The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy

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  • Опубліковано 22 гру 2024

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  • @RyeedAglan
    @RyeedAglan 3 роки тому +2471

    LOL, Godel, Noether, and Hilbert are drawn so adorable

    • @epicstuff7522
      @epicstuff7522 3 роки тому +5

      @@DyslexicMitochondria Hey bro I watch ur videos. Love your channeI

    • @geekoutnerd7882
      @geekoutnerd7882 3 роки тому +10

      Ohhh it was Hilbert! I was thinking Russell.

    • @sidcord7118
      @sidcord7118 3 роки тому +21

      Who is LOL?

    • @marisaliu3236
      @marisaliu3236 3 роки тому +2

      Definitely one of my favorite videos animation wise

    • @sidcord7118
      @sidcord7118 3 роки тому +4

      @ʜᴏɴᴇʏᴘɪᴇ bruh i was just kidding 😆

  • @pratikshahgreat
    @pratikshahgreat 3 роки тому +2623

    "Jim is his own worst enemy, and enemy of my enemy is a friend. But...." - Dwight Schrute

  • @ossapinhosfazemhumah
    @ossapinhosfazemhumah 3 роки тому +3812

    "Its Gödels all the way down"
    most underrated joke in this entire series.

    • @thomasstambaugh5181
      @thomasstambaugh5181 3 роки тому +28

      Surely a reference to Terry Pratchett ...

    • @fmanda
      @fmanda 3 роки тому +53

      Came to the comments to say exactly this and was glad to see it was already taken care of.

    • @dineshgoswami6237
      @dineshgoswami6237 3 роки тому +35

      Came to the comments to say exactly this and was glad to see it was already taken care of. (2)

    • @devilvocano420
      @devilvocano420 2 роки тому +11

      @@dineshgoswami6237 wat does it mean

    • @joy1ess
      @joy1ess 2 роки тому +13

      @@devilvocano420 brief history of time.. stephen hawking

  • @cyberkebab
    @cyberkebab 3 роки тому +10350

    "Breaking math" is the most badass thing a person could ever achieve and you can't change my mind

    • @theocho
      @theocho 3 роки тому +405

      Can you prove that your mind cannot be changed?

    • @floweyfangirl69420
      @floweyfangirl69420 3 роки тому +72

      @George Khoory agreed, he just "broke" the concept of "math is this because of this, period"

    • @nailboard6492
      @nailboard6492 3 роки тому +35

      @George Khoory it is not 100% impossible to violate physics and you can't prove otherwise.

    • @NamNguyen-my4ec
      @NamNguyen-my4ec 3 роки тому +38

      @George Khoory imo just like math or any other things, physic is just knowledge and physic laws are just agreement that people agree upon based on what have been known, and because of the fact that there are things that we dont know that we dont know, physic laws maybe true today but not so true in the future. That why i thing it is definitely possible to violate a 'current' physic law

    • @SlowReverbDHH
      @SlowReverbDHH 3 роки тому +1

      How about breaking the jokers mind to where he snuffs it? 🚨🚨🚨

  • @vanditrikhi9984
    @vanditrikhi9984 3 роки тому +2996

    Isaac Newton dancing for Gödel is now ingrained in my mind.

  • @thealienontheinternet
    @thealienontheinternet 3 роки тому +1755

    Imagine if he didn’t manage to prove his Incompleteness Theorems because they turned out to be true but unprovable. Complete mindfuck that would have been

    • @patrickdelana8953
      @patrickdelana8953 3 роки тому +9

      =))))

    • @Sir-Taco
      @Sir-Taco 3 роки тому +209

      If they were true but unprovable that would have been proof right there, but that means it would be true and provable which makes it so there is no proof, which would make it true and unprovable

    • @divyaojha9149
      @divyaojha9149 3 роки тому +8

      @@Sir-Taco wow

    • @jonathancaz7012
      @jonathancaz7012 2 роки тому +44

      @@Sir-Taco if the only way to prove that some mathematical states are unprovable is itself unprovable, then you can't know if things are unprovable. You wouldn't know it was unprovable. It would just look like it was very hard to solve.

    • @benheideveld4617
      @benheideveld4617 2 роки тому +5

      I think that is the case! Incompletene or Inconsistent may be Unprovable truth.

  • @Morningstar_37
    @Morningstar_37 3 роки тому +18207

    A long time ago, the mathmaticians lived together in harmony. But everything changed when Gödel published his incompleteness theorem
    guys it's been over a year, PLEASE stop replying
    Guys it was funny for the last two years, but you can stop replying to this now.
    help

    • @nHans
      @nHans 3 роки тому +615

      Mathematicians have regularly suffered existential crises since the beginning of history: Zeno's Paradox, irrational numbers, Non-Euclidean Geometry, Russell's Paradox, Halting Problem etc. One would think they'd have gotten used to it by now.

    • @ELYESSS
      @ELYESSS 3 роки тому +47

      @@nHans isn't this video's paradox the same as the halting problem?

    • @nHans
      @nHans 3 роки тому +74

      ​@@ELYESSS There are similarities in the proofs, yes, in the sense that in both proofs, you use self-reference to create a paradox. The problems themselves are, of course, quite different and were solved by different people:
      • Incompleteness: Gödel
      • Halting Problem: Independently by Alonzo Church and Alan Turing.
      (I know that Turing's proof uses self-reference; unfortunately I'm not aware how Church solved it.)

    • @Founderschannel123
      @Founderschannel123 3 роки тому +18

      @@nHans okay lets say paradoxes have ruined mathematician lives and not even einstein would even safe math

    • @kanzleribrahim6596
      @kanzleribrahim6596 3 роки тому +114

      it all changed when the fire nation attacked

  • @neizanmendez6317
    @neizanmendez6317 3 роки тому +11167

    Gödel casually making mathematicians notice that they have wasted 20 years of their lives in an unsolvable problem while being that cute in the video lol

    • @DecemberGalaxy0
      @DecemberGalaxy0 3 роки тому +208

      At least they wouldn't spend more

    • @foxgaming76yt24
      @foxgaming76yt24 3 роки тому +26

      Yeah

    • @alvarezjulio3800
      @alvarezjulio3800 3 роки тому +155

      Godel was a Mathematician too. I understand that people might be confused about logic. But logic is just a branch of mathematics.

    • @nagavignesh9292
      @nagavignesh9292 3 роки тому +41

      Still , math works , we live in the world full of stuffs that uses maths directly or indirectly

    • @name5702
      @name5702 3 роки тому +28

      Life is a waste no matter what you do

  • @cullenmott7614
    @cullenmott7614 3 роки тому +373

    *"...and* *he* *was* *even* *less* *confident* *that* *Mathematics* *was* *the* *right* *tool* *to* *investigate* *this* *problem."*
    This is a big lesson on life. Sometimes you have to go outside the system to identify and solve the problems within it. And you have to have the courage to do so, even if doing so leaves you completely alone, and working completely alone for a long time. The moment you get to strongly suspecting there's a major problem in something that matters a lot to you, you should start looking into it and you should consider that the structure of the system in which the problem lies may be contributing to it. The work of Gödel is a textbook example of this: the structure of Mathematics disguises paradoxes within Axiomatic proofs. So Gödel divested a bit from Mathematics and went outside the field to (a part of Philosophy called) Logic to try to identify (and maybe even solve) these problems. And he ended up revolutionizing his field.
    Whether the system is mathematics, or the world-economy, or even your own government, if there are problems you're noticing more and more, you may have to go outside the system to truly understand what's going on. The system may be disguising or even contributing to these problems. And you may be the only one who can solve (or begin to solve) them because you may be the only person who is able to see them.
    And so the work begins, and in all likelihood it's going to be heavy. And as you work, you may have to endure a lot of push-back and isolation before you can make a big change happen. There's almost always consequences for people trying to fix the problems of the world. You should do it anyway. It's only through honestly representing your truth in the face of the falsehoods of your era that you and the world will know peace.

    • @naimejb7921
      @naimejb7921 2 роки тому +12

      brother, thank you for this comment.., I appreciate it... I

    • @cullenmott7614
      @cullenmott7614 2 роки тому +2

      @@naimejb7921 you’re welcome ✌️

    • @cullenmott7614
      @cullenmott7614 2 роки тому

      @Anonymous :)

    • @aslampervez2294
      @aslampervez2294 2 роки тому

      Thanks

    • @cullenmott7614
      @cullenmott7614 2 роки тому

      @Israel Gawiseb you’re welcome! Thank you for your kind words.

  • @AKHELUS.
    @AKHELUS. 3 роки тому +6951

    Mathematicians: trying to prove that all equations can only be true or false
    Gödel: hippity hoppity your certainty is my property

    • @eternallight88
      @eternallight88 3 роки тому +32

      Lol

    • @ameen2428
      @ameen2428 3 роки тому +31

      sheldon, is that you?

    • @historicwine1283
      @historicwine1283 3 роки тому +10

      Wrong

    • @AKHELUS.
      @AKHELUS. 3 роки тому +8

      @@historicwine1283 w-what's wrong 😟 😅

    • @BrunoTurcatti
      @BrunoTurcatti 3 роки тому +23

      @@AKHELUS. What Gödel says is that there are statements that cannot be proved, no that they are not true nor false.

  • @wiandryadiwasistio2062
    @wiandryadiwasistio2062 3 роки тому +4518

    heisenberg's uncertainty principle: *here comes trouble...*
    gödel's incompleteness theorem: *...and make it double!*

    • @mrsugar7528
      @mrsugar7528 3 роки тому +206

      Imagine a hero mathematician comes out of nowhere and just solves them like that russian mathemician dude who solved an equation which was very difficult and just went back into living his normal life as a regular dude

    • @randomname285
      @randomname285 3 роки тому +173

      To protect the world from assertations
      To confuse the people of every nation
      To denounce the evils of truth and false
      To extend all measurements containing faults
      Werner
      Kurt
      Team Uncertainty put error bars on the speed of light
      Surrender now or your certitude will be out of sight
      Meowth, is that right?

    • @chriskandt150
      @chriskandt150 3 роки тому +35

      That's a definite A for effort. Was not expecting this on the comments. Thank you

    • @JohnathanLeeSprite
      @JohnathanLeeSprite 3 роки тому +26

      @@mrsugar7528 Grigori Perelman on Poincare conjecture?

    • @Hak616
      @Hak616 3 роки тому +9

      @@JohnathanLeeSprite That guy looks like a hippie but takes life seriously. But he seemed like an idealist. He should have just taken the Fields Medal.

  • @bruceli9094
    @bruceli9094 2 роки тому +1007

    Godel almost received his Nobel Prize but his theorem was incomplete.

    • @cerendemir9977
      @cerendemir9977 Рік тому +120

      Nice one! But mathematicians don't get a Nobel, they have the Fields Medal.

    • @rince7A
      @rince7A Рік тому +6

      @@cerendemir9977 John Nash got it.

    • @cerendemir9977
      @cerendemir9977 Рік тому +37

      @@rince7A Yes, in economy

    • @phillustrator
      @phillustrator Рік тому

      ​@rince7A The Nobel Memorial prize of economics is a fake Nobel

    • @gotfan7743
      @gotfan7743 Рік тому +8

      @@cerendemir9977 Fields Medal was introduced in 1936 and Goedel who lived until 1978 did not receive it.

  • @DGHeina
    @DGHeina 3 роки тому +2110

    For those who don't quite understand what "It's Godel all the way down" means.
    The phrase "It's turtles all the way down" comes from an anecdote told in the opening to Stephen Hawking's A Brief History of Time(Edit: although it is not the anecdote's or the saying's real origin):
    A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise." The scientist gave a superior smile before replying, "What is the tortoise standing on?" "You're very clever, young man, very clever", said the old lady. "But it's turtles all the way down!"
    The phrase is used to describe any system that appears to have dependencies that never end. In the anecdote old lady said that the Earth is flat and is supported on the back of a turtle, but that creates the fact that the turtle needs something to stand on, so the lady says that "It's turtles all the way down" which means that every turtle stands on another turtle which stands on another turtle and so on.
    For another example, imagine accountability in a (hypothetical) police department. The citizens are policed by police, the police are policed by internal affairs, which might lead to the formation of an "internal internal affairs" to police internal affairs. Someone might describe this system of policing as "turtles all the way down", meaning that the system of policing never ends.
    So "It's Godels all the way down" means that even if someone tries to make unprovebly true statements new axioms it would create new unprovebly true statements and if someone tries to make them axioms there will be new unprovebly true statements and so on, like said in the video.
    I do hope it wasn't long enough for you to get bored, but detailed enough so that you now get what's going on. Have a great day.
    Edit courtesy of ​ @silver6054 : In the form of "rocks all the way down", the saying dates to at least 1838, when it was printed in an unsigned anecdote in the New-York Mirror. A version of the saying in its "turtle" form appeared in an 1854 transcript of remarks by preacher Joseph Frederick Berg addressed to Joseph Barker:
    My opponent's reasoning reminds me of the heathen, who, being asked on what the world stood, replied, "On a tortoise." But on what does the tortoise stand? "On another tortoise." With Mr. Barker, too, there are tortoises all the way down. (Vehement and vociferous applause.)
    - "Second Evening: Remarks of Rev. Dr. Berg"
    So, I suppose Stephen Hawking was just the more known person to popularize the saying. Thanks to @silver6054, again, for the correction.

    • @heyved11
      @heyved11 2 роки тому +71

      Well you'd like to know i generally don't read this much in a comment like you said "get bored" but I did find your explanation so intriguing... that look! I am even leaving a comment to notify you😂. It was good btw!

    • @TheNabominable
      @TheNabominable 2 роки тому +19

      Thanks for this explanation. I'm no mathematician (far from it) but very intrigued by them, and I found this video quite distrubing and interesting at the same time. Your comment just completed with a great metaphor the theory of unprovable axioms I wasn't sure to get properly. Plus the fact that's still very modern problem (flat earthers and so on).
      Thanks a lot a lot a lot (and so on ... :D )

    • @rohithkumarbandari
      @rohithkumarbandari 2 роки тому +9

      Kudos to your patience.

    • @rodrigoaguiar5208
      @rodrigoaguiar5208 2 роки тому +6

      Ty! You da real mvp

    • @jahnvijoshi690
      @jahnvijoshi690 2 роки тому +2

      Men thankyou thankyou thankyou thankyou

  • @ordisidro5927
    @ordisidro5927 3 роки тому +1715

    Teacher: Why didn't you show your complete solution?!?
    Me: well ma'am, according to the Incompleteness Theorem....

    • @tetrachart4156
      @tetrachart4156 3 роки тому +141

      Teacher : but that solution was proved to be proveable , i know since *I DID BY SOlVING IT*.

    • @Lone-Lee
      @Lone-Lee 3 роки тому +81

      Teacher: "So you've chosen death 💀"

    • @tetrachart4156
      @tetrachart4156 3 роки тому +10

      @@LolwutLol2000 They are an definition , not a statement.

    • @-Subtle-
      @-Subtle- 3 роки тому +50

      Teacher: I'm well aware of Godel. Too bad your oversimplified Ted-Ed video didn't teach you enough.

    • @countrymanrandylewis8463
      @countrymanrandylewis8463 3 роки тому +12

      @@-Subtle- you may have a brain but... I HAVE A GUN

  •  3 роки тому +77

    I remember when I was in thrid gradr my math books had written in the cover "Maths make sense". And as a kid that hated math, I spent time trying to figure out any mistake in it, something that didnt make sense. I actually did it a few times, buuut it was actually just me making mistakes, not maths. Well, glad to see one guy did my childhood quest

    • @zeronothinghere9334
      @zeronothinghere9334 10 місяців тому +2

      But it still makes sense

    • @benzene_sandwich
      @benzene_sandwich 5 місяців тому

      terrance howard is like that. Thought he was breaking math but he was actually just making mistakes.

  • @thetsarofall8666
    @thetsarofall8666 3 роки тому +913

    Me: can barely do fractions
    Gödel: *breaks the entire field of mathematics spine over his knee*

    • @vaivs7903
      @vaivs7903 3 роки тому +6

      He Bautista Bombed it on a table!

    • @dhruvakhera5011
      @dhruvakhera5011 3 роки тому +14

      Gödel is the bane to maths’s batman

    • @wilforddraper1894
      @wilforddraper1894 3 роки тому

      It's more like giving it the ability to turn invisible

    • @kohwenxu
      @kohwenxu 3 роки тому +1

      50 years of attempts, beginning with the work of Gottlob Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics just gets thrown under the bus when Godel finds his Theorem

  • @arjunpardal1144
    @arjunpardal1144 3 роки тому +1765

    “Someday a real rain will come and wash all the certainty off the streets “-Gödel

    • @mrrodriguezHLP
      @mrrodriguezHLP 3 роки тому +98

      I also like his other quote:
      "The accumulated filth of all their certainty will foam up about their waists and all the physicists and mathematicians will look up and shout 'Save us!' and I'll whisper 'No.'" -Gödel

    • @sinnerman-i4r
      @sinnerman-i4r 3 роки тому +3

      Certainly.

    • @mattphorwich
      @mattphorwich 3 роки тому +3

      Wow!! Profound!!

    • @asherwade
      @asherwade 3 роки тому +42

      "He didn't say that." - A. Einstein

    • @arjunpardal1144
      @arjunpardal1144 3 роки тому +18

      @@mrrodriguezHLP "I guess it comes down to a simple choice,really,get busy proving or get busy assuming"-Gödel

  • @Xx_Eric_was_Here_xX
    @Xx_Eric_was_Here_xX 2 роки тому +24

    not only do i appreciate the concise synopsis of the theorem, i also appreciate getting to see godel dancing around in glee

  • @itzjustnub5179
    @itzjustnub5179 3 роки тому +2944

    “The man who broke math”
    Me when the calculator says *syntax error*:
    *look what he needs to mimic a fraction of my power*

    • @amiqai
      @amiqai 3 роки тому +7

      +

    • @caseysimmons9578
      @caseysimmons9578 3 роки тому +9

      Lol

    • @caseysimmons9578
      @caseysimmons9578 3 роки тому +66

      My math teachers always hated my questions. Ignored them with nervous laughter or acted bothered or said something along the lines of "that's a whole other discussion." Yeah. They were afraid of me.

    • @notbob2334
      @notbob2334 3 роки тому +74

      @Franklin Roe how to make baby? I want baby

    • @crackaby7075
      @crackaby7075 3 роки тому +35

      @Franklin Roe why was six scared of seven?

  • @lardna
    @lardna 3 роки тому +1619

    After this video, only one statement comes to mind: "I understand nothing" -Michael Scott

    • @donttalkaboutmymomsyo
      @donttalkaboutmymomsyo 3 роки тому +18

      Who knew Michael Scott was a fan of Socrates all along?

    • @qinisodlamini1139
      @qinisodlamini1139 3 роки тому +14

      What is there to understand if there's nothing in the first place 😂😂

    • @ahnjoseph6875
      @ahnjoseph6875 3 роки тому +4

      @@nada__ that is actually a really good example!

    • @secretfolders6069
      @secretfolders6069 3 роки тому +7

      Its basically schrodingers cat,
      You cant prove its dead or alive until youve opened the box

    • @johncaiwa
      @johncaiwa 3 роки тому +20

      well they made a whole video about math without showing the equation they were talking about. no one could understand what is not shown

  • @scottrackley4457
    @scottrackley4457 Рік тому +24

    A math professor I had said this, "No arbitrary system of rules can explain itself without external input"

    • @-TheUnkownUser
      @-TheUnkownUser Рік тому

      Alfred Tarski.

    • @Johnny_Appleweed
      @Johnny_Appleweed 5 місяців тому

      Same reason you can't use a word in its own definition. It's circular logic.

  • @justcocomoran5233
    @justcocomoran5233 3 роки тому +4782

    You guys should make a video about me titled: "the man math broke"

    • @manuelmathew848
      @manuelmathew848 3 роки тому +153

      that would be pretty unremarkable seeing that since every highschooler can be called that

    • @spoopyscaryskelebones3846
      @spoopyscaryskelebones3846 3 роки тому +27

      @@manuelmathew848 not every :)

    • @manuelmathew848
      @manuelmathew848 3 роки тому +41

      @@spoopyscaryskelebones3846 goddamit

    • @lechihieu2606
      @lechihieu2606 3 роки тому +21

      @@manuelmathew848 more than 50% then 😂

    • @alt1f4
      @alt1f4 3 роки тому +11

      Underrated comment

  • @letsgetreal2501
    @letsgetreal2501 3 роки тому +1009

    Socrates: *How did this guy not get poisoned?*
    Gödel: Oh wait...

    • @prod.hxrford3896
      @prod.hxrford3896 3 роки тому +18

      Underrated comment

    • @maximumoverdrive2676
      @maximumoverdrive2676 3 роки тому +78

      @@Athlin I’m pretty sure Godel had a severe fear of being poisoned. He only trusted his wife to prepare him food and refused to eat once she died until he died of malnutrition. Something along those lines

    • @franxx941
      @franxx941 3 роки тому +17

      @@maximumoverdrive2676 Wow that actually a pretty terrible way to die. Was really someone after him have there been aptempt at his life or did he become paranoid?

    • @lechihieu2606
      @lechihieu2606 3 роки тому +47

      @@maximumoverdrive2676 he is just paranoid, his wife was not dead beside, she just has to go to hospital for six months because of stroke and he hadn't eaten much the whole time. When she came back, he was 30kg. She brought him to hospital immediately, but unfortunately, some week later...

    • @Yokuyin
      @Yokuyin 3 роки тому +29

      @@franxx941 He became paranoid after his close friend Moritz Schlick was murdered.

  • @BattyBest
    @BattyBest 2 роки тому +52

    Dude took "Math is just numbers" to a whole new level

  • @Pastamistic
    @Pastamistic 3 роки тому +3963

    I can't help but feel like this guy's motivation for developing this was nothing more than spite.

    • @doilyhead
      @doilyhead 3 роки тому +68

      No. The Cantor diagonal problem is another demonstration of the same thing. In some ways it's easier to understand, depending on how one learns.

    • @thv.taecrew
      @thv.taecrew 2 роки тому +16

      true lol. ig spite is one of the primary motivation huh

    • @yve4889
      @yve4889 2 роки тому +11

      Soo.. Does this mean I can write this in my upcoming math test?

    • @UserName-mf9db
      @UserName-mf9db 2 роки тому +5

      if the guy above the Yve named JiminTae was i, we would form the word "guy" with our names

    • @yve4889
      @yve4889 2 роки тому +5

      @@UserName-mf9db now that's another level of observation 😎😂

  • @Brandoon296
    @Brandoon296 3 роки тому +930

    “It’s Godels all the way down” BRILLIANT

    • @davidowen4816
      @davidowen4816 3 роки тому +9

      Yeah, I loved that too. There's a pompous mathematician friend of mine who I can't wait to use it on.

    • @scionyx
      @scionyx 3 роки тому +4

      Physics: Turtles.
      Coding: Hand Grenades.

    • @flowercities
      @flowercities 3 роки тому +7

      i'm smooth brain, could someone please explain this to me? i've heard of "turtles all the way down" but i have no clue what it means

    • @DGHeina
      @DGHeina 3 роки тому +155

      @@flowercities The phrase comes from an anecdote told in the opening to Stephen Hawking's A Brief History of Time:
      A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise." The scientist gave a superior smile before replying, "What is the tortoise standing on?" "You're very clever, young man, very clever", said the old lady. "But it's turtles all the way down!"
      The phrase is used to describe any system that appears to have dependencies that never end. In the anecdote old lady said that the Earth is flat and is supported on the back of a turtle, but that creates one fact: the turtle needs something to stand on, so the lady says that "It's turtles all the way down" which means that every turtle stands on another turtle that stands on another turtle and so on.
      For another example, imagine accountability in a (hypothetical) police department. The citizens are policed by police, the police are policed by internal affairs, which might lead to the formation of an "internal internal affairs" to police internal affairs. Someone might describe this system of policing as "turtles all the way down", meaning that the system of policing never ends.
      So "It's Godels all the way down" means that even if someone tries to make unprovebly true statements new axioms it would create new unprovebly true statements and if someone tries to make them axioms there will be new unprovebly true statements and that's a never-ending cycle, like said in the video.
      I do hope that wasn't long enough for you to get bored and give up, but detailed enough so that you now get what's going on. Have a great day.

    • @VGInterviews
      @VGInterviews 3 роки тому +21

      @@DGHeina very good explanation, thanks

  • @ByTobys
    @ByTobys 3 роки тому +23

    I love how it’s Hilbert (we must know, we will know) who’s walking down the cliff at 4:09

  • @kenanwisaksenahudawan4271
    @kenanwisaksenahudawan4271 3 роки тому +687

    now let me introduce myself. I, the man who's broken by math

  • @Yash-wm1nj
    @Yash-wm1nj 3 роки тому +292

    4:05 That man with the hat falling down the hill was the great mathematician David Hilbert....who asked 3 most important question about whether math is complete, consistent or decidable. Gödel answered the first question using his Incompleteness Theorem.

    • @julienbara3082
      @julienbara3082 3 роки тому +1

      thanks i wasn't able to remember his name ^^

    • @ronharleypantaleon1824
      @ronharleypantaleon1824 3 роки тому +4

      The one who raced with Einstein about the equation of TGR?

    • @vitorpremoli5379
      @vitorpremoli5379 3 роки тому +8

      @@ronharleypantaleon1824 yes, the same Hilbert

    • @comh33
      @comh33 3 роки тому +29

      Fun fact, It was later proven that math is also undecidable.
      What does this mean? It means that there are some equations, algorithms, processes, and changing arrays that we will never know whether or not they come to a conclusions or loop endlessly.
      I believe Gödel also proved that mathematics cannot prove its own consistency. In this context, consistency means 2+2=4, always, or that adding two numbers always creates a bigger number. While some basic statements are pretty much a no brainer, the foundations of mathematics cannot be used to prove their own validity or consistency.
      So at the very best, mathematics is either:
      1). Incomplete, Consistent (but we will never know), and undecidable
      2). OR incomplete, inconsistent, and undecidable

    • @benjamindrhee
      @benjamindrhee 2 роки тому +8

      @comh33 I believe that Gödel proved in his second incompleteness theorem that the statement "Mathematics is consistent" also falls into the category of statements that can not be proved but is true regardless, just like those self-referencing axioms which cease to be proved by it's axiomatic/logical system.

  • @TheVofD
    @TheVofD 3 роки тому +20

    Truly wonderful detail, that you used Hilbert as the character, desperatly tries to fix the towers. since he dreamed the most about a complete system of axioms! wonderfull

  • @coolaznboy98
    @coolaznboy98 3 роки тому +320

    Veritasium did an amazing job explaining Godel's Incompleteness Theorem. I highly recommend for everyone if they want a more in depth video of the theorem.

    • @briansammond7801
      @briansammond7801 3 роки тому +10

      Yes, Veritasium did a much more thorough and in-depth exploration of Godel

    • @TheMcKenzieHaus
      @TheMcKenzieHaus 3 роки тому +3

      Yeah I saw that! I was eating and I almost choked lol 😂 I was so shocked lol

    • @nHans
      @nHans 3 роки тому +45

      Veritasium misunderstood Gödel’s theorem-specifically the part where it states that certain *true* statements cannot be proved. Veritasium ignored the 'true' part, and went on to claim that any currently unproven statement like the Riemann Hypothesis might be unproveable due to Gödel’s theorem.
      But that's wrong. It has been known since Euclid's times that if you start with a finite number of axioms (or axiom schemas), there are always statements that cannot be proved using those axioms. What's new in Gödel's theorem is that even 'true' statements cannot be proved.
      Most people have difficulty understanding the concept of "true but unprovable." Is the Riemann Hypothesis true but unprovable? Or, like Fermat's Last Theorem, is it just a matter of time before somebody proves it?
      Veritasium unfortunately created some confusion in that matter. Some students came up to me and said that the Riemann Hypothesis cannot be proved because of Gödel's theorem, and referred to Veritasium's video as the source.
      Don't misunderstand me, I love Veritasium's videos in general. But Derek frequently gives in to hyperbole and click-bait titles, possibly because he depends on UA-cam ad revenue for a living. So he called Gödel’s theorem a 'fatal' flaw in mathematics. Well, that 'fatal' flaw has existed in mathematics since the beginning of time, but mathematics is still alive and going strong. Forget being dead, it's not even crippled. 🤣

    • @bumwau
      @bumwau 3 роки тому

      Agreed. Feel like TED-Ed took the idea after realising they've been a bit too bias politically lately...

    • @bumwau
      @bumwau 3 роки тому +1

      @@nHans That's not what interpreted but, as always, it was a long video connecting several ideas

  • @Irondragon1945
    @Irondragon1945 3 роки тому +112

    There's something cathartic about even mathematics, something we both create and discover, having equally mysterious side as discoverries in nature and space.

    • @TheMcKenzieHaus
      @TheMcKenzieHaus 3 роки тому +1

      Agreed, the inherent mystery of the universe, it’s pretty fascinating

    • @historicwine1283
      @historicwine1283 3 роки тому +3

      I think it shows that mathematics isn't merely invented

  • @y2kmedia118
    @y2kmedia118 11 місяців тому +17

    The art and sound design of this video is especially satisfying.

  • @faizanquraishi4126
    @faizanquraishi4126 3 роки тому +2616

    Mathematicians : *represent numbers by alphabets.*
    Godel: *turns alphabets back to numbers.*
    Math students: "Is this the power of a god?"

    • @mayankrathore8721
      @mayankrathore8721 3 роки тому +37

      Gaara from Naruto. I understood it. Nice one

    • @earthgrazer2164
      @earthgrazer2164 3 роки тому +21

      More like he turned it into holders of information that could be represented in language

    • @semerendocr
      @semerendocr 2 роки тому +4

      Isaacus Neuu­to­nus.
      /*Je­ho­va Sanc­tus Unus*/

    • @sukhmandersingh4306
      @sukhmandersingh4306 2 роки тому +6

      No the power of godel

    • @איןסוף
      @איןסוף 2 роки тому +4

      gödel was a mathematician himself

  • @shaheen4663
    @shaheen4663 3 роки тому +339

    Ah here is another video which makes me question everything I've ever studied

  • @sina8883
    @sina8883 21 день тому

    Thank you for this video! Most explanations I have heard of this are either so simplistic as to mean nothing, or so long, technical, and complicated as to be nonsensical to a non-mathematician like me. This was the perfect balance to explain this theorem in a intuitive way to a non-mathematician. Thanks again!

  • @Mkhehla
    @Mkhehla 3 роки тому +237

    "There are known unkowns, and there are unknown unknowns... Things we don't know that we don't know!!"

    • @stansantos4733
      @stansantos4733 3 роки тому +7

      May Rumsfeld Rest In Peace.

    • @rogerforsberg3910
      @rogerforsberg3910 3 роки тому +7

      @@stansantos4733 I believe that this came from Rumsfeld's recollection of the Allegory of the Cave from Plato's Republic (Book VII).

    • @topg2820
      @topg2820 3 роки тому +3

      He said it to justify the Iraq war, what he didn't tell people is that the 4th kind is also there, the unknown known which would have prevented the war, looking at what transpired since then we know the answer now, his statement was a very elaborate cope for an excuse to start a war in Iraq

  • @josephm.6453
    @josephm.6453 3 роки тому +340

    "Godel rocked a mohawk in real life". This statement cannot be proved

    • @temkin9298
      @temkin9298 3 роки тому +11

      To which reality are we talking about?
      ~ A random dimensional hopper

    • @interdimensionalgoober8769
      @interdimensionalgoober8769 3 роки тому +6

      It cannot be proved that the original poster of this comment was referring to the mathematician Godel, and was instead referring to any other person named Godel.

    • @YEC999
      @YEC999 3 роки тому

      No it's just nonsense

  • @AhmedSheheryar
    @AhmedSheheryar 2 роки тому +7

    "The worst he can say is my formula is wrong"
    This person:

  • @eladblaier898
    @eladblaier898 3 роки тому +588

    Hilbert: "math is complete"
    Godel: "I'm about to end this man's whole career"

    • @ghasthordegd1201
      @ghasthordegd1201 3 роки тому +10

      It "Godels" all the way down

    • @calhackit9806
      @calhackit9806 3 роки тому +1

      maths is complete, just it's only internally consistant.
      all this is nonsense.

    • @liguow
      @liguow 3 роки тому +20

      @@calhackit9806 keep telling yourself that :)

    • @epicmarschmallow5049
      @epicmarschmallow5049 3 роки тому +22

      @@calhackit9806 mathematics is provably incomplete. That's what Goedels incompleteness theorems tells us. In order to make the claim "maths is complete", you need to disprove Goedel's theorems, rather than just decry them as nonsense

    • @jasonsmith4114
      @jasonsmith4114 3 роки тому +15

      @@epicmarschmallow5049 Yeah but that's too much work I'm not capable of doing, so I would rather call it nonsense on the internet.

  • @omarperez7415
    @omarperez7415 3 роки тому +184

    Gödel really went above and beyond to say "The situation here is that the question is badly worded" and he was right.

    • @princemachiavelli6570
      @princemachiavelli6570 3 роки тому +5

      Well Math is much more complicated than human language

    • @destractgodren5976
      @destractgodren5976 3 роки тому

      Yes I agree

    • @LLlap
      @LLlap 3 роки тому +2

      @@princemachiavelli6570 wait, whos language is it if not human?

    • @voltydequa845
      @voltydequa845 2 роки тому +1

      «Gödel really went above and beyond to say "The situation here is that the question is badly worded" and he was right.»
      ----
      Imho attention is to be paid to how HE worded it.

  • @Hallands.
    @Hallands. 2 роки тому +12

    It's good that you make it clear how Gödel’s self-referencing sentence is only interesting because it is stated in a language inside the system! But we also need to determine if there’s a possible flaw in translation from the verbal sentence "this statement is false" to the mathematical translation…
    And finally we need also determine whether an axiom kan be self-confirming or self-denying without creating a systemic paradox or placing itself in an order of axioms, not part of the class of all common axioms….

    • @edwinhuang9244
      @edwinhuang9244 Рік тому +3

      I'm sure the mathematicians thought about that.
      ANd it failed, which is why they called it a theorem.

  • @PrashantKg1996
    @PrashantKg1996 3 роки тому +141

    Glados: This sentence is false.
    Wheatley: True, I'll go true

    • @generalezaknenou
      @generalezaknenou 3 роки тому +5

      if you are in the danger of robots just close your eyes and shout out :

    • @jackbolitho7404
      @jackbolitho7404 3 роки тому

      This has absolutely nothing to do with the video, but I saw a dude with an Aperture Science t-shirt today and hope was reinstalled in me for the future

  • @Gala-yp8nx
    @Gala-yp8nx 3 роки тому +276

    Alternative Title: How to give a Mathematician an existential crisis.

    • @lonestarr1490
      @lonestarr1490 3 роки тому +12

      Kind of. But we settled to not think about it too often.
      I think most mathematicians have accepted by now that we do have no natural right to proofs. Everything we can prove is basically a miracle. And in some way, that makes it even more exciting. Now it's like, "Look, guys! I fought the universe and won!"

    • @_VISION.
      @_VISION. 3 роки тому +6

      @@lonestarr1490 I guess it depends on how fixated the mathematician is on the assumption that maths will help them find Truth.

    • @איןסוף
      @איןסוף 2 роки тому

      so gödel have himself an existential crisis, because he was also a mathematician

  • @MM-ux1kl
    @MM-ux1kl Рік тому +3

    I like that in 3:42 the human is Gilbert who believed that math is comple and you can prove every true statement, and now after discovering Gödels incompleteness theorem he suffers the most

  • @senkottuvelan
    @senkottuvelan 3 роки тому +215

    Now that's confusion ladies and gentlemen.

  • @nero1273
    @nero1273 3 роки тому +1611

    Mathematicians: You either die a hero or live long enough to see yourself become the villain.
    Godel: Yes

    • @robinbruce7838
      @robinbruce7838 3 роки тому +16

      That made me laugh way too hard 😂😂😂

    • @miglemaya3967
      @miglemaya3967 2 роки тому +1

      I see what you're doing there

    • @o_0_Lucifer
      @o_0_Lucifer 2 роки тому +1

      @@robinbruce7838 can u prove??😂😂

    • @pan_nekdo
      @pan_nekdo 10 місяців тому

      Fun fact (actually pretty sad): Goedel went insane. He thought everyone except his wife wanted to poison him. When she ^went to hospital for longer/died (I dunno what it was) he starved himself to death. 😊

  • @Daniel_Zalman
    @Daniel_Zalman 2 роки тому +7

    I don’t really understand what it is that Godel did at 1:53. Turning mathematical statements into random numbers is supposed to facilitate what, exactly?

    • @briansammond7801
      @briansammond7801 11 місяців тому +3

      1) He didn't convert mathematical statements into random numbers. He converted them into numbers in an orderly fashion according to a specific scheme that was universal. We all do the same habitually now via our computers, smart phones, etc., with ASCII, Unicode, binary, etc., for numbers and also image and video formats for photos (jpeg, png, etc.) and video (mp4, avi). All of these, at their very basis are just numbers. Everything in every computer essentially does the same thing that Godel did.
      2) Having converted everything to numbers, he described how to them put those numbers into specific functions (which also could be described as numbers). We do the same. We feed a .docx word document (which is really just a complicated number) into a word processor (Microsoft Word, which itself is just a complicated number) and we can manipulate it. Same with images and video. We take an mp4 (a complicated number) and feed it into VLC (a video program, also just a number) or UA-cam (a complicated number we get online) and watch a video.
      3) Godel did the same, but he fed his numbers into a specific function, which was a function that purported to be able to determine whether a given function was provable. He was able to show that a specific number that he could construct using his numbering scheme would have a numeric representation that corresponded to the mathematical statement that that statement itself was unprovable. The details of that are highly technical, just as the details of how Microsoft Word works, or UA-cam works, are also highly technical. I've greatly simplified things here, but I hope the analogy of how we now convert everything into numbers for computer purposes helps to illustrate how and why he did what he did.

    • @briansammond7801
      @briansammond7801 11 місяців тому

      If you want a more detailed explanation, search for Veritasium Godel, and watch Veritasium's video,which is titled "Math's Fundamental Flaw"

  • @BigMoistRat
    @BigMoistRat 3 роки тому +90

    Me trying to explain to my maths teacher why I didn’t do my homework:

  • @TristanSamuel
    @TristanSamuel 3 роки тому +588

    Me: "Breaks math"
    Teacher: That's cheating.

    • @lucypogcute
      @lucypogcute 3 роки тому +9

      Me: *dies*

    • @IスペクターI
      @IスペクターI 3 роки тому +4

      Gray: "we don't need Math where we're going"

    • @Noname-67
      @Noname-67 3 роки тому +8

      If you could mathematically break math, your teacher would impress (if only they're good at math, ofc)

    • @user-ox7gh1hz8d
      @user-ox7gh1hz8d 3 роки тому +2

      Me: Prove it!

    • @michagabo8819
      @michagabo8819 3 роки тому

      When things were at their very worst:
      2 Suns, Cross in the sky, 2 comets will collide = don`t be afraid - repent, accept Lord`s Hand of Mercy.
      Scientists will say it was a global illusion.
      Beaware - Jesus will never walk in flesh again.
      After WW3 - rise of the “ man of peace“ from the East = Antichrist - the most powerful, popular, charismatic and influential leader of all time. Many miracles will be attributed to him. He will imitate Jesus in every conceivable way.
      Don`t trust „pope“ Francis = the False Prophet
      - will seem to rise from the dead
      - will unite all Christian Churches and all Religions as one.
      One World Religion = the seat of the Antichrist.
      Benedict XVI is the last true pope - will be accused of a crime of which he is totally innocent.
      "Many events, including ecological upheavals, wars, the schism in My Church on Earth, the dictatorships in each of your nations - bound as one, at its very core - will all take place at the same time."
      1 November 2012
      The Book of Truth

  • @laumoreri5983
    @laumoreri5983 3 роки тому +6

    Gödel and Turing are my idols. Turing's Turing machine and Gödel's Gödel number both brought to life my adoration for mathematics and motivation to work in the field. Thx guys.

    • @goertzpsychiatry9340
      @goertzpsychiatry9340 3 роки тому

      ua-cam.com/video/voWzbIE6ZRs/v-deo.html

    • @stellaleicht4035
      @stellaleicht4035 Рік тому +1

      And both came about to prove hilbert wrong

    • @ronald3836
      @ronald3836 Рік тому +1

      @@stellaleicht4035 I also like the outcome of Hilbert's 10th problem: give an algorithm for finding the solutions of a diophantine equation (the integral solutions of a polynomial equation in multiple variables with integral coefficients).
      In 1970 Matiyasevich completed the proof that any computer program (any turing machine) can be encoded as a diophantine equation. Since there is no algorithm for the halting problem, the algorithm that Hilbert asked for does not exist.
      ChatGPT can be rewritten as a diophantine equation. Don't try this at home.

    • @EconAtheist
      @EconAtheist 6 місяців тому +1

      Two tragic figures, themselves.

  • @yourday1363
    @yourday1363 3 роки тому +16

    this channel educates us in a way that's so visually and mentally pleasing , i hope current educational systems would do something similar !

  • @brijeshsingh8460
    @brijeshsingh8460 3 роки тому +210

    In short: self reference with negation ruins everything

    • @Noname-67
      @Noname-67 3 роки тому +4

      There are more than that

    • @newname8988
      @newname8988 3 роки тому +8

      @@Noname-67 see halting problem

    • @tylee9373
      @tylee9373 3 роки тому

      I see no problem simplifying your shortcoming

    • @Wabbelpaddel
      @Wabbelpaddel 3 роки тому +3

      If a liar lies, does he say the truth?

    • @tranhuynhdung4809
      @tranhuynhdung4809 3 роки тому +10

      @@Wabbelpaddel no, he's lying, hence he is not saying the truth

  • @Cudddlefish
    @Cudddlefish 3 роки тому +121

    I feel this title is misleading: Godel didn’t “break” math any more than Ben Franklin “invented” electricity. Godel simply discovered a limitation that had always existed (which is still an incredible achievement btw since he had to construct the proof for that, as the video explains). Frankly, I find that to be far more disturbing: it means that one of the fundamental tools we use to understand the universe was inherently flawed from the outset.

    • @tolu619
      @tolu619 3 роки тому +19

      The title has to be designed to attract people to click on the video. The more people they can attract to watch, the more people they end up teaching. And if their goal is to teach, then clickbaity titles will enhance that goal

    • @briangruessner4453
      @briangruessner4453 3 роки тому +35

      Is the math flawed... or does it just accurately reflect the inherent uncertainty and incompleteness of reality?

    • @richard5808
      @richard5808 3 роки тому +14

      It's funny because it basically means that whatever we are "discovering" could be just an approximation or totally wrong. Similar to the nonsense predictions of the standard model, despite some accurate predictions.

    • @broncos435
      @broncos435 3 роки тому +14

      i think what people have to realize is that any tool created by imperfect humans is necessarily gonna be imperfect. people look at science and math as infallible, be-all, end-all solutions, but they're not. now, this isn't to say they're worthless and/or they're not the best tools we've got, but acknowledging that our tools are imperfect is ok (and necessary)

    • @goertzpsychiatry9340
      @goertzpsychiatry9340 3 роки тому

      ua-cam.com/video/voWzbIE6ZRs/v-deo.html

  • @akshatpathrikar7080
    @akshatpathrikar7080 3 роки тому +31

    my dude hilbert at the end was badly injured yet still happy to celebrate gödel's achievements.

  • @sznio
    @sznio 3 роки тому +25

    The animator did an extremely good job with this video.

  • @Thaddeus_Howe
    @Thaddeus_Howe 10 місяців тому +3

    I’m not a religious person, but when I first learnt about this and read more about Euclidean axioms, the more I started to believe that there is some higher power. Obviously it would be amazing if we could prove why axioms are true, but something tells me the reason of thing that makes them true is outside the capacity of human understanding.

  • @piruletaortizcasillas2730
    @piruletaortizcasillas2730 3 роки тому +13

    thank youuu Ted Ed! it was one of the best projects i've worked on, and im glad people are enjoying our animations :> !!!

  • @newname8988
    @newname8988 3 роки тому +81

    The dislikes may be by those who understand why this simplified version can be argued to be erroneous.

    • @kcb5989
      @kcb5989 3 роки тому +16

      That's precisely correct it was an oversimplified explanation and most people misunderstood it.

    • @guymanperson1
      @guymanperson1 3 роки тому +3

      The time it took to write this comment could have been used to create and post a copy-pasta of relevant URL links, search engine terms, and a brief message. With the assumption, of course, that spreading knowledge is your mission as opposed to self-admiration.
      *This copypasta was created on 7-4-2007. Please reuse.*

    • @darana1142
      @darana1142 3 роки тому +1

      @@guymanperson1 brilliant

  • @ShudoShrimp135
    @ShudoShrimp135 2 роки тому +1

    I know this is an old video but whoever did the sound design for this video deserves an award for all these sfx

  • @EvandroSchulz
    @EvandroSchulz 3 роки тому +427

    I'm a mathematician. The animation of the video was very cool. However, many concepts put in this video are difficult for a layman (including many mathematicians). This theorem does not deny or refute that 'a chair exists', he argues about some existing indeterminations when trying to 'prove' that 'a chair exists' within a very specific context. Basically, the foundations of the mathematical thought consist of axioms.
    Axioms is what is 'pure faith', but not a 'blind faith'. An almost religious essential of mathematics. For they are 'things' that you cannot prove exist by definition. But if you assume that they exist and are true, everything you build from them makes sense and is consistent. Mathematics is beautifully built on top of axioms. A specific part in which one seeks to prove this consistency in a specific context is about Gödel's Incompleteness Theorem argues.
    This question is addressed mainly regarding the philosophers of mathematics who question the consistency of demonstration methods. And maybe you ask yourself, why are people worried about this? Well, stop to think about computers. How does when entering your bank password, what makes the computer 'validate' that the number 6 you typed is actually 6, instead of 9?

    • @sivaprasath3638
      @sivaprasath3638 3 роки тому +25

      what exactly is "mathematician" ?
      I understand the context , but the phrase
      "I am a mathematician" , is that a job ?

    • @tanishqarora2647
      @tanishqarora2647 3 роки тому +21

      what do mathematicians do on a daily basis?

    • @MFSomething
      @MFSomething 3 роки тому +63

      @@tanishqarora2647 they do research and teach university classes usually

    • @madscientist1595
      @madscientist1595 3 роки тому +34

      @@tanishqarora2647 They eat math as breakfast. xD

    • @youknowwho9247
      @youknowwho9247 3 роки тому +46

      Strikes me as something that's only really a problem if you're on the side of the fence that claims that mathematics is discovered. If you take the view that mathematics is invented, then resting on unprovable axioms isn't really something that will shake your confidence too much. Seems a bit like building a house: If you put one brick on top of the other the right way, you'll end up with a building. This is true as long as bricks are what we mean when we say the word. We don't need to prove how they are what they are if all we care about is building a house.

  • @hexagon5610
    @hexagon5610 3 роки тому +47

    3:43 love the David Hilbert reference with "We must know, we will know!"

    • @segmentsAndCurves
      @segmentsAndCurves 3 роки тому +9

      Haha
      I feel bad for Hilbert.

    • @damnguen1726
      @damnguen1726 3 роки тому

      @@segmentsAndCurves i heard that he died before the discovery of Godel theorum

    • @WolfgangGalilei
      @WolfgangGalilei 3 роки тому +4

      @@damnguen1726 nope: Hilbert died in 1943 while Gödel published his paper in 1931 a darn full dozen years of misery for ma mann Hilbert

    • @damnguen1726
      @damnguen1726 3 роки тому +1

      @@WolfgangGalilei tks, for Hilbert that is depressing

  • @KSJR1000
    @KSJR1000 Рік тому +3

    It'd be nice to have a better explanation of the jump at 2:17

  • @cooperlucas2846
    @cooperlucas2846 3 роки тому +562

    “If it’s not true and not false, what is it?”
    Me: “A waste of time”

    • @royroos8036
      @royroos8036 3 роки тому +8

      Exactly. Whats even the use of this? Waste of time indeed

    • @mycrowsoffed
      @mycrowsoffed 3 роки тому +28

      @@royroos8036 Maybe, maybe not.

    • @creetan9997
      @creetan9997 3 роки тому +13

      Nothing. It doesn't exist. Something that doesn't exist is neither true or false because there is nothing there for it to be true or false.

    • @MartinPoulter
      @MartinPoulter 3 роки тому +49

      @@royroos8036 The insights discussed in the video led to, among other things, the development of computers, so it seems the exact opposite of a waste of time.

    • @lechihieu2606
      @lechihieu2606 3 роки тому +1

      @@MartinPoulter but why we need a computer?

  • @soggy9648
    @soggy9648 3 роки тому +35

    I had to do a study on godel. He's a legend. Dude made Einstein doubt his relativity theory. And then was so convinced someone was going to poison his food that his wife was the only person to make him food. When his wife went into the hospital he starved to death

  • @patrickwilson1804
    @patrickwilson1804 Рік тому +1

    I love how Hilbert is shown without reference expect with his iconic hat and him trying to formalize a system but running into trouble.

    • @-TheUnkownUser
      @-TheUnkownUser Рік тому

      They should have added Bertrand Russell.
      Principia Mathematica is dense that it needs some recognition.

  • @youraveragepersonwalkingth6850
    @youraveragepersonwalkingth6850 3 роки тому +139

    Whenever something is unprovable true I’m just gonna call it tralse

  • @mr.recorder4781
    @mr.recorder4781 3 роки тому +191

    Gödel: Exists.
    *Every Mathmatician*: Years of academy training wasted!

    • @Undubbedriah
      @Undubbedriah 3 роки тому +1

      LOLLLL

    • @kohwenxu
      @kohwenxu 3 роки тому

      Meanwhile 50 years of attempts on trying to make mathematics complete basically is wasted
      All the attempts at the proofs, just to have someone find that it is impossible for mathematics to be complete
      F for the mathematicians who tried

    • @איןסוף
      @איןסוף 2 роки тому

      gödel was a mathematician himself

  • @jolness1
    @jolness1 6 місяців тому +1

    The art style is freaking adorable.

  • @odd1ty612
    @odd1ty612 3 роки тому +33

    Random Mathematician guy: damn I love how everything in math can be proven using a set of basic axioms
    Godel: yeah about that...

    • @kohwenxu
      @kohwenxu 3 роки тому

      Yea just going to calmly destroy 50 years of work at trying to find that everything in math is provable with sets of axioms

    • @איןסוף
      @איןסוף 2 роки тому

      i wouldn't say that gödel destroyed mathematics, in fact he actually expanded mathematics

  • @akankshasharma7498
    @akankshasharma7498 3 роки тому +43

    Every once in a while, all we need is a wild Gödel to show up

  • @pablosampaio3694
    @pablosampaio3694 3 роки тому +2

    The video has some inaccuracies to make these deep results more accessible. For example, in 3:31, I think instead of "in every axiomatic system", it should be "in every axiomatic system (1) that cannot prove contradictory results and (2) that has the power to express certain amount of arithmetic".

  • @RobinDSaunders
    @RobinDSaunders 3 роки тому +45

    A few thoughts from someone with a strong interest in this area:
    Mathematical statements don't always have to be either true or false - it's common to assume they are, but can be useful to reject that assumption. For a simple example, statements about variables can be thought of as potentially true or false, but not necessarily either one.
    Another example is axioms. These don't have to be "undeniable" - they're more like conditions specifying the sort of situation we're currently interested in. If these axioms apply to a situation, and we use these deductive rules, then this result follows. In particular circumstances an axiom might be true, or false, or indeterminate.
    Going in the other direction, a statement might be provable but false, e.g. if the axioms are inconsistent. It might seem silly to use inconsistent axioms, but Gödel in effect proved that any list of axioms sufficient for ordinary arithmetic is potentially inconsistent: it can only be proved consistent if we add in extra axioms - and then proving this new list consistent would require extra axioms on top of those, and so on.
    One reason these ideas are important: when a statement can't be proven using particular axioms, it can often be thought of as being false for some models of those axioms. If you think it should actually be true, that's because you're implicitly assuming extra axioms that you weren't stating... and there's no way to list all the axioms you'd want without also including some that you don't want.
    On the other hand, only a tiny fraction of these "philosophically significant" axioms are needed for the vast majority of results used in science, technology and engineering. That's why most of the people interested in this stuff are logicians or philosophers.

    • @irrelevant_noob
      @irrelevant_noob 2 роки тому

      Yet that is why we have SPECIFIC TERMS for those: predicates (depending on variables) aren't statements, axioms aren't statements. ;-) Although intuitively i'd include "we don't know yet" in the neither true nor false category, but i feel neither you nor the video are considering those.
      As for the "provable but false" i think you're misinterpreting something. If you have a proof, then the conclusion is by definition true. Even though its negation would ALSO be true (because the system is inconsistent/contradictory), that doesn't mean the non-negation wasn't true.
      By the way, something feels off about your explanation of the "potentially inconsistent" arithmetic... Those phrases seem to be more about completeness than about consistency. 🤔

  • @nxpy6684
    @nxpy6684 3 роки тому +81

    Gödel: Comes up with a code language to write to his girlfriend
    The entire math community: "Why do I hear boss music ?"

    • @kohwenxu
      @kohwenxu 3 роки тому +3

      50 years of attempts at proving mathematics is complete:
      Sir, I don’t feel so good…

    • @lordbanetheplayer8844
      @lordbanetheplayer8844 3 роки тому +1

      @@kohwenxu *Mr. Stark, I don't feel so good...

    • @איןסוף
      @איןסוף 2 роки тому

      except, gödel is a part of the math community

  • @danielleanderson3971
    @danielleanderson3971 2 роки тому

    So this is what has plagued me for 5 years. Thanks for helping me confirm it wasn't just me, TED.

  • @johnbeltran2736
    @johnbeltran2736 3 роки тому +434

    "He translated mathematical statements and equations into code numbers". Ah, there's the problem.

    • @annaclarafenyo8185
      @annaclarafenyo8185 3 роки тому +15

      That 'translation' is just unicode.

    • @davidwight5974
      @davidwight5974 3 роки тому +45

      right thats what i was thinking once you change the equation into code it no longer is an equation cant change it into something completly different and act like its the same thing

    • @annaclarafenyo8185
      @annaclarafenyo8185 3 роки тому +142

      @@davidwight5974 That's not the important part of what Godel did, it's just stupidly obvious, so that's what incompetent popularizers present. The main idea of Godel is that a system of proving things about numbers can prove things about computer programs, because computer programs are sequences of large numbers (the content of your computer's memory is a gigantic number with as many binary digits as you have bits inside your computer and hard-drive). So any statement about computer programs is a statement about numbers.
      Further, you can program a computer to reason mathematically, and use a mathematical system.
      So now you can write a computer program which prints its own code out into a file, then starts looking in the mathematical system for a proof of "the code in that file never stops". If it ever finds this proof, it stops.
      Since the code in the file is its own code, the computer program is looking for a proof of "I do not stop running" at which point, it stops running. If it finds the proof, it stops running and makes the mathematical system into a liar. If it doesn't find the proof, that means it runs forever, and the mathematical system never proves this true fact, so it is incomplete.
      This is the entire proof of the theorem. The only slightly tricky bit is showing that a program can print out its own code into a file. That's a bit tricky, but not THAT tricky, it's an exercise for first year programming students.

    • @HarshDude126
      @HarshDude126 3 роки тому +40

      @@davidwight5974 Actually, you can, because symbols and concepts are two different things. Mathematical concepts can be expressed in different ways. An equation is just one way. Just because you change those symbols to something else doesn't mean the original concept has changed.

    • @pcky6646
      @pcky6646 3 роки тому +3

      Are you saying that his own arguments are also an axiom?

  • @SomePeopleCallMe
    @SomePeopleCallMe 3 роки тому +10

    The narrators pronounciation of "Gödel" is incredibely spot on, it sounds exactly like an austrian would say it

  • @gravysamich
    @gravysamich 6 місяців тому

    This is the first video on the incompleteness theorem that actually makes sense to me. I understood the idea before, but most explanations i have seen don't really feel like they actually say more than the fact that mathematics is incomplete because reasons.

  • @jamesyan3510
    @jamesyan3510 3 роки тому +97

    Smarter than me, that's for sure.

  • @3.05.10연병수
    @3.05.10연병수 3 роки тому +48

    This reminds me a lot of Heisenberg's Uncertainty Principle in science. Although they belong to different fields, both of them shows that we can't prove or know everything. They made me reflect on myself and think if I had been too proud of myself and acted like a know-it-all before. Thanks for the good video. Keep up the good work!

    • @goertzpsychiatry9340
      @goertzpsychiatry9340 3 роки тому

      ua-cam.com/video/voWzbIE6ZRs/v-deo.html

    • @Cooososoo
      @Cooososoo 2 роки тому +1

      No one can gain full knowledge only Allah knows everything he made this world

    • @randomhuman5525
      @randomhuman5525 Рік тому +8

      @@Cooososoo Who knows allah?

    • @TheFeatInk
      @TheFeatInk Рік тому +2

      Neither of them show that, otherwise how could we know the theories themselves?
      Heisenbergs principle could never invalidate the truth of causality for instance, as is often erroneously said, since one must presuppose causality in order to take any meaningful scientific measurement.

    • @starfishsystems
      @starfishsystems Рік тому +3

      ​@@Cooososoo
      You've first got to show that allah exists before you can use it as an explanation for anything.
      No one has done the existence proof yet. We'll wait until your paper is accepted for publication before discussing further.

  • @jinyoungkim4532
    @jinyoungkim4532 Рік тому +3

    This is such an excellent intuitive explanation of a complex idea!

  • @johnchessant3012
    @johnchessant3012 3 роки тому +11

    Let's be honest, you made up the pun "it's Gödel's all the way down" first and it was so good you had to make the rest of the video :)

  • @faceless-x-abberation8400
    @faceless-x-abberation8400 3 роки тому +5

    That was extremely interesting lol. I happened to pass by it while scrolling for something to watch. The video played with no sound, but had subtitles. So I sat and watched the whole thing while reading along to what was being said. Thanks for the video.

  • @PaulEverhart-k8g
    @PaulEverhart-k8g 2 місяці тому

    You’ve made this topic accessible to everyone!

  • @plankdorodo3122
    @plankdorodo3122 3 роки тому +6

    Please, call whoever drawed this more times. It's absolutely adorable!

  • @Racoonma392
    @Racoonma392 3 роки тому +26

    That is like "What will happen if Pinocchio says, 'My nose will grow now'?

    • @raulzaha3096
      @raulzaha3096 3 роки тому +8

      Nothing. His nose grows when he lies, not when he says something that's false. It wouldn't be lying unless he knew the truth value of a statement and chose to say the opposite. Since he can't know the truth value of "My nose will grow now" because it's a paradox, nothing would happen, because he's not lying.

    • @afridnishad6617
      @afridnishad6617 3 роки тому +2

      @@raulzaha3096 but since his nose didnt grow that statement now became a lie didnt it? "my nose will grow now" seems like a bold statement. its true he doesnt know the outcome but isnt stating things you dont know about *boldly* considered a lie if it doesnt happen. Like how politicians saying "we will reach that goal" completely and reassuringly without knowing the outcome or without any plan is a lie if it didnt happen, people will say hes a liar. Your thoughts?

    • @raulzaha3096
      @raulzaha3096 3 роки тому +5

      @@afridnishad6617 I think my point is that there has to be intet behind it for it to be a lie. If a politician promises something and fails to deliver the difference is in whether he/she tried. Being incompetent doesn't make you a liar, even if you are a confident one.
      My specific point is a logical one though. Since the statement is a paradox it means it's neither true nor false, or more definitely it's unknowable, which means he cannot intentionally state the opposite, even if the truth value would somehow reveal itself later.
      A good example is the statement "There is no life in the Proxima Centauri star system". Even if it turns out that there is life there, his nose wouldn't grow, because he can't have known. If it did grow, that means Pinocchio can uncover all the truths of the universe by simply stating them, which I don't think is the point of his curse.

    • @DaAcoustikChicken
      @DaAcoustikChicken 3 роки тому

      @@raulzaha3096 "He cannot intentionally state the opposite, even if the truth value would somehow reveal itself later."
      Sure, but he would therefore be intentionally presenting a current unknown as a bold truth. Why would you boldly state there is no life in the Proxima Centauri star system without any proof? (proving the inexistence of something is a whole other topic, but the point remains). Seems to me that Afrid's point is that claiming unknowns as absolute truths could be considered a lie, which I guess is ultimately a semantics argument on what exactly means to lie (as well as what are the exact mechanics behind Pinocchio's whole shebang).
      It comes from the Skeptics school of thought that we shouldn't be claiming assumptions as absolute truths, even if it's something so obvious as the sun rising the next morning (we don't really KNOW it will happen, we just reasonably expect it to happen the same way it has happened every day for past eons).

    • @HarshDude126
      @HarshDude126 3 роки тому +1

      @@raulzaha3096 You're wrong, he does know the truth value of that statement. The truth value is false, because there is no reason for his nose to grow "now".

  • @DrCorndog1
    @DrCorndog1 3 роки тому +1

    "It's Gödels all the way down."
    All right, that earned a like from me.

  • @rayansuryadikara9353
    @rayansuryadikara9353 3 роки тому +63

    "I used the math's logic to destroy the math's logic."

  • @1337CodeMaster
    @1337CodeMaster 3 роки тому +18

    This animation was amazing even when compared to the usual high level on this channel! Well done.

  • @merlinquark5659
    @merlinquark5659 2 місяці тому

    I love this! As a philosopher of physics, I have come to realise there are fundamental physical laws which cannot be explained by science, but must be taken for as a given (or an axiom), which fits very well with Gödels logic. The fact that logical systems, such as maths and physics cannot be fully self contained/provable, gives good evidence that there must be something metaphysical which grounds them, such as the Logos, or the mind of God

  • @segmentsAndCurves
    @segmentsAndCurves 3 роки тому +12

    4:51
    Hilbert: This young boy doesn't know what he has done.

  • @chaoticcranium
    @chaoticcranium 3 роки тому +24

    I remember learning about this concept through the lens of the Halting Problem for Turing machines, which is an example of a statement that is undecidable by computers, just as there are unprovable concepts in mathematics. It's also similar in that it's an example of a program trying to ask a question about its own state (ie, will it terminate or not), just like Godel using self referential statements here. Kinda blew my mind.

  • @XShollaj
    @XShollaj 3 роки тому +4

    Love how Godel rocks that Mohawk, sort of like a punk rock mathematician

  • @michaelnelson2976
    @michaelnelson2976 3 роки тому +13

    Ahh! This is always the case, but what a good mini lecture with such lovely art to back it up! I am always amazed!

  • @surojpaul14
    @surojpaul14 3 роки тому +11

    In physics Heisenberg and in math Gödel both bring some uncertainty in our life,, and that's how nature works. Kudos both

  • @SphuranaOfficial
    @SphuranaOfficial Рік тому +2

    The first statement is insubstantial. It’s like just saying “I’m lying”

  • @campa_t
    @campa_t 3 роки тому +9

    this guy has the clearest voice in the universe

  • @praevidens
    @praevidens 3 роки тому +19

    The artsyle is godly as always.

  • @johng6586
    @johng6586 2 роки тому +3

    When you can step back a frame reference and look from a more higher level/simpler view the details start to get blurred but new trends emerge on a higher level.
    Mom and Dad have a joint bank account. They make 1 and 2 and a combination of 3(sided combination like 70/30 of the way its used. etc.)

  • @adarshsingh2777
    @adarshsingh2777 3 роки тому +64

    If Gödel had seen this drawing of himself, he wouldn't have starved himself to death.
    Let me predict that Ted-Ed is going to bring Undecidibility to us soon...
    These will make kids love maths more or "less".

    • @ThoraninC
      @ThoraninC 3 роки тому

      Oh yeah gib me an animation of my boy Turing plz