6:30 im not convinced we couldn't find the same number anywhere on the list, can someone explain me please?

Black Bird

@Релёкс84 ah thanks if it's simply defined as such then yeah sure

Релёкс84

It is defined as being different from every number on the list. So if it ever were on the list, it would be different from itself, which is quite impossible.

M V

6:36 Why does the proof concludes this as you will also get an index for this new number?

talenttrading

Hilbert is wrong.

David

my brain hurts

adkr grc

no jocking the fatal flaw is that humans invented math, so stupid to believe that the universe all of the sudden after billions of years will just follow those rules.

Anton Hanzelik

thank you well, much obliged

Прохор Шляхтун

8:00 "informal leader of the formalists" - sounds funny

Necrotic Uterus

who wants to be a next gen parasite ?

VERICHO

But if an axiom is unprovable, then how can we know that it is true???

JCGoogle

The fatal flaw of this "Math Has Fatal Flaw" video is math only has a fatal flaw when you put fatally flawed restrictions on it. For example - the barber at 9:35. The paradox comes from the set of fatally flawed rules. Why must the law say the barber can't shave himself? Most of the range in which these kinds of questions are being asked is beyond where it provides practical usage. For example Newtonian physics works within a certain range and then at some point General Relativity has to be used. Or you can make up a problem that is not answerable,.. such as,.. "Which came first the chicken or the egg?" or more specifically "which came first, the chicken or the chicken egg?" The question has a fatal flaw that restricts the answer to only two possible outcomes,... when neither is true. On an evolutionary scale, the egg obviously came before the chicken. But regarding the chicken and the chicken egg question,... the answer is they both came at the same time. The chicken evolved with the chicken egg simultaneously. This kind of metaphorically proves that if something proves to be unprovable, we just may be using the wrong tools to prove it or asking a fatally flawed question.

Amine Amine

Maybe contradictions are true 🤔

László Tungli

Next possibility: Does not exists such a g. Proove: g contains g, it means, that g must be more complicatued than g. It is'nt true. Indirect proove.

Beldan4

and this is why people take such comfort in systems of faith. Even within the video itself, if people couldn't have faith (faith is the assurance of things hoped for, the evidence of things not seen) that one day the hole would be filled, they would go crazy thinking existence shouldn't be possible.

Tamás Hercz

So Math itself is infinite. And Math's infinity could be only proven by Math itself. So Math's biggest flaw is itself being a Self Reference Paradox.

Eleanor Lim

i come here when i'd like to get a migraine

Yankosh Badal

The statement below 👇 is false The statement above ☝️ is true

Golden Eagle Arbor

Sounds like a lot of these guys could of had better lives if they hadn't of taken "show your work" a little to seriously

Hamoudi

"Not all infinities are the same size" 😂😂 Size means that something is finite and has dimensions to it. This statement is so wrong in itself!

Hamoudi

In an infinite row of numbers, you cannot know for certain that the new number you created will not appear, how in the world one could prove of disapprove this? Nobody could write down an infinite list to begin with.

John Counts

Or there’s a third alternative to the goodle number g It’s not a number, thus it doesn’t prove anything at all other than that this guy was a tad off his rocker

John Counts

But………….if the index were made with the real numbers instead then you could make the same statement in reverse…….

Astig Astig

Hello, veritasium. So if 'trueness' and 'provability' are seperate, then: (While A= a statement) A can be true but can be unprovable A can be true and provable A can be false but unprovable A can be false but provable So if the statement with godel code 'g' is false, could it be that it's not a contradiction? Since if that statement is false, then it means that proving is true. Proving it as false means it's guaranteed that it's false, and there's no contradiction. Proving is proven to be true. Can anyone share some insights pls

Eli C

wait what did you use to play the game of life like that? or is it all just animation?

Crypt0tagz

Cool but ........ what???

HLQA

Godel's parents were cousins?

YordanYoshi

Seeing the game of life run itself honestly amazed me

유리 큐브 제왕Lord of the Glass Cube

God: "hehe, I remember when I was in pre-school"

maruftim

Prinkipia or prinsipia..?

Nathan5791

Hilbert - I'm gonna probably win a nobel price for this... Godel - Hold my beer while I wreck this guy's whole career!

Darren A

How the hell are there over 6K dislikes of this video? What morons.

Lelsewhere Lelsewhere

The whole Turing punishment is very misunderstood. Like, imagine if a person created a device that could do the 1940s equivalent of cracking nuclear launch codes. Now imagine that person keeps getting drunk, keeps picking up (very socially unacceptable) prostitutes, is continually arrested for causing disturbances with these prostitutes, and continually bailed out by his friends in government. Do you think this person can keep the device or its workings secret? Do you think that such a person could be blackmailed into giving his secret device up? Do you think this person's role in the device could be found out, and that could then be manipulated or kidnapped and forced to work with "the enemy"? That's almost literally the situation. I mean, if he just chose to live a quiet (though yes, somewhat secret and discrete) life with one gay lover, I'm 99% certain the government would have looked the other way, as they usually did with important men who were gay, even back then. Or do you think that there were no important, secretly gay (but known to the higher ups) men ever in Britain before the 1990s...

Joao Caetano

Very good video. Good work. When I was in basic school I wasn't very good. In secondary school I realised that some subject could actually help me understand, improve and do the tasks of the most ordinary routines of live. By "college" I notice that I was pretty much surrounded by knowledge and everything around me was almost cared for a different subject. Math's there too. But then a teacher had a sort of a public discussion about life and death and for a few days I had my brain wrap and immersed in all the knowledge I have learned so far. Unsurprisingly I came to the conclusion that every subject has its own development but as close that you get to the edge of it's circle the more that knowledge gets more and more mixed with the circule that is next to it. I think that this video is all about that as well. Is about the filosophy of maths. A sort of grey area between maths and filosophy. Or perhaps other subject. I always been convinced that is an area development by the best of maths but also by the ones more unsettled with the 1+1 is 2 who seek arguments in different subjects is order to support their ideas using their knowledge from another subject. It will certainly leave me unsettled for a few days.

Brother Malachai

The thing is, if our math is so incomplete and flawed, how do any of our modern equipment work...at all? Since it's so dependent on accurate mathematics?

Origami Master

It's more of that it was shown that there is no fundamental system of mathematics which can be complete or decidable and completeness is not provable. So, it's not that every computation in mathematics is wrong just that there are some statements that are unprovably true and that not every problem is actually solvable. It could be proven that it is not (such as the twin primes conjecture). Tbh this is what I understood from the video I could be misinterpreting.

Matt C

Can someone please explain to me again the part when Derek says that proving the g card makes no proof? I understood everything else but i keep watching this statement and I can't seem to get it

Ronnie Alacre

9:55 is a meme.

Jeremy Cheeseman

Isn't the diagonalization proof wrong? It seems that in order for it to work you must get to the end of an infinitely long list, which can't happen since it is infinite in length, right?

Mama Di

I need to say: this video makes me cry. About math, but so touching and personal at the same time. Thanks for that.

Reprint001

"This... is the game of life. Running... on the game of life." My response. "F..k". I don't know why that was my response.

Truth Holder

@Veritasium, please consider doing an episode that goes through the proof of why computers use binary. In one of my college courses that was mathematics for computer science majors, we had to go through that proof. The number that came out was the natural logarithm e (2.71828). The following class when we went over the assignment those of use who came up with e (and were also confused as to how you can have a computer system that is BASE e) got our answer. He explained that e was correct, but would be impossible to physically do that. So he rounded both up and down. Hence, BASE 2 and 3 were both good number bases for computers. Few people know this because we also think of computers as using 1s and 0s, or electronically positive voltage or zero voltage (usually +5v and 0v). But there have also been tertiary chips using +1, 0 and -1. (negative voltage, 0v and a positive voltage) Thus, both are equally efficient but the industry rapidly settled on binary.

Maynard Hahn

That does not make sense

Truth Holder

When I was in a computer class back before PCs, we had to write a game of life program in FORTRAN. There's much truth out there that can't be proven by the means we know now. That's the beauty of truth. Someday we'll learn the proof, but for now, we're just not there. It's also the beauty of live (as in real life) itself. :) Also, my first degree was in mathematics. In one of my classes we had to show examples where x divided by 0 is undefined. So many people think it is infinity, but that's not always the case. I sure wish I would have saved that paper. But, after many decades, all that stuff is somewhere in a landfill.

Nimish Shukla

Those who know how much wealth they have in bank should not stay here too long or else they might not have any brains left after trying to make sense of this maths-match here.

Hiten Doshi

I didn't quite understand Bertrand Russell's argument about R - why if it does not contain itself then it must contain itself, and vice-versa. Can anyone explain the logic? Thanks.

GriZz gamer

I love this guy's videos.

Yakubu Mshelia

I stopped understanding this video at 14:00 😂🤣

Ol' Smokey

Math doesn't factor in chaos and consciousness probably because it can't and reality has shown time and time again to contain both.

First Name Last Name

How the hell did anyone come up with this and actually understand it. It’s so fascinating but I don’t understand it lol

Blue Paint

press f for the barber

LunDruid

Whenever I hear about this kind of thing, it really makes me wonder if part of the problem is that our current understanding and use of mathematics is fundamentally flawed even right down to arithmetic. But then I realize how impossible it would probably be to come up with some system of math that didn't work with arithmetic as a base, and my head ends up hurting.

thefran901

Hilbert: "Math is complete." Gödel: "Well actually..." Hilbert: "Ok, fair enough, but math must be consistent." Gödel: "Ackchyually..." Hilbert: "Oh for fuck's sake, but it has to be at least decidable!" Turing: "LOL"

Thefreakofthehill

As a object oriented programmer used to deal with composite objects that has references of itself, the R set blow my mind

Kshitiz Gupta

Man I have seen this thrice now

Priyanshu Goel

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

Umang Ravaiya

29:56 which music is this??

kerry mackey

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

Andrew

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

36nibs

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

BonnetBee

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

Nature

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

Loruo Ditlhong

Lol just found the meaning of consciousness

Gerardo Contreras

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

BBucky98

Me too the card part is too much

Scots Diesel

So sad about Alan Turing...❤️

mohamed mada

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

Moyprod

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

Andre

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

mohamed mada

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

Andre

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

johnnytheprick

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

Alex Huggett

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

Justin Hamlin

♾️

Adhithyan Sreedharanarayanan

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

Adhithyan Sreedharanarayanan

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

𝕲𝖆𝖇𝖗𝖎𝖊𝖑

We are immortal until the day we die

Rex Dalit

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

One Issue Voter

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

TravelBig

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

Andre

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

flow_mang

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

Jayden Maree

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

Grant Currin

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

TerrorHuhn

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

Sasa Radetic

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

Willie Theron

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

Andre

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

Eddie RUKidding

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

Andre

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

Irfan Kanth

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

Irfan Kanth

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

Chase Thompson

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

Christopher Suiter

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

MdSteel7

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

Brendan Shimizu

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

Clark Massey

This is excellent.

Alaine Ninmah

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

Helgali

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

Giovanni A

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

Dane Norman

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

Snarckys

you are intelligence and wit made man, no doubt

Catchafire2000

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

Naveenmurugu KANNAN

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

kirdiekirdie

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

School of Grok

Sublime.

Sharan Pant

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

phil longneck

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

Poke fun at idiots

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

Vedant Jagtap

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

Keven Gagner

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

ludovic gauchet

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

6:30 im not convinced we couldn't find the same number anywhere on the list, can someone explain me please?

@Релёкс84 ah thanks if it's simply defined as such then yeah sure

It is defined as being different from every number on the list. So if it ever were on the list, it would be different from itself, which is quite impossible.

6:36 Why does the proof concludes this as you will also get an index for this new number?

Hilbert is wrong.

my brain hurts

no jocking the fatal flaw is that humans invented math, so stupid to believe that the universe all of the sudden after billions of years will just follow those rules.

thank you well, much obliged

8:00 "informal leader of the formalists" - sounds funny

who wants to be a next gen parasite ?

But if an axiom is unprovable, then how can we know that it is true???

The fatal flaw of this "Math Has Fatal Flaw" video is math only has a fatal flaw when you put fatally flawed restrictions on it. For example - the barber at 9:35. The paradox comes from the set of fatally flawed rules. Why must the law say the barber can't shave himself? Most of the range in which these kinds of questions are being asked is beyond where it provides practical usage. For example Newtonian physics works within a certain range and then at some point General Relativity has to be used. Or you can make up a problem that is not answerable,.. such as,.. "Which came first the chicken or the egg?" or more specifically "which came first, the chicken or the chicken egg?" The question has a fatal flaw that restricts the answer to only two possible outcomes,... when neither is true. On an evolutionary scale, the egg obviously came before the chicken. But regarding the chicken and the chicken egg question,... the answer is they both came at the same time. The chicken evolved with the chicken egg simultaneously. This kind of metaphorically proves that if something proves to be unprovable, we just may be using the wrong tools to prove it or asking a fatally flawed question.

Maybe contradictions are true 🤔

Next possibility: Does not exists such a g. Proove: g contains g, it means, that g must be more complicatued than g. It is'nt true. Indirect proove.

and this is why people take such comfort in systems of faith. Even within the video itself, if people couldn't have faith (faith is the assurance of things hoped for, the evidence of things not seen) that one day the hole would be filled, they would go crazy thinking existence shouldn't be possible.

So Math itself is infinite. And Math's infinity could be only proven by Math itself. So Math's biggest flaw is itself being a Self Reference Paradox.

i come here when i'd like to get a migraine

The statement below 👇 is false The statement above ☝️ is true

Sounds like a lot of these guys could of had better lives if they hadn't of taken "show your work" a little to seriously

"Not all infinities are the same size" 😂😂 Size means that something is finite and has dimensions to it. This statement is so wrong in itself!

In an infinite row of numbers, you cannot know for certain that the new number you created will not appear, how in the world one could prove of disapprove this? Nobody could write down an infinite list to begin with.

Or there’s a third alternative to the goodle number g It’s not a number, thus it doesn’t prove anything at all other than that this guy was a tad off his rocker

But………….if the index were made with the real numbers instead then you could make the same statement in reverse…….

Hello, veritasium. So if 'trueness' and 'provability' are seperate, then: (While A= a statement) A can be true but can be unprovable A can be true and provable A can be false but unprovable A can be false but provable So if the statement with godel code 'g' is false, could it be that it's not a contradiction? Since if that statement is false, then it means that proving is true. Proving it as false means it's guaranteed that it's false, and there's no contradiction. Proving is proven to be true. Can anyone share some insights pls

wait what did you use to play the game of life like that? or is it all just animation?

Cool but ........ what???

Godel's parents were cousins?

Seeing the game of life run itself honestly amazed me

God: "hehe, I remember when I was in pre-school"

Prinkipia or prinsipia..?

Hilbert - I'm gonna probably win a nobel price for this... Godel - Hold my beer while I wreck this guy's whole career!

How the hell are there over 6K dislikes of this video? What morons.

The whole Turing punishment is very misunderstood. Like, imagine if a person created a device that could do the 1940s equivalent of cracking nuclear launch codes. Now imagine that person keeps getting drunk, keeps picking up (very socially unacceptable) prostitutes, is continually arrested for causing disturbances with these prostitutes, and continually bailed out by his friends in government. Do you think this person can keep the device or its workings secret? Do you think that such a person could be blackmailed into giving his secret device up? Do you think this person's role in the device could be found out, and that could then be manipulated or kidnapped and forced to work with "the enemy"? That's almost literally the situation. I mean, if he just chose to live a quiet (though yes, somewhat secret and discrete) life with one gay lover, I'm 99% certain the government would have looked the other way, as they usually did with important men who were gay, even back then. Or do you think that there were no important, secretly gay (but known to the higher ups) men ever in Britain before the 1990s...

Very good video. Good work. When I was in basic school I wasn't very good. In secondary school I realised that some subject could actually help me understand, improve and do the tasks of the most ordinary routines of live. By "college" I notice that I was pretty much surrounded by knowledge and everything around me was almost cared for a different subject. Math's there too. But then a teacher had a sort of a public discussion about life and death and for a few days I had my brain wrap and immersed in all the knowledge I have learned so far. Unsurprisingly I came to the conclusion that every subject has its own development but as close that you get to the edge of it's circle the more that knowledge gets more and more mixed with the circule that is next to it. I think that this video is all about that as well. Is about the filosophy of maths. A sort of grey area between maths and filosophy. Or perhaps other subject. I always been convinced that is an area development by the best of maths but also by the ones more unsettled with the 1+1 is 2 who seek arguments in different subjects is order to support their ideas using their knowledge from another subject. It will certainly leave me unsettled for a few days.

The thing is, if our math is so incomplete and flawed, how do any of our modern equipment work...at all? Since it's so dependent on accurate mathematics?

It's more of that it was shown that there is no fundamental system of mathematics which can be complete or decidable and completeness is not provable. So, it's not that every computation in mathematics is wrong just that there are some statements that are unprovably true and that not every problem is actually solvable. It could be proven that it is not (such as the twin primes conjecture). Tbh this is what I understood from the video I could be misinterpreting.

Can someone please explain to me again the part when Derek says that proving the g card makes no proof? I understood everything else but i keep watching this statement and I can't seem to get it

9:55 is a meme.

Isn't the diagonalization proof wrong? It seems that in order for it to work you must get to the end of an infinitely long list, which can't happen since it is infinite in length, right?

I need to say: this video makes me cry. About math, but so touching and personal at the same time. Thanks for that.

"This... is the game of life. Running... on the game of life." My response. "F..k". I don't know why that was my response.

@Veritasium, please consider doing an episode that goes through the proof of why computers use binary. In one of my college courses that was mathematics for computer science majors, we had to go through that proof. The number that came out was the natural logarithm e (2.71828). The following class when we went over the assignment those of use who came up with e (and were also confused as to how you can have a computer system that is BASE e) got our answer. He explained that e was correct, but would be impossible to physically do that. So he rounded both up and down. Hence, BASE 2 and 3 were both good number bases for computers. Few people know this because we also think of computers as using 1s and 0s, or electronically positive voltage or zero voltage (usually +5v and 0v). But there have also been tertiary chips using +1, 0 and -1. (negative voltage, 0v and a positive voltage) Thus, both are equally efficient but the industry rapidly settled on binary.

That does not make sense

When I was in a computer class back before PCs, we had to write a game of life program in FORTRAN. There's much truth out there that can't be proven by the means we know now. That's the beauty of truth. Someday we'll learn the proof, but for now, we're just not there. It's also the beauty of live (as in real life) itself. :) Also, my first degree was in mathematics. In one of my classes we had to show examples where x divided by 0 is undefined. So many people think it is infinity, but that's not always the case. I sure wish I would have saved that paper. But, after many decades, all that stuff is somewhere in a landfill.

Those who know how much wealth they have in bank should not stay here too long or else they might not have any brains left after trying to make sense of this maths-match here.

I didn't quite understand Bertrand Russell's argument about R - why if it does not contain itself then it must contain itself, and vice-versa. Can anyone explain the logic? Thanks.

I love this guy's videos.

I stopped understanding this video at 14:00 😂🤣

Math doesn't factor in chaos and consciousness probably because it can't and reality has shown time and time again to contain both.

How the hell did anyone come up with this and actually understand it. It’s so fascinating but I don’t understand it lol

press f for the barber

Whenever I hear about this kind of thing, it really makes me wonder if part of the problem is that our current understanding and use of mathematics is fundamentally flawed even right down to arithmetic. But then I realize how impossible it would probably be to come up with some system of math that didn't work with arithmetic as a base, and my head ends up hurting.

Hilbert: "Math is complete." Gödel: "Well actually..." Hilbert: "Ok, fair enough, but math must be consistent." Gödel: "Ackchyually..." Hilbert: "Oh for fuck's sake, but it has to be at least decidable!" Turing: "LOL"

As a object oriented programmer used to deal with composite objects that has references of itself, the R set blow my mind

Man I have seen this thrice now

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

29:56 which music is this??

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

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

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

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

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

Lol just found the meaning of consciousness

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

Me too the card part is too much

So sad about Alan Turing...❤️

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

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

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

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

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

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

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

♾️

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

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

We are immortal until the day we die

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This is excellent.

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

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

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

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

you are intelligence and wit made man, no doubt

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

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

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

Sublime.

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

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

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

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

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

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

this seems like nonsense