*PRONUNCIATION: I believe I have anglicized the pronunciation, and I'm aware the 'correct' way is "geu-del." Will attempt it next time!* Try brilliant.org/Newsthink/ for FREE for 30 days, and the first 200 people will get 20% off their annual premium subscription
I'd love to watch a video from you on Georg Cantor - my personal number 1 greatest mind in history. Other 2 great names to discuss would be Archimedes and Leonardo Da Vinci.
...and its not pronounced prin"k"ipia but prin"s"ipia...... like principle.... earcancer olé... sorry i ment earcanker olé... seriously. Reading a text Chatgpt printed is seriously not too hard.
I think pronouncing Godel as "Girdle" is the correct pronunciation of a german name. But I'm no expert. Everybody loves to correct me when I mis-pronounce Porsche as "Por-sss", rather than "Porsha" My revenge: I ask, *Hey, What's the Capital of Italy* - The typical reply: *Rome, of course, you idiot ... Don't try to change the subject.* My reply: *Don't you mean "Roma" ... I've been there, I know how the Romans pronounce it*
You forgot to mention, that Godel's incompleteness theorems were not less a surprise to him, than to anyone else. He, like others, was trying to complete Hilbert's project and make mathematics one beautiful complete system, where every statement is provable from a set of axioms. His discovery of the inherent incompleteness of mathematics came as a shock to himself, as much as for the rest of the world. Thinking about all consequences of it, likely contributed to his mental state. It would be great to see a video from you about Georg Cantor.
@@chickenlover657 It is not implicit and it's a very important detail. Without it, someone unfamiliar with Godel's story might get impression that Godel has been working to disprove the possibility of Hilbert's project.
Gödel was also a Platonist, and first-order logic was already showing cracks in its ability to be absolute. For example, Skolem showed 10 years before Gödel's incompleteness theorems that every countable first-order theory, if it has a model, also has a countable model. I don't think Gödel would have found his result terribly contradictory with his philosophy. At least to a modern Platonist like Hugh Woodin, the mathematical universe is so large that we cannot capture it with "small" truths.
A mathematician friend told me that a better layman explanation of the theorems is: "a complete and consistent axiomatic system does not exist..if its complete then its not consistent and if its consistent then its not complete"
Define a 'consistent axiomatic system', grazie mille! Aah, you mean an axiomatic system cannot be both sound ('consistent') and complete. If it is"complete" it is not sound; all its theorems are provable - and then ”some”. A bit like Heisenberg's uncertainty principle Nasty.
Yeah. He should mention a few details: a countable one and one allowing to express the natural numbers. Those are very significant restrictions. They are stated right at the start of the actual proof. Not mentioning them is leading to a wide philosophical overinterpretation of the result.
It similar in concept as Heisenberg's Uncertainty Principle in physics, which states that we cannot know both the position and speed of a particle like a photon or electron with perfect accuracy. The more we nail down the particle's position, the less we know about its speed and vice versa.
My understanding of Goedel's Incompleteness Theorem is not as the narrator states, "There are mathematical truths that cannot be proved in any mathematical system." Rather, it is that there are mathematical truths in any mathematical system that cannot be proved within that axiomatic system. A significant difference!
@@ronald3836 I'd say "somewhat ambiguous" rather than "less accurate," as one of the plausible meanings of the statement goedelite quotes is equivalent to the 0:26 statement.
I think they did a good job with their explanation. Yours while more accurate would most likely confuse the audience. These kinds of videos are just an introduction to things, if you want to actually learn how something works, you actually have to study the subject.
I think I have it in wavy hands form as: Final musings? If there exists a statement that has a valid proposition in math logic with result x such that x is simultaneously both true and false then that is of the form (x∈X) ∧ (x∉X) and thus defines null set in math logic side of the proposition. Nothing can really be implied about the statement side other than it has a proposition form that defines null set in math logic space. Thus math logic side IS complete as null set can be reached through valid propositions however nothing can be deduced or inferred about statement side as that side is full of grammar, syntax, inflection and inference that are beyond definition in present scope of logic to define. For example This statement is false? This statement is true! This statement is false. This statement is incomplete, ... This statement is incomplete, This statement is incomplete (yes that is a statement side comma or just simply a non-terminating statement like This statement is incomplete (if you see what I mean) perhaps better as This statement is incomplete ... ) Therefore: Math logic just had its infinity over infinity and zero over zero and combinations thereof moments and has been found wanting?
When doctors treated ulcers in 1950s they really didn't know what they were doing. They thought it's a stress and lifestyle disease. Actually the most common cause of ulcers is infection of the stomach by bacteria called Helicobacter pylori (H pylori). Then one guy figured it out and was ridiculed by all the other doctors for years. My grandfather died because of that. There was knowledge of how to treat this condition, but mainstream doctors blocked it. The same thing happened about 100 years earlier with the discovery that germs cause diseases. Interestingly that happened in the Vienna General Hospital (Ignaz Semmelweis)
and the patron saint of both the standard theory & scientism, Carl Sagan, joined in on ruining the career of at least one scientist with valid alternative theories. modern science is mostly about ego & funding not truth/discovery despite how it's marketed to the public
I have had ulcers and a new one and both of mine are emotionally based. They tested me for the bacteria and it came back negative. Being an alternative therapist and coming from a personal development background, most dis-ease is emotionally and/or energetically based.
I love that Dr Godel's story illustrates how complex the human mind is. The man who was loved by everyone, and was far more accomplished than almost any other person, was haunted, hunted, and ultimately destroyed by that mind. But due to mathematics, he is immortal.
Its shocking how unhindged many people on the brink of genius really turn out to be... or are driven to... Albert wasn't exactly a saintly man himself after all.
that is not complex. they already did compiler ,electronics,nukes,rockets and medicine which is very very complicated than some random math theorem. get over the hype!
Even today people with amazing abilities show that at the same time they can be nuts. My favourite example is Michael Jackson. A look at his videos and a listen to "Thriller".
Gödel also developed the ideas of computability and recursive functions. It's equivalent to lambda calculus and/or turing machine. It's a bit unfortunate it wasn't mentioned. He was genius.
Gödel has some pretty dope contributions to math, outside of completeness theorem. Gödel contributed towards Axiomatic Set Theory, which other authors in their field are equally impressive in my opinion. Grothendieck, Russel, and Zermelo are among my favorites. A lot of the abstract thinking comes from a broad pool of mathematicians… these guys are geniuses, even though we like to make fun of some of their work occasionally haha.
It's also interesting how he travelled from Austria to America, since it was extremely difficult, westwards instead of eastwards, due to travel restrictions. It's amazing he could do it, given his state of mind.
Not really that difficult. Switzerland is right on Austrian border, from there air travel to Spain (or just directly into Spain by train but Swiss way is safer) and easy further travel to america...
The German exit certificate required the Godels to travel through Russia to Japan via the trans-Siberian railway, then by boat to America. This was in late 1939, early 1940. (Ref: Rebecca Goldstein's book "Incompleteness: The Proof and Paradox of Kurt Godel").
The other day I was walking down a street in Vienna - it was Florianigasse - and there I saw a plaque next to the door of a house that indicated that Kurt Gödel had lived there for some years. So I asked myself on how many places Kurt Gödel might have lived, because this was already the fourth place which I found in Vienna.
An uncle of a was victim of 50’s ulcer surgery. He struggled with eating for the remainder of his life because the surgeons removed most of his stomach.
But what is this obsession with Bach? This dude compare's the theorem(s) also to Bach's fugues - apparently owing to the dude some Hofstader (?) who wrote the GEB.
@@jackquinnes GEB is about self referential systems and how those ideas relate to consciousness. The three titular figures, while all operating in a different field (gödel in math, escher in paintings and bach in music) each created works with a heavy sense of self reference. Other systems which reflect the same overarching theme also show up such as DNA in biology and computer programs. For anyone who wants to read it, if you find it too dense I recommend only reading the achilles and the tortoise sections on the first go. Theyre very fun to read and give a much easier to digest overview of the ideas in the following chapter.
@@guy3717 Thanks. Guess I will - finally - give it a read. First I should go to the trouble of getting my hands on it ofc. - ’Self/selfhood is deeply mysterious’, wrote young Ludwig Wittgenstein himself (a pun intended). Yes, it might be - and it really is if you think about it, philosophically.. For better or worse. Wittgenstein had in mind this undoable kernel of consciousness we know as transendental ego. ’Self’ is in this ”original sense” (just)! the self-reference of human mind. It is what it is. As elusive as intriguing. Is it an i”illusion” then? Just ”unprovable truth” if we are willing to play the game.
Read it. Was a fascinating book. And it won the National Book Award, so it’s not just a math book and it’s definitely not boring. It Han an elegance to it. The writer manages to weave a compelling narrative on Bach canons and fugues and Eschers art and then walk you through symbolic logic and keep it all very interesting. At times it’s difficult to grasp but worth the effort to work through it.
@@Newsthink Well the images you used where a bit odd IMHO and suggest he came like most people via ship over the Atlantic. Personally I think how Gödel came to the US would have been really worth mentioning. 🙂
The course in Mathematical Logic I took that covered the incompleteness theorem was simultaneous with his death. My instructor cancelled class the day he got the news and we had a very non-mathematical discussion about the will to live the next class session.
Such a heart breaking fate. Every time I watch Vertisaium's "Math's Fundamental Flaw", when it gets to Godel and Turing's fate and their death, I just start crying... This made me even more emotional. So much catastrophe endured by this great man...
@@internetgevalletje I don't know if that proves it but otherwise you are right. It is a craft rather than a science. On the other hand as we are not and cannot be scientific realists either, so what we find in science is also partly "invented", not "discovered". I donno. Lol.
@@internetgevalletjewhy what does OP's reaction to learning about those mathematicians' fates have to do with the invented/discovered thing is this some pop culture reference
I remember hearing of his passing in High School - but had not heard of his troubled mental state... How sad for one so gifted to have perceived himself to be So Very Troubled - and To Not Be
Nice video, thank you. I hadn't known about the personal tragedy of Gödel's private life, though the mind-bending subject itself and the hostility from respected colleagues must have pushed him close to the edge.😥
I used to be under the impression that he died young like Alan Turing, so knowing that he died at age 71 doesn’t lessen the sadness of the thought of such a great man putting himself through such suffering, it does make it far more acceptable knowing he lived a long and full life.
My theory is that Goedel wasn't paranoid, but he knew people might think he was. He didn't care, because he was depressed. He was depressed because he couldn't get Einstein to stop searching for the Theory of Everything (TOE), that Goedel was strongly believed to be futile, according to Goedel's own Theorems. Einstein was working on TOE every day, until he died. Depression kills your appetite first, then it kills you.
@@ChristoferKelly here's my guess on the bridge. The ToE would be a single set of statements that can provide truth or falsehood to any provided statement about the physical world, a full and complete explanation of all concepts within the cGh cube. His theorems state that such a concept, at least in mathematics, can't exist. Whatever axioms you select (for a consistent system), these axioms necessarily come with undecidable statements. A ToE should, by its idealisation, have no undecidable statements.
@@xinpingdonohoe3978 I wonder whether you're over-generalising Gödelian incompleteness that, to my mind, doesnt obviously extend from the mathematically formal systems of first-order logic onto physical reality, which appears to have at least some components that are described by physical theories that aren't axiomatisable, e.g. quantum mechanics, which operates probabilistically and attributes uncertainty as an innate attribute of our reality. A Gödelian framework presupposes a deterministic system. But I'm only thinking out loud here, so I might be wrong or my reasoning might be flawed. It's an interesting idea to contemplate on, though.
Thanks for the interesting and informative video on Kurt Gödel. It is not uncommon for those of superior intelligence to be afflicted with paranoia and self-doubt.
True. But we "normal" people often dismiss this as a psychological malady, when in reality, it may be that we of "normal" intelligence just haven't considered the human condition deeply enough to perceive our utter moral depravity.
I discovered Gödel’s paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, which contains Gödel's incompleteness theorems, at the age of 14 and my mathematical trajectory was changed forever. It was a total revelation. Somehow everything else in mathematics (and theoretical physics) became pale in comparison and almost trivial to grasp.
You should also talk about Godel's *completeness* theorem, and the difference between the two theorems. That will give people a much better sense of what mathematical logic about.
One of the greatest 'truths' I've encountered was by a handyman - he said "I believe that whatever you do will kill you in the end" - and so by the same reasoning - Kurt Godel died from his own faulty logic when he refused to eat.
What's the correct pronunciation? My set theory lecturer also pronounced it as the narrator in this video, but with the "r" much shorter and softer. More like "Gedl".
@@thembadube9589 Well, there's no 'r' in there at all. The umlaut creates as vowel sound we don't quite have in English, but its definitely not an 'r' sound. A lot of English speakers slip and 'r' into 'Goebbels' for some reason too and call him 'Gerbels'.
@@xinpingdonohoe3978 No. It does not exist in English. I'm sure there's a phonetic representation in a dictionary somewhere, but how that would help an American I don't know. Ask a German when you meet one in the flesh. ;)
For any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function directly. For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that forms a set out of selecting one sock from each pair without invoking the 'Axiom of Choice' [axiomatic set theory]. - Bertrand Russell. Choice exists as a function in constructive mathematics, because a choice is implied by the very meaning of existence. Gödel's 1958 paper, 'Dialectica' showed type theory may be used to provide consistent proofs in arithmetic, linking Gödel's consistency proofs to Russell's type theory. This was after Allan Turing, had made good use of Gödel's encoding, together with Church encoding. Gödel's first and second incompleteness theorems, in theorems VI and XI, provided 2 previously undefined techniques; I. "rekursiv" (primitive functions) "primitive recursive functions", recast "This statement is not provable" sentence as self-referential formal arithmetic. II. Gödel numbering, as standard logic based on prime factorisation, uses Gödel encoding. The word foxy is represented by 102111120121. The logical formula x=y => y=x is represented as; 120061121032061062032121061120. The fundamental theorem of arithmetic, states that any number (and, in particular, a number obtained in this way) can be uniquely factored into prime factors, so it is possible to recover the original sequence from its Gödel number. They may be classed the same as hereditarily finite set. Gödel sets can be used to encode formulas in infinitary languages. Kurt Gödel (1931), "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I". Monatshefte für Mathematik und Physik 38: 173-198. doi:10.1007/BF01700692. Available online via SpringerLink.
In any formalized (i.e., rigorously defined) system based on logic, with a finite number of axioms, you'll always have two problems: (1.) *Completeness* - there will always be true statements about the system that cannot be proven true within the system. And (2.) *Consistency* - there will always be true statements about the system that contradict with other true statements about the system. *Conclusion* - Any set of axioms that defines a complete and consistent system must have an infinite number of axioms. Let's assume that Olive Null, the smallest infinity, which is known as the set of (positive) integers to keep the "first pass" simple. Axioms are simply the assumptions one starts out with. In *Euclidean Geometry* starts with just 5 explicit axioms: a line is assumed to be straight and runs for ever, two lines cross in one and only one point, etc. Descartes (father of cartesian coordinates) realized that our actual assumptions can be quite subtle, as in *I think, Therefor I am* - such assumptions are called *implicit assumptions* - i.e., you make implicit assumptions without even realizing it, perhaps hundreds, thousands, billions, etc .. Einstein literally thought his way out of the *3-dimensional box* that we all implicitly assume to be all there is. He gave us the 4 dimensional space-time, but he had help perfecting it from a former Professor (name to be added later), and the previous work by Riemman (spelling?). Einstein taught us that the 4th dimension is *Time* - So, aren't axioms basic knowledge? If so, knowledge could be infinite, but if so, it would have to rest on an infinite set of implicit axioms. If knowledge is infinite, we'll all have a chance to discover something. However, that has to include wrecking a lot of old models, as Einstein did. Sorry to ramble.
This is incorrect. Gödel's incompleteness theorems don't claim that all mathematical systems are both _incomplete_ *and* _inconsistent_ as you've stated; rather, they pertain strictly to mathematical systems axiomatised using first-order logic that encode the arithmetic of natural numbers (or, enough of it, at least). The incompleteness theorems then state that any such system cannot be both _complete_ *and* _consistent._ Therefore, if one of these systems is mathematically consistent, it cannot also be complete; nor can the system prove its own self-consistency. The incompleteness theorems don't preclude systems of infinitely many axioms, only that the system be _effectively_ axiomatisable, i.e. every statement that can possibly be expressed by the system can be enumerated (listed) using an algorithmic procedure-essentially, this simply demands that everything be deducible purely from the axioms alone. Since every algorithmic procedure can be formulated recursively, it's entirely possible to enumerate a system of infinitely many axioms given an infinite amount of time. Thus, your stated conclusion from your first paragraph won't be applicable. And, indeed, Euclidean geometry is both complete *and* consistent, as are many other non-Euclidean geometries, including that of Minkowski space-time, which Einstein uses in his theories of relativity. This doesn't contradict Gödel's incompleteness theorems, however, because those geometries don't encode a sufficient amount of arithmetic, so these systems fall outside the remit of these theorems. Lastly, you might need to think about how you define _knowledge_ relative to the extent that you wish to apply mathematical rigor. There's mathematical knowledge, which we already know has limitations. And this is separate to scientific knowledge, which pertains to our understanding of the real world, and necessitates (or so it seems) that our science be founded upon empiricism, *_not_* logic. Therefore, even if there were infinitely much to know about, our empirical-based science has fundamental limitations: firstly, within the domain of all possible knowledge, there will be regions that cannot be accessed through the scientific method, and will remain _unknowable;_ secondly, that which is potentially _knowable_ also can never be *known* in the same sense that is is possible _to know_ something mathematically. Empirical science doesn''t prove things to be true, because it fundamentally cannot; rather, it accrues evidence, based on mathematical models, which then is used to refine those models to produce a slighlty clearer impression of reality. But, we can never truly _know_ reality.
@@ChristoferKelly Nice of you to split hairs with me. From a practical point of view, if what you're saying is correct, then it doesn't do you much good, which is no better than what I put forth. About the only thing you can say about math (as a whole or in part) is that it's inconsistent. Even Peano ASrthmetic was found to have contradictions, like 20 years ago, so I doubt the Geometries are immune to hiccups. You said, *The incompleteness theorems then state that any such [complete] system cannot be both complete and consistent.* So, if it's complete, then it cannot be consistent. When we posit a system that's inconsistent, how can we possibly say the system is complete? I'd say that given any inconsistent system, you can both prove *and* disprove any crazy statement, whether it's true or not. I'd be more interested to hear what you think of implicit axioms. As an undergrad, I studied *Set Theory and Metric Spaces* in the late 1970s, so I'm a bit rusty now, but I vaguely recall an amazing journey through *Zorn's Lemma, the Axiom of Choice and the Well Ordering Principle*. I'm sure the curriculum has been streamlined and improved since then.
@@neilanderson891 _When we posit a system that's inconsistent, how can we possibly say the system is complete?_ That is actually a good question. I would say, for completeness' sake, we should call such a system not only "complete" but "complete trash". I agree that it is quite confusing that it is even mentioned on equal footing with a system that is consistent, but not complete. But Gödel's incompleteness theorem only applies to axiomatic systems of a certain mimimal complexity. Any axiomatic system that can formulate the arithmetic of integers is complex enough to fall within the scope of Gödel's incompleteness theorem. There are simpler systems which cannot describe arithmetic of the integers, which are consistent and complete.
@@neilanderson891 This wasn't a matter of splitting hairs since your description of Gödel's incompleteness theorems is very different (and incorrect) to what I tried my best to elucidate for you. But you then go on to misquote me, by inserting the word _"[complete]"_ at a point that completely changes the meaning of what was originally stated. I'm perplexed why you went and did that. Peano Arithmetic has *not* been shown to be inconsistent. While it hasn't been formally proven to be consistent either, it is strongly believed to be so, and no contradictions have ever been found. But consider the implications if it were found to be inconsistent: it would mean all statements in arithmetic, including their own negations, could be proven. This is the law of *_deductive explosion._* Rather than doubting the consistencies of certain geometries, you could simply read up on it. As I stated, Euclidean geometry is known to be both complete _and_ consistent, but as it doesn't encode a sufficient amount of integer arithmetic, Gödelian reasoning is not applicable. Anyway, I was pointing out your errors in good faith, because it seemed like they would be something you'd genuinely be helped by. Sorry if you felt attacked. I don't know what an _implicit axiom_ is. Are you able to provide a definition for me ?
Kurt Gödel’s Incompleteness Theorems transformed mathematics by revealing its inherent limitations. He showed that in any consistent system, some truths cannot be proven, challenging the quest for absolute certainty. Gödel's insights reshaped logic, philosophy, and computer science, emphasizing the complexity and beauty of truth. His work inspires us to embrace curiosity and explore the unknown, even when answers seem elusive. Gödel's brilliance reminds us that the pursuit of knowledge is as much about asking profound questions as it is about finding solutions, making his legacy a cornerstone of modern intellectual thought.
I remember when I first read Godels theorem I was in shock, I did not believe that it could be true yet his argument was absolutely sound and relatively simple.
I like Godel's theorem. It scratches the surface of a place from which we can catch a glimpse of the divine: You cannot know certain things about a cave if you are limited to being inside the cave. You can only claim the cave is all there is. There are assumptions in science that cannot be proven by science. You can only claim that science is the only way to know anything. And there are certain truths about the universe that we cannot prove from inside the universe. You can only claim the universe is all there is.
You only need to be able to grasp the scope of the fields we ponder. If you can see that the rabbit hole we can go down will have Alice-like properties, that's probably sufficient.
Wonderful and powerful video about the human life of these extraordinary men and women. Every man shud have an Adele in life... Strong pilar of support.
The loophole in the American Institution was almost used by someone whom everybody knows. Such an accurate and deep prediction that can only come from a paranoid. That's the reason why the human species survived the catastrophies where others couldn't. We have the paranoids as our lighthouses for the unexpected possibilities. Human brain's full radar function was assigned to this unfortunate man.
The point isn't that some things are 'out of reach'. The point is that human reasoning can surpass any formal system by showing that a statement in the system is true even though it can't be proved either way in the system. Its a statement of optimism and wonder that elevates the meaning of mathematics and logical thought.
Funny how so many brilliant Mathematicians became crazy, literally. Altho I didn't finish my Pure Maths undergrad as my life took a different direction, I still love maths, yet I also am very spiritual, so the Incompleteness Theorem makes total sense to me. My thinking and sense of humour are also very different from the norm, and just to add, according to the Myers-Briggs personality test, I come under the smallest group of personalities out there, not that there is an out-there other than what is projected from within me....just to play with some non-conventional maths and physics or Carl Jung's work.
I do not want to take your joy, but this tests, usually, aren't precise enough. If you do the same test two times, it could be to persons. Be carefull, you can be a Pisces person (/irony on).
Who else has a feeling of connectedness between Kurt Gödel's Incompleteness Theorem, Turing's Halting Problem and Heisenberg Uncertainty Principle. It's like the universe's way of saying you can't get everything.
Heisenburg Uncertainty Principle is the most surprising to me. It's not just a statement on hypothetical computers and abstract logic. It's real world stuff, and it's not a failure of our measuring devices. There's a fundamental sense of undecidability to everything that makes up our universe; it's not deterministic.
An interesting aside is when Euclid's geometric axioms were shown to be just one of several geometries that were logically consistent. About 50 years earlier I think. Poincare' said it didn't matter which was true pick one that's convenient. Now we have 2, then 3 truths (geometries) that were logically consistent. This must have shook science profoundly.
Sometimes people simply know they will die, and they stop eating because they can feel it. They feel their body giving out, and they know prolonging it is somewhat futile and could lead to exponentially more severe suffering right at the end.
He didn't "stop eating because he knew he would die", he starved himself down to 30 kgs over the course of a month because he was paranoid of being poisoned. His suffering and death were not caused by reasonable decisions, but a mental health crisis.
@@julioaurelio My theory is that Goedel wasn't paranoid, but he knew people might think he was. He didn't care, because he was depressed. He was depressed because he couldn't get Einstein to stop trying to find (and prove) the Theory of Everything (TOE), that he was certain was futile, according to Goedel's Theorems. Einstein was working on it every day, until he died. Depression kills your appetite first, then it kills you.
It did occur to me that an additional hurdle was growing up in a well to do family. Back then they would have had servants. I doubt it ever occurred to him to shop for and cook his own meals. Mind you paranoia could even make him believe the ingredients had been poisoned in the shop. Its a terrible condition. unresponsive to logic.
@@helenamcginty4920 I wrote you a long comment about the interesting connection of this topic with the topic of split consciousness in people who have undergone corpus callosotomy. And again, as in many cases, the comment was deleted by an automated system. An amazing situation - as if we returned to the times when this outstanding scientist lived, except that the iron cage is now with amenities...
If I'm not mistaken, there are 2 main theorems by Godel. One says that you can't prove all statements that are true in a system where these statements were built. You may be able to outside of that system. The second says that it's impossible to build a complete axiomatic system with a finite number of axioms. There will always be statements within that system that will require another axiom (that is, it can't be proved from the finite set of axioms). It also says that it's impossible to build such a system that is free of paradoxes. Sorry, but I can't remember all the details.
I was always amused that Godel's theorem that there will always be theorems in (carefully defined, but highly plausible) logical systems which are true, but for which no proof can exist in that system itself was proved by Godel.
He proved we'll never create self-aware technology, because we can't find the line that exists between intelligence and consciousness, the line where knowledge becomes understanding. We can't encapsulate metaphysics within itself.
But, unfortunately, the illuision of self-aware tech will be commercially sufficient. Pretending to have courage is indistinguishable from having courage.
@@rideon6140 Until 'artificial intelligence' figures out how to keep oil flowing from the bottom of the Gulf, to the refineries, and then to the generators and cooling pumps in the hydroelectric and nuclear power stations that run the world, all such talk is cartoonish at best. It has ALWAYS been mental masturbation worrying about this shit.
Yes, it helps ... but not necessarily. For example, as a result of his walks with Einstein, Goedel found another exact solution of Einstein's GR field equation: the so-called Goedel metric. So he had interests outside logic. His level of "nutiness" was simply too high....unlike Nash, another nut. Any human being would find his self-starving illogical: I am afraid of dying of poisoning so I will not eat!!!! PS Apparently the judge in the citizenship ceremony also realized that things were derailing. He went along with the speeding up to prevent Goedel to literaly perjure himself or refusing to take the oath.
Godel's story has some parallels with both John Nash & Srinivasa Ramanujan, brilliant minds with lives full of sadness & some tragedy. Even further back the famous Norwegian Neils Abel led a life of hardship, but still maintained a very high standard of original work, only to be recognized after his early death. This is why (as an amateur Math Graduate) I highly RESPECT Mathematicians & what they do, as sometines the price of progress comes at a cost that these men (& women eg. Hypatia, Katherine Johnson) willingly pay....🤔
Even though this is not Godel's loophole, some believe it could be one possible flaw in the Constitution that he may have noticed. en.wikipedia.org/wiki/G%C3%B6del%27s_Loophole#:~:text=G%C3%B6del's%20Loophole%20is%20a%20supposed,legally%20turned%20into%20a%20dictatorship.
@@hoi-polloi1863 in some senses, that's pretty useful. If a president implemented a law ξ, such as "No persons under 35 years may enter any form of lоvе-bаsеd rеlаtiоnshiр." and then a law ψ saying "Law ξ may not be changed or nullified.", a different president would have to break law ψ to be able to remove law ξ, so the ability to pardon himself means that previous presidents can't set "trap card" laws to protect their work.
@@Newsthink I appreciate the effort to replicate the correct German pronunciation, but without saying it in a German accent it just sounds odd. I assume this is an AI voice, hence the lack of attempt at replicating the German accent when it came to names and places, but you tried to force it to approach proper pronunciation by spelling names phonetically?
@@Newsthink Oh, that's much worse🤷♂Pronouncing foreign words correctly (even names) is merely a search on TH-cam away. If you're invested enough to make content like this, you should have the attention to detail to do cursory searches for correct pronunciation at least.
Yes you are. Those four words, in a vacuum, are the statement. We can label it, if it helps. Say γ="This statement is false" is a statement, a collection of words that forms an idea, with the axioms being the rules of the English language and the basic ideals of logical reasoning. γ cannot be decided to be true, and γ cannot be decided to be false.
objective facts exist? we observe phenomena and our perceptions are subjective so, ... objectivism is based on an omnipotent God sitting outside the universe watching it all... it's a bit of a medieval paradigm. bronze age paradigm, really. 🫤
The first incompleteness theorem states that for every logical system there are statements that cannot be proved false or true, that’s why no system can be “complete”. The second incompleteness theorem says that a formal system cannot be used to prove its own consistency even if it is assumed to be consistent. These two theorems mean that every formal system has to base on axioms that have to be chosen by social agreement, terminating any possibility of absolute objectiveness.
As in the case of John Nash, Kurt Goedel's mental state at the end of his life shows that a balance between the head and the heart is essential to a healthy like and a will to live. An overdeveloped intellect can lead to the type of paranoia from which they both suffered. Conversely, too much heart uncontrolled and tempered by the intellect as is the case with poets like John Keats or Syvia Plath, for example, can lead to serious depression and even suicide.
Safe and stable world in 21 century. Everything went awry very fast. And even if reasons are obvious, nothing can be done, especially because those who try to say it out loud are immediately silenced and branded as agents of the enemy.
If I'm not mistaken, there are 2 main theorems by Godel. One says that you can't prove all statements that are true in a system where these statements were built. You may be able to outside of that system. The second says that it's impossible to build a complete axiomatic system with a finite number of axioms. There will always be statements within that system that will require another axiom (that is, it can't be proved from the finite set of axioms). It also says that it's impossible to build such a system that is free of paradoxes. Sorry, but I can't remember all the details.
I hate to break this to ya, but there have recently been some breakthroughs in the value of pi. I can't give away a lot of details, but circles are about to get a hell of a lot rounder.
@@solconcordia4315 I thought you were mistaken. I thought you thought that e^(i/π)=-1. But you were using TeX notation. Why? This is TH-cam comments, not Overleaf.
This is actually really sad; but not uncommon. Many "brilliant" minds are plagued with "abnormal" thinking. It's probably because that person's mind doesn't think about things the way most people do; and that extends even into every aspect of their personal life.
Mathematical systems are not fixed objects. They can be expanded, redefined, refined, etc. to solve previously unsolved problems which are interesting and important. 😊
Aristotle grasped this concept in the metaphysics, when saying that there are truths that cannot be demonstrated, that is, conversely, that each demonstration is based on a bigger proposition that cannot be proved.
Do you have any further details on Gödel’s detection of a logical loophole in the Constitution making dictatorship possible? This would make a very interesting post given our current political environment.
I'm not sure, since I'm not American, but a quick glance at the 12th amendment is interesting. If I'm understanding what they're saying, if everyone from all but one of the states dies, except a single individual from a different state, then that different state person running for president or vice-president would necessarily win. Or perhaps it's that the 25th amendment needs only a majority of presidential officials, along with the vice president, to write a letter to the senate and the speaker of the house of representatives, to be able to immediately enact a legal coup to make the vice president into the new president with immediate effect.
*PRONUNCIATION: I believe I have anglicized the pronunciation, and I'm aware the 'correct' way is "geu-del." Will attempt it next time!*
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I'd love to watch a video from you on Georg Cantor - my personal number 1 greatest mind in history. Other 2 great names to discuss would be Archimedes and Leonardo Da Vinci.
...and its not pronounced prin"k"ipia but prin"s"ipia...... like principle.... earcancer olé... sorry i ment earcanker olé... seriously. Reading a text Chatgpt printed is seriously not too hard.
I think pronouncing Godel as "Girdle" is the correct pronunciation of a german name. But I'm no expert.
Everybody loves to correct me when I mis-pronounce Porsche as "Por-sss", rather than "Porsha"
My revenge: I ask, *Hey, What's the Capital of Italy* - The typical reply: *Rome, of course, you idiot ... Don't try to change the subject.*
My reply: *Don't you mean "Roma" ... I've been there, I know how the Romans pronounce it*
@@neilanderson891 No R, dummy. It's porsha, not porss. Same with Goedel.
@@kalleburberry2586 actually, it IS pronounced prin-KEEP-ia
You forgot to mention, that Godel's incompleteness theorems were not less a surprise to him, than to anyone else. He, like others, was trying to complete Hilbert's project and make mathematics one beautiful complete system, where every statement is provable from a set of axioms. His discovery of the inherent incompleteness of mathematics came as a shock to himself, as much as for the rest of the world. Thinking about all consequences of it, likely contributed to his mental state.
It would be great to see a video from you about Georg Cantor.
And Hilbert's truth was broken for ever... He would become another person after Godel's discovery.
It's implicit, genius.
@@chickenlover657 It is not implicit and it's a very important detail. Without it, someone unfamiliar with Godel's story might get impression that Godel has been working to disprove the possibility of Hilbert's project.
It would help if I even had the slightest idea what you were talking about.
Gödel was also a Platonist, and first-order logic was already showing cracks in its ability to be absolute. For example, Skolem showed 10 years before Gödel's incompleteness theorems that every countable first-order theory, if it has a model, also has a countable model.
I don't think Gödel would have found his result terribly contradictory with his philosophy. At least to a modern Platonist like Hugh Woodin, the mathematical universe is so large that we cannot capture it with "small" truths.
A mathematician friend told me that a better layman explanation of the theorems is: "a complete and consistent axiomatic system does not exist..if its complete then its not consistent and if its consistent then its not complete"
Define a 'consistent axiomatic system', grazie mille! Aah, you mean an axiomatic system cannot be both sound ('consistent') and complete. If it is"complete" it is not sound; all its theorems are provable - and then ”some”. A bit like Heisenberg's uncertainty principle Nasty.
Yeah. He should mention a few details: a countable one and one allowing to express the natural numbers. Those are very significant restrictions. They are stated right at the start of the actual proof. Not mentioning them is leading to a wide philosophical overinterpretation of the result.
It similar in concept as Heisenberg's Uncertainty Principle in physics, which states that we cannot know both the position and speed of a particle like a photon or electron with perfect accuracy. The more we nail down the particle's position, the less we know about its speed and vice versa.
Stop talking out of your Yanus. Just get lost! And take your "mathematician friend" with you!
axiomatic: self-evident or unquestionable. it is axiomatic that Californians should stop moving to Texas, especially Austin
My understanding of Goedel's Incompleteness Theorem is not as the narrator states, "There are mathematical truths that cannot be proved in any mathematical system." Rather, it is that there are mathematical truths in any mathematical system that cannot be proved within that axiomatic system. A significant difference!
But the narrator states exactly what you want her to state. See 0:26.
edit: OK, later statements are less accurate ;-)
@@ronald3836 I'd say "somewhat ambiguous" rather than "less accurate," as one of the plausible meanings of the statement goedelite quotes is equivalent to the 0:26 statement.
I think they did a good job with their explanation. Yours while more accurate would most likely confuse the audience. These kinds of videos are just an introduction to things, if you want to actually learn how something works, you actually have to study the subject.
Thank you!
This whole script of the video seems AI generated.
I'm not a mathematician but have a book on his incompletness theorem. The logic of his proof amazed me.
I think I have it in wavy hands form as:
Final musings?
If there exists a statement that has a valid proposition in math logic with result x such that x is simultaneously both true and false then that is of the form (x∈X) ∧ (x∉X) and thus defines null set in math logic side of the proposition. Nothing can really be implied about the statement side other than it has a proposition form that defines null set in math logic space.
Thus math logic side IS complete as null set can be reached through valid propositions however nothing can be deduced or inferred about statement side as that side is full of grammar, syntax, inflection and inference that are beyond definition in present scope of logic to define.
For example
This statement is false?
This statement is true!
This statement is false.
This statement is incomplete, ...
This statement is incomplete,
This statement is incomplete
(yes that is a statement side comma or just simply a non-terminating statement like This statement is incomplete (if you see what I mean) perhaps better as This statement is incomplete ... )
Therefore: Math logic just had its infinity over infinity and zero over zero and combinations thereof moments and has been found wanting?
Incorrect
When doctors treated ulcers in 1950s they really didn't know what they were doing. They thought it's a stress and lifestyle disease. Actually the most common cause of ulcers is infection of the stomach by bacteria called Helicobacter pylori (H pylori). Then one guy figured it out and was ridiculed by all the other doctors for years. My grandfather died because of that. There was knowledge of how to treat this condition, but mainstream doctors blocked it.
The same thing happened about 100 years earlier with the discovery that germs cause diseases. Interestingly that happened in the Vienna General Hospital (Ignaz Semmelweis)
If I remember correctly the dude that discovered H. Pylori drank some of the bacteria to give himself an ulcer then cured himself.
and the patron saint of both the standard theory & scientism, Carl Sagan, joined in on ruining the career of at least one scientist with valid alternative theories.
modern science is mostly about ego & funding not truth/discovery despite how it's marketed to the public
like IVERMECTIN in covid nonsense
I have had ulcers and a new one and both of mine are emotionally based. They tested me for the bacteria and it came back negative. Being an alternative therapist and coming from a personal development background, most dis-ease is emotionally and/or energetically based.
Medical doctors are still clueless and it’s even worse today
I love that Dr Godel's story illustrates how complex the human mind is. The man who was loved by everyone, and was far more accomplished than almost any other person, was haunted, hunted, and ultimately destroyed by that mind. But due to mathematics, he is immortal.
But how about people who have to vomit because of the sight of worn slippers? You didnt think about that, did you?
what is there to love about that
Its shocking how unhindged many people on the brink of genius really turn out to be... or are driven to... Albert wasn't exactly a saintly man himself after all.
that is not complex. they already did compiler ,electronics,nukes,rockets and medicine which is very very complicated than some random math theorem. get over the hype!
Even today people with amazing abilities show that at the same time they can be nuts. My favourite example is Michael Jackson. A look at his videos and a listen to "Thriller".
Gödel also developed the ideas of computability and recursive functions. It's equivalent to lambda calculus and/or turing machine. It's a bit unfortunate it wasn't mentioned. He was genius.
I was a math major at UCLA and remember hearing talk of his death in the department in 1978.
Gödel has some pretty dope contributions to math, outside of completeness theorem.
Gödel contributed towards Axiomatic Set Theory, which other authors in their field are equally impressive in my opinion.
Grothendieck, Russel, and Zermelo are among my favorites.
A lot of the abstract thinking comes from a broad pool of mathematicians… these guys are geniuses, even though we like to make fun of some of their work occasionally haha.
It took >360 pages to get 1+1=2.
@@jacksonwilloughby7625 this isn't actually accurate. 1+1=2 is fairly straightforward, the 360 pages is other stuff related i believe
@@akiya9216 Most of it is definitions and axiomatizing up to that point. So its just a joke on a technicality.
The name of that favourite of yours is RusseLL ...
His theorem became indirectly the nail on his coffin. What a sad story about a brilliant mind.
Thank you for bringing a fantastic inside about Kurt Gödel . He's one of my favorite mathematicians .
Thanks! This video has inspired me
Reallly appreciate it, thank you so much!
It's also interesting how he travelled from Austria to America, since it was extremely difficult, westwards instead of eastwards, due to travel restrictions. It's amazing he could do it, given his state of mind.
Not really that difficult. Switzerland is right on Austrian border, from there air travel to Spain (or just directly into Spain by train but Swiss way is safer) and easy further travel to america...
The German exit certificate required the Godels to travel through Russia to Japan via the trans-Siberian railway, then by boat to America. This was in late 1939, early 1940. (Ref: Rebecca Goldstein's book "Incompleteness: The Proof and Paradox of Kurt Godel").
@@KuK137 As gárreme too2238 explained, that route wasn’t an option for Goedel, so he travelled through the much longer and difficult westward way.
(Eastward)
@@jjeherrerado you mean... eastward??
The other day I was walking down a street in Vienna - it was Florianigasse - and there I saw a plaque next to the door of a house that indicated that Kurt Gödel had lived there for some years. So I asked myself on how many places Kurt Gödel might have lived, because this was already the fourth place which I found in Vienna.
It does make sense though. I have known people who lived in more than twenty places when they were students.
A well-researched documentary. As a mathematician grappling with health issues, I can clearly identify with this. Thank youl.
An uncle of a was victim of 50’s ulcer surgery. He struggled with eating for the remainder of his life because the surgeons removed most of his stomach.
Godel, Escher, Bach, is a good (if dense) read. It's probably the best explanation of incompleteness theorem I've come across.
Its not just an explanation, he actually proves the first theorem in the axiomatic system he develops throughout the book.
But what is this obsession with Bach? This dude compare's the theorem(s) also to Bach's fugues - apparently owing to the dude some Hofstader (?) who wrote the GEB.
@@jackquinnes GEB is about self referential systems and how those ideas relate to consciousness. The three titular figures, while all operating in a different field (gödel in math, escher in paintings and bach in music) each created works with a heavy sense of self reference. Other systems which reflect the same overarching theme also show up such as DNA in biology and computer programs. For anyone who wants to read it, if you find it too dense I recommend only reading the achilles and the tortoise sections on the first go. Theyre very fun to read and give a much easier to digest overview of the ideas in the following chapter.
@@guy3717 Thanks. Guess I will - finally - give it a read. First I should go to the trouble of getting my hands on it ofc. - ’Self/selfhood is deeply mysterious’, wrote young Ludwig Wittgenstein himself (a pun intended). Yes, it might be - and it really is if you think about it, philosophically.. For better or worse. Wittgenstein had in mind this undoable kernel of consciousness we know as transendental ego. ’Self’ is in this ”original sense” (just)! the self-reference of human mind. It is what it is. As elusive as intriguing. Is it an i”illusion” then? Just ”unprovable truth” if we are willing to play the game.
Read it. Was a fascinating book. And it won the National Book Award, so it’s not just a math book and it’s definitely not boring. It Han an elegance to it. The writer manages to weave a compelling narrative on Bach canons and fugues and Eschers art and then walk you through symbolic logic and keep it all very interesting. At times it’s difficult to grasp but worth the effort to work through it.
By saying that his "doctors didn't know what they were doing" - Kurt Godel demonstrated that he was even smarter than we thought.
@7:20 Gödel fled to the US via Soviet Union (Trans-Siberian Railway) and Japan. He arrived by ship in San Francisco in March 1940. Not in New York.
He arrived by train from San Fran to New York
@@Newsthink Well the images you used where a bit odd IMHO and suggest he came like most people via ship over the Atlantic. Personally I think how Gödel came to the US would have been really worth mentioning. 🙂
@@tetsi0815 ChatGPT has its inherent flaws, especially as what comes to things of special human interest.
The course in Mathematical Logic I took that covered the incompleteness theorem was simultaneous with his death. My instructor cancelled class the day he got the news and we had a very non-mathematical discussion about the will to live the next class session.
I studied his proofs in depth in two different courses.
I knew little about the man.
Thanks for the video.
Such a heart breaking fate. Every time I watch Vertisaium's "Math's Fundamental Flaw", when it gets to Godel and Turing's fate and their death, I just start crying... This made me even more emotional. So much catastrophe endured by this great man...
You being emotional is the proof math is invented, not discovered. Thank you.
@@internetgevalletje I don't know if that proves it but otherwise you are right. It is a craft rather than a science. On the other hand as we are not and cannot be scientific realists either, so what we find in science is also partly "invented", not "discovered". I donno. Lol.
@@internetgevalletjewhy
what does OP's reaction to learning about those mathematicians' fates have to do with the invented/discovered thing
is this some pop culture reference
I remember hearing of his passing in High School - but had not heard of his troubled mental state...
How sad for one so gifted to have perceived himself to be So Very Troubled
- and To Not Be
Nice video, thank you. I hadn't known about the personal tragedy of Gödel's private life, though the mind-bending subject itself and the hostility from respected colleagues must have pushed him close to the edge.😥
I used to be under the impression that he died young like Alan Turing, so knowing that he died at age 71 doesn’t lessen the sadness of the thought of such a great man putting himself through such suffering, it does make it far more acceptable knowing he lived a long and full life.
Thank you Newsthink for this excellent biography of Kurt Godël’s life.
My theory is that Goedel wasn't paranoid, but he knew people might think he was. He didn't care, because he was depressed. He was depressed because he couldn't get Einstein to stop searching for the Theory of Everything (TOE), that Goedel was strongly believed to be futile, according to Goedel's own Theorems. Einstein was working on TOE every day, until he died. Depression kills your appetite first, then it kills you.
How do Gödel's theorems relate to TTOE ?
@@ChristoferKelly here's my guess on the bridge. The ToE would be a single set of statements that can provide truth or falsehood to any provided statement about the physical world, a full and complete explanation of all concepts within the cGh cube. His theorems state that such a concept, at least in mathematics, can't exist. Whatever axioms you select (for a consistent system), these axioms necessarily come with undecidable statements. A ToE should, by its idealisation, have no undecidable statements.
@@xinpingdonohoe3978 I wonder whether you're over-generalising Gödelian incompleteness that, to my mind, doesnt obviously extend from the mathematically formal systems of first-order logic onto physical reality, which appears to have at least some components that are described by physical theories that aren't axiomatisable, e.g. quantum mechanics, which operates probabilistically and attributes uncertainty as an innate attribute of our reality. A Gödelian framework presupposes a deterministic system. But I'm only thinking out loud here, so I might be wrong or my reasoning might be flawed. It's an interesting idea to contemplate on, though.
@@ChristoferKelly I'm not sure I understand your question, but I'd say that Godel thought Einstein was tilting at windmills.
@@neilanderson891 Pretty much all physicists thought that. Even Einstein said it probably wasn't going to work. He just liked working on it.
Will always be the man who broke math. RIP kurt.
Cindy Pom did it again. Fantastic video on a complicated subject and man. Thank you. BTW, nice explorer 😉.
Thanks for the interesting and informative video on Kurt Gödel. It is not uncommon for those of superior intelligence to be afflicted with paranoia and self-doubt.
True. But we "normal" people often dismiss this as a psychological malady, when in reality, it may be that we of "normal" intelligence just haven't considered the human condition deeply enough to perceive our utter moral depravity.
@@lastchance8142
Excellent point.
I discovered Gödel’s paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, which contains Gödel's incompleteness theorems, at the age of 14 and my mathematical trajectory was changed forever. It was a total revelation. Somehow everything else in mathematics (and theoretical physics) became pale in comparison and almost trivial to grasp.
I nerded out way too much when Godel made a cameo in Oppenheimer (2023).
Rarely we have such deep minded humans ❤
Social systems obscure, ignore and destroy so many genius minds all over the world 😢😢😢
I love how in math and science, a person's feelings about the matter bears no significance to reality.
You should also talk about Godel's *completeness* theorem, and the difference between the two theorems. That will give people a much better sense of what mathematical logic about.
One of the greatest 'truths' I've encountered was by a handyman - he said "I believe that whatever you do will kill you in the end" - and so by the same reasoning - Kurt Godel died from his own faulty logic when he refused to eat.
Why is the logic faulty? Say health runs on [0,1], with x
Probably so focused in his mind that he forgot his body: sleep, food etc...
...and then you are in a downgoing spiral...
I winced every time she said "Girdle"
What's the correct pronunciation? My set theory lecturer also pronounced it as the narrator in this video, but with the "r" much shorter and softer. More like "Gedl".
@@thembadube9589 Well, there's no 'r' in there at all. The umlaut creates as vowel sound we don't quite have in English, but its definitely not an 'r' sound. A lot of English speakers slip and 'r' into 'Goebbels' for some reason too and call him 'Gerbels'.
@@thembadube9589 The proper vowel does not exist in English. But it would be closer to just say 'Goh-del.'
@@AllAhabNoMoby is the ö not similar to the first vowel of burger, but more rounded?
@@xinpingdonohoe3978 No. It does not exist in English. I'm sure there's a phonetic representation in a dictionary somewhere, but how that would help an American I don't know. Ask a German when you meet one in the flesh. ;)
For any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i.e. set) of shoes; this makes it possible to define a choice function directly. For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that forms a set out of selecting one sock from each pair without invoking the 'Axiom of Choice' [axiomatic set theory]. - Bertrand Russell.
Choice exists as a function in constructive mathematics, because a choice is implied by the very meaning of existence.
Gödel's 1958 paper, 'Dialectica' showed type theory may be used to provide consistent proofs in arithmetic, linking Gödel's consistency proofs to Russell's type theory. This was after Allan Turing, had made good use of Gödel's encoding, together with Church encoding.
Gödel's first and second incompleteness theorems, in theorems VI and XI, provided 2 previously undefined techniques;
I. "rekursiv" (primitive functions) "primitive recursive functions", recast "This statement is not provable" sentence as self-referential formal arithmetic.
II. Gödel numbering, as standard logic based on prime factorisation, uses Gödel encoding.
The word foxy is represented by 102111120121.
The logical formula x=y => y=x is represented as; 120061121032061062032121061120.
The fundamental theorem of arithmetic, states that any number (and, in particular, a number obtained in this way) can be uniquely factored into prime factors, so it is possible to recover the original sequence from its Gödel number. They may be classed the same as hereditarily finite set. Gödel sets can be used to encode formulas in infinitary languages.
Kurt Gödel (1931), "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I". Monatshefte für Mathematik und Physik 38: 173-198. doi:10.1007/BF01700692. Available online via SpringerLink.
In any formalized (i.e., rigorously defined) system based on logic, with a finite number of axioms, you'll always have two problems: (1.) *Completeness* - there will always be true statements about the system that cannot be proven true within the system. And (2.) *Consistency* - there will always be true statements about the system that contradict with other true statements about the system. *Conclusion* - Any set of axioms that defines a complete and consistent system must have an infinite number of axioms. Let's assume that Olive Null, the smallest infinity, which is known as the set of (positive) integers to keep the "first pass" simple.
Axioms are simply the assumptions one starts out with. In *Euclidean Geometry* starts with just 5 explicit axioms: a line is assumed to be straight and runs for ever, two lines cross in one and only one point, etc. Descartes (father of cartesian coordinates) realized that our actual assumptions can be quite subtle, as in *I think, Therefor I am* - such assumptions are called *implicit assumptions* - i.e., you make implicit assumptions without even realizing it, perhaps hundreds, thousands, billions, etc ..
Einstein literally thought his way out of the *3-dimensional box* that we all implicitly assume to be all there is. He gave us the 4 dimensional space-time, but he had help perfecting it from a former Professor (name to be added later), and the previous work by Riemman (spelling?). Einstein taught us that the 4th dimension is *Time* - So, aren't axioms basic knowledge? If so, knowledge could be infinite, but if so, it would have to rest on an infinite set of implicit axioms.
If knowledge is infinite, we'll all have a chance to discover something. However, that has to include wrecking a lot of old models, as Einstein did. Sorry to ramble.
This is incorrect. Gödel's incompleteness theorems don't claim that all mathematical systems are both _incomplete_ *and* _inconsistent_ as you've stated; rather, they pertain strictly to mathematical systems axiomatised using first-order logic that encode the arithmetic of natural numbers (or, enough of it, at least). The incompleteness theorems then state that any such system cannot be both _complete_ *and* _consistent._ Therefore, if one of these systems is mathematically consistent, it cannot also be complete; nor can the system prove its own self-consistency. The incompleteness theorems don't preclude systems of infinitely many axioms, only that the system be _effectively_ axiomatisable, i.e. every statement that can possibly be expressed by the system can be enumerated (listed) using an algorithmic procedure-essentially, this simply demands that everything be deducible purely from the axioms alone. Since every algorithmic procedure can be formulated recursively, it's entirely possible to enumerate a system of infinitely many axioms given an infinite amount of time.
Thus, your stated conclusion from your first paragraph won't be applicable. And, indeed, Euclidean geometry is both complete *and* consistent, as are many other non-Euclidean geometries, including that of Minkowski space-time, which Einstein uses in his theories of relativity. This doesn't contradict Gödel's incompleteness theorems, however, because those geometries don't encode a sufficient amount of arithmetic, so these systems fall outside the remit of these theorems.
Lastly, you might need to think about how you define _knowledge_ relative to the extent that you wish to apply mathematical rigor. There's mathematical knowledge, which we already know has limitations. And this is separate to scientific knowledge, which pertains to our understanding of the real world, and necessitates (or so it seems) that our science be founded upon empiricism, *_not_* logic. Therefore, even if there were infinitely much to know about, our empirical-based science has fundamental limitations: firstly, within the domain of all possible knowledge, there will be regions that cannot be accessed through the scientific method, and will remain _unknowable;_ secondly, that which is potentially _knowable_ also can never be *known* in the same sense that is is possible _to know_ something mathematically. Empirical science doesn''t prove things to be true, because it fundamentally cannot; rather, it accrues evidence, based on mathematical models, which then is used to refine those models to produce a slighlty clearer impression of reality. But, we can never truly _know_ reality.
@@ChristoferKelly Nice of you to split hairs with me. From a practical point of view, if what you're saying is correct, then it doesn't do you much good, which is no better than what I put forth. About the only thing you can say about math (as a whole or in part) is that it's inconsistent. Even Peano ASrthmetic was found to have contradictions, like 20 years ago, so I doubt the Geometries are immune to hiccups.
You said, *The incompleteness theorems then state that any such [complete] system cannot be both complete and consistent.* So, if it's complete, then it cannot be consistent.
When we posit a system that's inconsistent, how can we possibly say the system is complete? I'd say that given any inconsistent system, you can both prove *and* disprove any crazy statement, whether it's true or not.
I'd be more interested to hear what you think of implicit axioms.
As an undergrad, I studied *Set Theory and Metric Spaces* in the late 1970s, so I'm a bit rusty now, but I vaguely recall an amazing journey through *Zorn's Lemma, the Axiom of Choice and the Well Ordering Principle*. I'm sure the curriculum has been streamlined and improved since then.
I dated Olive Null.
@@neilanderson891 _When we posit a system that's inconsistent, how can we possibly say the system is complete?_
That is actually a good question. I would say, for completeness' sake, we should call such a system not only "complete" but "complete trash".
I agree that it is quite confusing that it is even mentioned on equal footing with a system that is consistent, but not complete.
But Gödel's incompleteness theorem only applies to axiomatic systems of a certain mimimal complexity. Any axiomatic system that can formulate the arithmetic of integers is complex enough to fall within the scope of Gödel's incompleteness theorem.
There are simpler systems which cannot describe arithmetic of the integers, which are consistent and complete.
@@neilanderson891 This wasn't a matter of splitting hairs since your description of Gödel's incompleteness theorems is very different (and incorrect) to what I tried my best to elucidate for you.
But you then go on to misquote me, by inserting the word _"[complete]"_ at a point that completely changes the meaning of what was originally stated. I'm perplexed why you went and did that.
Peano Arithmetic has *not* been shown to be inconsistent. While it hasn't been formally proven to be consistent either, it is strongly believed to be so, and no contradictions have ever been found. But consider the implications if it were found to be inconsistent: it would mean all statements in arithmetic, including their own negations, could be proven. This is the law of *_deductive explosion._*
Rather than doubting the consistencies of certain geometries, you could simply read up on it. As I stated, Euclidean geometry is known to be both complete _and_ consistent, but as it doesn't encode a sufficient amount of integer arithmetic, Gödelian reasoning is not applicable.
Anyway, I was pointing out your errors in good faith, because it seemed like they would be something you'd genuinely be helped by. Sorry if you felt attacked.
I don't know what an _implicit axiom_ is. Are you able to provide a definition for me ?
His solution to Einstein's equations of General Relativity, which allows closed time-like loops, was also amazing.
It comforts me to know i’m not alone in my thinking. There have been others before me.
Lol. Alright.
Gosh. If you are feeling anything like K. G., please seek help. We wouldn't want to lose you.
Kurt Gödel’s Incompleteness Theorems transformed mathematics by revealing its inherent limitations. He showed that in any consistent system, some truths cannot be proven, challenging the quest for absolute certainty. Gödel's insights reshaped logic, philosophy, and computer science, emphasizing the complexity and beauty of truth. His work inspires us to embrace curiosity and explore the unknown, even when answers seem elusive. Gödel's brilliance reminds us that the pursuit of knowledge is as much about asking profound questions as it is about finding solutions, making his legacy a cornerstone of modern intellectual thought.
The best and most respectful bio of Kurt Godel on You Tube....Could you do one on William Feller??
You have captured Gödel quite concisely. Thank you.
His bizarre descent into apparent madness has always fascinated me.
I remember when I first read Godels theorem I was in shock, I did not believe that it could be true yet his argument was absolutely sound and relatively simple.
Striving to attain is itself fulfilling as per Albert Camus in the Myth of Sisyphus. Goëdel likely suffered from some form of depression.
I like Godel's theorem. It scratches the surface of a place from which we can catch a glimpse of the divine:
You cannot know certain things about a cave if you are limited to being inside the cave. You can only claim the cave is all there is.
There are assumptions in science that cannot be proven by science. You can only claim that science is the only way to know anything.
And there are certain truths about the universe that we cannot prove from inside the universe. You can only claim the universe is all there is.
I’m kind of surprised no one has made a movie of this. It’s a tragic story but very interesting.
How smart do I have to be to realize how stupid I am?
Time to use incompleteness theorem to answer this question
@@chechennel4817 Good luck.
About this much: ||
You only need to be able to grasp the scope of the fields we ponder. If you can see that the rabbit hole we can go down will have Alice-like properties, that's probably sufficient.
I was mediocre at best at math in high school, but I love a lot this channel
Wow, i like this narrator of yours she has a calm and good narrating voice :) and the information given is also quite amazing. Thanks for this video
Most likely it's a ROBOTIC VOICE!
@@artmanrom she's literally in the video narrating , not an ai
"... narrator of yours". What does that mean?
@@thembadube9589 well by "yours" I was referring to the people behind the camera
Wonderful and powerful video about the human life of these extraordinary men and women. Every man shud have an Adele in life... Strong pilar of support.
Awesome❤
This is a wonderful measured and accessible presentation. Thank you. I benefited from this.
Alej
The loophole in the American Institution was almost used by someone whom everybody knows. Such an accurate and deep prediction that can only come from a paranoid. That's the reason why the human species survived the catastrophies where others couldn't. We have the paranoids as our lighthouses for the unexpected possibilities. Human brain's full radar function was assigned to this unfortunate man.
The point isn't that some things are 'out of reach'. The point is that human reasoning can surpass any formal system by showing that a statement in the system is true even though it can't be proved either way in the system. Its a statement of optimism and wonder that elevates the meaning of mathematics and logical thought.
Funny how so many brilliant Mathematicians became crazy, literally. Altho I didn't finish my Pure Maths undergrad as my life took a different direction, I still love maths, yet I also am very spiritual, so the Incompleteness Theorem makes total sense to me. My thinking and sense of humour are also very different from the norm, and just to add, according to the Myers-Briggs personality test, I come under the smallest group of personalities out there, not that there is an out-there other than what is projected from within me....just to play with some non-conventional maths and physics or Carl Jung's work.
what is your personality type?
@@anonymousinfinido2540 From memory it is INFP
@@erniesulovic4734 myself infj 😅😂
@@anonymousinfinido2540 Cool 🙂
I do not want to take your joy, but this tests, usually, aren't precise enough. If you do the same test two times, it could be to persons. Be carefull, you can be a Pisces person (/irony on).
Arigato!
Thanks for the context and the devotion!
"You're a mathematician Harry"
"I'm a what?"
This was excellent
Thank you
It's not pronounced like 'girdle'. There is no implied 'r' sound using an umlaut.
Hey baby girl
Wonderful video. Thanks for posting. A nice twist in the end as usual. 😅
Who else has a feeling of connectedness between Kurt Gödel's Incompleteness Theorem, Turing's Halting Problem and Heisenberg Uncertainty Principle.
It's like the universe's way of saying you can't get everything.
Heisenburg Uncertainty Principle is the most surprising to me. It's not just a statement on hypothetical computers and abstract logic. It's real world stuff, and it's not a failure of our measuring devices. There's a fundamental sense of undecidability to everything that makes up our universe; it's not deterministic.
An interesting aside is when Euclid's geometric axioms were shown to be just one of several geometries that were logically consistent. About 50 years earlier I think. Poincare' said it didn't matter which was true pick one that's convenient. Now we have 2, then 3 truths (geometries) that were logically consistent. This must have shook science profoundly.
'The beginning of knowledge is an act of faith.'
-
St. Augustine of Hippo
thank you for providing
Ms Cindy you a wonderful voice for your presentations. Its soothing and it keeps our attention. Nice work....
Sometimes people simply know they will die, and they stop eating because they can feel it. They feel their body giving out, and they know prolonging it is somewhat futile and could lead to exponentially more severe suffering right at the end.
He didn't "stop eating because he knew he would die", he starved himself down to 30 kgs over the course of a month because he was paranoid of being poisoned. His suffering and death were not caused by reasonable decisions, but a mental health crisis.
@@julioaurelio My theory is that Goedel wasn't paranoid, but he knew people might think he was. He didn't care, because he was depressed. He was depressed because he couldn't get Einstein to stop trying to find (and prove) the Theory of Everything (TOE), that he was certain was futile, according to Goedel's Theorems. Einstein was working on it every day, until he died. Depression kills your appetite first, then it kills you.
It did occur to me that an additional hurdle was growing up in a well to do family. Back then they would have had servants. I doubt it ever occurred to him to shop for and cook his own meals. Mind you paranoia could even make him believe the ingredients had been poisoned in the shop. Its a terrible condition. unresponsive to logic.
@@helenamcginty4920 I wrote you a long comment about the interesting connection of this topic with the topic of split consciousness in people who have undergone corpus callosotomy. And again, as in many cases, the comment was deleted by an automated system. An amazing situation - as if we returned to the times when this outstanding scientist lived, except that the iron cage is now with amenities...
This was a very interesting video for the still inexplicable disconnect between objective achievement and personal perception
I still didnt understand what godel did;
If I'm not mistaken, there are 2 main theorems by Godel. One says that you can't prove all statements that are true in a system where these statements were built. You may be able to outside of that system. The second says that it's impossible to build a complete axiomatic system with a finite number of axioms. There will always be statements within that system that will require another axiom (that is, it can't be proved from the finite set of axioms). It also says that it's impossible to build such a system that is free of paradoxes. Sorry, but I can't remember all the details.
I was always amused that Godel's theorem that there will always be theorems in (carefully defined, but highly plausible) logical systems which are true, but for which no proof can exist in that system itself was proved by Godel.
I was guessing this was going to be Kurt Godel! Guess I was correct
He proved we'll never create self-aware technology, because we can't find the line that exists between intelligence and consciousness, the line where knowledge becomes understanding. We can't encapsulate metaphysics within itself.
But, unfortunately, the illuision of self-aware tech will be commercially sufficient. Pretending to have courage is indistinguishable from having courage.
@@rideon6140 Until 'artificial intelligence' figures out how to keep oil flowing from the bottom of the Gulf, to the refineries, and then to the generators and cooling pumps in the hydroelectric and nuclear power stations that run the world, all such talk is cartoonish at best. It has ALWAYS been mental masturbation worrying about this shit.
He definitely would have believed in the simulation theory.
Such an amazing story, thank you.
Most men at the pinnacle of brilliance are really nutty... Paul Erdos, Nikola Tesla, Godel...... etc. etc.
Me.
Yes, it helps ... but not necessarily. For example, as a result of his walks with Einstein, Goedel found another exact solution of Einstein's GR field equation: the so-called Goedel metric. So he had interests outside logic.
His level of "nutiness" was simply too high....unlike Nash, another nut. Any human being would find his self-starving illogical: I am afraid of dying of poisoning so I will not eat!!!!
PS Apparently the judge in the citizenship ceremony also realized that things were derailing. He went along with the speeding up to prevent Goedel to literaly perjure himself or refusing to take the oath.
John Nash
He must have had schizophrenia. Paranoia is one of the main symptoms of schizophrenia, and many people with schizophrenia are said to be geniuses.
Also Bobby Fischer, the American chess genius who took on the Soviets in their game and showed them how to play it correctly.
Thanks! Do video on alexnet 2012 victory kickstart of artificial intelligence revolution. These 3 guys are from your Toronto backyard!❤
Really appreciate your support. Thank you so much!!
Goudel, not Gurdle
Godel's story has some parallels with both John Nash & Srinivasa Ramanujan, brilliant minds with lives full of sadness & some tragedy. Even further back the famous Norwegian Neils Abel led a life of hardship, but still maintained a very high standard of original work, only to be recognized after his early death.
This is why (as an amateur Math Graduate) I highly RESPECT Mathematicians & what they do, as sometines the price of progress comes at a cost that these men (& women eg. Hypatia, Katherine Johnson) willingly pay....🤔
i for one want to know more about the dictator loophole he spotted.
Even though this is not Godel's loophole, some believe it could be one possible flaw in the Constitution that he may have noticed.
en.wikipedia.org/wiki/G%C3%B6del%27s_Loophole#:~:text=G%C3%B6del's%20Loophole%20is%20a%20supposed,legally%20turned%20into%20a%20dictatorship.
I wonder if it's related to the President's power to issue pardons. He can do whatever he wants, then immediately pardon himself! ;D
maybe trump can use it when he is re-elected. then we can be shielded from the looney tunes democrats for at least 20 years.
@@hoi-polloi1863 in some senses, that's pretty useful. If a president implemented a law ξ, such as "No persons under 35 years may enter any form of lоvе-bаsеd rеlаtiоnshiр." and then a law ψ saying "Law ξ may not be changed or nullified.", a different president would have to break law ψ to be able to remove law ξ, so the ability to pardon himself means that previous presidents can't set "trap card" laws to protect their work.
Interesting report about a great mystery in maths. Thank you.
"Court Girdle"??🤦♀
That's the closest pronunciation to the German name
@@Newsthink I appreciate the effort to replicate the correct German pronunciation, but without saying it in a German accent it just sounds odd.
I assume this is an AI voice, hence the lack of attempt at replicating the German accent when it came to names and places, but you tried to force it to approach proper pronunciation by spelling names phonetically?
@@WahrheitMachtFrei. I’m literally a real human being and I appear on camera. This is not AI 😂
@@Newsthink Oh, that's much worse🤷♂Pronouncing foreign words correctly (even names) is merely a search on TH-cam away. If you're invested enough to make content like this, you should have the attention to detail to do cursory searches for correct pronunciation at least.
@@user-uz4vr7to8c So you can recognise that the pronunciation is "lacking", but I should apologise for rationalising *why* it should be so mangled?
Love hearing your voice
Why does English pronunciation of his name put an ‘r’ there ?
The 'ö' in German has a sound that is similar to an 'r'. You can hear the same sound in 'Göring."
*American.
Why does *_her_* English pronunciation add an "r" where there isn't one ? That is the correct question.
The example proposition given “This statement is false” can be neither evaluated as true or false as we aren't told what the statement is.
That's what I think... It's a mess, isn't it?
Yes you are. Those four words, in a vacuum, are the statement. We can label it, if it helps. Say γ="This statement is false" is a statement, a collection of words that forms an idea, with the axioms being the rules of the English language and the basic ideals of logical reasoning. γ cannot be decided to be true, and γ cannot be decided to be false.
@@xinpingdonohoe3978 We end up with the Quine-Duhem thesis, isn't it? The meaning of the sentence can change when you change your web of beliefs:)
The Movie I.Q. with Tim Robbins, Meg Ryan, and Walter Matthau showed a friendly relationship between Albert Einstein and Kurt Gödel.
Genius. There's a book by Jim Holt " When Einstein walked with Godel".
My fav qualitative interpretation of Gödel’s theorem: every objetive fact is, at its heart, subjective
Proof that Mathematics can also be poetry
objective facts exist? we observe phenomena and our perceptions are subjective so, ...
objectivism is based on an omnipotent God sitting outside the universe watching it all... it's a bit of a medieval paradigm. bronze age paradigm, really. 🫤
Hmm... are you sure about that interpretation? After all Gödel never challenged the axioms (ie facts) that make up a formal system.
That's not what the Incompleteness Theorem says.
The first incompleteness theorem states that for every logical system there are statements that cannot be proved false or true, that’s why no system can be “complete”. The second incompleteness theorem says that a formal system cannot be used to prove its own consistency even if it is assumed to be consistent. These two theorems mean that every formal system has to base on axioms that have to be chosen by social agreement, terminating any possibility of absolute objectiveness.
Gurdle?
As in the case of John Nash, Kurt Goedel's mental state at the end of his life shows that a balance between the head and the heart is essential to a healthy like and a will to live. An overdeveloped intellect can lead to the type of paranoia from which they both suffered. Conversely, too much heart uncontrolled and tempered by the intellect as is the case with poets like John Keats or Syvia Plath, for example, can lead to serious depression and even suicide.
1906, "The safe and stable Austro-Hungarian Empire"
Give it a few years, YIKES
Safe and stable world in 21 century. Everything went awry very fast. And even if reasons are obvious, nothing can be done, especially because those who try to say it out loud are immediately silenced and branded as agents of the enemy.
If I'm not mistaken, there are 2 main theorems by Godel. One says that you can't prove all statements that are true in a system where these statements were built. You may be able to outside of that system. The second says that it's impossible to build a complete axiomatic system with a finite number of axioms. There will always be statements within that system that will require another axiom (that is, it can't be proved from the finite set of axioms). It also says that it's impossible to build such a system that is free of paradoxes. Sorry, but I can't remember all the details.
Pi is exactly 3
But only for small values of pi...
I hate to break this to ya, but there have recently been some breakthroughs in the value of pi. I can't give away a lot of details, but circles are about to get a hell of a lot rounder.
3 is between e and pi but where should i and -1 be, if e^i\pi = -1 ?
@@hoi-polloi1863 Wow, that's very funny. Thanks for the good laugh.
@@solconcordia4315 I thought you were mistaken. I thought you thought that e^(i/π)=-1. But you were using TeX notation. Why? This is TH-cam comments, not Overleaf.
His final work proved that “you can be book smart and still fail in life” damn!
Oh wow, he predicted the rise of the Trump cult?
This is actually really sad; but not uncommon. Many "brilliant" minds are plagued with "abnormal" thinking. It's probably because that person's mind doesn't think about things the way most people do; and that extends even into every aspect of their personal life.
Mathematical systems are not fixed objects. They can be expanded, redefined, refined, etc. to solve previously unsolved problems which are interesting and important. 😊
Aristotle grasped this concept in the metaphysics, when saying that there are truths that cannot be demonstrated, that is, conversely, that each demonstration is based on a bigger proposition that cannot be proved.
Do you have any further details on Gödel’s detection of a logical loophole in the Constitution making dictatorship possible? This would make a very interesting post given our current political environment.
I'm not sure, since I'm not American, but a quick glance at the 12th amendment is interesting. If I'm understanding what they're saying, if everyone from all but one of the states dies, except a single individual from a different state, then that different state person running for president or vice-president would necessarily win.
Or perhaps it's that the 25th amendment needs only a majority of presidential officials, along with the vice president, to write a letter to the senate and the speaker of the house of representatives, to be able to immediately enact a legal coup to make the vice president into the new president with immediate effect.