## 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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## Kommentarer

The imaginary tub medicinally identify because apology pertinently undress over a foolish creature. squalid, tacky nest

What's wrong with mathematicians??

The problem I have with uncountable infinities as a set is... the logic is entirely flawed and incorrectly applies a practice onto the set w/o any verification what so ever on duplications. The problem comes in if you suddenly forget how many numbers are only 1 digit off. If I had ... as an example 0.100.... & 0.200.... in an infinity set. each with many numbers between them... if I added 1 to the tenths digit, as a new number I would suddenly have a duplicate. --- the blind acceptance that when in regards to infinity ... we forget about the fact every combination would exist, and we would suddenly forget about morphing one number into another. Honestly I think this 'larger infinity' is a childs fuckup. Though an unprovable theories is the problem when dealing with 'infinity' it self, or something close enough to it that, that frankly reliable enough is the only efficent & effective way one can work, and without means, ability, or time to do infinite things.. it can't be proven.. that at some point, 'true enough' is all you are going to get.

## Andre

## 17 minuter sedan

_" the logic is entirely flawed"_ It is not. _"applies a practice onto the set w/o any verification what so ever on duplications."_ There are no "duplications" _" if I added 1 to the tenths digit, as a new number I would suddenly have a duplicate."_ Nobody does this, bnecause ALL digits are changed. _"Honestly I think this 'larger infinity' is a childs fuckup. "_ Because your understanding of the proof is flawed.

This may be a silly question and may have been answered in the detailed description of Cantor's works, but when we assign a real number to each natural number in a grid, is the grid populated on the assumption that the list is complete? You mention at 5:01 that "the key is to get them all, with no duplicates". If that is the assumption, then wouldn't the list that was just created actually be finite? In other words, if the grid of natural numbers and their corresponding real numbers was a truly infinite list then the new real number obtained by adding one to each successive decimal place across the diagonal can just be assigned to the next available natural number, can it not?

This may be a silly question and may have been answered in the detailed description of Cantor's works, but when we assign a real number to each natural number in a grid, is the grid populated on the assumption that the list is complete? You mention at 5:08 that "the key is to get them all, without duplicates". If that is the assumption, then wouldn't the list that was just created actually be finite? In other words, if the grid of natural numbers and their corresponding real numbers was a truly infinite list then the new real number obtained by adding one to each successive decimal place across the diagonal can just be assigned to the next available natural number, can it not?

The self is unknowable, attempting to know self is the illusion of duality. All phenomena and non-phenomena are the same wholeness and the same lack of wholeness. Existence isn't inherent.

"There will always be true statements that are impossible to prove... That, Is life!"

So the intuitionists were right. shut up and solve real math. What we got out of that debate was someone disproving the formalists.

The mind can only know so much for now

Lesson 1: invent math Lesson 2: be confused it doesn't make sense.

Here is a mind blowing revelation. Chuck Norris was able to count up to infinity. He did it twice!

with the bit about cantor, wouldn't that "new" real number between 0 and 1 just have a natural (index) number 1 greater than that of the "last" one?

If math was taught to me like this, maybe I wouldn't be a human smart enough to pass the MENSA test, with an above average IQ, who is unable to help my daughter pass freshman algebra without re-teaching every lesson to myself beforehand and almost crying several times from the effort.

Thats how Truth works. It never changes since it is characterised by consistency, so it has an attribute of infinity: and the infinite cannot be traversed. Simply put, you will never conclude that anything isnt absolutely consistent: you will only be able to say you have nothing to prove otherwise thus far.

I love this video, it aligns with new views and algorythms I been playing with, old questions pondered during childhood, I come back to it and get different incites.

Wait a minute... 3 or more and it dies, 2 or more and it comes to life, 1 or less and it dies... I feel like this problem somehow can be tied to 3x + 1 after just watching that video first. I don't know how yet. This video just started.

“Math has a fatal flaw” Tell me something I didn’t know

Listening to this as if I understand every word but in reality not even one.

29:40 But will the game of life running itself end?

## The_Circuit

## 6 timmar sedan

That said maybe I grossly misunderstood the game of life and the video

## The_Circuit

## 6 timmar sedan

Maybe it will even create itself again and again Or maybe it will create itself creating itself, and it ends in one of the creations but the ones the came before continue, or somehow the ones after

Infinity is a useful fiction. Like many things in math.

Why is smoke coming out of my brain?

NP = P!

all of these questions will stop once you realize all of these things are made up. 1 is not a thing without the thing the number is meant to count, we are putting way too much power into these numbers and use them to theorize about the creation of the earth, when all we need to know is how many people we have and how many things we need to keep those people alive and happy.

The fatal flaw with maths is , if infinity + n = infinity, , infinity - n = infinity , so 2πr for example (or any Symbol equation ) never gives correct answer because the error in π is infinitly small, which will always be equal to infinity . So all your algrebra , all of it has an error of infinity . Because maths is a made up subject , but this is very useful , if you can understand it , extremely so . I wont tell you how , I dont expect you to ask , but the Taoists revered idiots like me , qe are the super intelligent , we know we are idiots , you dont . Infinity is the only number or answer you need , which mean you have to rhink fir yourself , impossible for a mathmetician . Im not a maffmotician , not thick enough .

So talking about 3n+1: How 'bout this is one of those "its true, but i cannot be prooven" things? Or does this needs to be proven? Which would be a proof of a proof. Like in .... "SELECT * FROM [....] WHERE (SELECT [...]" etc...?

The sooner you guys accept that everything has been intelligently designed by the Creator, and so intricately beyond our comprehension, the sooner you'll stop driving yourselves crazy.

Ahh, the good ol' numbers...

"make mathematicians mad" sounds a bit like "make 0 a even number". Is impossible, because one cannot just change the inner nature of a number. But looking at that from the point of view of someone who likes fiction a lot, I can imagine the change that would mean for humanity if someone tomorrow puts together another artificial language which has noting to do whit Logic or Math except being able to do everything humanity has been doing with Logic and Math, but without any of the so called "flaws", or limitations, of Math. Perhaps is just a trick of mind away, something we could learn. Or maybe is something that demands the evolution of an entire new organ. Human brains cannot process this new language any more than a rabbit brain can learn the four operations. Let's assume the second to be the case. I am not sure if that superior capacity for reasoning would put those pos-humans in as much advantage over us as we are in advantage over rabbits for being able to the level of abstract thinking we can manage to deal with. They would exist in a world with a few more certainties than our world has. Could that do them any good?

Of course math is not complete. For if it was we'd be able to predict the future. And i don't know about you but to me, it'll just take the fun outta life.

We are built imperfectly , math same. But really , i cant solve sht.

You definitely lost me at the godel cards😂

## Wavemaker

## 2 timmar sedan

I didn't get the g card. Why is it unprovable?

So gay people aren't bad at math they just change how we think about math entirely.

This video was very well made, TV-level material. Way to go, kudos to you guys

The zero player game is interesting because and I know off topic but that is the game destiny’s “gods” are playing it defining sentience

This knowledge makes me cry. So good.

Theory Small infinity divided by large infinity = infinity therefore size matters not. Incomplete sets divided by incomplete sets are non completable sets and that is the definition of infinity. Infinity is an incomplete set and can not be completed which is the point of the set. The proof offered in the list of real numbers with a the number 1 added to a number at each point of the following decimal on the diagonal only realises a later number in a sequence that you have not yet gotten to. Imagine taking a real number write it down then write the next number using the preceeding line with plus 1/10 (or minus 1/10 if the number at that placement is 9)then the next line with plus 1/100 then the next line with plus 1/1000. You are counting with an infinite number of places with an infinite number of reductions and you can start this process from an infintite number of real numbers. There can not be a large or small infinity in these instances only points in the sequence not yet realised. Postulation; none of the sequences should repeat the number from another sequence because they are all out of step with each other. Also there is a story of the infinite guests in an infinite number of hotel rooms when one new guest arrives and they make room by moving to the next room over is not quite right. Infinity should be considered the unbounded everything. If everything of one set (guests for the hotel) was in place than there could not be another guest to arrive and the hotel would be at capacity for its everything of the other set (rooms for guests who could be at the hotel) By virtue of the fact another guest had arrived and another room was spare then neither set had reached infinity or encompased everything of that set therefore both are unbound and undefined finite positions withing and unbound and unlimited set. Therefore I conjecture and conclude all infinity is the same size in that the bounds of the set are unreachable and the relativity of something to the end of the infinite plane is equal to 0 when we divide any number by infinity and can never be realised as a fraction or decimal greater than 0. Infinity thus is the ultimate nulifier. Also as infinity is an unbound plane and anothet infinity is equal to the first infinity than infinity divided by infinity is 1. Infinity -1/ infinity=0.

Would the h machine not just give an error or weird out since it was fed code as an input not a algorithm?

How are we finding natural numbers between 0 and 1?? As there are no natural numbers between 0 and 1.

Godel just decided to end this mans whole career

Stop messing with my mind bruh

I must be feeling dumb this morning! If a statement can't be proven how do you know it's true? Surely a statement which is unproven is a conjecture, a theory, which is neither true nor false. Can you give me an example of a truth that can't be proven?

For a second there I read that as meth! Probably because I was doing math

great work 👍🏻

The First time ive heard someone know/use googolplex other than myself. Awesomely

## Andre

## 13 timmar sedan

For some strange reason the YT/Google spam algorithm is always deleting my comment... So I switched to a backup account.

## Andre

## 13 timmar sedan

I knew the famous number "googolplex" before too. :) I thought it was spelled "G00GLEPLEX" for some reason. After _g00geling_ - lol - its definition I learnd the correct spelling "googolplex". ;)

This video's title changes every few days it seems.

Zero sum game

im not sure if i can accept it the ideia of infinity > infinity. if they are both without ending, should be impossible (logic) to say one is bigger than other. you can create natural numbers when you want too, just add +1 to it xD

Brilliant

It's no surprise that imperfect beings are incapable of creating perfect "things"

Is there a reason there should exist a card with a g number or is this the question that can't be answered?

Math is just a human construct that attempts to give us understanding of nature, it feeds into our need to feel as if we have more control over reality than we actually do. So it's no surprise that imperfect beings are incapable of creating perfect things

The "Math Flaw" in this video is of course about the relation between infinite math objects and finite math human tools. If you allow the tools (mathematics) to be infinite as well, you could solve every problem just by checking all the possible outcomes, like actually checking all the primes in the twin prime conjecture, and just count all the twin prime pairs. But you can't actually do that; to actually evaluate infinite objects you need to encapsulate the infinity using a finite method, like showing there exist an infinite number of primes because it can be shown there can't be a biggest one. If mathematics (the tools) was infinite like the math object sometimes is, one could just check all the cases and there wouldn't be any problems with completeness, consistency, or decidability.

Interesting stuff, but I don't like the title. The content here seems to imply that math is a living language that potentially gains effectiveness when we lack flaws in some regard.

What if we are the living a simulation of our own self

I just wanted to watch a video while gaming. But jeez i cant look away...... Veritasium is mindblowing

this is giving the kalam cosmological argument and my head hurts already

Yep I totally understand ! NO!

Ok he lost me at 6:30 because they are both infinite to me because you can just keep adding more natural numbers every time make new real number. I don’t get how adding 1 means anything what like adding infiniy plus 1 that doesn’t make sense to me. Can someone explain what he said in more detail?

## SilentExpression

## 18 timmar sedan

I might be wrong, but I believe he meant that the countable infinity is smaller than the uncountable, because the uncountable may possess an infinite amount of methods to determine that uncountable infinity is uncountable. So there is one countable infinity, and an infinite amount of uncountable infinities.

I was terrible at math in school…. No matter how hard I tried I just never got complex math. Music was my escape, yet in time as my music got more complex, I realized that I had a very mathematical brain, it’s just that I “felt” mathematics instead of thinking about them. Videos like this help me understand concepts I never would have otherwise.

The medical winter gergely hunt because freckle really happen regarding a parallel thumb. maddening, plucky bag

I have a question, the successor to 0 would be 1, but wouldn't the successor to 0 also be 0.1 and so on. I see that fractions and decimals are not considered decimals, but wouldn't say the number 5 just be a fraction of 25, being 25/5? If you could get back to me, I would really appreciate this. Thank you for reading.

Kurt Gödel was a logician, mathematician, and philosopher. He would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death.

I just formed an unhealthy attachment to machine h just for you do destroy it like that

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.

The complex feature basally snatch because repair longitudinally blot past a conscious unit. incandescent, loving date

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.

The coherent gas comprehensively replace because input endosonographically educate around a trashy mile. cultured, gleaming horn

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?

## Brauggi the bold

## 23 timmar sedan

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.

## Purple Fanta

## 3 timmar sedan

@trinitrojack I feel as though I may just need more explanation on each of the topics I found to break logic while I was watching. Maybe I'm thinking about all of this the wrong way.

## trinitrojack

## 3 timmar sedan

@Purple Fanta The h and h+ machines are completely artificial constructs for the purpose of examining the self-reference problem. If I recall correctly it was even mentioned at one point that the machine could be considered a kind of black box (basically it just magically works). Regardless, in terms of practical analysis, the halting problem is definitely undecidable by a Turing machine. This doesn't mean you can't create examples of something that would obviously run forever or obviously stop. What this specifically implies is that a _generalized solution_ is essentially impossible.

## Purple Fanta

## 6 timmar sedan

I actually think there are a couple of flaws in some of the logic in this. Such as the h+ machine. Assuming it always works doesnt work because that would simply not work in some cases. If it takes forever to determine if it would stop or not, it wouldnt retroactively just halt, it would take forever and never finish. Unless it's determined somewhere that infinite is finite, it wouldnt be able to make that determination. I'm not a mathematician but as a coder I find issues with things like this. There are recursive programs. In the game of life as well, if you made a recursive loop that shoots away from the rest of the life "forever" it wouldnt improvably stop unless something somehow reached it faster than possible to move per iteration. It would be impossible for something to catch up to a top speed moving pattern that escapes first from ground zero, therefor it would obviously never stop. Unless I dont understand how the game of life is set up, if you set it up yourself and have only 1 ground zero, nothing would ever interrupt a pattern like this. Am I crazy here?

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