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. — https://ve42.co/Dunham2013
Conway, J. (1970). The game of life. Scientific American, 223(4), 4. — https://ve42.co/Conway1970
Churchill, A., Biderman, S., Herrick, A. (2019). Magic: The Gathering is Turing Complete. ArXiv. — https://ve42.co/Churchill2019
Gaifman, H. (2006). Naming and Diagonalization, from Cantor to Godel to Kleene. Logic Journal of the IGPL, 14(5), 709-728. — https://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. — https://ve42.co/Lnrt2010
Attribution of Poincare’s quote, The Mathematical Intelligencer, vol. 13, no. 1, Winter 1991. — https://ve42.co/Poincare
Irvine, A. D., & Deutsch, H. (1995). Russell’s paradox. — https://ve42.co/Irvine1995
Gödel, K. (1992). On formally undecidable propositions of Principia Mathematica and related systems. Courier Corporation. — https://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. — https://ve42.co/Russel1910
Gödel, K. (1986). Kurt Gödel: Collected Works: Volume I: Publications 1929-1936 (Vol. 1). Oxford University Press, USA. — https://ve42.co/Godel1986
Cubitt, T. S., Perez-Garcia, D., & Wolf, M. M. (2015). Undecidability of the spectral gap. Nature, 528(7581), 207-211. — https://ve42.co/Cubitt2015
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Executive Producer: Derek Muller
Writers: Adam Becker, Jonny Hyman, Derek Muller
Animators: Fabio Albertelli, Jakub Misiek, Ivy Tello, Jonny Hyman
SFX & Music: Jonny Hyman
Camerapeople: Derek Muller, Raquel Nuno
Editors: Derek Muller
Producers: Petr Lebedev, Emily Zhang
Additional video supplied by Getty Images
Thumbnail by Geoff Barrett
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48 Replies to “Math’s Fundamental Flaw”
I'd say a lot of "this infinity is bigger than that" and "prove my completely inapplicable endless equation" is more philosophy blending into math than actual math.
i come to this video every so often ,its my favorite veritasium video
The game of life… running on the game of life 😂😂😂😂😂
Science made easy and interesting. Great topics every time.
Saying the key to cantors theory is writing down ALL real numbers but then adding one which isnt in the list feels like cheating… the list is not complete if you can add a number.
Isn't that the whole problem with "infinity" ?
Was that from covid or with covid?
Im 11F and i don't know how, I understand more than 90% of this
I genuinely appreciate how well he can take topics that are on some mind breaking difficultly levels to understand and makes the mysterious appear crystal clear.
If they think some dude playing with math is a corruptor of the youth wait until they see the 2025 update
“[A]s we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns — the ones we don’t know we don’t know.” Donald Rumsfeld
Thanks for posting! I need help: My Tron Wallet has some Tether, and I know the recovery phrase:
cleanpartysocceradvanceauditcleanevilfinish -tonightinvolvewhip-action-. What’s the best way should I proceed with moving them to Coinbase Exchange?Won't (X=7) V (Y=6) get to a different godel number than (X=7 V Y=6)? I mean… isn't the card correspondent to a proposition dipendent on the way you compose propositions (and godel numbers) to get to that final equation/proposition?
🤯
Cantors diagonalization proof is BS. The same proof can be applied to counting numbers. It's all BS.
Perfect.
20:50 why can't g be considered an axiom? Those are true with no proof
Makes sense… It's like manifestation, using your consciousness to manipulate energy so as to materialize something, even the Mathematics. So, the consciousness that is trying to understand mathematics is the one that created it. Paradox is the basis of this universe, no one can tell if the chicken came first or the egg…
19:48 Such a CHAD move, Gödel! Well played!
Math is just a game we
made up in our little monkey brains.
The idea that those tiny little brains could
understand the entire universe is laughable.
All of this is mostly possibe because of hardly known mathematician Emmy Noether.
yet number is completely, consistently & decidably in existence
1:13 Conway lost the game of life 💀😂
loser
this was good but not great
u missed some stuff
too much salt n pepper
Boring
1:15 😂🤦♂️
The first 45 seconds conjecture is wrong. Math is a decimal and the pattern isn't just going to change. You can see that in both directions of the decimal equally.
"Number g."
I believe that the nature of the issue is linguistic in nature.
If we can determine the final outcome of a pattern, then we can know beforehand from the pattern what the outcome will be, but we cannot use a pattern to make a prediction if we are not familiar with the properties of the pattern.
Therefore, it may be possible to compile all of the possible patterns into group theories of their own but there's no way to know whether or not we will be able to compute the final answer and so therefore the question will be unresolved until we have observed the outcome. However, if we observe enough outcomes, it may be possible to determine a pattern of everything by grouping groups of patterns into groups of patterns. It would at the very least be a powerful tool in simplification, but either the result should be irrational and therefore unpredictable or rational and/or self repeating.
Therefore, we could potentially prove true or false, but there's a chance that we can't.
There's a 50% chance that we might never know everything that will happen in a 4 dimensional plane intuitively. I don't hate those odds.❤
The size of natural and real numbers is effectively the same since the infinite 9 recurring after the decimal point is implicit as potential action.
thank you
I watch this once every couple months to try and understand it
The fatal error with Cantor's proof is that there is no such thing as a countable infinity. You can't just assume that you can create an infinite set. The proof, whilst it looks convincing, is in error as it fails to realise that what he is doing is a continuous operation and that if taken far enough, the real numbers on the right that have had one digit altered, will in fact, begin to appear on the list. It is not a viable proof of anything, other than Cantor, and many other people including highly educated mathematicians, do not understand what infiniy is. The claim that one infinity is larger or smaller than another, is only logical when describing functions that are divergent or convergent, and convergant infinities approach a limit and are therefore measurable in one sense, but not truely infinite in the sense of the divergent series. They are two similar yet seperate concepts, but ultimately, the divergent series is incalculable and therefore making any quantitative statement about such a set is meaningless.
You know, hypothetically, a monkey could have typed this video out.
Why must "There is no proof for the statement with Godel number g" be assigned to "card number g"?
Other than to forcing the self reference, that is. Or is that the whole point? Seems like you could just have a rule that bans self reference like they did with set theory?
Theoretical mathematics can lead to insights in apparently unrelated concepts and fields. For example, Godel's work led to Douglas Hofstader using it as a way to illustrate the self-referential 'strange loopiness' of the human mind, the "I", or what may be called the soul.
So basically… can math prove itself? No. But math can prove other things, math just can’t prove itself.
Cantor's diagonalization proof seems wrong from a logical perspective. You're monkeying with the list with what looks like (to me) logical fallacies. It presupposes that it is possible to have a complete list of infinity, but moreover, it plays trickery with that completeness to claim a number that's not already on the list is somehow excluded from what is, by definition, a complete list. But he gets away with it because the seeming random disorder of the numbers listed between zero and one blind us to his trickery. By ordering the numbers between zero and one in actual parity with a corresponding number, i.e. 0.1 for 1, 0.2 for 2, .01 for 10, and so forth, in a mirrored ordering from how the decimal system is written, we achieve an exact correspondence with a unique identity for every number on the list, and by performing the diagonalization tactic on that list in reverse diagonal on the real number list, each newly generated number on the supposedly already complete list receives a corresponding, unique identity to match it. Ergo, the number of numbers between zero and 1 is equal to the number of integers.
7:00 Okay well I think this is stupid. Even if you did the canted method to get a new real number it’s still just another new real number and all you have done is add one to your list and added one more index. I don’t see how that proves that one is larger than the other, it isn’t like you couldn’t create another index for that real number to lie at. You always could. If someone could explain to me how that somehow makes one larger than the other I’d love that.
Mathematics as an abstract exercise was always dancing on the edge of the cliff, since funding required a certain level of deception. But most of the more famous problems are a function of word games. Smarter people than me can tell you the terminology which properly describes "different sized infinities". But set theory itself is a nonsensical pursuit if you accept the implied premise that set analysis involves answering questions about puzzling realities. Set theory could be used to describe marketing, analytical chemistry, DNA mapping, song writing, airplane design, dating norms, learning curves, and pimple infections. In other words, EVERYTHING. So, what the hell is set theory but a new name which now carries the stigma of word game parody?
Flaw? Abstraction reflecting abstractively? Since when did abstraction our brains cause either the universe let alone geology? I love math, way too much math is causing duration,distance,.and the universe fantasy gibberish. Hell its clueless about evolution what it is even at that point. I blame genetics at some point. At least geology isnt so infantile lame.
Can you please make a playlist of all your videos?
[2:255] GOD: there is no other god besides Him, the Living, the Eternal. Never a moment of unawareness or slumber overtakes Him. To Him belongs everything in the heavens and everything on earth. Who could intercede with Him, except in accordance with His will? He knows their past, and their future. No one attains any knowledge, except as He wills. His dominion encompasses the heavens and the earth, and ruling them never burdens Him. He is the Most High, the Great.
I'm autistic
I think I nice thing to take from this video too is that if someone feels stuck or trapped and the solution is undecidable, for instance "Will I be okay tomorrow and every day after that?", it can seem like there's no point in trying because you can't prove that everything will be okay. However, trying is what allows each of us to experience everything we do and if we all stopped trying to solve this undecidable problem, we wouldn't be here at all. If anyone is reading this and struggling, remember this, while we can't know that everything will be okay, its the process of trying to be okay that allows us to experience a reality in which things end up okay after all. Just keep trying.
when a cell multiplies, it does it by dividing. What we are told are opposite operations (one undoes the other), are in fact not at all. The act of division is the mechanism by which our cells multiply. Therein lies the physical manifestion of the hole at the bottom of mathematics. It just so happens that all of life emanates from this hole.