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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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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.

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

_"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.

Can you not? Math is hard already stop giving them ideas on making it harder I'm still studying just do it after I graduate ty

I don't have to know any of this. I'm richer than yall could ever be. But I only know arithmetic. Chile imma count my money now. Also I don't actually have physical money anymore

There is no preferable side of the equal sign.

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

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

At 17:17, where do the prime numbers come from? 2, 3 and 5. Is it just because those are the first three prime numbers, and the equation holds three elements, or is it something else?

It’s boring

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

Trying to use math to prove math is like trying to use light to prove light. As you would say, this is a self reference. The light needs the dark to be proven. Math is part ‘logos,’ the logical foundation of all that exists. It’s part of the source of all things.

Godel, Escher, Bach

This guy is a total moron and he has no idea why. Everything he says is crap. The universe does not work the way he says it does.

The Barber Paradox is not a paradox. Just because something is illegal doesn't mean it's impossible to do. Just because it's illegal for the Barber to shave someone who shaves themselves doesn't change the fact that he is probably going to shave himself It's just an imperfect law. And just because each set supposedly cannot contain itself doesn't change the fact that R (itself) was made up. It exists as an idea. And ideas are infinite

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

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

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

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

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

Godel grief

What a good ole number

Math is just a Riddle.

My life is a lie

The set that contains everything can't contain nothing

Wow

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

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

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

Sounds like you need a second barber lol.

Yes, but the machine h is flawed in this situation as there is no paradox output. Even if we add the rules to h+: > if h outputs paradox then halt, > if h outputs halt then go into a loop, > if h outputs loop then create a paradox, and pass h+ into h+ as both instructions and input, h+ will just halt. If it halts, h is correct, because, to get to the point of halting, h has to detect there is a paradox. The fact that h+ has stopped does not mean that h is wrong as a paradox occurred before halting. Reasons for this: > h would be wrong to output halt as a paradox occurred before halting. > h+ will not go into an infinite loop as I have already pointed out that h would be wrong to output halt. > h+ will not create a *second* paradox because, as stated above, there is no chance of h+ entering a loop. If you then pass that h+ into h, then the output will still be paradox. Now, there is complete decidability. Btw, I'm not entirely sure if this is correct and I would be highly interested in arguments against it: "If you think that something is true, you should try as hard as you can to disprove it." - Derek Muller.

3:20 me: you forgot about the youtube algorithm, why was this recommended to me and why am i watching it

Stopped watching after narrator started bs claims touring was the brains behind cracking enigma when in reality he just improved the already working system designed by rejewski, rozycki and zygalski

30:05 goosebumps 🤯

Since the "rules" are made up, you can't use these examples as proof of anything other than math needs rules to work (i.e. its all made up and not a naturally occurring truth).

There is something inherently limiting in the universe by itself and even more in our brains. We are not infinite, our body and brains have limits and that is probably our Turing limit. The universe limits and wether it is infinite or not (and if yes, can someone tell me how it is growing ?) Is the universe limit. I think the universe is finite but infinitely divisible like the distance between 0 and 1. But I am bad at doing maths so I can't prove it xD

Bravo

the cards literally went above my head.

ma head HELP

The set of all sets is empty. This axiom resolves all contradictions of the set theory . Because the empty set containes themself. :-)

@Дмитрий Косолобов >> This leads to a much much weaker theory that cannot express all mathematic What do you mean "all mathematics"? As far as I know there is impossible to build "all mathematics". Gödel is the witness. :-) I look at mathematical theory from practical point of view. My approach makes fractals very simple objects and allows to solve differential equation in integers. My next goal is clear theoretical turbulence theory without any empirical assumption. It is impossible with traditional approach. :-)

@Андрей Иванов Apparently, you try to define a set theory in which only a predicate "A is a subset of B" is defined (in the traditional set theory there is a predicate "A contains B"). This leads to a much much weaker theory that cannot express all mathematic (the traditional set theory can do this). So, such theory is of no use in the discussed context (the interest is to find a theory that is consistent and can express all or at least most of mathematics). Or maybe I misunderstood something in what you said.

@Дмитрий Косолобов >> empty set is denoted by {} Element can be denoted as {a}. What the difference between {} and {a}? You try to explain me traditional approach. Change definition and system of axioms and you eliminate all contradiction in set theory. :-)

@Андрей Иванов You didn't understand, it seems. What you say is wrong: the empty set is not an element of every set; what is true is that the empty set is a _subset_ of every set. This is what confuses you. If the empty set is denoted by {} and a set of elemens a,b,c is denoted by {a,b,c}, then {{}} denotes a set that contains the empty set, but the empty set {} does not contain {}.

@Дмитрий Косолобов According to definition empty set is an element of every set without exception. So empty net contains itself by definition. I don't see reason to argue about definition. :-)

I dont understand why you allow self reference as an axiom, in the first place. Why the barber is not a woman? Or a special citizen that has other laws.

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