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

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

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

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

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

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

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

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

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

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

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

*"...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.

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

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

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

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

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

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

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

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

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

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

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

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

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….

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

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

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

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

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

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.

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

Thank you so much for uploading this video. It is helping me get through the pandemic!

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

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

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.

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

“It’s Godels all the way down” BRILLIANT

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.

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.)

Not only love the topic but also the animation and the music 💜

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.

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

This channel is just awesome.

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

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

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

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

Beautifully animated!

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

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

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?

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.

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

Now that's confusion ladies and gentlemen.

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

Wow! He was 5 year earlier than Alan Turing solved the Halting Problem. Which actually is the same kind idea. That's awesome!

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

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

i really loved the sound effects too..!!!

The sounds in this one are so satisfying

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

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

the sounds are so lovely to my ears

Love the animations here :)

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

In short: self reference with negation ruins everything

hats off to the animators and the sound effects team

Your video is really cool and amazing! I want to be good at math like you!

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

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

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

Thank you very much!

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

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

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.

I have understood nothing but it's a very interesting matter 😉

0:22 It's Tralse

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

thanks for breaking my brain, TED Ed

i love watching this video whilst not understanding most of it

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

The animator did an extremely good job with this video.

great animations!

He translated it and doubted math but he didn't doubt his translation. Truly a "mastermind"

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?

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

Excellent!

for some reason this video made me so happy

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

Smarter than me, that's for sure.

I didn't fully understand it but sounds very important

I love all the references to other mathematicians like Noether and Hilbert.

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.

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

"It's Gödels all the way down" is my new favorite phrase

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

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

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

Thank You ❤️🙏

Great video! As useful as mathematics is, we can only prove what our limited human minds can process. But there is truth outside of our ability to understand. There must be a handoff between mathematics and philosophy.