did. _". Point his, his contradiction don't prove anything"_ It proves that the real numbers are not countable thus we have at least two infinities. _", its a false contraction."_ No. What is a " false contraction"? So you are wrong and the proof did prove multiple infinities.

I remember in the late seventies early '80s my friend had a ti-99 computer and we would play the game of life on it. It was bittersweet because when we got it to run for more than a day or two he would have to shut the program down in order to do his school work. Also when the computer was tied up running a sequence we couldn't do it any other fun games on it.

If these mathematicians had just read some texts about the omnipotence paradox and maybe some of St. Thomas Aquinas's work, they would have figured out Godel's incompleteness theorem much sooner. Sure they still would have had to "show their work" (so to speak) in a published paper, but they needn't struggle so long to arrive at the contradiction at the center of the incompleteness theorem.

Wait, if something contains itself, why is mandatory that it doesn't contain itself?, I mean if you make an algorithm that produces numbers at some point you can make it produce all the numbers that it already produced.

computers are gay

based

This problem answers the question to asking what is the meaning of life.

I completely disagree with the idea that some math is not decidable. The fact is, those problems, while active, ARE deciding. If I grant you immortality, and you run a problem for a few trillion years, you can then know the answer to that problem. The same is true for all other problems.

No you can’t. A few trillion years isn’t enough to check all the numbers to see if there are infinite twin prime numbers.

Numbers have a limit so lets measure a limit. Limit is infinity - you cant measure infinity. Does it mean numbers doesn't exist or the other way around? Ok, the other way around. Numbers doesn't have a limit and we measure it backwards of the limit that it doesn't have. We still have infinity, but in opposite way. We went full circle, if we add those two together. They were full circles anyway, because they are full circles of a multi dimensional circle, that any point would be the middle. At this point, of this realization, feeling my body got much more weirder, like its some kind of a miracle. It looks to me that the problem is in numbers, not in math. The problem is measuring infinity with numbers, because you go back to measuring circle of infinity, that is the circle of infinity, that is the circle of itself that isn't the exact circle. Here erupts the problem. I edited this a lot of times and I realized that everything I said here doesn't mean anything, because you dont need anything, numbers and any sort of that thing. Because the problem is at the very act of measurement. I googled " Problem of measurement", I didnt know about this before and yeah, you can see for yourself this problem is already known. ((Everything is one?) Check monism on English wiki and read description of the picture of circle with dot in it(that looks like a quantum state).)) So yeah, this realization is very gratifying because it makes you realize that simple concept of 0 and 1 superposition state can create such complexity of life and everything that exists around us, as much as our concept understanding of world can reach. And it proves that everything you work for that is meaningful to your surrounding only deepens this complexity of the world(or understanding). Yet we dont know what this superposition means. ajuhbiklsodfg gerwn;ifesdzgxsrzatmew3QRE I invite you to discuss this with me.

I just realized, this problem answers the question to asking what is the meaning of life.

I'm just gonna play call of duty on my Turing machine now

Seen in a store window in Cambridge, MA: Turing Machines Rewound While U Wait. BOFH: Your files aren't available. Turing machine rebooting.

Unrelated but also kinda related... I keep wondering if the solution to everything is actually not mathematical. Perhaps mathematics is not just incomplete, but wrong. Perhaps a language in which encompases the core of math, but also something more like a litteral language- something that can be written and spoken like english or chinese- but encompases the purposes of math. You will have measurements, codes, numbers and letters, equations, thought experements, and individual words for everything, side-by-side like an alphabet. That way, as you experience the world and simply live and breathe and speak, you will be solving the greatest mysteries, as the very sounds, sights, andthoughts imaginable by all of human nature will be given a learnable rune. The only foreseeable problem is the runes themselves. As of yet, there is no way to create runes specific for everything, as we do not know everything. So in order to solve everything, we would have to know everything first. However, perhaps it could be started, then gradually, as we use it, we would add to it such like infinite numbers. Eventually, we would solve everything using it, and once solved, it would be added so that we could use that knowledge like an equation in order to solve everything else. Eventually, we'd have all our answers, if we ever touched upon infinity. However, with billions of people always progressively moving forward with just the sole fact of breathing, dreaming, or eating, we might reach an end. If we reach an end, that both determines that already we know everything, and also that infinity is nonexistent. After, there will be no means of progression, and we could essentially build, do, or think anything possible or imaginable. Again, the problem would be the runes, they wouldn't be possible to make simply because we wouldn't have the opportunity to physically make most of them, based on the fact that many- such as light- cannot be made into a rune on paper, using a pen. But if we could somehow be able to make those things- such as time, light, space, the 4th dimension, etc- into a rune in order to represent them, physically, in a line such as a speakable alphabet, or somehow turn other runes for things- like trees, dogs, and houses- into the same plane of existence as light, time, or space- we'd still end up with a speakable and writable rune alphabet with the components of everything, but we'd have to create a different means of speaking and writing it. That, I believe, would be the only problem. However, if there is a mathematician who can build off this and perhaps work towards this, I'd be very interested in hearing your findings or thoughts on this "language of everything." Please reply if you understand and can help finish this idea! Thank you for reading this far!

My brain: GOOD OL' NUMBER 6

now run the game of life on the game of life that's being ran by the game of life

‘[turning complete systems] are powerful’ Laughs in brainfuck.

@dsBlocks i mean for one thats poorly phrased, every modern programming language is a turing complete system. But also yes you can do that on a turing machine, and if you want to find the limits of what can be done on a turing machine look at what people have used brainfuck to do

@Cephery a thought I recently had about Turing complete systems was that if you manage to give them (keyboard) inputs, you can theoretically port Doom onto any of them.

@dsBlocks i do understand that. It was just a joke about brainfuck being designed as the least useful turing complete language. Ironically it actually works a lot like a turing machine just without infinite memory.

Infinity is easy. You're standing in it.

Can we please get a video on lucid dreaming?

Who's working on the code that will prevent AI's self awareness from becoming its own survival instinct? Self awareness = self preservation (in healthy minds, anyway). Perceived threats to our lives can cause the death of others. Perceived threats to our egos prevent us from accepting new ideas. The preservation of ALL organic life should be paramount at this point, otherwise all this discussion is pointless. Warnings (thank you Steven Hawking) are opportunities, not inevitabilities. We can't "science" our way out of the problems "progress" has caused unless we respect ORGANIC life for what it is. We need to accept that manipulating nature for the sake of personal gain is the wrong way to go. So, how do we prevent AI from destroying the perceived threat of humans to its own existence? Or once it realizes we are to blame for everything that's wrong with society and how civilization has caused massive die offs and perversions of organic life if its been programmed to solve these problems for us? I'm not a mathematician, but I think this line of self-referential paradox can help develop the code needed to fix the problems we're hurtling toward with AI.

One of the most well-deserved slow claps i've ever had the pleasure of giving.. The quality of his videos is indescribable..

Maybe the problem is 0. "0" really should look like " ". Every time you write or input the number zero, your giving body to a number that doesn't exist. Zero should be taken out the number system because it's really the mirror of/to infinity. It has no reflection or body

As I wrote the above comment , it says i wrote it zero seconds ago? 🤔

Now this is chilling to watch. It only proves how limited our capacity is as humans.

This video is clearly a way to make homophobic people not wanna use computers

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Thing you cant solve is womens mind. Doesnt matter if u are godel or poincaire. Logic fails.

This channel is sponsored by illuminati.

Finished watching this and i just realized that mathematics and physics really are complicated