As a working mathematician, the scariest part of incompleteness is that when I can't solve a problem, I don't know if the problem I'm working on is just really hard... or if it's actually impossible.

​@G no solution is a solution.

BRUH im only a high school student and i have the exact same problem sometimes!

These were incredible They wrote a whole book to say 1+1=2 Unbelievable am in awe

When u need 5 pages on ur math but u only can do addition for some reason so I write a complex version of saying 1+ the other 1 equals once 1and 1 come together to make two

​@Ali Eser "how fed up the authors were with the amount of rigor and pointless proofs" refers to the authors' thoughts, not explicitly implying that it is useless. Hence the end of the sentence, they still acknowledged

​@Mark Mcgoveran which is also unprovable

@Paragjyoti Deka that was provable

@Paragjyoti Deka Huh?

The moment he showed the game of life running inside the game of life, I was totally blown away. Such a mind bending topic to contemplate.

This reminds me a lot of "minecraft running on a computer made entirely in minecraft" Was blown away by that one as well

@nag 007 it was my A level project, so I think 1980/81.

@Al Banana which year did you do it?

@Al Banana That sounds great! If only this video was available back then, then the teacher would have definitely given you the best grade. The game of life is awesome.

I wrote an implementation of Game of Life as an A level project on a Commodore PET. I had to use machine code as BASIC was too slow. I got a bad grade compared to others in the class who wrote simple stock entry systems, as the teacher didn't understand what I was trying to do.

There's no one's story that saddens me most in the scientific world more than Alan Turing's, a brilliant and exceptional genius, who literally saved the world, but was very simply born both at the right and wrong time 😔😔😔

@JJ _The_Ent wow

id make a small note; its always… morbidly amusing to me that, if memory serves, theres 2 ways to violate and essentially “break” the gross indecency law listed in the video. 1) sexual acts in public and 2) acts with a minor (regardless of publicity) they convicted him of committing sexual acts in public… after sending 2 separate private investigators to find said acts, and after 2 weeks of searching, they finally convicted him of violating the public part of gross indecency after finding an adult male life partner… in his house… “public”… sure-

and at the wrong place.

@Supersonic Tumbleweed no you don’t lol

I first watched this video around when it came out. Now I am taking a final exam on logic and computability in 8 hours and I am back watching this to study. I didn't realize how much I'd learned until I realized all the topics here are familiar already. Still, it is an absolutely amazing explanation.

@Christopher I think of meditative states as an updated form of consciousness relative to the previous form that the subject was familiar with. usually short lasting, and misrepresented. Math is s form of representing that update as an equation solving for values, as is language, where compositions are equations solving for meanings.

Listen to Alan Watts. He's more wrong than right, but that's to be expected when talking about an indescribable reality. The key to understanding lies in Eastern Philosophy. You can't grasp it, and you can't not grasp it. Those who know don't know, but those who don't know know. It comes from the knowledge of the meaning of words. Words are limited in their scope, We don't have words for metaphysical concepts, or non-conceptual realities - we can only reason within the framework of concepts, which is a something which is contrasted by a something else. Which - in a non-dualistic, non-conceptual reality means we are all very literally quite screwed, because we only deal in concepts.

Same here! I watched it back then as well and today I understand those concepts from my university course in logic for computer scientists. Hope your exam went well btw :)

What was done to Turing was just horrible. The horror is compounded by just how his brilliance could have gone on to do more. To be so incredibly brilliant and then defeated by the stupidity of others is just heartbreaking.

As a mathematician I haven't seen a more elegent presentation of these concepts,especially Godel's theorem. Amazing job thank you.

@Henning Breede «@VoiceOver No, this is the sole exception. I clicked on the youtube video because it was recommended and after reading the comments I'm not very motivated to watch it either. It doesn't seem to do a good job at addressing common misconceptions.» ---- Wonderful! I started to do the same - reading comments to decide if worth watching - with (pseudo)sci videos. I was doing the same regarding users cinema reviews — that from the affinity with (the style & semantics of) comments I could find the affinity with the movie. Clickbaiters became very smart in serving confusion as it was science and/or deep thinking. The psychotrick is simple: satisfy the ego of those that need to believe they understood (something).

@Aisha I never got "very good" at math. Finished a degree and had an accepted pre-print, though a couple details needed to be worked out, in a way I have no doubt I could've, but immaturity got the better of me and I never did it. Either which way, it was in this tiny, tiny sliver of mathematics that no one would've cared about the paper. The subject had to do with what are called c* algebras. All the same, I can definitely tell you, the first way is do every problem in the textbook for whatever class you are currently in and the next up till precalculus. Then for Pre-calc, do every problem in the book for 2 textbooks. For calc, do every problem in the book as well. Now is where it gets tricky. Read some approaches to set theory that are aimed at rigor but not necessarily for logic students or anything. Perhaps like a discrete math book or something. Then, work through Spivak's calculus book. Do as many problems as you can. You will have to really learn the subject from a formal perspective and probably get help at other points. What is imperative is that you learn delta-epsilon proofs beyond a shadow of a doubt. After that, we're getting into tougher territory. I think probably the next idea would be to go through Munkre's topology and really work out mosts of the problems and for each theorem, after you read the proof, make sure you could generally recreate them, except for the ones munkres specifically mentions. I guess then you'll be in a pretty good spot to start learning a variety of any higher level math like actual analysis, topology, or algebra. Algebra has basically not been touched upon till now, but that's because, while the original objects are much easier to understand, their nuance is immediately difficult and requires, in my experience, quite a bit of mathematical maturity. Anyway, personally, I was told to go through Rudin's principles of mathematical analysis and do the following. For every theorem in the book, read the theorem and prove the theorem without looking at the provided proof. Really, really try. Not like, try for an hour then finally look it up. You should spend 2-3 days on a proof and if then you don't get it, you can look at it. Obviously then, you should be able to do that proof so at the end of the chpater, make sure of that. Also, do the problems, you can do a general selection, colleges usually have their psets when they hadd rudin as their book. You can follow those. Unless you're a genius or at least absolutely obsessed with mathematics (probably unhealthily so), this will probably take you through high school. At which point you can speak to professors who will be approximately a billion times better than me at instructing how to go from there. Good luck with high school and with your mathematics journey!

Any tips on becoming good at math as a high schooler?☹️

@Dayton Robar What's naturally good? Opinions are endless.

Presentation is everything for people that are not naturally good at math.

the sheer amount of information about mathematics is overwhelming for my little brain

It is truly a gigantic field. Most maps of mathematics stop short of the connections between the fields it lists; let alone all the unique problems in the fields and what they’d imply if they are or aren’t true.

@Skwalka Damn

There is wider breadth between points of our understanding in mathematics than the distance between 1920's Jazz and soufflé baking tips.

Keep at it my friend 💪🎯

This video is actually a masterpiece.

Some statements are not TRUE!!! For every # you can come up in set (0,1), you can always find new # in the set of N (natural) #''s.

@RC Barrientos you're very welcome!

@RC Barrientos Please watch the video from 30:00 to 32:40 to understand why it is important that mathematicians go into researching these abstract concepts to gift us with the fruits of our modern joy. They are putting in hard work to lay foundation of future that only gets realised one or two generations later and they do all of this knowing very well that the true recognition for their work will probably not get realised in their lifetimes. Its a highly selfless act.

@RipJawSquad Ah yes, That actually makes sense and I hadn't thought of this. Thanks for pointing it out;)

@RC Barrientos but then you realise that solving "unsolvable"problems like the one in the video only enriches the quality of mathematicians' thinking and problem solving skills, often leading to world changing inventions like the computer.

I am really not good with maths and mathematical concepts and yet I still wasn't fully lost during this video, which is consider as a great achievement

@jimjimjimjim jim like which bits?

big up yourself!! I had to replay parts a bunch, and I'm still a bit lost.

One of the clearest and most concise explanations of completeness, consistency, Goedel's theorems, and undecidability I've ever seen. Makes great use of the video medium in a way that a book can't.

This video is literally a masterpiece. I loved it when it came out, but I keep coming back to watch it again every few months because I randomly remember just how good it is

I don't know why but I love the idea of mathematicians gathered in a room yelling and hurling insults at one another

It’s called college 😂

@RedDragonZ81 jokes on you that's me completing the square

"you are the '-1 + 1' in a equation!"

I'm in love with science just for this reason

I loved this soooo much, it hit on 4 thinkers whose work I'm fairly familiar with (Cantor, Gilbert, Goedel, and Turing) and just brought their work (and how they all intertwine) to life. This was just brilliant. I really enjoy that while a lot of this is math, that it's also just as much philosophy, just so good.

Do you mean Gilbert or Hilbert?

my interest in mathematics has been certainly increased after discovering this channel. Thanks to the team.

As a mathematician I am deeply disturbed but also inspired by this.

This video explores some really mind-blowing concepts.

Did this become recommended suddenly? 4 days ago comment?

Fr

"There will always be true statements that can never be proven " wow

@Bumble But a fact describes that something has been proven

That in itself is a paradox lmao

true relative to something

@Mihail Milev not really. For instance it has been shown that the Continuum Hypothesis in set theory cannot be proven nor disproven in set theory (ZFC set theory in particular) assuming that set theory is consistent, in a similar manner as to how Godel proved his incompleteness theorems. The Continuum Hypothesis is not a “weird” statement at all and is relevant to the study of infinite cardinals.

If you're a mathematician and you are labelled a "corrupter of the youth", you are doing something very right.

@Quetzalcoatl something else that rubs ke the wrong way is going to the extend to ruins someones livelihood by not letting them get a job for a theory they thought up

@Linus Fu doesnt warrant not letting him get a job tho

@Quetzalcoatl I'm not necessarily arguing against you here, I'm just pointing out that just because someone is right about most things, doesn't mean that they are right about most things. Btw, I like the name.😉

Mathematicians: arguing over a barber Engineers: just hire one more 🍷🗿

In particular, barber #2 must be a lady

- You don't understand the problem! - Perhaps. But I can provide a solution.

I have came across Russel's paradox before but this video helps me to understand the paradox for the first time. Wow!

When I first learned about Gödel I was sorta crushed, I had always learned and sorta expected that math was almost this pillar of pure objectivity.

Ever since school and until my uni years, whenever a teacher told us to do a free topic presentation, I always picked a Turing related story. Either his inventions, significance or even his death. What a genius he was.

His story is emotionally impactful. His work was remarkable. There is no important figure in recent history that I think about more often than Turing. He met a tragic end, but he will forever be a hero.

@Wolfette Plays ah yes, either undertale or the holocaust

I always did one on Hitler or Toby Fox. Just to break the mold

I’m hopeless at maths but I love hearing it explained in this way. I see beautiful things here, and that game looked like sparks burning and fizzling out. I often see natural shapes, so to know nature in effect follows maths principles is so satisfying to me as a believer in God, who gives us all this. It’s fantastic!

👆👆Thanks for watching and congratulation 🎊you have been selected among my shortlisted winners. Telegram only to claim your prize 🎁🥰✔️

So basically... Can math prove itself? No. But math can prove that math can't prove itself.

No it can't.

@Wesley P «... anyone who knows about computers would understand why we could never make such a machine.» -- True. To simulate dynamics of the universe you would need at least another ten of them. Anyway there are also limits due to anthropocentrism, that make us think that an eventualy self-aware computer would heat itself wasting energy in prediction calculations (because the computer would be affected by the human's existentialism syndrome). ---- « but it breaks apart instantly when trying to imagine what a 4d object looks like. we just cant. computers can! » -- Wrong. "Imagine" verb cannot be applied to computers. They can only hold representations, and do whatever functional calculus on them.

@Charles Trudel «Well, as a fan of computer theory, ...» ---- Think about a theorem proving system. Of course there could be theorems that could need too long time. But a decent theorem proving system would refuse nonsense paradoxical statements like that of "you, system, cannot prove yourself", could be with something like "Cannot detect any sense, in that statement, within my domain". Now, written this, I got another hint (beyond Godel's trick of generating nonsense statements by recurring to cheap encoding / transliteration of whatever) — out of domain (semantics).

What about the statement "The frogs defecate axioms that can prove themselves"? The point, for eventually hit by the parrot syndrome, is: if Godel's statement can be valid, the frog's too can be valid. And also the statement "The statement that math cannot proof itself is an utter nonsense dressed as pompous stunt".

«So basically... Can math prove itself? No. But math can prove that math can't prove itself.» ---- Nonsense hidden under rhetoric dressing as paradox. Same as "Can our language prove that the meanings of the words are right? " Math is an instrument that encodes quantities and relations. If the math needs improving its coherence, for the sake of being / remaining useful, it can be reformed/reformulated. All the rest is hot talking the hot talk.

I will pretend that I have understood 15% of this video and I will be proud of it. I am so smart now! a masterpiece of a video! Derek never disappoints

My god. You speak to the depths of my mind when you make videos this deep. I literally am tryna work and listen at the same time and I'm so engrossed in it it's just so intricate. I wish your videos didn't end man! I absolutely love learning science

This is truly an amazing piece of work! The complexity of the math that you’ve made understandable to the average person is impressive. Thank you for this !

That's how a teacher should teach 🥰 I understood everything, 'Putting everything so smoothly and easily and at the end connecting everything with emotions of those great minds' just a masterpiece of teaching May God bless you for spreading these amazing concepts

The chill running down my spine when he showed the game of life reproducing itself... I had already seen the video before and wanted to come back with different eyes and I didn't remember that part. Just awe inspiring job

Teacher: Your math is flawed. Student: No, math itself is flawed.

Reality is often disappointing

No really many things we take in school are not proved

You missed the point of the video kiddo

My son tried that line in calculus, disputing his teacher. Was not spoken to, by the teacher, the rest of the year. Hes44 and just fine ❤️🇨🇦❤️

Fascinating, I remember these names from studying Computer Science a few years ago. Very nostalgic!

Very nicely done. Small question/quibble, I have read elsewhere that Hilbert did not himself attend the 1930 Solvay conference, which was instead run by von Neumann. There is also the interesting story that von Neumann returned from the conference where he learned of Godel's first theorem and proceeded to prove the second theorem. He wrote to Godel, who responded that he had also proved the second theorem, at which point von Neumann decided to leave the further development to Godel and moved on to different things. I read that when Godel died, he left a large collection of unpublished papers that mathematicians are still trying to understand.

This is one of the best videos Ive ever seen on Youtube. Thanks so much for this, you actually managed to make math interesting and fun😄😄❣️

Great video, the history of maths puts everything into context. This would have been good to see in school.

Can we just appreciate how well animated and produced this video is? God, so much effort.

@Peter Codner just to tell her that incompleteness theory is agreed everywhere and it is a breakthrough and no way to compare it with the electricity video of this channel which oversimplified some aspects of the experiment although it was a nice one.

@Ward Fadel So, or therefore, what?

Far simpler clearer and quicker to advance the axiom that a mirror cannot reflect itself.

@Fred Esch nice 👍

The chart scene looks lile Flash MX discontinued

Science is so interesting because it's information that helps you understand the world as a whole versus just our speech such as English or physical education in school. This information sparks other ideas and curiosities in our minds and it's crazy where it takes us. Science has to be the most important branch of education in my opinion. You making these videos is a huge step in the right direction as far as humanity as a whole. The more curiousity the more we strive for an answer and that forces evolution in the best way. Thank you so much Veritasium! I'm looking forward to another compelling video. 200/100 💯

Philosophy is the most important branch of education, not math/science. English, and language in general, is VERY important, because you deconstruct all symbolic notation, which is what mathematics is, with cognitive tools that have evolved linguistically. It would be impossible to learn mathematics without already learning how to read and write. Think about that. And many of our greatest thinkers were very well-read in literature, music, Latin, etc. You should want to be as widely read, and widely exposed as possible, those things inform, enrich and enhance science and mathematics by giving your imagination more disparate stimuli to work with. Philosophy, and thinking philosophically, is what has guided our greatest thinkers: Archimedes, Plato, Leibniz, Socrates, Aristotle, Einstein, Newton, Galileo, Cantor, Maxwell, Poincare, etc. Without Philosophy, and specifically the insights into the philosophy of mathematics made by Frege and others, Godel would not have been able to create his masterwork in mathematical logic, for the very study of mathematical logic is itself a PHILOSOPHICAL discipline. The smartest people to ever live were, at their core, philosophers, not scientists. Turing was a philosopher - and if you need any proof of this, all you have to do is look at the Turing Test. It's reasoning is fundamentally philosophical even though it can be framed mathematically. Most of the greatest insights into mathematics and science, specifically physics (the king of the sciences), were philosophical questions that had to be answered in the language of mathematics and physics (e.g. Einstein's equivalence principle for instance). We should be teaching critical thinking and logic and REAL history so that students understand the real things that have happened that have led to contemporary life.

The quality of his videos have gotten SO good. its unfortunate the viewership isn't as high

There should be a weighting determined by the quality of discussion. Imagine if all viewers had their own Godel number, some 'quality' weighting, that would be fun.. :)

Two years ago, I took the final exam of my computer ethics class. One essay question was to explain whether the ethics of actions could be algorithmically determined & why or why not. I answered no, but if this video were uploaded by that point & I’d seen it and gone into the exam with the knowledge I currently have, I might’ve been able to realize that such a question was the halting problem of ethics, if such study is Turing complete, and could’ve mentioned that in my response.

Your videos add so much to my life. Thanks a lot for making these. I am a Computer Science professor and love to show your videos to my students.

so in my country there is literally no value for how and why things happen, doesn't matter if its biology or maths. I remember studying in 4th standard and endless questions would arise everytime we studied maths or anything simple yet complex. I stopped asking because the teachers referred to them as "academic distractions". But I'm glad i never gave up on em

I'm 75, female; I am grateful that I have had enough education to have at least heard of the people you reference. Awed that you explained it all so well that I could not stop listening. Lastly, so proud to have lived this era from beginning to undecidable end.

I'm 104, male. I'm grateful I watched this video

"Education" is a rather vague portmanteau word into which any number of sins and evils can be crammed, just as useless information is rammed down the throats of small beings who would rather play or do some useful work, but No, they must be "educated" whatever educated means, but let us just call it bullied.

"Reference" is a noun in pure English, not a verb; one can no more reference than one can parent or debut- except in that dialect of pure English that is American. If the salt has lost its savour, wherewithal shall it be salted?

@capratchet this is honestly might be the most beautiful way I've seen the edutube community described and encouraged yet. cant wait to share a classroom with everyone else too.

One of the many privliges of getting a comp sci degree was having to study all of this stuff. I loved math before, calculus especially, but getting into set theory/number theory/automata theory made me fall in love with it all over again.

True. Learning computer science gave so much insights into mathematics, information and languages.

Thanks for the great video! It reminds me of another limitation in quantum mechanics, the uncertainty principle, which sounds to me like we can know something but we can never know everything. However, I really like your conclusion. Such limitation improves us or even inspires us. Thanks!

Wow, I love this! I was trying to make a number system for myself using logic similar to what you're describing Godel's as. I think I will go back to it now.

I love the content you produce and this video in particular. Somehow, it gives me hope. Thank you for doing what you do.

👆👆Thanks for watching and congratulation 🎊you have been selected among my shortlisted winners. Telegram only to claim your prize 🎁🥰⬆️

Your channel is awe inspiring and I wish I'd had access to it years ago. But I studied pre Internet. It's very spiritual to encounter such problems.

I suspect that for many people, making this video might be considered a lifetime achievement. But for Derek, just one more brick in his incredible, historic castle of outstanding teaching.

If only he is my math teacher or history teachers

I suspect that you are one of his groupies.

Derek: "... But for me, it was Thursday..."

look closer at the bricks composing the castle what are the bricks composed of.

Yeah fr

After studying all these concepts years ago, this video showing up randomly on my list tonight was lots of sweet memberberries. Mixing in the historic conflicts makes this so much better than any lecture I can remember.

I so deeply appreciate your videos, Derek. Thank you so much for these.

Would love to learn more about Goodall’s second theorem

I recently discovered this channel and am obsessed. I hate math but I watched the whole video with my jaw dropped. As a teacher I definetely plan to show some of your videos to my students. I recommended your channel to my fellow teachers. Maybe they will find inspiration in this as I have. Great work! Keep them coming! 🎉

@Potatoes and Ducks he didn't say he was math teacher tho He could be teacher of other subject he may love and tell about this video to his students ? Idk, I am only making a guess

Wait, you hate maths but are a maths teacher?

Man every concepts you explained are so on point and easy to understand I am computer science graduate. But still found your proofs enlightening

If there was an Oscar for YouTube videos, I have absolutely no doubt this would be nominated. Well done sir!

@Jean le Ronde d'Amelbert lol, here's my like sir

@Gabriel Carvalho you can like it now :)

@pottyputter05 I commented without much thought but I absolutely agree. Some (emphasis on some) of the content on YouTube is absolutely on par with Oscar nominated films, especially some of the lower budget ones

So we have the rewind or whatever it is but we don’t have YT oscars? Ricky we need you

I was going to like your comment, but it says 404...

Self reference seems to often plague logic problems (at least those that are popularly discussed on YouTube), but are there any systems which allow for a kind of logical superposition wherein we can consistently allow for something to simultaneously occupy two logical states that would otherwise be binary in other systems? And are these systems useful?

15:15 - The fact that it was Neumann who paid attention doesn't surprise me. The guy was genius among geniuses. In fact upon hearing Gödel he quickly realized and proved inconsistency by himself. Only when he took it to Gödel, he was informed Gödel already submitted work on that also.

You should make a Riemann Theormen video. It would be really cool. Loved these one.

This was absolutely excellent. I arrived here after watching your film about sucking liquid up a straw, that resonated with me as I'm a firefighter in the UK and we have to learn about suction lift etc. Delighted to have found your channel.

This video was awesome! I'm an economics major, but this makes me want to take computer science.

This is one of the best videos on this channel ever. My brain hurts a little, but I thoroughly enjoyed the experience.

762 pages just to prove 1+1=2? Did no one teach these blokes finger counting in childhood.😂😂😂

Can't agree more.

Took words right out of my mouth.

I can’t imagine this 30-minutes video covers one of my major course about finite-state and Turing automatons in college. Natural language, primitive recursive functions and state machines are always my favourite topic!

I didn't understand a thing but this video was still interesting and entertaining. Thank you Veritasium as always.

This is also philosophically/religiously enlightening: That there might not be such a thing as "fate" or destiny or predetermination; even if certain rules and parameters of our existence are preset from the start, we cannot know how we'll end up. And that's actually kinda liberating to think of. 🥰 Thanks

Whether there is a definite answer to whether a given machine halts or not if we can’t ever tell is actually a hot topic. There is this notion of constructive/intuitionistic logic which is really useful in a lot of cases (Curry-Howard Isomorphism comes to mind) where you do away with the law of the excluded middle. Infinity is weird.

@Tomas even if you don’t use a deterministic interpretation of quantum mechanics. You only proved that the processes that world operates with is probabilisitic not free will means anything.

@Tomas Well sorta, he is right in saying we cannot KNOW how we'll end up, but there would still be a deterministic and definite answer. We can never know if a program will halt or not, but there is an objective answer to that question: it either will or won't, it cannot be neither nor both. Also, there are deterministic interpretations of quantum mechanics, such as many worlds.

This video doesn't show that actually. I'm assuming you refer to the halting problem. Although you can't compute whether a program halts, the setting is still deterministic. On the other hand quantum mechanics does show that things can be truly random and cannot be predicted

This video is astonishing. The ending just make me crumble a tear, and I don't know why. I feel like maybe it just made me realise how much we just don't know about things around us. We use phones or computers without even realising ever how much hard work and dedication it took from other people why might never know ever existed. I would like to believe that we just simply cannot be thankful enough of them - but maybe we can express some respect by trying not to be ignorant about it, and atleast once - stop for a moment, and think of them, remember the names that we *can* remember to. Thank you veritasium.

The end made me tear up too... I guess we are weird... 😂

Let's take a few minutes of apreciation for the fact we can watch this content for free, specially because the quality is so HIGH. Veritasium and Micheal Stevens are the moral purpose of youtube.

I swear this is my favourite video on the internet. I've watched it easily 20 times

There was a brief moment while reading Hofstedter's *Gödel, Escher, Bach* where I felt I truly understood the concepts... This video brought me right back to that feeling! Very well written, presented, and produced! BRAVO!

​@sdfsdgsdfsdf23423423 As I understand it, you're saying we should skip over any theorems that are self-referential; if you tried to tell a judge "I'm innocent because this sentence is false", they would just fine you for contempt of court. Given an arbitrary theorem, it's not actually possible to tell whether the theorem is self-referential (it's undecidable). You could be trying to solve some innocent-looking arithmetic problem, but secretly the problem encodes the rules of arithmetic itself. That's all Gödel numbers are, after all. We don't introduce the self-reference ourselves, it's simply something that can crop up in the problems we try to solve. If the problem itself has a paradox, we can certainly ignore the paradox like you suggest and keep trying to solve it, but it's much better to realise that no solution ever will exist. That way we can go spend our time on something more fruitful.

@Rob Inson I agree. If you didn't already understand Godel's work, Hofstader's book would just confuse you.

@Leah C Checkout Babbage, and others, which were develloping computers regardless of Godel and their math plays. Turing was just one of the many who dabbled into computing.

I have seen this video many times, and I absolutely love it. Great Job, Derek!❤🧡💛👍👍👍👍

Great video! More please. And let me leave you with this thought. If mathematics has a hole whose center is self reference, then perhaps black holes are the representation of that concept in the physical world.

It felt like watching a movie about finding mathematics's true identity

I love your channel so much! Keep up the great work!

Turing's legacy lives on in everyday parlance among software engineers. When talking about a new novel programming language we always ask "Is it Turing complete?" Which asks if any system of logic can be expressed in it. HTML and XML for example are not Turing complete, where as JavaScript and c++ is.

As someone who majors in mathematics while minoring in computer science, this video is absolutely awesome. I've learned about a lot of these things in isolation, but this really connects them all.

@Peter Codner Horrible attempt at sounding smart. Did you feel bad this video didn't including anything about your field of expertise which you could flex. Sit down buddy, you've successfully cringed me out.

@jonasba276 Nice etymology: Middle English (in the sense ‘stupid’): from Old French, from Latin nescius ‘ignorant’, from nescire ‘not know’. Other early senses included ‘coy, reserved’, giving rise to ‘fastidious, scrupulous’: this led both to the sense ‘fine, subtle’ (regarded by some as the ‘correct’ sense), and to the main current senses.

​@Peter Codner Damn you sound like a nice guy

Apparently you decided to skip English , in the pure form of which there are no verbs to major or to minor and thus no gerunds thereof since they are adjectives. Of course there is nothing to prevent you from inventing your own exclusive-to-you language save perhaps that you will be its only speaker.

You like conflating. Well that sums up this whole video. Have fun!

All I can say, is I wish All Teachers were like you. Imagine what the next Generation could do with Teachers like this.

"Even a perfect, complete description of the microscopic interactions between a material's particles is not always enough to deduce its macroscopic properties." This single sentence makes me feel like I'm standing on the edge of a precipice

Gotta love quantum shaboingery

Amazing man! Some of the best learning content!! Just inspirring 🔥🔥👋🏽

This might be the best Veritasium video ever. Pure quality.

I'm literally learning Gödel's completeness rn in class, and this video made everything 100% easier

My first instinct would be to assume that the battle between Intuitionists and Formalists couldn't have been that dramatic. But then I remembered that there was an actual riot featuring thrown chairs and fistfights on opening night of Stravinsky's Rite of Spring over the timbre of the bassoonist. Man, nerds back in the day were hardcore.

@deadpxn That was wrong sorry

@c2onoui c2onoui No worries 😊

@deadpxn So you have nothing to back your comment or what.. if no have a nice day then😊

@c2onoui c2onoui I forgive you

@c2onoui c2onoui I'd have written out a paragraph or two if I wasn't calm

This is exactly the thing thats insanely insane, yet it exists.... I came once across the problem of the set that includes itself. I couldnt, obviously, solve it. I thought a few times whether the math is perfect or not, and now I got the prove of it. U have even one more insane question: does exist an reality, where 1/=1, or where geometry works differently? I always think of this and also stop everytime because it seemingly pointless xd-. Btw amazing video as always! :)

Freshmen with Mathematics major here, truly insightful and thought provoking video by you ;)

When I first learned about this theorem an analogy came to my mind: "Math is like a toolbox full of tools. These tools can fix/build many things except the toolbox they are in." But I don't know if it is correct.

Excellent video. Amazing for 30 min. What are the implications for any Science Theory based on some Math Model? Always some things it will not understand? I suspect so. Thanks much Derek.

One of the most well researched videos I have ever seen on this channel!!! Bravo!!!

I’ve just finished a 3 month university mathematics module on Gödels theorems and you have managed to summarise the whole thing in impressive detail in just 30 minutes. Well done sir.

Hasan Tınaz I’m sure you’re not… Desire and determination is all you need to study maths (and it looks like you already have the former). Don’t get frustrated and, if you want to, you’ll get there eventually :)

@Pink Lady Wow you are an impressive learner! Half of the students on my course don’t show as much interest as you. I am in my third year of a four year masters course in mathematics and it looks like a lot of the stuff you’ve been learning so far is at a year 1 & 2 level. Logic is certainly something you should be reading into if you haven’t already (which is the subject area of this video). Otherwise I’d say you have a very solid base range of mathematical knowledge. As I said I haven’t read many books since all of my modules have lecture notes that are enough (they are really just a subset of the optional further reading books). All of the modules have optional reading lists of 5-10 books and I’ve taken about 25 modules so there’s a lot to choose from! So I can recommend some if there is a subject area that you are specifically interested in, otherwise there will be too many to list here. Topology is on of my favourites, it is a subject I have just studied this year also. It draws on lots of different areas of maths (set theory, geometry, calculus, analysis, logic, group theory, metric spaces, combinatorics, graph theory, vector spaces, and probably more!) so it is definitely a later one to get into.

Hasan Tınaz I keep a study diary on all the tutorial videos that I watch in Youtube, Udemy, etc. A study diary hugely helps.

Hasan Tınaz Been there and done that. New maths is always a struggle. It is one step at a time. I treat each information like a gold nugget. Gather them and clump all the gold nuggets together and that is a gold bar, which is knowledge. Many gold bars gathered — a talent. I oftentimes learn math by brain-picking nerds social media like Twitter, Quora, Facebook groups, etc.

Anil Vips First off, I am a traditional artist with a growing interest in 3D realism & animation including physics simulations, which require lots & lots of writing maths inside node compositing & scripts. I am only intermediate in mathematics. I am not in college, but I can self-study at home, no problem. Only two years ago, I took up studying mathematics at home, as I had needed it for 3D realism, also for programming & scripting. Also, I have just recently taken up physics and engineering as well. I study those a little and maths more. Initially, two years ago, I had to relearn highschool maths at home as I had forgotten half of it. After having learned them off, I moved onto reading undergraduate books on calculus, analysis, linear algebra, set theory, number theory, abstract algebra, discrete mathematics, etc. I know just roughly 75% of them. I don’t know what other undergraduate maths I am missing on the list. Well, I will get there anyhow. At home, I have roughly 25 paper books in my collection, those on maths & physics. Half that number in Kindle. Of paper books, I have only one on calculus (metric version) by Ian Stewart, and I don’t know what next calculus books to read after that. Three books on linear algebra. One on discrete mathematics. One on vectors and tensors by Dan Fleisch. One on algebraic number theory. I’m always curious to know what books that undergraduate students read in college, especially in their first year & second year, and what they read thereafter. I understand just a little of topology, but I don’t know what prerequisites to study before moving to topology, category theory, and suchlikes. I regularly go exploring their internal topics inside Wikipedia, Mathematics Stackoverflow, MathOverflow, Quora, etc. I read what others are studying in colleges, what books they read for studies.

Cantor’s set theory is so fascination I studied it for 18 weeks

One of my favorite Veritasium videos, thanks so much!

I watched this video many times even though that I knew before about all information it presents. It's well presented and perfectly explained. Great work!!!

👆👆Thanks for watching and congratulation 🎊you have been selected among my shortlisted winners. Telegram only to claim your prize 🎁🥰✔️

Yes I must agree with many responses. Precise and Powerful: This is more than magic from the easy to understand machines. Thank you mate.

a truly fascinating yet troubling video. I'm fully absorbed with this problem.

Ironic that Godel's death was the result of a self-referential paradox: he died in order to not die

Beat me to it. I nearly did a spit take when I got to that part of the video.

Why isn't this pinned.....??!?

This is one of best comments for me

As master Oogway once said :" One often meets his destiny on the road he took to avoid it."

😂😂

Watching this video again, And i still can't help but be blown away by it, It is sad though, that Turing never got to see his legacy, Same with Gödel, Rest in peace, You did truly change the world

05:58 But that only means, that not all possible unique numbers have been listed in this mapping process. For example if you make a finite mapping process of the combination of only the literals "1","2" and "3" then the set of all unique combinations will be complete and look like below. For that reason you will not be able to add any new combination with that technique in the video (Cantor's "Diagonalization Proof") in this set of finite literals... 123 & 132 & 312 & 321 & 213 & 231 In other words you can not list all numbers in a mapping process of infinite numbers and for that reason you will always be able to add a new number through "Cantor's Diagonalization Proof" in any set of infinite numbers. If we would be able to list every unique entry of a set of infinite numbers, then this set of infinite numbers, wouldn't be infinite.

In a world dominated by big corporations, nuclear and pinpoint missiles, this field feels as though it is the individual person's last chance at control

This comment sounds like it's coming from a dark place. I hope you are ok.

Would love to watch a video showing how you do all the research for the video, like selecting topic, research methodology, etc.

yeah me too

This one was more difficult to understand and also just amazing 🤯

This video is a masterpiece. The content, the animations. Everything is out of this world

This is a dream for every math teacher

Cardboards were very much from this world ;)

The use of mathematical symbols as buildings is *chefs kiss*

basically what people dont like is that there are things here in maths that cannot exist but are needed to form what exists and for some reason it actually makes sense and when mathematicians wanna fight they just show what doesnt exist and hopscotches with it kinda like negative numbers, negative squares etc...

As a high schooler and prospective engineering student. I am glad to not be getting a math degree this is just too much I would like to work within the practical real world thank you.

25:00 I love this. It explains the dimensions of thought processing patterns

H is valid 🤣

That was fascinating to watch (it won't surprise anyone if I say I have never really been into math) but from the way it has left my brain feeling I wonder if it was simply the case, he had no choice if you have a brain that works like his once you go down the rabbit hole sadly there is no turning it off. I am now going to take some time with the 'He is his own worst enemy' thing and the one posted below about 'the next sentence is true'. All along I have been thinking the answer was 42, what a fool I feel.