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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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 _"some* contradictions don't prove anything."_ *ALL* contradictions prove the assumption wrong - as it is the case here _"How can a number between 0 and 1 not be in the interval that include all number between 0 and 1 ?"_ That was the assumption. The contradiction proves this wrong.

@Andre *some** contradictions don't prove anything. Like the ones set up to contradict themself. for the 6:23, How can a number between 0 and 1 not be in the interval that include all number between 0 and 1 ?

_"__6:23__ - not true. infini = infini, "_ No. _"this theory doesn't prove anything and make no sense "_ It is no "theroy" but a proof. And as a proof it proves that there are multiple infinities. _"contradictions don't prove anything except that u've made something that contradict itself. "_ They do. They are called "proof by contradiction". _"if u set it up so it can't work then it won't, logic"_ This has nothing to do with "logic".

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

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.

And why I am a mystic who loves logic. Subscribed.

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.

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

By design, science only disproves, consequently autoreference is only scientifically useful when it disproves. Illusory truth is a contradiction in terms. Finite minds cannot process infinity in any true form, insanity is the demonstrated result of relentless attempts in spite. Formalist is a euphemism. 0 doesn't exist. An infinite set is a contradiction in terms. g is irrelevant. No one can stand on the shoulders of giants if midgets are permitted to anklebite them down, let alone encouraged. Believing a lie defiles your mind, and the more obvious the lie the greater the defilement.

yo does anyone remember that tile game from a dong like 10 years ago

Can we just appreciate the animation quality and hard work he put?!!

i want to watch the conways game of life in conways game of life

Moobs

Brain just melted 😂 smart guy!👍

About Cantor's Diagonalization Proof (I am not a mathematician, I have read some Wikipedia articles, but couldn't understand them fully, so what I say here might be bollocks): If we create another real number by adding one to next digit in all the real numbers, then I see two problems: 1. Here is the contradiction without using natural numbers at all: earlier we assumed the list was complete, and contained every real number between 0 and 1, so if we can create another one, the assumption was wrong. 2. We can do the same operation on the natural numbers on the left: add one to next digit in all following numbers (padding all with zeros on the left), and look at this: we have got another natural number, different from all previous ones. This will be our newly created index for the created real number. Isn't this a proof the number set sizes are equal? Of course this suffers from the same contradiction as in #1, but I wonder, did no one think about this simple thing before?

@Kazimierz Król well, the fact that naturals have finitely many digits and decimals have infinitely many digits implies that one set is larger than another, but not trivially. it needs to be proved, which is exactly what the diagonal argument does. for example the set of all rational numbers has the same size as the set of natural numbers, and yet most rational numbers have infinitely many digits when represented by decimals.

Thanks. So can't the conception that there is more real numbers between 0 and 1 than natural numbers be derived just from the fact that the number of digits must be finite for naturals, while can be infinite for reals? The diagonal proof does not seem necessary at all.

1. it's a proof by contradiction. assume the reals can be listed => derive contradiction => the assumption that the reals can be listed was wrong (which we wanted to prove in the first place) 2. the diagonal argument fails for the natural numbers because it produces an infinite string of digits which is not a natural number. natural numbers have only finitely many digits.

I bet Hilbert was looking over at Gödel thinking like... “You really going to build your entire career around saying I’m wrong? That’s rude, what’d I ever do to you?”

And now I’m wondering if Hilbert DID do something to Gödel to spark that amount of determination 😂

The thing about mathematics is that numbers and the values they represent always get bigger & bigger much like ego... until they don't... almost as if self aware, they change their values.

i feel like the barber reference isnt that accurate, if he cant shave him self and the barber cant shave him. he would be exiled or go to jail? not welcome in that"set". No?

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My brain hurts

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

Glad there is a mathematical answer to why I can't figure out the truth about Covid!

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

only a mathematical system that has no axioms, can ever be truly complete, consistent and decidable. Axioms are the evil that corrupts everything ;-)

Now my head hurts.

give it a few decades-current math is wrong-it will change but still just be humans way of measurement.

And here I read this as Meth Has A Fatal Flaw.

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.

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.

@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.

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.

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.

I want to launch my own branch of math! How do you do that

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.

9:25 need explanation

Having 2 apples is impossible. Identical numbers are not possible. What makes numbers and apple differ is the environment around them :9

1:25 i forgor :skull:

none of this can be proven to be true.. concurrent quantum states prevent this.

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

just learned set theory for my computer science degree really interesting stuff

@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.

Russell's paradox is a violation of the law of excluded middle. Ergo the law of excluded middle is wrong. Ergo superposition.

The voice is good though I'd never understand the contents...

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._

"This is the game of life, running on the game of life." Then proceeds to slow zoom out. Wow my mind literally was blown.

I got chills

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

Mind Blown

The game of life animations at the start are awesome - does anyone know how they were made?

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25:57 and it vanishes in a puff of logic

somehow i heard "godel" as "good old" for almost the entire video

The language of God...the Creator/Designer, and mind of ALL information. The language of God...Mathematics. Amazing...isn't it? ☀️

_"The language of God...the Creator/Designer,"_ There is no god and no creator.

More like word salad

Yeah, but it is neither a flaw, nor is it fatal.

@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.

@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.

@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?

@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.

Not being able to prove consistency is something I would call a "flaw".

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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" 😆

Excellent Video ! Feels like one of your best

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.

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.

This video and his “how a infinite hotel ran out of rooms” video match up I just thought about it

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

@Peter Shmain _"given that the list is static, then you should not be able to create a new number from the decimal side either, "_ No. Cantor has shown that you can do this. That is the whole idea behind it. _"thus defeating the point if being static,"_ No. _"because then logic stands to reason "_ What "logic"? _"So if you can take a decimal and add 1 to each digit, why can't you take the index number and add 1 to it?"_ All indexes are taken. The list is static.

@Peter Shmain the list is not an equation, you're not "doing" anything to it. it's an assignment of a decimal to every natural number (which can be pictured as an infinite list indexed by the natural numbers). for any natural number n there's a given decimal d_n. one such assignment would be d_n = 1/n, which in the "infinite list" representation looks like d_1 = 1/1 = 1.0000... d_2 = 1/2 = 0.5000... d_3 = 1/3 = 0.3333... ... this is just an example. clearly a lot of decimals are missed, for example 2/3 never appears. the diagonal argument proves that we can find a missing number for any such list, therefore it's impossible to come up with a formula for d_n which would account for all decimals.

@Tom Svoboda given that the list is static, then you should not be able to create a new number from the decimal side either, thus defeating the point if being static, because then logic stands to reason that what you do to one side of an equation, you do to the other side, correct? So if you can take a decimal and add 1 to each digit, why can't you take the index number and add 1 to it? Maybe im missing something here?

there is no new index. the list is not being generated, it's static and completed from the moment you consider it. all the indices (the natural numbers) are already taken.

_"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.

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

Best video I've watched in a looooooooong time.

So is the fact that not all true things can be proven also unprovable?

It is the opposite.

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.

MEGALIKE 👍

Thank you for this waste of time.

More like insufficient computational ability to prove true, but can never be prove false.

Absolutely nothing to do with "computational ability"

What does it say about me that i read the title as "Meth has a fatal flaw" ??

That's the reason I love watching this channel. It forces me to "THINK"

I don't know if there is truth to be found studying mathematics, but there is much beauty to behold.

Godel hurts my damn head. How would you come up with that.

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.

1 + 2 = 4

Lobachevsky and Bolyai, Gauss is at most the third.

Perhaps maths really is the language of reality in so much as they are both paradoxical

I knew it.. Remember "Computer Code Discovered In Superstring Equations" ?

We don't know what we don't know.