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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### Math Has a Fatal Flaw

Publicēšanas datums 22 mai 2021

komentāri

## AnthonyMain

So I've watched a few videos on Godel now and this is the only one that took the time to explain the system and legitimize the method Godel used so lay people could actually understand and appreciate its implications. Props Veritasium!

## Cmace

Pov : I got too high and thought I solved the universe

## WolferDub Jazz

but isnt this 'mathematical black hole' countered by probability which is a mathematical concept.... Im not the smartest but if im not mistaken, the incompleteness of math is a possibility itself meaning that math is complete if you consider the worst probability -- its incompleteness. I think the main concern of math is not exactly incompleteness but rather assumption--the fact that something is impossible to acheive is rather expected but the fact that some principles of math can be assumed shows that they are in fact complete to a certain degree.

## Novice IV

Greetings,

A message from the Gospel in The Christian Holy Bible. This is the message of forgiveness of sin through the sacrifice of Jesus Christ. In first Corinthians, chapter 15 verses 1 to 4, the apostle Paul states:

“Moreover, brethren, I declare unto you the gospel which I have preached unto you, which also ye have received, and wherein ye stand; by which also ye are saved, if ye keep in memory what I preached unto you, unless ye have believed in vain. For I delivered unto you, first of all that which I also received, how that Christ died for our sins according to the scriptures; and that he was buried, and that he rose again the third day according to the scriptures”.

Romans chapter 3 verse 23, informs us,

“For all have sinned and fall short of the Glory of God”.

Ultimately it is a redeeming plan set by God, and for anybody who will faithfully trust in his Son will be saved.

Romans chapter 6 verse 23, says,

“For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord”.

A sinner must recognise the despairing of her or his guilt before God in order for forgiveness to take effect.

Every believer must understand in order to inherit the kingdom of God as sons of the king, the reality and foundational truth of sin must be acknowledged, for all who approach the throne of God for salvation.

God promises us eternal life, for anybody who departs from sin through the saving faith in Jesus Christ.

Jesus lived a perfect and sinless life which no person could live. Jesus’s sacrifice on the cross ultimately paid the debt we owe to God for our sin.

Sin against an infinite God requires an infinite sacrifice; and Jesus died a substitutionary death for the sinner.

In short, the works of Jesus paid the penalty of sin through an infinite sacrifice. Only the creator of life has the power to resurrect life after death and is essentially the proof of the power of God.

Jesus Christ offers his salvation as a free gift and only be received by faith, despite any works or worthiness on our part.

The scripture in Romans informs us,

“If you declare with your mouth, Jesus is Lord, and believe in your heart that God raised him from the dead, you will be saved”.

In conclusion, the resurrection of Christ is to bring everlasting life and salvation to anybody who follows him. The relationship between God and man can be restored through his saving faith.

## Alexander Koravit Caszo

maths after inventing germans: $$$

## Amanda Capsicum

Set of all sets contains itself factoid actually incorrect: Cantor Georg, who lives in a cave and contains 110,000 sets per day is an outlier and should not be counted

## Dila Afdhol Santosa

Math is simple. It is about, what you can do with math.

## Fred Ouellet

This is the most insane made up autistic s... I never heard of lol. At this point there is no logic behind since a long time they just make things up and trying to solve them. And plot twist, this guy don't even know what he's talking about

## Aaron Fidelis

When you have numbers and signs substituted for an index number, you know the results are gonna be reality-breaking.

(Well… maybe not. But I ain’t got the time to do that myself. I’m going to let Veritasium explain it to the public.)

## Matej Pranjić

Cool, reminds me of many other paradoxes in life/universe/existence/reality

## Phúc Hồng Nguyễn

I can't understand what you're saying but I agreed that math have some flaws since there are reasons why I believed that "perfection" is impossible.

## Chef Cheffington

Math is racist according to some people in America.

## nzriot

Math doesn't have a fatal flaw. You do. The number of real numbers between 0 and 1 are infinite already (equal to the number of decimal points you can put behind a number).

thus, even though there are an infinite number of whole numbers as well ~~~ Real numbers reach infinity before the whole ones even get off the starting grid (because you can put an infinite number of decimal points behind zero).

LOGIC.

## Apollo

This is one of the most comprehensive and interesting videos I've ever seen on math.

## nzriot

Numbers only ever go up and one can never count to infinity, and numbers that get up that high are quite useless for anything.

I would have thought that is just logic. How is some magic number way up in the clouds towards infinity going to possibly be of any usefulness?

## Wassim AOUNI

We can't know and we won't know, simply because this is our own brain that is making these rules and these connections, it's with our own individual logic, and crazy enough this logic works perfectly with the worlds around us. This is why questions like why are we here, and is god real, can never be determined, the only way to know is to use your brain, since like I said our brain's logic seems to work perfectly with the world around us.

## DanDCool

19:37 wdym find he wrote some card u can't prove what does it have to do with anything else :/

## DanDCool

20:10 if you say no proof is wrong it mean there IS proof

Edit: 20:05 you say it's fault so there is a prove but when did u prove there is no proof bruh what does it even have to do with the a number

## soflogator

I swear it just feels mathematicians just make things up at some point

## Fred Ouellet

absolutely what I was thinking lol

## DanDCool

18:27 -1?

## DanDCool

17:39 why add prime numbers lol

## YamatoHD

This is why there is no Nobel prize for math, so far we don't know if even math is connected to science

## Dr Chaow

Meth

## DanDCool

12:58 and that ladies and gentlemen is called stupidity

Just go have fun in your limited life instead of proving your mom and your dad are your mom and your dad :/

## DanDCool

12:23

Yeah no if you change ur organs which impossible with our logic u can go immortal

Obvious af honestly

Or just go freeze and never wake up

Technically not ded

## DanDCool

11:15

Why are some people s m a r t

Yeah if the color is unique and nowhere else no u can't and if it's not you can as long as the other pieces match the parts

I mean if it's infinite u always find a tile that can fit but a certain amount? No

And if it's infinite u can always expand the puzzle but how can u complete infinity bruh

## DanDCool

10:07 add a rule lol java brrrr

if self is barber let him shave already u stupid b

## DanDCool

9:30

By R do you mean like Real numbers or just M that's everything (as i saw in a book)

Cuz it should have everything but everything doesn't have to be sets

Simple

Like imagine all numbers were 1 to 10

R= 1 to 10

And wouldnt need to also have a set with 1 to 10

But cuz the guy said it should then whatever paradox go brrr?

## DanDCool

8:01 bruh Infinity doesn't exist how can 1 be bigger than another

Just cuz from 0 to 1 there is more real deal numbers than natural skin lol numbers doesn't mean real is more

Simply doesn't end

He do be stupid for even trying to figure something out of infinity that wont even end

## Petit Neveu

Just wow.

## DanDCool

6:06 cuz you can never complete and infinite list? Bruh

## DanDCool

5:45 the stupid infinite hotel thing? Brrrr woosh

## DanDCool

5:16 they wont end and infinite doesn't exist so you can't even say this

Simply they both never end but in a certain area real numbers are more compiled

## DanDCool

4:23 E

## DanDCool

4:10 ah yes school

## DanDCool

3:05 yes it does work for looped patterns :/

## DanDCool

1:37 and keeps flickering dead alive lol

Edit:w oh ok

## DanDCool

Or the fact that from 1 to 9 it's 1 digit then 10 is 2 digits so we have like 9 positive 1 digit numbers instead of any other number?

## Nicole

1:25 i forgot

## Nufta1

How can a statement be true if its not proofable? Its only true if there is no counterexample. But if we had a counterexample, it wouldnt be true. On the other hand, if we dont have a counterexample, we either checked every number/found a proof, making it proofable. And as long as we didnt do that the statement wasnt even true in the first place. Or is there a problem in my argumentation?

## Siderminer KGL

S+ presentation!

## Justin Silva

I very much appreciate the time and effort you put into these videos, because I'm pretty bad at math but the examples really help! The history behind math is cool too, would've much rather had a mathematics theory class in HS rather than Trig😂

## rainbowroad the killjoy

Okay John Green, we get it. Some infinities are larger than other infinities, geeze

## Femi Oye

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

## yazeed aziz

What's wrong with mathematicians??

## Kerns Noel

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

_" 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.

## Toolingaround

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?

## guitargod7682

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 5:02 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 would have to exist in the list somewhere because without the existence of this new real number in the list, we haven't captured every possible real number between 0 and 1, which is the basis for this list

## Nate

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.

## Treqzie Ps4

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

## Chimerawizard

So the intuitionists were right. shut up and solve real math.

What we got out of that debate was someone disproving the formalists.

## Reylan David

The mind can only know so much for now

## Wes

Lesson 1: invent math

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

## Wavemaker

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

## Frank Lincoln

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?

## Bethany Gee

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.

## PHS Link

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.

## Omar McCord

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.

## Mason DeRoss

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.

## General Kenobi

“Math has a fatal flaw”

Tell me something I didn’t know

## Boo Shush

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

## The_Circuit

29:40

But will the game of life running itself end?

## The_Circuit

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

## The_Circuit

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

## Bruce Fulton

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

## a clown

Why is smoke coming out of my brain?

## xCherish

NP = P!

## LORD MAV B.M.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.

## Marcus Alexander

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 .

## Marco L

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...?

## Peppers&Eggs

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.

## Gastón Pössel

Ahh, the good ol' numbers...

## Corvo Branco

"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?

## dafullclip

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.

## Yesha Croe

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

## koen armstrong

You definitely lost me at the godel cards😂

## Wavemaker

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

## Claudia Wilson

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

## Kobor Kutya

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

## utsu002

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

## Sir Eren

This knowledge makes me cry. So good.

## G M

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.

## Conor

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

## Siddhanta Patro

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

## Dan Han

Godel just decided to end this mans whole career

## Gooby Pls

Stop messing with my mind bruh

## Ian Brooke

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?

## Nichi 1909

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

## Mohammed Alhajjo

great work 👍🏻

## minecreeper

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

## Andre

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

## Andre

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". ;)

## WaaDoku 【和ァ独】

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

## gina888warhol1

Zero sum game

## Helio Domingues

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

## Viet Food Hackers

Brilliant

## Err Moc

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

## xrhsthsuserxrhsths

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

## Err Moc

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

## MisterrLi

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.

## SilentExpression

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.

## Juan Camilo Zapata

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

## A D

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