@@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.
@@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.
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?
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.
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.
"You were the chosen one! It was said that you would solve Hilbert's program, not destroy it! Bring consistency to mathematics, not leave it in darkness!"
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 love how Numberphile always breaks down these incredible, mind-bending concepts, so that we ordinary people can (seem) to understand it. And at the same time, not illegitimately oversimplifying it. I bet that's a hard job and you have to know your stuff in and out.
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
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.
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!
9:16 "and that's exactly what Gödel wanted". From what I have read this is not the case - Gödel was actually attempting to confirm Hilbert's "consistency agenda", not destroy it. He was quite upset at his own discovery.
@@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.
It is so lucky that the godel's imcompleteness theorem has a proof. What if it is true that some truths could not be proved and this theorem is one of them itself...
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.
For 7:52 onwards, the way Godel writes this is rather elegant. He suggests the existence of a number G that when decoded yields "G is unprovable under the axioms".
I know it's a bit of a cliché, but if you haven't read Gödel, Escher, Bach, then go find a copy. Even if you don't possess a mathematical background, the book goes slowly and deeply enough to give you a real appreciation of the theorem (as well as fun hypotheticals about what it could entail for minds and the real world).
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
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 TH-cam 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
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.
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.
Wow, this guy explains things in a way that is really really understandable, unlike many mathematicians that are impossible to follow or to understand. Congrats to his being extremely skilled with words.
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!
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..
I think the vid is really a bit misleading. What Goedel's Theorem says is this. In a sufficiently rich FORMAL SYSTEM, which is strong enough to express/define arithmetic in it, there will always be correctly built sentences which will not be provable from the axioms. That, of course, means their contradictions will not be provable, either. So, in a word, the sentences, even though correctly built, will be INDEPENDENT OF the set of axioms. They are neither false nor true in the system. They are INDEPENDENT (cannot stress this enough). We want axioms to be independent of each other, for instance. That's because if an axiom is dependent on the other axioms, it can then be safely removed from the set and it'll be deduced as a theorem. The theory is THE SAME without it. Now, the continuum hypothesis, for instance, is INDEPENDENT of the Zermelo-Fraenkel axioms of the set theory (this was proved by Cohen). Therefore, it's OK to have two different set theories and they will be on an equal footing: the one with the hypothesis attached and the one with its contradiction. There'll be no contradictions in either of the theories precisely because the hypothesis is INDEPENDENT of the other axioms. Another example of such an unprovable Goedelian sentence is the 5. axiom of geometry about the parallel lines. Because of its INDEPENDENCE of the other axioms, we have 3 types of geometry: hyperbolic, parabolic and Euclidean. And this is the real core of The Goedel Incompleteness Theorem. By the way... What's even more puzzling and interesting is the fact that the physical world is not Euclidean on a large scale, as Einstein demonstrated in his Theory of Relativity. At least partially thanks to the works of Goedel we know that there are other geometries/worlds/mathematics possible and they would be consistent.
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.
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.
@@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.)
Gödel's Incompleteness Theorem absolutely amazed me when I learnt it as an undergraduate and I still think it is quite possibly the most amazing thing I know. I actually felt fairly upset to realise that not everything that is true is provably so. I've got used to the idea in the intervening 30 years.
This video was extra GREAT. Time ago I gave a very shy look into the formal Goedel theorem and I was rejected right away. I would have never imagined someone could have given such an heuristc presentation of the matter and being understanble, enjoyable too. THANK YOU and VERY WELL DONE. CONGRATS !!! This was so explicating ...THANK YOU !
this makes me think of what Kant said about using the reason outside of the experience as it would be independent of it, meaning that when we use logic or mathematics as if they were independent of a possible application to objects of experience reason is bound to build things that could or could make no sense. Is like thinking that because words could be used to describe things, if you use words to describe a thing that thing should exist, just because it can be described, which is not true as we could all agree
i find the Incompleteness Theorem to be satisfying. I feel it implies there's no bound to imagination. I feel it implies reality is not a consequence of logic, and logic/mathematics is not the means to comprehend reality .. merely a means to model parts of it to some degree of accuracy. I feel it implies reality will always be beyond anything that can possibly be comprehended.
I've noodled over this paradoxical logic for a while and from logical analysis it shows that there are true statements that cannot be proven. But from the physicist inside me this paradoxical logic is exactly how you accomplish positive feedback systems. Best example I can think of is creating an oscillator with 3 NOT gates in parallel and tying the output back to the input. From formal analysis this circuit creates an astable output, and this output is actually very useful. I think these axiom statements referencing the axioms have similar properties. These statements shouldn't be tossed aside as logical fallacy, but instead used as a tool to explore the chaotic side of logic and proofs.
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'
It is worth noting that the Goldbach conjecture and the Riemann hypothesis are Pi_1 sentences, that is, they can be formulated as "For all integers n, P(n)" for a predicate P(n) that is recursive in Peano Arithmetic(this is pretty trivial for Goldbach and pretty deep for RH). All our major methods of proving a statement independent of Peano Arithmetic prove the stronger claim that that the statement is independent of PA+all true Pi_1 sentences. Therefore, any method that would prove either of these undecidable would need to be fundamentally new from our current methods of proving undecidability. (Edits for grammar/spelling.)
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.
Doesn't the paradox of "smallest number that cannot be defined in 20 words" simply stem from the ambiguous meaning of "define"? More precisely, at which level of abstraction does this "define" work?
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.
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.
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.
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.
That last part of how the inability to prove the Riemann Hypothesis would prove the Riemann Hypothesis is SO fascinating and mind-blowing at the same time! This was great to watch. Thank you so much for putting in the time and effort to make it! :)
thoroughly enjoyed maths during my uni days many years ago but vids like this remind me why I never became a mathematician ...... its like logically and rationally working your way into insanity ! :)
I wish they'd discussed examples of actual mathematical statements that are not provable. E.g. continuum hypothesis? Got close with Riemann, but I wish they'd showed more about the paper that proved some "interesting" statements that are unprovable.
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.
The Hilbert's problem on consistency of Mathematics could also be tackled using the ideas introduced by Alan Turing (Turing Machines).[Entscheidungsproblem]
Does anyone else watch Numberphile's videos without understanding what's going on? My takeaway from this video was that I can take a notecard and write "The statement on the other side is False" and on the other side, "The statement on the other side is True." And proceed to give this card to a friend
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."
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.
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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 TH-cam 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
I think if you let the problem sit at the level of real time then it is always a choice between coherance vs completeness. This is very relatable from an applied science/engineering point of view. Every model and algorithm applied to real problems will become less valid as it becomes too empirical or too linear/rational, even though both those qualities will register as 'true' in themselves.
I have always likened Godel's Theorem to the notion that there will always be questions we can ask that we can't answer based on what we now know. Maybe that's an overly simplistic analogy in some ways, but to me, it captures the 'flavor' of it in an philosophical way. For me, it fits in with my ideas of what mathematical 'reality' means to me. Obviously, I don't think there are "mathematical entities" out there somewhere. There are, problems, facts, proofs, etc., but the abstract ideas themselves are things of human creation. To me, the question of whether we discover or invent is just misguided. We discover - but what do we discover? Why, rules, of course. Math is a set of rules. Much of the math we know of pertains to our actual universe, simply because that's what we're trying to describe. But really, math could "correctly" describe many things that don't apply here, like objects in 83 dimensions (or some such.) Since it is the rules we're discovering (and not always ones that apply to our reality!), we can easily see that, like any game, there will always be questions we can ask that we can't answer according to our current knowledge of the rules. tavi.
Thank you very much for the video. I'd found many others, but this is absolutly the best one!
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Yay. This gentelman, Mr. Gödel, was born in the city I now study in. I was looking at the plaque with his name on the house he was born in the other day. :-)
The first verbal paradox isn't a paradox. It says the statement on the other side is false. So that means "the statement on the other side is true," is false. Then the loop stops there. The guy is assuming that if the statement is false it means the inverse of the statement is true, which may not be the case. So the loop would end immediately after flipping the card.
I’m 16 and fascinated by maths, I first heard of Marcus Du Sautoy a few days ago when I started listening to his BBC podcast called “A Brief History of Mathematics”. Prior to his I’d not seen or heard of him or his work, but now it seems I can’t avoid him, shows, books, podcasts etc. It seems that if it’s related to maths, he’ll be there
The statement you made about the Riemann Hypothesis reminds me of a trick I sometimes used to solve puzzles like Sudoko. We know those puzzles have an intended solution, so if you make a change that makes the solution become ambiguous, then you know that change was incorrect, and thus you learned something.
Our mathematics is a reflection of our limited understanding of the laws of nature. Gödel's Theorem also warns us to avoid getting caught up in incomprehensible and infinite aspect of the universe, especially through using counting. We are not meant to comprehend the infinite, rather the infinite is a source of other more practical yet still profound ideas. E.G. Einstein.
Nice! As I read about this theorem first time, it also seemed unnerving. But now I really think it' a very educated statement about limitations of truth and knowledge. It really also allows some type of belief and that gives math a sympathetic imperfection, including knowing about it, being relaxed about it and not be arrogantly claiming to be the only way to achieve all truths.
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.
Godel, Russell, and Turing are in their own class when it comes to Hocus Pocus (you could throw Cantor in there too). They went full-throttle at the limits of sanity. Russell it seems was the only one who came back to earth unscathed
In mathematics, there is no formal definition of what a set is. Informal definitions say a set is a collection of things, but that is just using a synonimous word (collection). However, it should be obvious that a set of just one member is not distinguishable from that member, and this means that everything is a set of itself, whether it is or not a set of other things. There are no sets that do not contain themselves.
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.
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.
For y’all out there interested in Philosophy, there’s some really interesting work out there talking about the nature of subjectivity that mirrors this systemic uncovering of unavoidable contradictions. Hegel is the one who really got this idea on the map-if I’m remembering right, his doctoral thesis was a demonstration that only contradictory statements can be true or false, to oversimplify (I’m just recalling that bit of information from a lecture by the philosopher Mladen Dolar called “Substance is Subject”). To get into this, the easiest entry point right now, imo, is watching Slavoj Žižek’s public talks, and then his lectures. He’s a contemporary Hegelian philosopher, also drawing off of the psychoanalyst Jacques Lacan (who focuses a lot on this fundamental contradiction of subjectivity as well). At any rate, I just think the idea of incompleteness being intrinsic to set-theory based ontologies is really cool, and just wanted to put this out there for anyone into in philosophy :)
Okay so I never knew something, and learning it blew my mind. The way I understand it, all mathematics is created by observing variations in the interaction between axioms. So in a way, axioms are like primary colors, from which every possible color derives. Therefore mathematics might actually just be a perception of some force in the universe in a way our brains can understand.. Amazing.
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.
Wonderful video, however I must note that it is very unfortunate that at around minute 3 to 4 the words "complete" and "consistent" are not sharply separated. The notion of "consistent", which Hilbert and Gödel would have called "widerspruchsfrei" does not entail that every statement expressible in the formal system can be proved to be either true or false (that adjective would have been "complete"), but rather that no contradiction can ever be deduced from the axioms. Yes, Gödel indeed shows that we may expect NEITHER of the two, but these are two different theorems and it would be nice not to mix up the two. Another important point to keep in mind here is that this only counts for theories, or, formal systems "strong enough" or "large enough" to "talk about itself" in the way Gödel's manages to by devising his code (which, using prime numbers, must at least have recourse to addition and multiplication, if I remember correctly from Douglas Hofstadter's must-read-bestseller on this topic "Gödel, Escher, Bach") Smaller theories may be complete indeed, such as the theory of integer addition or the theory of Boolean logic without quantification.
This reminds me of something I heard Robert Anton Wilson say: "The map is not the territory." He was using it in reference to our perceptions, and that all we know is the sensory information we receive about "the objective world", but we cannot ever truly know any objective world (assuming one exists), because all we can ever know is what we sense of that world, and our senses are limited and flawed (for instance, we can only see part of the spectrum of light, and can be fooled by optical illusions). For anybody not following the analogy, the map is the sensory information our nervous systems receive, and the territory that our senses map is the physical "objective" world. He went on to say that if we ever did have a complete map, it would be useless, because it would just be a copy of the thing that it was describing- maps are useful because they don't give a complete picture, and only shows info we find useful- it will show you where roads are, but not the position of any individual tree in a forest. Applying that concept to this subject, using math to describe the natural world is much the same- it will always be incomplete in its description of the world. If it were complete, it would essentially just be a copy of our world (albeit written in our mathematical code, instead of the code of molecules and DNA, for instance). In other words, you cannot completely describe anything without essentially making an exact copy of it (and even then, it couldn't occupy the same physical space as the original, so it would still necessarily be different in that way). So I don't think that we will ever have mathematics that can completely describe the world- or any other language, for that matter. The best that we can hope for is a fairly accurate description of the world (although even that is a lofty goal, given that the universe is ever-changing, and that both our instruments and personal senses can be flawed), and a description that is useful to us humans. Again, going back to the map analogy, we don't need the map to show every individual tree or blade of grass- we just need it show show our current location, our destination, and possible routes to that destination (and maybe some landmarks to help navigate). Instead of looking for unsolvable problems, why try to solve the problems that we know it can solve?
Lol I took a math logic course back in college and I can proudly say that I understood none of the stuff when we spent like NINE lectures on Godel's incompleteness theorem.
He made an interesting transition toward the end of the video, he started using the word 'undecidable' which is more associated with Turing. So my question is, what is the relationship between incompleteness and undecidability.
The thing that strikes me about what he was saying about the Riemann Hypothesis is that if you could indeed show it to be *unprovable* (and hence true, since the falsity of the Riemann Hypothesis would be necessarily provable), then would this not be a _proof_ of the truth of the Riemann Hypothesis (and therefore the statement is *provable* )? To me, this seems to lead to a contradiction in its own right.
The video didn't get into this, but there's a big difference between "formal logic" and "meta-logic". Formal logic is where you have a list of axioms and a list of rules of inference, and you are bound to using only these specific axioms and specific rules of inference. Meta-logic is when you're talking about a system of formal logic, usually from outside of that system (often you won't have as strict of rules imposed on meta-logic as you do in the formal logic you're talking about). So if someone "proved the Riemann Hypothesis unprovable" (and really he should also state unrefutable as well), then really, this is a meta-theorem about a particular formal logical structure. This means that there is no formal proof within a particular mathematical logical system (probably ZFC), but using a broader scope and not just relying on the axioms of a particular mathematical system - in this case, using the rules of logic itself, one could show that the Riemann Hypothesis has to be true.
No contradiction because when you say that Riemann Hypothesis is unprovable, it means that you can't say it's true or false, since you can't prove that it is false it must be true ( but you can't prove it also) so you add it to mathematics as an axiom and now it's true in the context of this new realm.
Tim Weaving this does seem to be a contradiction, which means if the hypothesis is unprovable, it should be also impossible to prove that it is unprovable.
That's what I had initially thought too kimitsudesu, but as MuffinsAPlenty points out, a distinction needs to be made between the "meta-logic" used to show the undecidability of a statement within some formal system, and the language of the formal system itself. If it could be shown that the Riemann Hypothesis was undecidable, then it asserts that no proof can be found _within_ said formal system. Though the Riemann Hypothesis has this additional property whereby if the statement was _false_ then it need be *provably* so, thus implying the truth or the Riemann Hypothesis. Though this is not a 'proof' in the normal sense as we do not abide by the strict logical rules of the system in which we are working, so as darkwachu says, we can simply add it as an axiom and continue as normal!
Either way, if the Riemann Hypothesis is unprovable, that can only be shown by stepping outside the formal system. I believe we can say with certainty that we cannot prove it is unprovable inside the system itself.
Does this not hinge on Gödel's system being "perfect"? Is there not some level of arbitrary human influence on the way we relate the axioms to each other, or even the way we codify them to their numbers in his system?
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.
This issue is also what caused Russell and White to go mad on Principia Mathematica. It's the ultimate heartbreak: saying that Mathematics are the only thing truly consistent and certain in this world, only to then learn our needed conventions blew pure logic out of the water and now not everything true can be proved. Heartbreak.
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Gõdel, L'Hôpital, and Bernoulli walk into a bar. Gödel looks around and says, "This joke might be funny, but we can't tell because we're in it."
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.
I heard that if you say "This sentence is false" three times in a mirror, you'll cause a stack overflow in the simulation...
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
And thanks to Gödel, my life was bettered by the inclusion of "Option E: Non of the above"
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
Me: I have a proof of the theorem
Teacher: Ok so what is it
Me: There is no proof
Teacher: NOICE.
Do not try and prove the theorem. That's impossible. Instead... Only try to realize the truth.
There is no proof.
I have a truly marvelous proof of Goldbach’s Conjecture, however there is not enough space in the TH-cam comments section to contain it
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.
It funny to see Brady's evolution through the videos in numberphile, he started as a mathematical potato but now is asking really smart questions..
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
"You were the chosen one! It was said that you would solve Hilbert's program, not destroy it! Bring consistency to mathematics, not leave it in darkness!"
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“
I love how Numberphile always breaks down these incredible, mind-bending concepts, so that we ordinary people can (seem) to understand it. And at the same time, not illegitimately oversimplifying it. I bet that's a hard job and you have to know your stuff in and out.
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.
For anyone who is interested, Hofstadter's book 'Gödel, Escher, Bach' goes into these concepts a little more - best book I have ever read.
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!
9:16 "and that's exactly what Gödel wanted".
From what I have read this is not the case - Gödel was actually attempting to confirm Hilbert's "consistency agenda", not destroy it. He was quite upset at his own discovery.
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.
It is so lucky that the godel's imcompleteness theorem has a proof. What if it is true that some truths could not be proved and this theorem is one of them itself...
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
For 7:52 onwards, the way Godel writes this is rather elegant. He suggests the existence of a number G that when decoded yields "G is unprovable under the axioms".
I know it's a bit of a cliché, but if you haven't read Gödel, Escher, Bach, then go find a copy. Even if you don't possess a mathematical background, the book goes slowly and deeply enough to give you a real appreciation of the theorem (as well as fun hypotheticals about what it could entail for minds and the real world).
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 TH-cam 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 Riemann Hypothesis can't be unprovebly false"
"Hold my Beer"
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.
Wow, this guy explains things in a way that is really really understandable, unlike many mathematicians that are impossible to follow or to understand. Congrats to his being extremely skilled with words.
This, for some reason, always make me think of that one sentence from The Boondocks: "The absence of evidence is not the evidence of absence."
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.
Tried to understand forgot who I am
TurboCMinusMinus You are Turing Incomplete because I said so
LSD is helluva drug...
if someone proves that the Riemann hypothesis is undecidable, would the person still win a million dollars? Asking for a friend.
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.
I think the vid is really a bit misleading. What Goedel's Theorem says is this. In a sufficiently rich FORMAL SYSTEM, which is strong enough to express/define arithmetic in it, there will always be correctly built sentences which will not be provable from the axioms. That, of course, means their contradictions will not be provable, either. So, in a word, the sentences, even though correctly built, will be INDEPENDENT OF the set of axioms. They are neither false nor true in the system. They are INDEPENDENT (cannot stress this enough). We want axioms to be independent of each other, for instance. That's because if an axiom is dependent on the other axioms, it can then be safely removed from the set and it'll be deduced as a theorem. The theory is THE SAME without it. Now, the continuum hypothesis, for instance, is INDEPENDENT of the Zermelo-Fraenkel axioms of the set theory (this was proved by Cohen). Therefore, it's OK to have two different set theories and they will be on an equal footing: the one with the hypothesis attached and the one with its contradiction. There'll be no contradictions in either of the theories precisely because the hypothesis is INDEPENDENT of the other axioms. Another example of such an unprovable Goedelian sentence is the 5. axiom of geometry about the parallel lines. Because of its INDEPENDENCE of the other axioms, we have 3 types of geometry: hyperbolic, parabolic and Euclidean. And this is the real core of The Goedel Incompleteness Theorem. By the way... What's even more puzzling and interesting is the fact that the physical world is not Euclidean on a large scale, as Einstein demonstrated in his Theory of Relativity. At least partially thanks to the works of Goedel we know that there are other geometries/worlds/mathematics possible and they would be consistent.
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
So basically Godel said "This statement is false" and broke maths.
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.
Gödel's Incompleteness Theorem absolutely amazed me when I learnt it as an undergraduate and I still think it is quite possibly the most amazing thing I know.
I actually felt fairly upset to realise that not everything that is true is provably so. I've got used to the idea in the intervening 30 years.
And now I have an existential crisis. Thanks
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?
This video was extra GREAT. Time ago I gave a very shy look into the formal Goedel theorem and I was rejected right away. I would have never imagined someone could have given such an heuristc presentation of the matter and being understanble, enjoyable too. THANK YOU and VERY WELL DONE. CONGRATS !!!
This was so explicating ...THANK YOU !
The smallest number which cannot be defined in less than twenty words, which needs these extra words to even exist.
If I knew in high school that these sort of questions were what constituted 'Math', I may have had a different major!
this makes me think of what Kant said about using the reason outside of the experience as it would be independent of it, meaning that when we use logic or mathematics as if they were independent of a possible application to objects of experience reason is bound to build things that could or could make no sense. Is like thinking that because words could be used to describe things, if you use words to describe a thing that thing should exist, just because it can be described, which is not true as we could all agree
"Oops, I forgot to eat."
-Gödel's last words.
Maybe his only dish was pepperoni?
Dark...😏
Too soon...
Don’t understand?
Wasn't it Hilbert who died from starvation?
The best thing about infinity is it will keep mathematicians busy forever.
Just do a supertask
i find the Incompleteness Theorem to be satisfying. I feel it implies there's no bound to imagination. I feel it implies reality is not a consequence of logic, and logic/mathematics is not the means to comprehend reality .. merely a means to model parts of it to some degree of accuracy. I feel it implies reality will always be beyond anything that can possibly be comprehended.
I find your statements highly pretentious.
@@MAC0071234 lol yeh.
Shoutout to the subtitler who kept spelling "Godel" as "girdle".
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've noodled over this paradoxical logic for a while and from logical analysis it shows that there are true statements that cannot be proven. But from the physicist inside me this paradoxical logic is exactly how you accomplish positive feedback systems. Best example I can think of is creating an oscillator with 3 NOT gates in parallel and tying the output back to the input. From formal analysis this circuit creates an astable output, and this output is actually very useful. I think these axiom statements referencing the axioms have similar properties. These statements shouldn't be tossed aside as logical fallacy, but instead used as a tool to explore the chaotic side of logic and proofs.
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'
I want more gödel videos, I can never have enough of it.
THEN DON'T HAVE ENOUGH OF IT
It is worth noting that the Goldbach conjecture and the Riemann hypothesis are Pi_1 sentences, that is, they can be formulated as "For all integers n, P(n)" for a predicate P(n) that is recursive in Peano Arithmetic(this is pretty trivial for Goldbach and pretty deep for RH). All our major methods of proving a statement independent of Peano Arithmetic prove the stronger claim that that the statement is independent of PA+all true Pi_1 sentences. Therefore, any method that would prove either of these undecidable would need to be fundamentally new from our current methods of proving undecidability.
(Edits for grammar/spelling.)
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.
the person explaining the paradox is awesome!! very beautifully explained. I would have been blessed to have him as my teacher!RESPECT!!!~~~~~
Doesn't the paradox of "smallest number that cannot be defined in 20 words" simply stem from the ambiguous meaning of "define"? More precisely, at which level of abstraction does this "define" work?
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.
Excellent! Thanks for pointing out the common quality of paradoxes is their self-referencing.
Now you can finally do one on galois theory
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! :)
If there is a lurking bogeyman in mathematics, it is Gödel's Incompleteness Theorem.
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.
I was supposed to have lunch with the family, but now my brain is just pudding...
@Maiahi I was thinking the same, LOL
@Maiahi If you don't eat your meat,how can you have any pudding?
That last part of how the inability to prove the Riemann Hypothesis would prove the Riemann Hypothesis is SO fascinating and mind-blowing at the same time! This was great to watch. Thank you so much for putting in the time and effort to make it! :)
thoroughly enjoyed maths during my uni days many years ago but vids like this remind me why I never became a mathematician ...... its like logically and rationally working your way into insanity ! :)
I wish they'd discussed examples of actual mathematical statements that are not provable. E.g. continuum hypothesis? Got close with Riemann, but I wish they'd showed more about the paper that proved some "interesting" statements that are unprovable.
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.
I love this guy, he really gets to the heart of the problem in a relaxed way
The Hilbert's problem on consistency of Mathematics could also be tackled using the ideas introduced by Alan Turing (Turing Machines).[Entscheidungsproblem]
Does anyone else watch Numberphile's videos without understanding what's going on?
My takeaway from this video was that I can take a notecard and write "The statement on the other side is False" and on the other side, "The statement on the other side is True." And proceed to give this card to a friend
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 TH-cam 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
I think if you let the problem sit at the level of real time then it is always a choice between coherance vs completeness. This is very relatable from an applied science/engineering point of view. Every model and algorithm applied to real problems will become less valid as it becomes too empirical or too linear/rational, even though both those qualities will register as 'true' in themselves.
I have always likened Godel's Theorem to the notion that there will always be questions we can ask that we can't answer based on what we now know. Maybe that's an overly simplistic analogy in some ways, but to me, it captures the 'flavor' of it in an philosophical way. For me, it fits in with my ideas of what mathematical 'reality' means to me. Obviously, I don't think there are "mathematical entities" out there somewhere. There are, problems, facts, proofs, etc., but the abstract ideas themselves are things of human creation. To me, the question of whether we discover or invent is just misguided. We discover - but what do we discover? Why, rules, of course. Math is a set of rules. Much of the math we know of pertains to our actual universe, simply because that's what we're trying to describe. But really, math could "correctly" describe many things that don't apply here, like objects in 83 dimensions (or some such.) Since it is the rules we're discovering (and not always ones that apply to our reality!), we can easily see that, like any game, there will always be questions we can ask that we can't answer according to our current knowledge of the rules. tavi.
this gentleman can express these ideas very clearly. thanks for that
Did you and CGP Grey plan on releasing videos within a minute of each other? 😄
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.
Fascinating talk. Thank you!
Best general explanation I've heard on it. Thanks.
Thank you very much for the video. I'd found many others, but this is absolutly the best one!
Yay. This gentelman, Mr. Gödel, was born in the city I now study in. I was looking at the plaque with his name on the house he was born in the other day. :-)
The first verbal paradox isn't a paradox. It says the statement on the other side is false. So that means "the statement on the other side is true," is false. Then the loop stops there. The guy is assuming that if the statement is false it means the inverse of the statement is true, which may not be the case. So the loop would end immediately after flipping the card.
otherwise this probably one of the most interesting videos i watched.
Thank you once again for such quality content.
I’m 16 and fascinated by maths, I first heard of Marcus Du Sautoy a few days ago when I started listening to his BBC podcast called “A Brief History of Mathematics”. Prior to his I’d not seen or heard of him or his work, but now it seems I can’t avoid him, shows, books, podcasts etc. It seems that if it’s related to maths, he’ll be there
The statement you made about the Riemann Hypothesis reminds me of a trick I sometimes used to solve puzzles like Sudoko. We know those puzzles have an intended solution, so if you make a change that makes the solution become ambiguous, then you know that change was incorrect, and thus you learned something.
Similar trick for solving hashi bridge puzzles, I had to do it a few times for very hard ones
The thing with the card at the start can be done much simpler. Simply with one sentence, namely:
'This sentence is false'
Wouter Dijkslag yep.
Our mathematics is a reflection of our limited understanding of the laws of nature.
Gödel's Theorem also warns us to avoid getting caught up in incomprehensible and infinite aspect of the universe, especially through using counting. We are not meant to comprehend the infinite, rather the infinite is a source of other more practical yet still profound ideas. E.G. Einstein.
So "This page intentionally left blank" is true or false?
That's more of a perceived paradox which is resolved by identifying the equivocation in the definition of "blank".
YES... FINALLY A VIDEO ABOUT GODEL
Ah, a classic. Waiting for Godel.
I see what you did there! ^^ +1
Nice!
As I read about this theorem first time, it also seemed unnerving. But now I really think it' a very educated statement about limitations of truth and knowledge. It really also allows some type of belief and that gives math a sympathetic imperfection, including knowing about it, being relaxed about it and not be arrogantly claiming to be the only way to achieve all truths.
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 love this guy's delivery
This is unbelievably well explained, I love it!
I´ve been wating for this so FREAKIN long!
Godel, Russell, and Turing are in their own class when it comes to Hocus Pocus (you could throw Cantor in there too).
They went full-throttle at the limits of sanity.
Russell it seems was the only one who came back to earth unscathed
Yes you were definitely correct to put Cantor into the mix
In mathematics, there is no formal definition of what a set is. Informal definitions say a set is a collection of things, but that is just using a synonimous word (collection). However, it should be obvious that a set of just one member is not distinguishable from that member, and this means that everything is a set of itself, whether it is or not a set of other things. There are no sets that do not contain themselves.
Wow! Very clear! Thank you so much for this explanation.
The only thing I understood from this video was nothing.
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.
If you can't answer a Math problem in exam, just answer "Incompleteness Theorem". Guaranteed 💯!
Just multiply both sides by zero. Booom
If you exclude assertions with referential cycles, does the Incompleteness theorem still apply to the remaining non-cyclic assertions?
Great video! Marcus du Satoy is a master explainer.
This comment is true but unprovable.
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)
"This statement is false" feels like a prime number in the complex plane.
The Goldbach conjecture can be proved by showing that it is not provable.
MIND BLOWN !!!!
For y’all out there interested in Philosophy, there’s some really interesting work out there talking about the nature of subjectivity that mirrors this systemic uncovering of unavoidable contradictions. Hegel is the one who really got this idea on the map-if I’m remembering right, his doctoral thesis was a demonstration that only contradictory statements can be true or false, to oversimplify (I’m just recalling that bit of information from a lecture by the philosopher Mladen Dolar called “Substance is Subject”).
To get into this, the easiest entry point right now, imo, is watching Slavoj Žižek’s public talks, and then his lectures. He’s a contemporary Hegelian philosopher, also drawing off of the psychoanalyst Jacques Lacan (who focuses a lot on this fundamental contradiction of subjectivity as well). At any rate, I just think the idea of incompleteness being intrinsic to set-theory based ontologies is really cool, and just wanted to put this out there for anyone into in philosophy :)
One of the implications from Godels work is that we can never know all there is to know about the universe.
The precompiler for C/C++ taught me about passes. If you break logic into passes it’s very difficult to have these simple paradox’s .
Okay so I never knew something, and learning it blew my mind. The way I understand it, all mathematics is created by observing variations in the interaction between axioms.
So in a way, axioms are like primary colors, from which every possible color derives.
Therefore mathematics might actually just be a perception of some force in the universe in a way our brains can understand.. Amazing.
Honestly this topic is so mindblowing.
The "incompleteness" in the thumbnail is incomplete.
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"...
Wonderful video, however I must note that it is very unfortunate that at around minute 3 to 4 the words "complete" and "consistent" are not sharply separated. The notion of "consistent", which Hilbert and Gödel would have called "widerspruchsfrei" does not entail that every statement expressible in the formal system can be proved to be either true or false (that adjective would have been "complete"), but rather that no contradiction can ever be deduced from the axioms. Yes, Gödel indeed shows that we may expect NEITHER of the two, but these are two different theorems and it would be nice not to mix up the two.
Another important point to keep in mind here is that this only counts for theories, or, formal systems "strong enough" or "large enough" to "talk about itself" in the way Gödel's manages to by devising his code (which, using prime numbers, must at least have recourse to addition and multiplication, if I remember correctly from Douglas Hofstadter's must-read-bestseller on this topic "Gödel, Escher, Bach") Smaller theories may be complete indeed, such as the theory of integer addition or the theory of Boolean logic without quantification.
Superb! Precise & Intriguing..!
This reminds me of something I heard Robert Anton Wilson say: "The map is not the territory." He was using it in reference to our perceptions, and that all we know is the sensory information we receive about "the objective world", but we cannot ever truly know any objective world (assuming one exists), because all we can ever know is what we sense of that world, and our senses are limited and flawed (for instance, we can only see part of the spectrum of light, and can be fooled by optical illusions). For anybody not following the analogy, the map is the sensory information our nervous systems receive, and the territory that our senses map is the physical "objective" world. He went on to say that if we ever did have a complete map, it would be useless, because it would just be a copy of the thing that it was describing- maps are useful because they don't give a complete picture, and only shows info we find useful- it will show you where roads are, but not the position of any individual tree in a forest.
Applying that concept to this subject, using math to describe the natural world is much the same- it will always be incomplete in its description of the world. If it were complete, it would essentially just be a copy of our world (albeit written in our mathematical code, instead of the code of molecules and DNA, for instance). In other words, you cannot completely describe anything without essentially making an exact copy of it (and even then, it couldn't occupy the same physical space as the original, so it would still necessarily be different in that way).
So I don't think that we will ever have mathematics that can completely describe the world- or any other language, for that matter. The best that we can hope for is a fairly accurate description of the world (although even that is a lofty goal, given that the universe is ever-changing, and that both our instruments and personal senses can be flawed), and a description that is useful to us humans. Again, going back to the map analogy, we don't need the map to show every individual tree or blade of grass- we just need it show show our current location, our destination, and possible routes to that destination (and maybe some landmarks to help navigate). Instead of looking for unsolvable problems, why try to solve the problems that we know it can solve?
r.i.p my brain
😂😂🤣
lllllllllllllloooooooooooooooollllllllllllllllllllllll
Lol I took a math logic course back in college and I can proudly say that I understood none of the stuff when we spent like NINE lectures on Godel's incompleteness theorem.
He made an interesting transition toward the end of the video, he started using the word 'undecidable' which is more associated with Turing. So my question is, what is the relationship between incompleteness and undecidability.
So well explained!
It makes me wonder whether math or language is more to blame
Merv McRough no one to blame, it’s just how the world is
Seeing as maths is a human translation of the nature of their surroundings, this is a quirk of the linguistics in both cases.
Math is a language. A formalized language, but indeed a language.
If you haven't already read, the brown and blue books- Wittgenstein
Right? Maybe our way of coding information is what causes most of these problems.
I really love the southern English pronunciation of Kö' Gödel
"Gurdle" . Brain the size of a planet, and here I am, protecting women's chastity....
That final point about the RH blew me away
Mathematics is so vast, and so perfect that it even allows for statements that contradict themselves.
The thing that strikes me about what he was saying about the Riemann Hypothesis is that if you could indeed show it to be *unprovable* (and hence true, since the falsity of the Riemann Hypothesis would be necessarily provable), then would this not be a _proof_ of the truth of the Riemann Hypothesis (and therefore the statement is *provable* )? To me, this seems to lead to a contradiction in its own right.
The video didn't get into this, but there's a big difference between "formal logic" and "meta-logic". Formal logic is where you have a list of axioms and a list of rules of inference, and you are bound to using only these specific axioms and specific rules of inference. Meta-logic is when you're talking about a system of formal logic, usually from outside of that system (often you won't have as strict of rules imposed on meta-logic as you do in the formal logic you're talking about).
So if someone "proved the Riemann Hypothesis unprovable" (and really he should also state unrefutable as well), then really, this is a meta-theorem about a particular formal logical structure. This means that there is no formal proof within a particular mathematical logical system (probably ZFC), but using a broader scope and not just relying on the axioms of a particular mathematical system - in this case, using the rules of logic itself, one could show that the Riemann Hypothesis has to be true.
No contradiction because when you say that Riemann Hypothesis is unprovable, it means that you can't say it's true or false, since you can't prove that it is false it must be true ( but you can't prove it also) so you add it to mathematics as an axiom and now it's true in the context of this new realm.
Tim Weaving this does seem to be a contradiction, which means if the hypothesis is unprovable, it should be also impossible to prove that it is unprovable.
That's what I had initially thought too kimitsudesu, but as MuffinsAPlenty points out, a distinction needs to be made between the "meta-logic" used to show the undecidability of a statement within some formal system, and the language of the formal system itself. If it could be shown that the Riemann Hypothesis was undecidable, then it asserts that no proof can be found _within_ said formal system. Though the Riemann Hypothesis has this additional property whereby if the statement was _false_ then it need be *provably* so, thus implying the truth or the Riemann Hypothesis. Though this is not a 'proof' in the normal sense as we do not abide by the strict logical rules of the system in which we are working, so as darkwachu says, we can simply add it as an axiom and continue as normal!
Either way, if the Riemann Hypothesis is unprovable, that can only be shown by stepping outside the formal system. I believe we can say with certainty that we cannot prove it is unprovable inside the system itself.
Does this not hinge on Gödel's system being "perfect"? Is there not some level of arbitrary human influence on the way we relate the axioms to each other, or even the way we codify them to their numbers in his system?
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.
This issue is also what caused Russell and White to go mad on Principia Mathematica. It's the ultimate heartbreak: saying that Mathematics are the only thing truly consistent and certain in this world, only to then learn our needed conventions blew pure logic out of the water and now not everything true can be proved. Heartbreak.
The smallest positive integer that is the sum of two cubes in two different ways is 91 = 3^3+4^3 = 6^3+(-5)^3.
this is probably the most mind blowing thing i've ever learned
Why? I found it very interesting, but not mind blowing. So I probably lost the point.
At 6:45, that's not ASCII, it's windows 1252...
Right. Ö is also coded as hex 0xd6 in standard ISO 8859-1 (Latin 1)
True, but he's using "ASCII" in the same sense as we use "Kleenex."
@@davidchristian3612 ahem, I don't use Kleenex, I use Puffs Ultra Soft & Strong. I sure hope he doesn't use Kleenex as well, that would be SO uncool
great analogy between Gödel and primes...
Thanks so much for the video. I was always amazed by Kurt Godel.