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., \u0026 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., \u0026 Whitehead, A. (1973). Principia Mathematica [PM], vol I, 1910, vol. II, 1912, vol III, 1913, vol. I, 1925, vol II \u0026 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., \u0026 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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Reacties 

  1. Kshitiz Gupta

    Kshitiz Gupta

    32 minuten geleden

    Man I have seen this thrice now

  2. Priyanshu Goel

    Priyanshu Goel

    48 minuten geleden

    This self referencing is the cause of me not understanding flipflops.

  3. Umang Ravaiya

    Umang Ravaiya

    2 uur geleden

    29:56 which music is this??

  4. kerry mackey

    kerry mackey

    3 uur geleden

    The secretive invoice optimally taste because milk lately carry unto a shocking dock. beneficial, soft retailer

  5. Andrew

    Andrew

    6 uur geleden

    Language is not a math! Car can be different pronounce and some people thinking about car call it a motor :-)

  6. 36nibs

    36nibs

    8 uur geleden

    How fortunate is it that you can apply these mathematical principles to other systems

  7. BonnetBee

    BonnetBee

    8 uur geleden

    A potentially stupid question: Why would you bother to create H+? Why wouldn’t you just create H and call it a day? And would the answer still be the same if you did? 🤔

  8. Nature

    Nature

    8 uur geleden

    Why do people assume they need to count every vain of each leaf on a tree to know it's a tree? If red or green round things grow off twiggs attached to branches, it's a red or green "Apple" "tree". You do not have to count how many apples have worm holes to know it's an Apple tree, and worms like Apple too.

  9. Loruo Ditlhong

    Loruo Ditlhong

    11 uur geleden

    Lol just found the meaning of consciousness

  10. Gerardo Contreras

    Gerardo Contreras

    13 uur geleden

    I swear he makes this sound so simple but at the same time my brain cannot comprehend anything he is saying

    • BBucky98

      BBucky98

      5 uur geleden

      Me too the card part is too much

  11. Scots Diesel

    Scots Diesel

    14 uur geleden

    So sad about Alan Turing...❤️

  12. mohamed mada

    mohamed mada

    15 uur geleden

    Okay, that's the same explanation for the hotel with infinite numbers, My question is what if Cantor's"Diagonalization proof" is Wrong? What if in the set of infinite numbers there are infinite numbers with all the infinite possibilities of the diagonalization proof? I mean what if there are indeed numbers that are greater (and less) than the numbers in all indexes of all numbers with all different possibilities (antidiagonal)? Given the nature of infinity, this is a legit question. P.S I'm not trolling, I truly need an answer.

    • Moyprod

      Moyprod

      9 uur geleden

      @mohamed mada Not, it is not the same as Hilberts hotel. There you have only 1 infinity. The infinity of natural/rational/integer numbers. In Cantors diagonalization argument occur 2 infinities. One is bigger.

    • Andre

      Andre

      13 uur geleden

      @mohamed mada _"You didn't quite capture the essence of my question,"_ I did: Cantor was not wrong. _"I didn't say that there could be a number greater than another number in one set "_ You said: "I mean what if there are indeed numbers that are greater (and less) than the numbers in all indexes of all numbers"- There are the natural numbers. Those DEFINE the term "countable". It doesn't matter if there are other numbers. There are of course. The rational numbers for example. But those are not really more. And there are the real numbers. Those are "more". So we are already talking about this. _"My question simply is what if such a number already exists"_ There exists no such number. A number is not a set. And there is no natural number "greater than all natural numbers". _"and our list which contains infinite numbers having infinite possibilities?"_ All possibilities do not contain all real numbers in [0,1] as the proof has shown.

    • mohamed mada

      mohamed mada

      14 uur geleden

      @Andre You didn't quite capture the essence of my question, I didn't say that there could be a number greater than another number in one set even though in an infinite set of numbers that could easily happen, I mean since the diagonalization proof says that in the list a different number would be created changing the index of each number increasing it by one ergo it won't belong to our list. My question simply is what if such a number already exists and our list which contains infinite numbers having infinite possibilities?

    • Andre

      Andre

      15 uur geleden

      _"My question is what if Cantor's"Diagonalization proof" is Wrong?"_ It is not. Next question. _"what if there are indeed numbers that are greater (and less) than the numbers in all indexes of all numbers with all different possibilities "_ A number cannot be greater than all numbers. This is trivial to prove. _"Given the nature of infinity, this is a legit question."_ No, it is not.

  13. johnnytheprick

    johnnytheprick

    15 uur geleden

    I couldn't agree more, it's convoluted, isn't taught in ways most understand, and it doesn't olve man's greatest problem: stupidity.

  14. Alex Huggett

    Alex Huggett

    15 uur geleden

    This is so much better explained than everything else that it's the only one that actually says the problem how it is

  15. Justin Hamlin

    Justin Hamlin

    15 uur geleden

    ♾️

  16. Adhithyan Sreedharanarayanan

    Adhithyan Sreedharanarayanan

    15 uur geleden

    I have a Doubt Derek. If the Machine h has to produce some kind of output, It has to first run a code and an input for which the sequence of the output may or may not terminate. Then the machine h+ comes into play, Which implies that the sequence of the program and the input which was already coming is inverted completely which follows a loop around the machines h and h+ and I don't see why this creates a contradiction(like if the machine h gives out the output that the first inserted program and input produces a sequence that terminates. Then the not gate put inside h+ reverses the sequence and terminates it which in turn produces a inverted loop. In which the steps of procedure are inverted with respect to the steps mentioned before.)Which implies that the machine h when working in conjecture with the machine h+ never produces a stable output when it is fed with h+'s program code and the recurring input. Then I don't see why they simply assume that the machine h is impossible to make.

    • Adhithyan Sreedharanarayanan

      Adhithyan Sreedharanarayanan

      15 uur geleden

      This can go in either one of two ways. 1) Either I am Stupid.2)Or You are a Genius to Understand this.

  17. 𝕲𝖆𝖇𝖗𝖎𝖊𝖑

    𝕲𝖆𝖇𝖗𝖎𝖊𝖑

    16 uur geleden

    We are immortal until the day we die

  18. Rex Dalit

    Rex Dalit

    16 uur geleden

    One issue here is that any "solution" for Russell's Paradox is probably translatable to a homologous "solution" for Godel's Incompleteness Theorem. My understanding is that there have been at least 3 PhDs generated in the last 100 years purporting to "solve" Russell's Paradox, in 3 different ways. Thus one (I) would expect that there are at least 3 corrective counter-theorems to Godel's Incompleteness Theorem. Hilbert might, and probably would, content himself with these, since if memory serves, these are more or less formal ways of encapsulating and manipulating Russell's Paradox. [Note that I would conjecture that homotopy theory implies there exits an (uncountable) infinity of such solutions to Russell, translatable to an infinity of solutions to Godel incompleteness. Cheer up, dead Hilbert, no need to twist in your mathematical grave; your glorious formal headstone still marks your intellectual location for would-be visitors.]

  19. One Issue Voter

    One Issue Voter

    17 uur geleden

    The problem with infinities is a problem of the human brain, not of mathematics. Consider : humans cannot truly understand what infinity is because our brains are finite. Stop considering infinity as a static object, and think about them more as multi-dimensional like space-time. The set of real numbers is infinite, as is the set of integers, but the real number set "grows" faster than integers. So they are both infinite but not the same size at the same point in some (newly defined) dimension.

  20. TravelBig

    TravelBig

    17 uur geleden

    5:47. I think if we can get this new real number then we didn't actually write all the real numbers in the first step (we wrote all the real numbers minus one).

    • Andre

      Andre

      16 uur geleden

      _" then we didn't actually write all the real numbers in the first step (we wrote all the real numbers minus one)."_ And that is the reason why it is not possible and therefore there are more real numbers than natural numbers.

  21. flow_mang

    flow_mang

    18 uur geleden

    Turing was essentially killed for being gay. Lot's of wonder and beauty presented in this video balanced with some dark stuff.

  22. Jayden Maree

    Jayden Maree

    18 uur geleden

    I love how much I hate this, but I also hate how much I love this.

  23. Grant Currin

    Grant Currin

    19 uur geleden

    4:06 I wonder how to search through all the comments and find out why a set of nothing is inside the set of everything?! >search:?

  24. TerrorHuhn

    TerrorHuhn

    19 uur geleden

    i'm new here.. why the heck are you explaining math in the middle of nowhere? :D

  25. Sasa Radetic

    Sasa Radetic

    19 uur geleden

    That is why man as a part of creation, will never be the creator god of everything.

  26. Willie Theron

    Willie Theron

    21 uur geleden

    Many here: This is complicated, Me, halfway in, oh, that guy was wrong, 6 hours later I disproved the statement. Pushing a video about it this weekend, will edit it in here. Im a programmer, not a mathematition. but I find myself baffled that nobody could write the counter hypothesis. It seems so basic, yet, nobody could see it. Anyways, I will link my counter hypothesis here in a day or two, Seems like a great first video.

    • Andre

      Andre

      15 uur geleden

      _"Me, halfway in, oh, that guy was wrong, 6 hours later I disproved the statement. "_ Then you made a mistake. Because all of this is well known and proven.

  27. Eddie RUKidding

    Eddie RUKidding

    23 uur geleden

    Great posting Maths has many inconsistencies, even 0 to the power of 0 is not complete, mathematics as we have derived it is not a complete system Incidentally Tuning, although how great he was did not invent the computing used at Bletchley Park (that was Tommy Flowers for Colossus and Max Newman for the earlier "Heath Robinson") and the digital computer inventor was John Vincent Atanasoff in the 1930's.

    • Andre

      Andre

      16 uur geleden

      _"even 0 to the power of 0 is not complete,"_ That is no "inconsistency". _"Maths has many inconsistencies,"_ Can you show a single one? Until now nobody has found one - and if there is one, it would be a nightmare.

  28. Irfan Kanth

    Irfan Kanth

    23 uur geleden

    One plus one is one ... Does that make sense ??

  29. Irfan Kanth

    Irfan Kanth

    23 uur geleden

    Is there any difference in mathematics and Russian as a language !!

  30. Chase Thompson

    Chase Thompson

    Dag geleden

    26:40 so if it gives the wrong answer each time just make it pick the opposite of the outcome

  31. Christopher Suiter

    Christopher Suiter

    Dag geleden

    Just sharing my thoughts. I am not a mathematician and I don't consider myself a scientist. I am just a thinker with a reasonable grasp on science and math. Math is philosophical in nature, as is every other educational subject. The formulas we use and the results we get are real, but the subject itself is philosophical. We can explain how squaring "i" (referring to imaginary numbers) will produce a -1, but there is no practical way to show this in an observable way. Quantum physics often does this as well because, while the math does work on paper, there is no way to really show the process in action from start to finish. We can only show the math, the solution, and an example or depiction of it all, but we cannot show something observable in real time for every equation. For example, how worm holes work. We cannot show it in action and, despite being mathematically "possible", it is still not observable. The simple workaround to the problem of this is to accept that Math is just philosophical -- it is nothing more than a the study of how we use the "tools" of math to answer problems. And, as a tool, math is immensely important. Having said all of that, I think it's also important to accept that since we cannot understand certain concepts without math, (such as infiniti, which is actually impossible to understand without a simple definition or symbol), we must also be unable to understand other things about the universe around us. We simply lack the ability to understand everything. That's the simple reality. Because of that, the second best thing we can do is try to understand it "to the best of our ability" which is what science is for. My primary point is that we lack the ability to understand everything in the universe, therefore we may observe things to which there is no way to conceputalize, rationalize, measure, or study because we simply lack the ability to do so. Is it possible that AI could solve things that we cannot? Possibly, but we won't know until it happens. I believe the "big bang" happened at the very end of the "great ending" as the universe simply explodes, maxes out, and implodes constantly. On top of that, everything is infinitely smaller or larger. We are made up of cells, our cells are made up of atoms, atoms are made up of smaller particles such as protons, neutrons, and electrons, and those are made up of smaller particles, and so on and so fourth. Everything gets smaller and smaller and a single cell may be equivalent to a "universe." Within the cell, atoms act as galaxies and electroncs act as solar systems and within those electrons there are particals that act as stars and so on and so fourth. Everything is infinitely smaller but we cannot observe it. And our entire universe may be equivalent to the atom of another reality. When we destroy an atom, we effectively destroy a universe... but new universes (particles) are created from it. Upon thinking about it like this, I have realized that time is very relevant but everything must also be infinitely linked as well. The only way to make it sort of make sense is to imagine that everything is connected interdimensionally. Not only that, but these connections, if we could travel them, could theoretically let us travel "through dimensions". I don't think we could ever go "back in time" as time isn't real, but we could speed up or slow down ourselves as we relate to specific places. If you traveled "down" a dimension and stayed there for 30 years, you would be 30 years older but maybe only 5 minutes had elapsed at home. You could also travel "up" a dimension and stay there for 5 minutes before going back home. Thirty years may have elapsed since you left even though you have only aged 5 minutes relatively speaking. But you cannot go "back" in time as time is just a concept and not a timeline. As such, there is no way to travel back in time and warn your family about something. You can, however, travel to a lower dimension and find a civilization of primitive people and educate them before heading back to your own dimension. That civilization may only take days (relative to you) before they have evolved into smarter beings and they can travel to your dimension (higher than theirs) and educate you now that they are the more advanced people. Then we get into the question of multiple universes. If multiple universes are real, and these run linear to our own, and all outcomes from all universes are possible as there is an infinite number of universes, then we may be able to jump through dimensions to communicate with other universes. In the end, we are, in fact, living in a simulation as is everything else. The simulation is real, but it is impossible for us to observe because we are a part of it. And to exit the simulation would require you to not exist which means you wouldn't be able to observe anything. Even traveling through dimensions or other universes is still part of this simulation. Something "above" the simulation must exist and we are absolutely unable to understand it and we never well. Even if it transcended to our simulation and educated us about the truth, we would be bound by our inability to understand it and we would be skeptical of this truth. I think the only way to ascend out of this simulation is through death.

  32. MdSteel7

    MdSteel7

    Dag geleden

    What did I just watch? I'm totaly amazed.

  33. Brendan Shimizu

    Brendan Shimizu

    Dag geleden

    "there is a hole at the bottom of mathematics" was a better title, in my opinion. Ive watched this start to finish many times, just because I find the discoveries so compelling and exciting. It truly feels like an exploration to the bottom of logic, much in the way that physics feels like an exploration to the bottom of reality. Really love the flow and the stylistic presentation. Been a fan for a minute, excited to see more

  34. Clark Massey

    Clark Massey

    Dag geleden

    This is excellent.

  35. Alaine Ninmah

    Alaine Ninmah

    Dag geleden

    I see math as a language rather than numbers and equations. And its a language that is ever 'evolving' yes evolving si the wrong word just can't think of the right word.

  36. Helgali

    Helgali

    Dag geleden

    I'm always sad about Alan Turing when I think about him. Humanity would advance so much more if there weren't things like racism, homophobia, sexism etc etc etc. :/

  37. Giovanni A

    Giovanni A

    Dag geleden

    I'm really into both mathematics and the office in this period 😂😂 I wouldn't have ever thought of that connection even after watching this video

  38. Dane Norman

    Dane Norman

    Dag geleden

    1931: "On formally undecidable propositions of Principia Mathematica and related systems" - Math Scholars 2021: Math is RACIST! - Math Professors

  39. Catchafire2000

    Catchafire2000

    Dag geleden

    Just absurd that because a man is gay, he was unaccepted... Although his invention helped to end the war!!!

  40. Naveenmurugu KANNAN

    Naveenmurugu KANNAN

    Dag geleden

    U found now only but I knew it during math class itself

  41. kirdiekirdie

    kirdiekirdie

    Dag geleden

    First time I watch this channel, the video and animation quality is top notch!

  42. School of Grok

    School of Grok

    Dag geleden

    Sublime.

  43. Sharan Pant

    Sharan Pant

    Dag geleden

    Can someone explain or refer me to a text explaining as to how the statement for Godel Number g is g?

  44. phil longneck

    phil longneck

    Dag geleden

    The unadvised bowling cumulatively sprout because centimeter frustratingly nod apropos a tacit siberian. oval, murky tornado

  45. Poke fun at idiots

    Poke fun at idiots

    Dag geleden

    There's some real smart arses commenting on this video. It proves some intelligent people do watch NLid

  46. Vedant Jagtap

    Vedant Jagtap

    Dag geleden

    At the end of the video u will realise that all our lives and this universe is a simulation with hyper realistic graphics and emotional ,thrilling story .

  47. Keven Gagner

    Keven Gagner

    Dag geleden

    Best video I've seen about maths in a while amazing work

  48. ludovic gauchet

    ludovic gauchet

    Dag geleden

    Quit your job as a barber, shave yourself, start new job as a barber. Solved!

  49. Dank Hill

    Dank Hill

    2 dagen geleden

    this seems like nonsense

  50. TinkyWinkyGaming

    TinkyWinkyGaming

    2 dagen geleden

    are there any more triple primes like 3, 5, 7 ?

    • TinkyWinkyGaming

      TinkyWinkyGaming

      2 dagen geleden

      oh right yeah one of them would have to be a multiple of 3 or 5

    • Релёкс84

      Релёкс84

      2 dagen geleden

      No, there are not. You can try to prove it if you want, it's actually rather easy. However there is a slightly different notion of prime triplets of the form (p, p+2, p+6) which are much more numerous.

  51. Stephen Appiah

    Stephen Appiah

    2 dagen geleden

    1. is math complete: is there a way to prove every true statement? does every true statement have proof? 2. is math consistent: is it free from contradictions? meaning one cannot prove A and not-A as true statements. 3. is math decidable: is there an algorithm that can always determine whether a statement follows from the axioms. no

    • Andre

      Andre

      Dag geleden

      For 2. it is unknown and hopefully not a "no". PS: And Gödels Theorems are not really meaningful for the "whole math" but just for formalized systems that are part of it. Maybe there are other such systems that can prove the consistency of the others - but they can't do it for themself and we are again stuck.

  52. Hovant

    Hovant

    2 dagen geleden

    Why do you retroactively change video titles so much?

  53. kartoffelmozart

    kartoffelmozart

    2 dagen geleden

    whenever i see somebody trying to explain the gödel incompleteness theorem i always get the feeling they are either leaving out something of huge importance that makes the proof work, or that gödel's theorem is really weak and it's getting far too much attention :p i mean my money is on the first one, but im getting slightly annoyed by people being super excited about showing a proof of the theorem, and then showing something really weak that gives a vague idea of what gödel was doing

    • Brauggi the bold

      Brauggi the bold

      2 dagen geleden

      Yes, the usual popscientific presentations of Gödel's theorems leave out large parts of the actual work that goes into the proof. In particular, they leave out almost all of the nitty gritty math. Given your name, I assume you speak german?

  54. M Mitchell

    M Mitchell

    2 dagen geleden

    Just one question: What?

  55. rahool jain

    rahool jain

    2 dagen geleden

    I am wondering if I watched this video or not? It's undecided!

  56. Y

    Y

    2 dagen geleden

    Turing was killed with a poisoned apple by MI6 (British secret service) he had takem to one bite out of it before he died... Steve Jobs made the 🍎 with a bite out of it the iconic symbol for the mac book as a sort of quiet homage for Alan's sacrifice

    • Poke fun at idiots

      Poke fun at idiots

      Dag geleden

      An inquest ruled that it was suicide, although this has been contested more recently, with Turing expert Prof Jack Copeland attributing his death to the accidental inhalation of cyanide fumes during an experiment.

  57. secondsun24

    secondsun24

    2 dagen geleden

    Wow. This is one of your best works, which is incredible since a lot you've published had been so carefully laid out and thought provoking. I love how you give body and life to complex topics so many of us don't have the time, expertise, or, often, the inclination to delve into in a formal way. Well done and thank you.

  58. Honest Ranking

    Honest Ranking

    2 dagen geleden

    Maths are a language. It can only be used to describe simplicity. You can't fully explain a very complex idea with words, and you can't explain the complexity of nature with any vocabulary. Maths are an attempt to describe something complex, only an attempt.

  59. Игорь Картохин

    Игорь Картохин

    2 dagen geleden

    19:46 what will stop me to write "1 = 0"? as i understand, its 7656 and this is incorrect. 2^7 * 3^6 * 5^5 * 7^6 ~= 3,4306448 × 10^13, thats number of incorrect card and who will stop this card?

    • Andre

      Andre

      2 dagen geleden

      _"thats number of incorrect card and who will stop this card?"_ Just because there is a card does not mean it is true or a proof.

  60. TheFinalChapters

    TheFinalChapters

    2 dagen geleden

    "There is no proof for the statement with Godel number g" is an infinite statement. Infinity itself results in paradoxes like the above, and only proves that infinite statements are not complete. Same for the turing box h. If you pass h into itself, or some similarly looping program, this inner h must then create another h inside itself, which then must create another h inside itself, infinitely. Except, this *can* be detected, and given the infinite loop nature, h will correctly report that it is an infinite loop. The only paradox, again, comes from infinity. In other words, there can still be completeness, decidability, and likely consistency for anything that does not involve infinity.

    • Andre

      Andre

      Dag geleden

      @Brauggi the bold There are some really strange ideas what "complete" means around here. A lot of people believe that an "infinite list" (what serves only as an illustration anyway) is never "complete", because there is always something "missing". Well... there is nothing "missing" of course and infinities do not match well with real life experiences. And a "complete statement"... what is a "not complete statement"? Nobody knows...

    • Brauggi the bold

      Brauggi the bold

      2 dagen geleden

      ​@TheFinalChapters - " write out g in its entirety" g in its entirety is a monstrously large equation and could probably fill several book pages. Your request is hardly reasonable. Still, there is nothing infinite about it. - without using anything but the initial "axioms". It's very clear that you're struggling to understand how mathematical statements are built and you try to fake having a clue. Hint: Axioms do NOT serve as the atoms. - "But if it's running h, then h must then run itself, which must run itself, which must run itself, etc." No, this is pure speculation on your behalf, guessing on how you would feel the proof works based on what you have seen on oversimplified NLid videos. - "'complete' was defined in the video." No, it was not. The video said what a complete formal system is. What you mean by "complete statement" remains a mystery.

    • Andre

      Andre

      2 dagen geleden

      @TheFinalChapters _""complete" was defined in the video."_ But not for statements. So I will repeat the question: What is a "complete statement" supposed to be? _" If there is no infinite statement, then write out g in its entirety"_ Gödel did it a bit differently than described in this video. _". If you pass h into itself, or some similarly looping program, this inner h must then create another h inside itself,"_ No. _" This is an infinite recursion disguised as a single statement."_ No. _"In other words, there can still be completeness, decidability, and likely consistency for anything that does not involve infinity."_ So for borings aspects only, But math always involve infinity - at least potentially. _"I'm only talking about the topics here,"_ No. You don't.

    • TheFinalChapters

      TheFinalChapters

      2 dagen geleden

      @Brauggi the bold If there is no infinite statement, then write out g in its entirety, without using anything but the initial "axioms". You can't, because the only way to do it is with g itself. "complete" was defined in the video. h determines if something concludes by running it. There is no other way to determine the result of an arbitrary program. But if it's running h, then h must then run itself, which must run itself, which must run itself, etc. This is an infinite recursion disguised as a single statement. I'm only talking about the topics here, not any other theories. To allow an infinite number of other theories into the discussion would be never-ending.

    • Brauggi the bold

      Brauggi the bold

      2 dagen geleden

      - "[...] is an infinite statement" There is no such thing as an infinite statement. - "infinite statements are not complete" What is a "complete statement" supposed to be? - "If you pass h into itself, or some similarly looping program, this inner h must then create another h inside itself" h reasons about its own Gödel number, not about itself. There are no copys created and no infinite regress is involved. - "there can still be [...] decidability, [...] for anything that does not involve infinity." That is demostrably wrong according to Trachtenbrot's theorem.

  61. Voltaire

    Voltaire

    2 dagen geleden

    True, False... Lie.

  62. April Fool

    April Fool

    2 dagen geleden

    That's the loophole in science that leaves room for god..... Ps.: No need to attack me, I don't believe in one ;)

  63. Kevin Scheuerman

    Kevin Scheuerman

    2 dagen geleden

    "Weird paradoxes that arise from self reference. There is a hole at the bottom of existence. We will never know everything with certainty. There will always be true statements that can't be proven." And then there are atheists.

  64. hsine benmessaoud

    hsine benmessaoud

    2 dagen geleden

    anyone else got a hard time to wrap his head around the h and h+ paradox ?

  65. Henry Glo

    Henry Glo

    2 dagen geleden

    9:00 I'm confused with the visual. "A set must contain itself." Does that mean that set A that contains another copy of itself? Set A = a, b, c, d. Then it must contain a, a, b, b, c, c, d, d?

    • Henry Glo

      Henry Glo

      2 dagen geleden

      @Релёкс84I get it now!!! Thank you!!! So it is not a sum!

    • Andre

      Andre

      2 dagen geleden

      @Henry Glo _"that's confusing. I'll just disregard the video's visuals."_ No.

    • Andre

      Andre

      2 dagen geleden

      @Henry Glo _"then what is it? Why can't it be a whole number?"_ Equal. Remember: {A,A} = {A}

    • Релёкс84

      Релёкс84

      2 dagen geleden

      @Henry Glo Think of a set in terms of computer science. A set is not the sum of its elements: instead, it is itself a single element, that is defined as containing other elements: there may be no elements contained by it, there may be infinitely many, and as it turns out the set itself can be one of these elements - but it is not a copy, it is the set itself. This works well in programming languages, so it's not a paradoxical concept - just a counterintuitive one.

    • Henry Glo

      Henry Glo

      2 dagen geleden

      @Lucas yes! That set must contain itself confuses me. Why should it contain itself? Isn't that the same as saying I have another me inside me?

  66. ludovic gauchet

    ludovic gauchet

    2 dagen geleden

    You can’t have a complete infinite list! So to say the number is not on the list is wrong again.

    • Andre

      Andre

      Dag geleden

      @ludovic gauchet _" cantor says he did"_ No he did, because everybody can check his proof. _"even u will understand that if u think about it a bit longer"_ So you still did not understand his proof? Well, try harder. It is really simple. Clever people will get it in a few minutes, but take your time. _", always easy putting things into theory, "_ Gibberish will help you how? _"anything can be put into words and yet most of it is impossible and not reality."_ We have a proof in reality. What do you have?

    • ludovic gauchet

      ludovic gauchet

      Dag geleden

      @Andre cantor says he did, but does not make it so just cause he says he did, even u will understand that if u think about it a bit longer , always easy putting things into theory, anything can be put into words and yet most of it is impossible and not reality.

    • Andre

      Andre

      Dag geleden

      @ludovic gauchet _"ok, make me that list and show me it’s not on there,"_ Cantor did it. It is shown in the video. There is no need to show it again. And the proof is really simple. Even you will understand it, if you think a bit longer. _" let me know when u finish that list "_ The list is already finished. There is no need to do it again.

    • ludovic gauchet

      ludovic gauchet

      Dag geleden

      @Andre ok, make me that list and show me it’s not on there, let me know when u finish that list 👍

    • Andre

      Andre

      2 dagen geleden

      @ludovic gauchet _"the list goes on forever,"_ No. Therew is no "forever". The list is just infinite and was complete from the beginning. _"so how can u say the number is not on it,"_ Because there is a proof it is not. _" you would have to reach the end of infinite"_ No, you have not. _"Impossible. You can’t put a number on infinite"_ You can prove it is not on the list.

  67. Bix McGoo

    Bix McGoo

    2 dagen geleden

    This isn't even math it's just a stupid cipher.

  68. Learn IT

    Learn IT

    2 dagen geleden

    Why can't statements be prooved true and false both with respect to a frame of reference?

  69. Abhijit Zimare

    Abhijit Zimare

    2 dagen geleden

    If Godel knew how to cook, and people are trying to poison home is Godel trying to poison himself?

  70. ShiroFPS

    ShiroFPS

    2 dagen geleden

    Bro you look kinda like Sam Riegel

  71. Russ Jones

    Russ Jones

    2 dagen geleden

    The barber paradox is one of my favorites.

  72. Aditya Biswal

    Aditya Biswal

    3 dagen geleden

    After watching the whole video, I forgot what I was gonna do the whole day.

  73. Roger Terrazas

    Roger Terrazas

    3 dagen geleden

    I opened mtg arena and then saw this vid and decided to watch it before I played. I don't know how to feel.

  74. Gabber Piet

    Gabber Piet

    3 dagen geleden

    Life has a flaw. Arrogance

  75. D p

    D p

    3 dagen geleden

    first off gurdle, g is not a number 😒

  76. Philip Swensen

    Philip Swensen

    3 dagen geleden

    But... "0 = 1"... is actually true. Or, at least cannot be proven false. No, I am not joking. Just ponder any common "little 't'" with "subscript '0'" attached to it in any common financial formula long enough. And... for the F*cking record, there is no such thing as a negative number, unless designed by humans as mathematical convenience, such as a designation behind a debt owed, but a debt isn't paid in negative numbers. The only actual requirement for negative numbers that I see for humans, is still a convenience tool, however, one could not describe their three-dimensional coordinates in any other way than negative numbers with our current system.... once again, constructed by humans. The Earth doesn't create many natural Circles, a Hexagon is a much more efficient shape.

  77. Peter Hewitt

    Peter Hewitt

    3 dagen geleden

    Maths! It's short for mathematics. Bugs me sorry. Great vid

    • Poke fun at idiots

      Poke fun at idiots

      Dag geleden

      @Релёкс84 yeah bruv tiz

    • Релёкс84

      Релёкс84

      3 dagen geleden

      Well it's mathematics, not mathsematics innit

  78. grovermatic

    grovermatic

    3 dagen geleden

    19:44 Oh for fuck's sake.

  79. David G

    David G

    3 dagen geleden

    Wow. Just WOW!

  80. grovermatic

    grovermatic

    3 dagen geleden

    14:23 I'm sorry, but the animation of HIlbert here makes him look like he's saying the word "poop" over and over again.

  81. grovermatic

    grovermatic

    3 dagen geleden

    I have so many questions. Chief among them: When I hear the word "true" in every video about this, is it the same "true" as in Boolean logic? Like true/false in computer code and whatnot? Because _that_ I have at least a basic grasp on. I've watched this damnable video at least three times now. :-P

  82. grovermatic

    grovermatic

    3 dagen geleden

    _"Now you would think that given the simple rules of the game, you could just look at any pattern and determine what will happen to it. Will it eventually reach a steady state, or will it keep growing without limit? But it turns out, this question is impossible to answer."_ Sounds like Turing's halting problem.

  83. Senderles

    Senderles

    3 dagen geleden

    Ahh discrete mathematics i love it

  84. B B

    B B

    3 dagen geleden

    I bet that guy from Will Hunting could figure it out... man he smart!

  85. Ivan Yogovich

    Ivan Yogovich

    3 dagen geleden

    32:34 Well, Time to print out every frame of this video and turn it into a flip book just to make that reasonable expectation false.

  86. Revenant S.

    Revenant S.

    3 dagen geleden

    Always felt math and science are incomplete.

  87. Alexandra Monaco

    Alexandra Monaco

    3 dagen geleden

    Good and evil? Could it come down to liking the good part of his enemy that doesnt like himself and agreeing they both dislike that part of himself. They can agree to disagree. OR why does he have to be friends with the enemies enemy? Do they HAVE to like one another even though they have a common interest? I'm left with many more questions with this theory lol

  88. the Rabbit Whisper

    the Rabbit Whisper

    3 dagen geleden

    and in less then a minute, i was drowning, beyond my pay grade

  89. orbit the moon

    orbit the moon

    3 dagen geleden

    watched this at 2AM and kept me interested the whole time. this is freaking mind-blowing. a truly brilliant video!

  90. Martha Soon

    Martha Soon

    3 dagen geleden

    My question is, how do we get the real numbers before we get to the point of creating the infinite numbers?

  91. Alfonso J. Ramos

    Alfonso J. Ramos

    3 dagen geleden

    Is g a natural number? Given: - The sequence of symbols is encoded as the product of non zero powers of consecutive primes, each factor representing a symbol. - A number n is encoded as n successor symbols followed by the zero symbol. It follows that: - The encoding of 0 is greater than 0. - The encoding of n+1 is greater than the encoding of n. - Adding symbols can only increase the value of the encoding. Therefore: - The encoding of a number is greater than the number. - The encoding of a expression that includes a number is greater than that number. Thus, the encoding of any expression that includes its encoding, must greater than its encoding. And since there is no natural number m such that m is greater than m. We must conclude that g is not a natural number. This begs the question: Do all true expressions that have natural number encoding have proof?

    • Andre

      Andre

      2 dagen geleden

      @Релёкс84 This is correct. And there are videos of math lectures out there, that are more precise. But the general idea of the video is correct.

    • Релёкс84

      Релёкс84

      3 dagen geleden

      Gödel's proof is far, far more intricate than what is presented in this video. I do not claim to understand it, and if you want to take a dive you can look it up: it's quite long.

  92. alinonymous

    alinonymous

    3 dagen geleden

    Congrats for this truly great video! It makes me almost love that which I hate; uh, isn't that the point where human intervention fringes on divine? Keep the good work going (in the same admirable English, pls.).

  93. Alfonso J. Ramos

    Alfonso J. Ramos

    3 dagen geleden

    9:40 The Barber: I Sexually Identify as an Attack Helicopter, therefore I'm not a man, therefore the shaving law don't apply to me. Solved.

  94. Marco Moretto

    Marco Moretto

    3 dagen geleden

    Gödel, Escher, Bach is the book that got me in bioinformatics. Amazing video as always. I wonder if, speaking about recursion, you want to add something about RNA as a molecule that encodes instructions about itself (and how to duplicate like in Viruses).

  95. mthlay15

    mthlay15

    3 dagen geleden

    11:20 that's a veritasium logo, for sure

  96. Roman Vereb

    Roman Vereb

    3 dagen geleden

    Good ol' numbers.

  97. Tim Sagichnicht

    Tim Sagichnicht

    3 dagen geleden

    Damn this video give me goose bumbs even if i dont really understand all of it. Still thankful for your channel❤️

  98. Tim Sagichnicht

    Tim Sagichnicht

    3 dagen geleden

    I understand the digonalization proof but still dont understand why there are more real number then natural numbers. I mean yes with the digonalization proof there always is a new real number but you also can count one more to the natural number. I know the idea is to write down all the natural and real numbers and then use the digonalization proof.. but the problem is you can not write down all the numbers and thats why you can not do a hypothesis like this. You can try to explain to me if i‘m wrong but i think i actually understand it. I just dont male sense to me

    • Moyprod

      Moyprod

      2 dagen geleden

      The key point is the following. You actually have already used ALL natural numbers to count the real numbers on the list. So you already counted to infinity and beyond. So your "one more than infinity" number is already being used to count the "one more than infinity number" from the real numbers. You can not use that for counting the newly created diagonal number. Hope it makes sense.

    • Luzid

      Luzid

      3 dagen geleden

      Hey. I was also confused by this for some time until I learned the formality at uni. 2 Sets are the same size if you can find a bijection between them (1 to 1 correspondence). Lets assume we have such a bijection between the naturals (input) and the reals (output) and call that funktion f. That's exactly what the list is. So I can give you ANY natural number n and you can find the output f(n) as a real number. however. If you do the diagonalisation you found a number that NO natural number can map to with the use of f. Therefore f can't be bijective -> The set of real numbers is bigger than the set of natural numbers.

  99. Scott Wallace

    Scott Wallace

    3 dagen geleden

    I'd say that _fatal_ flaw is a bit exaggerated. Math still is, after all, good for counting apples, or getting a space ship to Mars. Pretty good for something dead.

  100. Mathieu Nunu

    Mathieu Nunu

    3 dagen geleden

    Hey, geeks! Cantor's Diagonalization Proof is a number built from an infinite set of numbers and thus can never be completed to then be checked against the list of Real Numbers in the Natural Number "Index." If you ever stop adding 1 to a decimal place to check the number against the list, it will exist. How am I wrong? I have to be wrong.

    • Andre

      Andre

      3 dagen geleden

      _" If you ever stop adding 1 to a decimal place to check the number against the list, it will exist."_ You don't have to stop. The constructed number is a real number. _"How am I wrong?"_ The list never has to be completed in reality. _"I have to be wrong."_ And you are. Listen again to Cantors Diagonalization Proof.

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