Math Has a Fatal Flaw

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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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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.

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

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

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

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

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

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

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

I can understand EVERYTHING here except ONE thing. Did the damn barber shave himself or not? pt2 required.

It's not a flaw. Nor is it fatal.

phfffff...XDD......you really want the answer? MATH IS FINITE, IT STARTS FORM 0 AND IT ENDS TO 9, EVERYTHING ELSE IT'S A LOOP ( think about when you draw a circle you start from point 0 and when the circle it's complete, it ends to point 9, after that it's repeated point 10 to point 19, etc.)

Next time I owe you $1001 I'll give you $1 and tell you to loop it around.

So... is math what it is or is it what we found it to be?

if there are true statements that cannot be proven then the fact that God exists must be a true statement that cannot be proven.

I've watched your videos for quite a while now and this is by far the best work you've done so far! Great job!

“There will always be true statements that cannot be proven” for all my skeptic friends who say they don’t believe in God without prove.

he speaks fondly of those German and European "Scientists" and about their "extraordinary discoveries", But when a scientist with an Asian name comes up, he just calls him a "mathematician". It is disgusting to see such things, trying to make people believe that Europe's position of power and wealth was all because of their intelligence, hardwork and education, when we all know that it was by means of force, war and slavery. If you weren't aware of this, then please check for the scientists who actually were the first to make these discoveries, and if you already knew about all this, then I hope you burn in hell.

then no math pls deleted it from syllabas

Turing did not "invent the computer". Turing did not "lead the team".

I don't think math will ever reach any pure truth like physics or chemistry. All notation, theories and symbols will never be able to truly describe the intricacies of the world accurately.

6:00 maybe I'm stupid but if 9 goes back to 8 and goes back to 9 because you add 1 then eventually the real number wil just oscillate between 8 and 9 right? Sry for bad English

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I did not know math is philosophical, and I'm an engineer

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this was fantastic

ALAN TURING WAS GAY?

Mathematicians 🤝 philosophers Giving peeps existential crisis

David Hilbert: "We must know, we will know" Derek Muller: "The truth is, we don't know, and sometimes, we can't know"

I understood about 40% of this video, still loved it.

I donno I see a glaring issue with referring the self and then not being able to solve the problem because the system is stuck in a loop that is inevitable because a loop was the start of the system querying.

So… the pot thinking about its self and can’t decide if it is or not because it thinking about it self

everything you say depends upon how you define the words you use to say it. These definitions are by definition unstable, and therefore you can never fully know the exact meaning of what was said, unless you deny all instability of definition or interpretation or usage, which all change according to need, subject, experience, and intention, so you ultimately cannot fully know what journey meanings will take. I wish anyone good luck trying to know answers in advance of their calculation and expression. They are by nature out our control, as is the entire concept of stable, complete, pure and truthful Mathematics. But without mystery, there is surely only death, no colour and no sound.

17:50 Wouldn't it be easier to just have the Gödel number of a sentence be the concatenation of the symbols that make it up as numerals in base ?

So 1 is equal to 2?

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Yeah, it's made up.

Bravo! What a nice presentation! Thank you, from the bottom of my heart, before someone discovers that there is a hole there!. You really know how to make understandable the unintelligible. Congrats!

21:00 this had me in a mental pretzel lmao 🤣

This video is Turing Complete; it halted at the end.

Is this why there are always gonna be bugs in the programs ?

Can you please do an episode on whether quantum computers have capabilities beyond a Turing machine? Perhaps it's an easy answer...

Countable infinites.....lol nah

it seems like the godel number g is only a paradox of "if it's true it's false, if it's false it's true" because it was given the godel number of g. but i guess eventually, if godel numbers are infnite, there would be a godel number just like it eventually. idk my brain is trying to understand lol

There is no such thing as something true you can't prove....

There is.

when i look up godel numbers, i unfortunately get some charts that show the same godel number meaning 2 different things.

Oh wow, nice to see the article by Tony Cubitt, David Perez Garcia and Michael Wolf here!

Im, sorry, I know this sounds a bit arrogent. But for the diagnalization proof. If you made your list in actual order. So 1 is 0.1, 2 0.2, 3 0.3, 10 0.0 then at 11 you get 0.01. Then you do that infinitly. Could you not just make this new number, follow the new index you would get, and then find that exact number having been made. Especially if 9 rolls back to 8. If you had a number of 0.999 then you could go to get 0.888 from the diagnal which from the previous organization you know 0.888 comes far before 0.999 comes, so it is already drawn. We know if the number has 9 anywhere in a number, then if the index is organized propperly, then we can know with 100% certinty that the number made has apeared.

The diagonal number you get from your list will be 0.211111... forever. The problem is that you're only listing numbers with finitely many digits, and so you're missing all real numbers with infinite digits, such as 1/3, pi/10 etc. and of course, 0.2111... as well. So everything's the same as in the video: the diagonal number is not on your list, nor are infinitely many numbers that are left out.

So "Because" doesn't count as an answer?

that would mean there is a stand alone loop in the number line that says "there is no proof that this is on the number line"

Regarding the diagonalization proof isn't there an infinite amount of real numbers? So the number you created with the diagonal has to occur anyways. Also it's still a number meaning that it still occurs between 0 and 1 so why could it not be identified as a natural number?

The diagonal number absolutely is à real number, nobody is debating that : but it's here to show that the list you made cannot contain all real numbers, since it will always miss at least this diagonal number no matter what (and in actuality it will miss infinitely many others as well, the diagonal one is just easy to find)

I read that meth has a fatal problem

A flaw lies in [his] introducing an index, he quantifies real numbers with a sequential relativity of intergers, they're distinct sets treated with the same dynamics questioning why they were originally distinguished

I wonder when Bertrand Russell’s paradox would be mentioned. He was so much more it’s nearly impossible to believe such a person existed. Great show

i wish youube did split the comments of ppl who disliked and ppl who liked, so we can read what both parties say about this, especially the dislikes

This video is so meaningful, I honestly even cried at the end. Thanks for putting that out in such a way. Amazing job, guys! Unfortunately the guy who created the computer’s language, killed himself because of our crap animal mentality. Something to think about as well.

This video stimulated my brain activity. ❤️

what about a quantum computer?

So this is where the Christopher Nolan gets ideas for his movies

conclusion: we are simulating ourselves

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!

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

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.

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maths after inventing germans: $$$

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

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

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

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.)

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

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.

Math is racist according to some people in America.

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.

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

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?

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.

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

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

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

absolutely what I was thinking lol

18:27 -1?

17:39 why add prime numbers lol

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

Meth

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 :/

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

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

10:07 add a rule lol java brrrr if self is barber let him shave already u stupid b

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?

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

Just wow.

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

5:45 the stupid infinite hotel thing? Brrrr woosh

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

4:23 E

4:10 ah yes school

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

1:37 and keeps flickering dead alive lol Edit:w oh ok

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?

1:25 i forgot

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?

S+ presentation!

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😂

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

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