Gödel's Incompleteness Theorem - Numberphile 

And Bernoulli start beating L'Hôpital for taking credit for his theorem.

@@justasaiyanfromearth5252 surely you mean *buying* credit

what was the joke? L'Hospital was sick and went to the Hospital?

@@ThomasJr the joke is that godel realises that he himself is in the joke

@@asamanthinketh5944 The point is obviously the incompleteness theorem. They can't tell because the system is not complete, and you need to leave the system in order to be able to make statements about it. L'Hopital needed a hospital.

Based

They patched it already, there's a recursion depth check. It stops at 42.

@@misiol smart

its rude to talk about stack overflow and to not ask stack overflow what it thinks about this

@@misiol I was going to like your comment but it has 42 likes :3

Do not try and prove the theorem. That's impossible. Instead... Only try to realize the truth. There is no proof.

Anyone here knows 'Genetically Modified Sceptic'? He's very recommendable.

The teachers: "It's a multiple choice question, just circle a letter" _Circles any letter "e" and declares it as "None of the above"_ The teachers: ö

@@loturzelrestaurant True, but in what ways is it relevant? Does he talk about Gödel's theorems in his videos? If yes, could you tell which one it is? Thanks.

@@loturzelrestaurant Yes, waste of time, go read books

Given that he is supposed to represent the viewer and ask the questions we would ask, it only make sense that his questions will become smarter as we get smarter.

To be fair, he's always asked good questions, the only difference is, he now knows how to better articulate these questions.

Yeah, he's basically being paid to audit an entire math curriculum...along with every other subject his work covers. And for every minute of footage that ends up in a video, there's at least a minute (and probably more) of material we don't see. He's a lucky guy.

Antonio Cunicelli? um abs de outro groselheiro.

haha, a groselha é densa na internet

Nice try Fermat XD

in 350 years someone shall proove it.

Well then, that probably does not count? Shouldn't a proof be reasonably accessible? What exactly is a proof? Is there a proof of what makes a proof to be a valid proof?

Did you forget you can put links on the internet *Fermat* ?

Nice try, Uncle Petros.

Just cherry pick your axioms and you are fine in your frame of reference. Even linguistics. Turtle realms all the way down. E.g. I accept self-referential paradoxes in my realm to exist or no self-referential paradoxes allowed in my realm. du -h --max-depth=1 /home/universe

I hate you!

It is a crime that this comment only has 155 likes.

only a sith deals in ZFC

„I brought improvable statements, contradictions and incompleteness to my ne axioms“ „Because of Goldbach?“ „You turned Hilbert against me“

Thanks, I've always wondered what mathematical horror looks like.

@@ronithazarika2042 Try reading the late work of Georg Cantor.

Huh yeah

No. The proof is not that some truths could not be proved but SPECIFICALLY complex truths are unprovable. If a system is simply enough truth is always there. But when a system becomes too complex truth and falsehood become fuzzy to assign so the theorem remains true - there is no paradox. It is about complex truths, etc.

Like np problem

interesting. tho he obviously meant that it proves the consistency or the lack of it, which was what he sought

well after his negative result he had a mental breakdown and went to a sanatorium. so I guess he didn't expect that deconstructive result:D

@@70ME3E no. he was trying to prove and find the final consistent system of mathematics.

@@_kopcsi_ That was because of the Nazis, not because of mathematics.

@@DukeOnkled well, probably you are also right. i mean a mental breakdown can and usually have more trigger factors and reasons. he had paranoid mental problems because of the Nazis, indeed. but before that he had already have mental problems following his negative results in logic and mathematics.

MU!

Great book! The comment about AI is not quite right I think. My recollection is that he gives a generous amount of space to present the view points of other philosophers who have attempted to use the Incompleteness Theorem to shoot down strong A.I. but he doesn't particularly support their lines of argument. In fact he is in the strong A.I. camp.

Tim Weaving Plus one for Gödel, Escher, Bach! Big thanks to Lev Grossman for bringing it to my attention via his excellent Magicians trilogy.

While GEB is excellent, for this particular topic I'd recommend Nagel and Newman's little book, _Gödel's Proof_.

I recommend the graphic novel "Logicomix: an epic search for truth" which begins at Bertrand Russell's work on set theory and introduces Gödel as well. Great read, and easily accessible!

what are you using to discover maths? you hsve no idea? what would you use to discover what you are using to discover maths?Cam a mirror reflect itself?

There is a quote attributed to Einstein that "if you don't understand something well enough to explain it in simple terms, then you don't understand it well enough" I think that holds very true for most things

sometimes they do illegitimately oversimplify but otherwise i completely agree!

@@dnuma5852 Yes. The first ”1+2+3+… = -1/12” -video being a textbook-example of that.

TurboCMinusMinus You are Turing Incomplete because I said so

LSD is helluva drug...

Johnny Coull It's also an amusing read on the way - jokes and puns make it read differently to the style of most maths books, as well as the links to Escher and Bach

Also snag his book Metamagical Themas

thx for the info

Incredibly enough, I discovered the proof of the CH by simply establishing 1-to-1 correspondence between positive decimal floats and positive decimal integers. For the last 125 years such a proof eluded the top set-theory experts of the planet. It was eventually abandoned and classified either as impossible or undecidable. Well, no more. Sit back and enjoy one of the most trivial and elegant proofs in set theory. It is published on RU-vid with the title : “Proving the Continuum Hypothesis”. I would appreciate if you watched and commented on this video. Your input will be given full credit.Tamas Varhegyi

This is a masterpiece

The thing is that all you need to prove that it is false is a counter example, as said in the video. While you might show that it is false without being able to prove it, it doesn't change the fact that there is a way to show that it is false that is proveable.

that sounds like the halting problem.

the finite process of finding a counterexample might still take more time than we actually have. a process can be finite but still impossible.

The idea is that the counter example fits in the current axioms. So, the Riemann Hypothesis cannot be unprovably false. We thus have 3 options : Provably false with a counter example, Provably True, Unprovably True.

Couldn't it still be false, but the counter example is some weird transcendental number which can not be constructed with the current axioms of mathematics? Just like you can not construct non-Lebesgue-measurable sets without choice.

That doesn't originate from The Boondocks. Carl Sagan is attributed with popularizing it, but it's been around for quite some time. Theists use it to cope with the fact that there's no evidence for the things they believe in.

@@piecrumbs9951 Limiting evidence to empirical evidence is the source of that fallacious "fact" that there is no evidence. Theologians argue with rational evidence about the nature of things ( Metaphysics ) starting with logical principles and causality principle. Before being skeptical with principles, it should be dealt with the internal consistency of arguments. If it's consistent internally, then according to that set of principles that the majority of intellectuals accept as neccessary : it's an evidence. If you start to become skeptical about logical principles you should start by bringing reasons for that skepticism!

@@dakyion ratio

​@@piecrumbs9951strawman

@@dakyion My man is *so* angry that his imaginary friend is imaginary.

Okay....

You're absolutely right, and I was about to write a comment on the same line. Without a clear and explicit reference to the concept of a formal system all that is said in this video is highly inaccurate, if not altogether wrong. For instance, he says that Gödel's statement is true, after saying that Gödel's Theorem states that it can't be proved either true or false. Without adding "formally" that doesn't really make much sense. He only talks about axioms, which are only a part of a formal system, and totally neglects talking about rules of inference, which are what the theorem really deals with.

Thank you.

You wrote: "So, in a word, the sentences, even though correctly built, will be INDEPENDENT OF the set of axioms". If by independent you mean 'logically independent', that is only a consequence of Gödel's theorem in first order languages, whose logic is complete. In second order arithmetic, the Peano axioms entail all arithmetical truths (they characterize up to isomorphism the naturals), so that no arithmetical sentence is logically independent of such axioms. It occurs, however, that second order logic is incomplete and there is no way to add to the axioms a set of inference rules able to recursively derive from the axioms all of their logical consequences. This is why Gödel's theorem holds in higher order languages too. In fact, this is how the incompleteness of higher order logic follows from Gödel's theorem.

hm. got it. thanks

Does your friend's name happen to be Grigori Perelman?

bakavasa: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo." -- Grigori Perelman. So no.

loved the asking for a friend bit, one can't ever be to safe. =D I guess you can win more then 1 million, there are a few millennium problems left that I believe to just be unprovable..

Would you win the full prize for a single counterexample, i.e., a non-trivial zero outside the critical line? Also asking for a friend.

Divergent Integral yes. It proves it false.

Lelouch Yagami Can you proof that you don't exist? I don't think so.

keep living!

Shardar 1 being able to prove that you DO exist depends on your definition of 'existing'

MineWarz I think, therefore I am (a brain in a jar).

Shardar 1 knew you'd answer that but how do you know you think and there's nobody just giving you your thoughts?

Yes. But the clever thing was that he did it mathematically, so that the mathematicians have to believe it.

Why do Brits make math plural?

@@KenJackson_US I'm not british, I don't know why I made it plural but it sounds right in the sentence like that

@@KenJackson_US Mathematics itself is plural, so why shorten it to a singular? (Perhaps it's plural because it's several disciplines, inc. arithmetic, geometry, etc.)

@@KenJackson_US What I was always told is Mathematics is plural, so maths is too.

Just do a supertask

Maybe his only dish was pepperoni?

Dark...😏

Too soon...

Don’t understand?

Wasn't it Hilbert who died from starvation?

I find your statements highly pretentious.

@@MAC0071234 lol yeh.

I think it does...

Congratulations, that's exactly what Godel says. In our current definition of "define", however complex it gotten, it still can't erase the inconsistency/paradox of this certain question. So you need to search for more guidelines so that we can answer this without such paradox.

wouldnt that be a sorites heap paradox?

No, actually it’s a lot more of a problem with the definition of the word “words”.

And what is a "word"? What if a number was a power of three, and you could get its description under twenty words by coining the word "powotri" or whatever? If you got enough people to use the word, it would become a real word, thus adding a new number (probably a bunch of them) and changing the set. "Autopower" could replace "n to the nth", "threeven" could replace "multiple of three", and then maybe you don't even need to invent new words: there's obscure words, words with extra meanings in mathematics (like kissing numbers and twin primes), words in other languages, just words and words forever. The set of all words isn't set in stone, so neither is the set of all numbers describeable in under twenty words. Words.

THEN DON'T HAVE ENOUGH OF IT

What I get from the description in this video is that "in system X (such as probably ZFC, or else possibly Peano arithmetic which I actually haven't looked up yet) a statement must be provably false or else it is true", and that if that is an accurate assessment then we may be well served by finding the *simplest* system (X, set of axia, etc) for which the above still holds that is available to us (for example, that has yet been formulated). If such a system is brain-bludgeoningly simple or low-entropy enough - yet still satisfies the above condition - then perhaps that would make undecidability testing easier. Is that by any chance what you are getting at (or comparable to what you are getting at) by saying that we would need a completely novel framework from which to assess the unprovability of these perennial favorites?

Not quite. It may help to realize that if any statement is undecidable then so is its negation, so there's no privileged statement to care about. It is true that for certain classes of statements, in certain axiomatic systems being undecidable implies they are true. The easiest example are the generic Pi_1 sentences mentioned above. One way of thinking of the Pi_1 sentences is those sentences which make a claim about every positive integer and where we can test that claim with a straightforward algorithm. So for example, Goldbach is Pi_1 because I can test "Does n satisfy Goldbach's conjecture" by checking if either n is odd or if for running through every integer p from 1 to n whether there is a value of p where p and n-p are both prime. But not all sentences have this form. The key insight for why Goldbach's conjecture would have to be true if it is undecidable is that if it were false we could then find a specific n where we could run our algorithm and find that it didn't work for that n. To see an example that is *not* of this form, consider the twin prime conjecture. This conjecture says that there are infinitely many twin primes, that is primes which are 2 away from each (examples are 11 and 13, or 29 and 31. A non-example is 23 since 21 and 25 are both composite). Now, let's say we knew somehow that the twin prime conjecture was undecidable. We could *not* make the same argument as with Goldbach because it might be false and we won't notice. Say there's some largest twin prime pair; there's no obvious calculation we can do with it to show that it is the largest, unlike with the Goldbach situation where when something is a counterexample we can do a straightforward check. It is true that(most?) of the conjectures we know and love are of the same variety of Goldbach's conjecture, but fact that we would need other techniques to prove their undecidability has more to do with the limitations of our machinery for proving things are undecidable (although one is certainly using the fact that Pi_1 sentences like Goldbach must be true if they are undecidable). As to proving undecidability in weaker systems, there's been a lot of work on that in the last 60 years or so. Robinson arithmetic is one such system en.wikipedia.org/wiki/Robinson_arithmetic . This system is in some sense the weakest natural system in which Godel's theorems apply. But this system is so weak that one cannot in it prove that addition is commutative. In this case there are a variety of statements which one can easily prove are undecidable in Robinson arithmetic. But many of our techniques for doing so are almost cheating- we can explicitly give examples of other very simple systems that are not the natural numbers which satisfy the axioms of Robinson arithmetic, so anything satisfied by one model and not another must be undecidable in RA.

It’s automated

Said the one who spelt it as "Godel" instead of "Gödel".

@@gabor6259 wow you really showed them

ElTurbinado: indeed the “manual” subtitles don’t have the same error there.

because they say „girdle“ not „gödel“ :)

I wonder if you could construct a system of paraconsistent logic that deals exclusively in contradictory statements similar to how you can create oscilators with logic gates. Where the conclusions that result from the statement being both true and false are proposed simultaneously at simply separate 'states' of the system rather than looking for only one conclusion, similar to how certain algebraic equations have multiple solutions. Why can't logical systems have multiple solutions? Edit: I looked into this and apparently this is 'dialetheic logic'

If you don't finish all your math homework, then Gödel's Incompleteness Theorem will get you in your sleep. And as your father I'm telling you this is true, even if you can't prove it.

Your metalogic has no power over me, father. Tarski's undefinability theorem will eat your theorem for breakfast.

727. Have fun with it my axiom.

A whole series on Field Extensions.

Well, maybe 3Blue1Brown is more suitable for the task.

Felipe Hindi shots fired

Hahaha! Well, I don't mean to say that 3Blue1Brown is better, just that they have different purposes/approaches with regards to spreading mathematical knowledge. :) I really like both channels.

Felipe Hindi If you like 3blue1brown, I bet you'll like Infinite Series too. Check it out! :)

Similar trick for solving hashi bridge puzzles, I had to do it a few times for very hard ones

@Maiahi I was thinking the same, LOL

@Maiahi If you don't eat your meat,how can you have any pudding?

That's more of a perceived paradox which is resolved by identifying the equivocation in the definition of "blank".

otherwise this probably one of the most interesting videos i watched.

I paused the video when he showed the snippets of the paper. It sounded like the statements they used in their proof were very complex and would not lend themselves to the simplistic explanations Numberphile viewers expect.

If you prove a statement is unprovable, then that in itself proves the statement (he tells us at 12:08)... So there is a contradiction there ;). Naming 'unprovable statements' is not possible I think. If you know a statement to be unprovable, then that proves the statement.

YCLP What he is saying there (and not saying well) is that if you have a statement which cannot be proved true or false, you can just accept it as an axiom. For the Reimanb hypothesis, the inportant thing wouldn't be that we can't prove it's true, it would be that we can't prove it is false. If we can't prove it is false, we can accept it as true. However, we would not know whetber or not it is true within the given axiomatic system or if we need to expand the axiomatic system to include it. All we would know, if we prove that it can't be disproved, is that there is nothing inconsistent aboit expanding the axiomatic system to include it. In other words, doing so wouldn't break math. If you can prove that a statement can neither be proved true nor false (the only exanple I know is Cantor's Continuum Hypothesis) then we say the statement in undecidable. In that case, you can make the statement a new axiom or not and you have no fear of inconaistency either way. If you make it a new axiom, it is no longer unprovable in your system. It is unprovable in the old system, but now in your new system, it is provable and there are some other unknown unprovable statements.

+YCLP That is only true for statements that are disprovable by finding a counter-example. Not every statement can be proved false by counter-example. For example something like "there are arbitrarily long gaps between consecutive primes." cannot be disproved by a counter-example.

anon8109 In the extra footage he makes this point more explicitly. You should check it out.

Imagine that there is a pyramid which groups people based on their IQ. The dumbest are at the bottom and the smarter towards the top. "It is lonely at the top."

CandidDate _Wisdom of Crowds™_ tho

If you got it, well, it would blow your mind. It did mine. It just means reality is ultimately unexplainable. You cannot create a mathematical theory of everything. It's beyond material science. The answers aren't in this physical universe.

Yes, me. Well, I actually don't watch the video, I read the comment section. It's far more entertaining.

Incredibly enough, I discovered the proof of the CH by simply establishing 1-to-1 correspondence between positive decimal floats and positive decimal integers. For the last 125 years such a proof eluded the top set-theory experts of the planet. It was eventually abandoned and classified either as impossible or undecidable. Well, no more. Sit back and enjoy one of the most trivial and elegant proofs in set theory. It is published on RU-vid with the title : “Proving the Continuum Hypothesis”. I would appreciate if you watched and commented on this video. Your input will be given full credit.Tamas Varhegyi

Wouter Dijkslag yep.

Ah, a classic. Waiting for Godel.

I see what you did there! ^^ +1

Yes you were definitely correct to put Cantor into the mix

nselchow0 Didn't Grey get the joy of the counter of Brady's videos since his last video being zero?

Shardar 1 Did u make a Game of Thrones reference....

Robb V. Not intended. I referenced René Descartes. A 17th century philosophist.

CGP Grey is trash, he's the epitome of internet faux-intellectual.

In the videos I've seen he just read wikipedia or someone else's article and animated it.

Logical humility to any science is what allows us to not get stuck in our ways of thinking. It breeds imagination: an often forgotten element of all discovery.

Olaf Doschke It's a limit on models of symbolic logic and proof methods. When you hear mathematicians say a statement is undecidable, they usually mean undecidable _in ZFC_, the standard set theory axioms most modern math is grounded on. There are lots of other possible non-trivial axioms that you can consider, and there is a whole branches of math called axiomatic set theory which studies this. I think the Gödel Incompleteness Theorems are often blown out of proportion these days when literally no mathematician sees it as anything but one more interesting result in axiomatics and formal logic.

Well, the same goes about the halting problem for developers.

I'm not 100% sure what you're asking in terms of "the way we relate the axioms to each other" and "the way we codify them to their numbers in his system". As with any form of communication, things can get misinterpreted. I don't know if that's relevant to what you're saying, but I should point that out. However, formal logic is designed to mitigate this issue was much as possible. By "how axioms relate to each other", are you talking about rules of inference and laws of logic which are assumed? If so, then yes, Gödel's proofs rely on these things. If you change allowable rules of inference (thus changing what counts as a valid proof), then the theorem might not be true for this new version of logic. That being said, Gödel's results are existence results, not construction results. Gödel's proofs are not designed to _find_ a sentence that is independent from the axioms, but rather to show that one must always exist. The specific encoding of logical symbols into natural numbers does not matter. As long as at least one way of encoding works, Gödel's proof holds, and shows that there must be some sentence independent of the axioms. I'm not sure if I've answered any of your questions, but I'm happy to revisit this if you would like more clarification.

Not true, provable. If a statement is true but not provable, it will not be divisible

Axiom Other.

This comment is false but provable.

This comment is provably unprovable.

Vodkacannon You are Turing Incomplete because I said so.

The comment of this reply has only false replies. (That's better)

YES, but the incompleteness is complete, completely present. "incompleteness" is lost in rendering. Hence, axiom, once a function is applied, you may not reference to the variables used in it before. Including linguistic (function) referencing.

The last word in this sentence is "omitted"...

Zoey Spencer the coolest takeaway from Godel's Incompleteness Theorem is that it proves that humans are incapable of knowing everything. There will always be knowledge that exists but is unreachable.

i mean it makes a much more specific statement about systems of formal mathematical reasoning. "there is a limit on the human species' capacity for knowledge" is a more general philosophical problem which seems intuitively true.

& that's already ε more than I did, for arbitrarily small ε > 0 ;)

dothemathright: No, that's a gross misrepresentation of the Incompleteness Theorem.

dothemathright: I didn't provide an explanation because I don't feel up to the task of explaining the Incompleteness Theorem in a youtube comment. Unlike you I only provide an answer if I actually have an answer to something.

Except that if you expand it to all of Brady's channels (not just Numberphile), that's not much different than saying "A CGP Grey video!" :P

"Gurdle" . Brain the size of a planet, and here I am, protecting women's chastity....