Absolutely phenomenal insights from Goldstein. I am so pleased that I watched this to the end. It is also spine-chilling to think that she also met Godel himself.
I have to think long and hard to come up with a book that has touched me more at just the time when I needed it most. Rebecca Goldstein is a true Renaissance woman.
Her intellectual courage is worth a ponder. She goes off the beaten path. She engages in novel writing. Her roots are mathematics. Her knowledge of logic is deep and genuine. I admire her even when I disagree or wonder what the point of a digression is. She is a force of nature. Just ask Steve Pinker 👍
"Just think, the definitions and deductive methods which everyone learns, teaches, and uses in mathematics, the paragon of truth and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?'' --David Hilbert
I think very highly of Dr. Rebecca Goldstein. I do wish she would speak without so many "um" "uh" as she speaks. It's almost like she is really shy, nervous, self-conscious, or something while she is a brilliant person. She should accept her self, and be speak with conviction. I know, easier said than done. About 35 mins in, she settles into the presentation and speaks in a clear and confidence manner. KG is legit af! Slays the scientism that permeates our current zeitgeist.
"Men... knockin' at balls" 1:08... Rebecca you missed your calling as an ESPN analyst. I'm with you though - Godel is so fascinating, and to me, his proofs and ideas are just as amazing as Einstein's. Godel,Turing, Russell, Einstein -- these were guys that were essentially 100 years ahead of their time. And people wonder why they were maladjusted. PS_ Loved your Godel book by the way
The Institute for Advanced Study was founded in 1930 by philanthropists Louis Bamberger and Caroline Bamberger Fuld, with the vision of educator Abraham Flexner as its founding director.
If the universe is a complete system then we can't ever prove everything. Mmmmm. Many thanks for a wonderful, insightful and most enjoyable lecture. 👍🏽😘
I have a hard time understanding how we can’t ever prove ‘everything’ if everything really means everything at all. Can we prove to have a conscious thought about not being able to prove everything?
I want this talk to last 8 hours. .25 speed for me. Rebecka is an amazing writer. My friend gave me _Plato at the Googlepex_ last year and I've been a huge fan since.
Good talk. I think proving that R is bigger than N is easy to explain. You just assume there is a pairing, which is just a function f(n) and then consider the real number a1.a2a3a4… where an differs from the nth digit of f(n). It’s a real number that is left out.
I used to take long walks with those guys in the thumbnail. I can't remember their names...but I think they both liked cursing a lot. Not me though. I was purely logical. Thanks for listening, friends
This is the most succinct point: Kurt Gödel's first incompleteness theorem violates the rules of correct reasoning and mainly contains a false dilemma-based reasoning error, but it also contains the circular, raisin-swelling, non-real Scotsman and the expert disguise reasoning error. Furthermore, the lack of novelty can also be criticized, since oxymorons and paradoxes have been discovered and known for at least 2,500 years. It can be rejected in science for several reasons, even 5-6. So, in reality, this is just an example of a paradox that is slightly different from the others. Thus, it is an important part of the evolution of human thinking and logic, but it can be outgrown and surpassed. It may remain as science history.
Logicians cant escape self reference. Indeed, Theory of Computing is impossible without recursion. So no, you cant just wall off Logic that wont conform to experience.
8:35 Hey! Wait a minute! Compared to anyone with a year's worth of college level math education, **I** am a layman, and yet I can STILL passably well explain the Incompleteness Theorem, even to another layman! OK, so maybe I'm a little half-step higher than the AVERAGE layman, but I still know what I am, and what it is and what it says. :-) Oh, OK, she straightened out that little bit right after I stopped the video to write this.
This statement seems obvious: "Every positive even integer can be written as the sum of two primes." It seems true, indeed must be true. Incompleteness says that there may be statements in math that cant be solved as true or false.
It's like an existence theorem in topology. It does not give us a means of identifying any such true but unprovable statement. But we know it exists. Or consider the set of all transcendental numbers. There are uncountably many, but we have a very hard time constructing one, or proving that a given one is indeed transcendental. The algebraic numbers are countable, and it's easy to give you an infinite subset of them, at the drop of a hat: the rational numbers are one; the square roots of the primes another (which has no numbers in common with the rationals), etc. This analogy isn't as good, but it gives you an idea.
@@l.w.paradis2108 Yes that makes perfect sense. Funny I had just spent an hour watching a video about sets of numbers, especially transcendentals today, so that really laid the groundwork for your explanation. It seems to me that this has potential implications for theology, but no one seems to want to bite. I guess people are uncomfortable with the idea. Except that interestingly Gödel actually had strong inclinations towards "proving" God. I don't know much about what he was trying to do or say, but I would like to know. I find it fascinating.
Gödel's theorems are often described as proving the existence of a "true but unprovable" theorem. As you ask, this statement doesn't make much sense. There is quite a subtlety here. Say you have any axiomatic system from which the arithmetic of the set of natural numbers can be fully described. Say, for example, the Peano axioms. (The goal of the Peano axioms was to pin down the set of natural numbers. But there are other axiomatizations capable of doing the same thing. For example, the Zermelo-Fraenkel axioms of set theory can fully describe the arithmetic of the natural numbers.) What Gödel showed is that the axioms you have prove that the Gödel sentence is equivalent to a statement which implicitly says that the Gödel sentence is unprovable. If we additionally assume that our axioms are consistent (so not proving a contradiction), this added consistency assumption is enough to prove the Gödel sentence (and, simultaneously, prove that the Gödel sentence is unprovable). So it's _true_ in the sense that it is provable from the axioms *_and the additional assumption that the axioms are consistent._* So the Gödel sentence is "true" (about the standard natural numbers) and unprovable (from your specific set of axioms) _if you assume that your axioms are consistent._ Most mathematicians believe that the Peano axioms or ZF axioms or what have you are consistent, so this seems like a reasonable assumption. (Also, this naturally leads to Gödel's second incompleteness theorem as a quick corollary. If the axioms could prove themselves consistent, then you wouldn't need a separate assumption of consistency, meaning the axioms themselves would be capable of proving the Gödel sentence and also the sentence which says that the Gödel sentence is not provable from the axioms. So if the axioms could prove themselves consistent, they would prove a contradiction, and hence be inconsistent. Therefore, no [reasonable] consistent axiomatization of mathematics can ever prove itself consistent.)
@@MuffinsAPlenty Believe it or not, that makes perfect sense. Now how does that extend beyond mathematics to philosophy? I believe Gödel said something along the lines of any system of logic cannot contain what is necessary to prove itself. It must be proven from another system beyond itself. Do you know what I'm talking about? Can you explain that please?
@@MatthewSchellenberg When people talk about "proving itself", I suspect they're talking about Gödel's Second Incompleteness Theorem, that reasonable axiomatizations of mathematics cannot prove themselves to be consistent. This stems from Hilbert's formalism. David Hilbert was a prominent mathematician in the late 19th and early 20th centuries. Essentially, Hilbert's view of mathematical truth was one where you could create a consistent formal axiomatization of mathematics. So for a system to "prove itself" would essentially be proving itself consistent. As far as Gödel's theorems applying to things other than mathematics, it's not clear to me how much further it can be taken. Technically, the formal logical system has to satisfy a couple of conditions for Gödel's argument to apply. 1. The axiomatization has to be _effectively generated._ This is a superbly reasonable condition, but it is technically a condition. An axiomatization is effectively generated if there is an algorithm which can determine whether or not any given sentence in the formal language is or is not an axiom. [There are actually counterexamples to the conclusions of Gödel's theorems if you take non-effectively generated axiomatizations. For example, you could take your axiom set to be the collection of all true statements about the natural numbers within the language of arithmetic. This axiomatization would be both complete and consistent. However, we wouldn't have a method for determining whether any given statement was an axiom. So it is, practically speaking, useless to have a non-effectively generated axiomatization - almost not even worth mentioning. But I think the silly counterexample shows that Gödel's theorems don't just "apply to everything" as some people like to claim.] 2. The axiomatization has to be capable of representing a sufficient amount of natural number arithmetic. You need to be able to carry out computations involving addition and multiplication of any two natural numbers. And your axiomatization has to be capable of _representing_ all of this computation. It is condition 2, the "sufficient amount of arithmetic" condition which is the big roadblock to applying Gödel's theorems outside of mathematics. There are multiple examples of formal axiomatizations which are effectively generated, complete, and consistent. They simply cannot represent sufficiently enough arithmetic. One example is Presburger arithmetic, which is an axiomatization of the natural numbers capable of representing natural number addition, but incapable of representing natural number multiplication. (You can still, indirectly, do multiplication as repeated addition, however you cannot encode multiplication as its own operation within Presburger arithmetic.) It turns out that Presburger arithmetic is effectively generated, complete, and consistent. There are also things called self-verifying theories in mathematical logic. Dan Willard has done some work on them. He has essentially created axiomatizations of the natural numbers which are too weak to fully encode addition and multiplication, but still describe a lot of addition and multiplication, and are effectively generated, consistent, and _capable of proving their own consistency._ All that being said, there are some limitations to where Gödel's incompleteness theorems apply, and I would be wary of people saying they have strong implications outside of mathematics. If you want to apply Gödel's incompleteness theorems outside of mathematics, you should check that you have an effectively generated axiomatization within first-order logic which is capable of representing natural number addition and multiplication. And as Presburger arithmetic shows, being able to _do_ multiplication (in a tedious way) is insufficient. There's also the question of infinities popping up. There are infinitely many natural numbers, and Gödel's proofs rely on this fact. So if you have an axiomatization for the "universe" (whatever that means) and it only allows for a finite amount of space, then you cannot represent the full natural number arithmetic within such a system. One place where it does have a sort of philosophical implication is that it counteracted a strong form of mathematical formalism within mathematical logicism. Mathematical logicism is a philosophical belief that mathematics can be reduced to logic. And formalism is a particular viewpoint within that, where mathematics can be reduced to formal systems. In other words, you _could_ (not that you _would_ but you could) do mathematics purely with symbol pushing where the symbols don't have any meaning. Because that's what a formal system is. It involves a bunch of meaningless symbols, and a couple logical symbols with logical meaning. And you can prove statements within your formal language from a set of axioms without even giving those symbols meaning, just by following strict formal rules of deduction. This may seem silly to you at first because no one actually does mathematics by pushing around meaningless symbols (for the most part). However, the belief that all of mathematics could be reduced to a complete, consistent, effectively axiomatized formal system would bring a strong sense of objectivity and reliability to mathematics. Because once you figured out how to prove something, you could formalize that proof (by converting it to meaningless symbols), and then have something like a computer check your formal proof to guarantee that you really did prove what you set out to prove. Gödel's theorems essentially proved, from a philosophical view, that mathematics cannot be fully reduced to formal logic in this way. After all, if you want to axiomatize mathematics within first-order formal logic, you surely will want to have an effective axiomatization and you will want it to be able to represent natural number arithmetic. And Gödel's theorems then, essentially, show that there will always be statements that this system cannot prove but, nevertheless, are _true_ when meaning is given to the symbols. So, no matter what consistent, effectively generated axiomatization of "mathematics" you fix, there will always be facts about "mathematics" which you can prove informally but which you will never be able to formalize into a checkable proof within that system. This doesn't mean that using formal systems is worthless. It led us to widespread acceptance of axiomatic frameworks (but perhaps not as formal) in mathematics, and tons of subdisciplines (e.g., abstract algebra and category theory) probably wouldn't exist in their current form without Hilbert's formalism having been a widespread philosophy for some time.
As godel, or exactly like godel, lol lol lol lolooo, what are you like jealous of godel, have some rage against him , was a mathematical paper of yours, close to your heart, rejected from being published..., Is there someone WHOSE ATTENTION AND APPROVAL MEANS THE WORLD TO YOU , UNCONSCIOUSLY, BUT YOU DON'T WANNA ACCEPT IT CONSCIOUSLY, AND YOU THINK GODEL HAS THAT ACCEPTANCE, SO YOU GIVE YOURSELF, BUT HONESTLY TO THAT PERSON OR PEOPLE, A GOOD ENOUGH - PRETENTIOUSLY INTELLECTUAL REASON ...., haha gotcha I mean that's a weird and kinda dumb comment. And don't think about why i wrote this comment, i enjoy judging people, who are unconsciously craving attention and approval.... like me, cuz i am a normal human being
"It's always NOW", QM-TIME Completeness Actuality of pure-math relative-timing reciprocation-recirculation ratio-rates of Bose-Einsteinian Condensation Quantum-fields Mechanism, aka Superposition Totality of instantaneous log-antilog time-timing here-now-forever,all-ways all-at-once sync-duration holography, Eternity-now. It is a scenario that every child must recognize and remember, teachers lead out a personal resonant complexity into a comprehensible Fluxion-Integral Temporal Calculus Condensation of intellectual alignment with the functional Universal standing wave-packaging phenomenon of e-Pi-i Superposition-point Entanglement in Time Duration Timing, aka Resonance. The task of demonstrating the possibility of Zero Kelvin positioned in the Universal flash-fractal exponentiation-ness superposition here-now-forever of nothing floating in No-thing Relativity is an aspect-version of Eternity-now pure-math musical format for mathematical reciprocation-recirculation, instantaneous Interval.., Disproof Methodology is essential knowledge of all Mathematics and by default => Disproof Methodology.. Actual Mathematicians. Disproof inclusion-exclusion Methodology Philosophy in the hands of Theoretical Physics demonstrates Einsteinian Relativity in Susskind's reasoning format that positioning Singularity-point Condensation Conception, Black Hole Singularity in/of Black-body Holographic Principle, ..is Lensing orientation-observation resonance bonding, a concept in/of QM-TIME Conception Totality of instantaneous pure-math absolute zero-infinity sync-duration vanishing-into-no-thing potential possibilities motion, of sense-in-common at trancendental center of relative-timing cause-effect observations.
Questioner: "Everything is incomplete, so why does it matter?" My take: This kind of things mattered to great western intellectuals because they were fighting collectively against the established Christianity in late 19th century. God is complete and consistent, so people believe in God. Those intellectuals aimed to replace God with science or the broader rationality. To compete with God, they rushed to show science or rationality or math is complete and consistent. Chinese, Japanese, Indians do not have the Christian God concept, so they never bother themselves, collectively as peoples, with this kind of questions, even though they are very smart, with very high IQs. Nowadays, western intellectuals have largely succeeded in driving God out of their circles, so they start to feel "Why ask such questions like completeness or consistency?" God is the final driving force behind the western thinking. By throwing away God, western intellectuals are losing the fundamental drive to ask deep questions. As in the old saying, "Be careful, you may get what you ask for."
I think I got it but... I still have a problem on getting the legitimacy of a statement that refers to its own true/false value (that is autoreferential in that sense). Shouldn't statements that refer to true/false values belong to a meta-language. Hadn't Russell created set theory to avoid such self-refering statements mess with theories?
It's the essence of the hard problem (at least that's what always comes to my mind). Materialism vs idealism: either mind gives rise to matter, matter gives rise to mind, or something else gives rise to both mind and matter. You can't have mind giving rise to mind or matter giving rise to matter. Any system that is used to describe/model other systems can not effectively describe/model itself. You could never build a computer capable of perfectly simulating itself; it's from a completely different substrate. That's why they are all "incomplete". That's all man. The concept is already kinda programmed into the human mind, we just lose a lot of it as we get older and incorporate more and more symbols and ideas. Most kids have it on lockdown, even if they can't express it - they show it. Godel's huge thing was that he was actually able to spell it out with maths.
@@MoiLiberty Thanks for your compressed form of my sentence. Of course the statement makes no sense , but so does it counterpart : "This statement is true". It doesn't add any information .. does it ?
I think no, godel's statement was about its provability and not its truthness. It was a clever way of using self referentiality avoiding paradoxes. That's why we say there's a difference between truth and proof
Well, what he showed was that, by simply using all symbols and rules of basic arithmetics, you can build a sentence which, translated to English, roughly says: "I'm not provable". And of course you can ask: Is that true? Here comes the problem. Suppose that A) it is true. So now your system contains a sentence which you know to be true but you cannot prove to be true ( because that's exactly what the sentence says about itself ) B) it's false. So, for what the sentence says, it IS provable and so you have now a sentence which can be proven by your system despite being false. So the conclusion is: either your system is incomplete because it cannot prove all truths that can be legitimately expressed in its language OR it is inconsistent because it can prove a theorem which is known to be false. In a sense, it is an extension of the famous lyar's paradox but it's more about demonstrability than it is about truth.
21:26 That's an obvious error. Assuming that philosophy handles meta-questions is a common but incorrect mistake. In reality, philosophy fails to answer these questions and they revert back to science. This includes questions about philosophy itself which cannot be answered philosophically. 51:20 The mention of the unreliability of intuitions is good. These are almost always incorrect. 55:00 Russell's paradox. This isn't a real paradox. 1:03:00 After listening to Goldstein fawning over this proof, it is quite disappointing to see a non-proof presented. The liar's paradox is not a real paradox. 1:05:00 It's hard to believe that someone sees this as a problem. It only exists in an informal system like philosophy where structure is lax, allowing equivocation. 1:07:00 Except the system presented isn't formal. 1:24:00 This is partly true and partly false. It is true that brains are not computers and this is provable. However, it does not seem to be the case that this is absent a formal system. Rather the formal system is not limited to math and logic (both of which are computable). 1:35:36 I can't really criticize Goldstein for being uncomfortable with Penrose-Lucas. She has nothing in her background that would allow her to formally analyze it. We are not delusional computers.
@@godofgodseyesshe’s talking to a lay audience. You expect her to only write out the incredibly dense and complex predicate logic? What a joy all those audience members would have!
Absolutely phenomenal insights from Goldstein. I am so pleased that I watched this to the end. It is also spine-chilling to think that she also met Godel himself.
I have to think long and hard to come up with a book that has touched me more at just the time when I needed it most. Rebecca Goldstein is a true Renaissance woman.
Her intellectual courage is worth a ponder. She goes off the beaten path. She engages in novel writing. Her roots are mathematics. Her knowledge of logic is deep and genuine.
I admire her even when I disagree or wonder what the point of a digression is. She is a force of nature. Just ask Steve Pinker 👍
I’ve got to remember how much valuable content like this is on TH-cam. Thanks so much!
Marvellous and inspiring. Do watch it all. What a fabulous role model is part of the presence for us. おめでとう御座います！Gratulieren! Félicitations!
"Just think, the definitions and deductive methods which everyone learns, teaches, and uses in mathematics, the paragon of truth and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?''
--David Hilbert
Where is that Hilbert quote from?
The lecture starts @9:40
Thanks dude save my time
Best comment.
I think very highly of Dr. Rebecca Goldstein. I do wish she would speak without so many "um" "uh" as she speaks.
It's almost like she is really shy, nervous, self-conscious, or something while she is a brilliant person. She should accept her self, and be speak with conviction. I know, easier said than done.
About 35 mins in, she settles into the presentation and speaks in a clear and confidence manner.
KG is legit af! Slays the scientism that permeates our current zeitgeist.
He was always at odd with the zeitgeist , that was so bad ass
She is very shy. And a genius. We throw that word around too much. Well, this time . . .
"Men... knockin' at balls" 1:08... Rebecca you missed your calling as an ESPN analyst.
I'm with you though - Godel is so fascinating, and to me, his proofs and ideas are just as amazing as Einstein's.
Godel,Turing, Russell, Einstein -- these were guys that were essentially 100 years ahead of their time. And people wonder why they were maladjusted.
PS_ Loved your Godel book by the way
The Institute for Advanced Study was founded in 1930 by philanthropists Louis Bamberger and Caroline Bamberger Fuld, with the vision of educator Abraham Flexner as its founding director.
Mir's conjuncture (continued)
Example: One is as general as it can get, zero point anything fallowed by anything is as specific as it can get. Deriving none existences off one is much easier done, deriving none existences off zero point anything fallowed by anything, is much harder done especially as a continuation, making such continuations eventually fail in showing a proof of existence.
The difficulty becomes difficulty as a consequence of the movement of that which is existing in a hierarchical descending and ascending way, together and often with and within a, any, and all possible combinations of corresponding, representing and derived none existences, making it highly and in some cases extremely difficult to establish which is which, being one is a proof and the other is not, yet could enable or disable one.
Pythagorean triangular numbers are probably the best and longest standing example of this, as a proof of that which exists and that which does not. Such methods I believe to have with and within it as a sequence of base existing consecutive numbers a, any, and all possible combinations of mathematical, and as consequence physics proof.
It is achieved through taking base groups of consecutive numbers well established as observance with and within reality, and derive another base groups of consecutive numbers as none existing reality, which none the less are well established as observance with and within the starting base within reality groups and numbers.
The complexity of such method involves in having complet visible and none visible functions interwoven with and within the whole process giving a, any, and all possible combinations of results wich are reality based and none reality based, which then in turn can, and will consequently trigger more groups and more and more.
The most impressive part, one which is the most fascinating, the most hidden and most powerful, involves in making thus having always a group made of many groups which are simultaneously real and none real as one whole real intact group, represented as a un interrupted linear multi directional sequence, and as a precise geometric shape of a un interrupted dimensional reality simultaneously, with the complete uncertainty and certainty of one being reality based while the other being none reality based.
In order to imprint the current thread I am fallowing, I have but to result in conjuring a proverb as one should in order to imprint anything, being that proverbs are the best way and method for memorising while simultaneously hiding in full plain sight that which is being memorised.
The proverb is this.. If and when one is shaving ones genitales, one has to make sure not to cut ones self. Because in having done so, and paid no attention to the process, one would have cut themselves.
There is only one way to shuffle cards in order to get complet randomness allowed in the first place off, by, for, from and to the actual number of cards being shuffled, which in the case of playing cards is the number fifty two.
This involves in proceeding the shuffling as a doing against a very hard, with the highest probable flatness possible surface, meaning the cards are shuffled against that which is way more solid and harder than their physical structural construct.
A even distance among each cards has to be, including off the surface, together with the downward force applied, and the gathering has to fallow the downward order of counting also, meaning from the top to the bottom.
The mathematics is four to a ten, making sixteen, leaving four possible cuts, archiving in turn the full achieved possible randomness allowed off, by, for, from and to the actual number of cards being shuffled.
Any and all other possible shuffles are not even close to a, any and possible combinations of randomness allowed by, off, for, from and to any specific number of cards, especially if and when used for divination, which means wether known or unknown, the set used is a very predetermined and calculated set up pattern, a true theatre.
Two is one, one is two, and two is obviously three. This is the simplest answer as proof and simultaneously the counter proof including being a standing paradox of both Mir's conjunctures, making it the top level of logic if and when invoked.
Before that it might be useful to recognise the most dangerous logical pattern in existence in regards to earth including all of its individual and combined systems. Above this logical pattern there is none more dangerous, so dangerous it is, yet evenly equaled as the top of a, any and all possible combinations logical patterns, that life itself is, has and will continue to try and escape it at all possible cost, simply because if and when not, life is no more, which is in itself the very function and the core foundation of it as a logical pattern, by now applied to almost everything by and off civilisation.
It, as a produced and existing logical pattern is a consequence of a paradoxical conscious and unconscious understanding of consciousness and unconsciousness as not paradoxical. More precisely a dogmatic belief in what is known as abstract, being alive, real and conscious. The logical pattern premise follows as such.....
(((( The only possible, and as a consequence of evolution most practical, way for and of life, which is highlighted in terms of human life, to continue existing, is to die, and having reached and achieved such a point, where life, all life is dead, this dead life would, should become conscious of such a dead existence and consequently continue to dead live forever.))))
There is nothing more dangerous than this most logical, at the highest possible level of pattern throughout the universe. It requires for all types of possible combinations of heirarchical level's and degrees to give up such levels and degrees, and reduce to a lesser and lesser dead level and degree of heirarchical existence, and from that possible of hardly existence, manifest the most formidable levels of consciousness instead.
Life will always run way from such a logical pattern, even if and when entrapped.
Prime numbers are always a triangle no matter, because all prime numbers are one, and in order to be one, can only do so by becoming two. All primes have a base which equals always to one with a selected direction equaling also to one, making the base always zero and the direction one, or the direction zero and the base one, which if and when combined are two.
Two is a prime, the first prime, because it gives two possible directions in order to choose one, becoming a one through the transformation of one direction as a base zero and one direction as a direction one. Nine is not a prime because it reverses the direction back to zero which becomes a prime on the next new base by transforming such a zero into one as a base, and giving it a new direction as a one, consequently making any and all prime's always have zero on the right side if and when integer sequenced, which is exactly what happens to two at the begging of a base, by becoming a zero one calculation.
Zero is a prime and a composite simultaneously, in being so a, any, and all possible combinations of practical uses in a measuring form cannot be derived from, on the literal other hand an abstract representation off and by such zero can be derived in the form of a one representation, the problem still remains as a consequence off, and by this new zero as a one, still being a zero, an abstract zero, making such form have a very limited use in measuring terms, giving rise to the necessity of a two representation, of such an abstract one, derived and representing a direct reality corrolation to zero.
Once at this point, a, any, and all possible combinations of uses in measuring terms begin to manifest as a consequence of the simple yet most complicated reality of the none existence of a, any, and all possible combinations of continuous uniterrpted flat surfaces in regards to such sequences, whereas zero and it's abstract one representation give, let alone show such none existing existence.
As a consequence a, any, and all possible combinations of prime's are found by base zeroing everything but a remaining one. What is not utilised, is the infinity of other prime's which spring up as a consequence of a, any, and all possible combinations of a sequence producing prime's, as a, any, and all possible combinations of new sequence to fallow off and by such prime.
Fallowing this prime principle, the fallowing two paragraphs and one short sentence can be understood much better.
[{\|/ What is faster than the "speed" of "light"? I don't know! I "also" don't know! I bet you are going to tell "us"! Of course I am! You are my mind, and you are my brain. And you are.......!? "I" "am" "language". !!!......
The speed of light is faster than the speed of light. "If" "where" "when" "how" the speed of light is, gets with and within there is "it", the speed of light is already there.
If not....! "You" "mind" "and" "you" brain" "drop" "the" "case".\|/}]
In reality not just the speed of light is already there, but it is simultaneously into the future through a, any, and all possible combinations of prime's, moreover also into the past through a, any, and all possible combinations of base zeroing, always before a, any, and all possible combinations of prime's.
Despite thought otherwise or not known...Prime's make chemistry, which in turn means prime's make biology, this means lost prime's is lost chemistry, lost chemistry is lost biology, lost biology is death, death is the most dangerous logical pattern in existence, and the difference between death consciousness, and life consciousness is ...... Existing
© Mir
March/2024
If the universe is a complete system then we can't ever prove everything. Mmmmm.
Many thanks for a wonderful, insightful and most enjoyable lecture. 👍🏽😘
I have a hard time understanding how we can’t ever prove ‘everything’ if everything really means everything at all. Can we prove to have a conscious thought about not being able to prove everything?
I just got her book on kurt and it was heavy at times but I'm pushing through
If you want to have a more enjoyable experience while watching this video, I recommend turning up the speed to 1.25.
I want this talk to last 8 hours.
.25 speed for me. Rebecka is an amazing writer. My friend gave me _Plato at the Googlepex_ last year and I've been a huge fan since.
😂😂😂
When I first read about Godel and his theorems, I immediately thought of Einstein. They both introduced revolutionary ideas.
Good talk. I think proving that R is bigger than N is easy to explain. You just assume there is a pairing, which is just a function f(n) and then consider the real number a1.a2a3a4… where an differs from the nth digit of f(n). It’s a real number that is left out.
The inconsistency of the Incompleteness Theorem consists in not recognizing it as Inconsistency Theorem.
How can we discover more maths?
I used to take long walks with those guys in the thumbnail. I can't remember their names...but I think they both liked cursing a lot. Not me though. I was purely logical. Thanks for listening, friends
Excellent video!!
formal system, completeness, consistency
This is the most succinct point: Kurt Gödel's first incompleteness theorem violates the rules of correct reasoning and mainly contains a false dilemma-based reasoning error, but it also contains the circular, raisin-swelling, non-real Scotsman and the expert disguise reasoning error. Furthermore, the lack of novelty can also be criticized, since oxymorons and paradoxes have been discovered and known for at least 2,500 years. It can be rejected in science for several reasons, even 5-6. So, in reality, this is just an example of a paradox that is slightly different from the others. Thus, it is an important part of the evolution of human thinking and logic, but it can be outgrown and surpassed. It may remain as science history.
Why are you telling us that you don't understand mathematics? We don't care. :-)
Logicians cant escape self reference. Indeed, Theory of Computing is impossible without recursion. So no, you cant just wall off Logic that wont conform to experience.
@@timdion9527 One can trivially avoid Goedel... by only reasoning about finite sets (computing with finite memory). That's boring, of course.
Nice to know that Second Order Theorems are consistent after all. Now I can make my universal theorem solver and launch an IPO.
@@timdion9527 I'll buy one share at twelve cents. :-)
8:35
Hey! Wait a minute! Compared to anyone with a year's worth of college level math education, **I** am a layman, and yet I can STILL passably well explain the Incompleteness Theorem, even to another layman!
OK, so maybe I'm a little half-step higher than the AVERAGE layman, but I still know what I am, and what it is and what it says. :-)
Oh, OK, she straightened out that little bit right after I stopped the video to write this.
This statement seems obvious: "Every positive even integer can be written as the sum of two primes." It seems true, indeed must be true. Incompleteness says that there may be statements in math that cant be solved as true or false.
Fabulous❤
3:07 "My laser 'snot working" -- Dr. Rebecca Goldstein
My Millennium Falcon is at the garage.
Question: What sense does it make to speak of things which are true but unprovable? How are we to decide the truth of the matter if it is unprovable?
It's like an existence theorem in topology. It does not give us a means of identifying any such true but unprovable statement. But we know it exists.
Or consider the set of all transcendental numbers. There are uncountably many, but we have a very hard time constructing one, or proving that a given one is indeed transcendental. The algebraic numbers are countable, and it's easy to give you an infinite subset of them, at the drop of a hat: the rational numbers are one; the square roots of the primes another (which has no numbers in common with the rationals), etc. This analogy isn't as good, but it gives you an idea.
@@l.w.paradis2108 Yes that makes perfect sense. Funny I had just spent an hour watching a video about sets of numbers, especially transcendentals today, so that really laid the groundwork for your explanation.
It seems to me that this has potential implications for theology, but no one seems to want to bite. I guess people are uncomfortable with the idea. Except that interestingly Gödel actually had strong inclinations towards "proving" God. I don't know much about what he was trying to do or say, but I would like to know. I find it fascinating.
Gödel's theorems are often described as proving the existence of a "true but unprovable" theorem. As you ask, this statement doesn't make much sense. There is quite a subtlety here.
Say you have any axiomatic system from which the arithmetic of the set of natural numbers can be fully described. Say, for example, the Peano axioms. (The goal of the Peano axioms was to pin down the set of natural numbers. But there are other axiomatizations capable of doing the same thing. For example, the Zermelo-Fraenkel axioms of set theory can fully describe the arithmetic of the natural numbers.)
What Gödel showed is that the axioms you have prove that the Gödel sentence is equivalent to a statement which implicitly says that the Gödel sentence is unprovable. If we additionally assume that our axioms are consistent (so not proving a contradiction), this added consistency assumption is enough to prove the Gödel sentence (and, simultaneously, prove that the Gödel sentence is unprovable). So it's _true_ in the sense that it is provable from the axioms *_and the additional assumption that the axioms are consistent._*
So the Gödel sentence is "true" (about the standard natural numbers) and unprovable (from your specific set of axioms) _if you assume that your axioms are consistent._ Most mathematicians believe that the Peano axioms or ZF axioms or what have you are consistent, so this seems like a reasonable assumption.
(Also, this naturally leads to Gödel's second incompleteness theorem as a quick corollary. If the axioms could prove themselves consistent, then you wouldn't need a separate assumption of consistency, meaning the axioms themselves would be capable of proving the Gödel sentence and also the sentence which says that the Gödel sentence is not provable from the axioms. So if the axioms could prove themselves consistent, they would prove a contradiction, and hence be inconsistent. Therefore, no [reasonable] consistent axiomatization of mathematics can ever prove itself consistent.)
@@MuffinsAPlenty Believe it or not, that makes perfect sense. Now how does that extend beyond mathematics to philosophy? I believe Gödel said something along the lines of any system of logic cannot contain what is necessary to prove itself. It must be proven from another system beyond itself. Do you know what I'm talking about? Can you explain that please?
@@MatthewSchellenberg When people talk about "proving itself", I suspect they're talking about Gödel's Second Incompleteness Theorem, that reasonable axiomatizations of mathematics cannot prove themselves to be consistent. This stems from Hilbert's formalism. David Hilbert was a prominent mathematician in the late 19th and early 20th centuries. Essentially, Hilbert's view of mathematical truth was one where you could create a consistent formal axiomatization of mathematics. So for a system to "prove itself" would essentially be proving itself consistent.
As far as Gödel's theorems applying to things other than mathematics, it's not clear to me how much further it can be taken. Technically, the formal logical system has to satisfy a couple of conditions for Gödel's argument to apply.
1. The axiomatization has to be _effectively generated._ This is a superbly reasonable condition, but it is technically a condition. An axiomatization is effectively generated if there is an algorithm which can determine whether or not any given sentence in the formal language is or is not an axiom. [There are actually counterexamples to the conclusions of Gödel's theorems if you take non-effectively generated axiomatizations. For example, you could take your axiom set to be the collection of all true statements about the natural numbers within the language of arithmetic. This axiomatization would be both complete and consistent. However, we wouldn't have a method for determining whether any given statement was an axiom. So it is, practically speaking, useless to have a non-effectively generated axiomatization - almost not even worth mentioning. But I think the silly counterexample shows that Gödel's theorems don't just "apply to everything" as some people like to claim.]
2. The axiomatization has to be capable of representing a sufficient amount of natural number arithmetic. You need to be able to carry out computations involving addition and multiplication of any two natural numbers. And your axiomatization has to be capable of _representing_ all of this computation.
It is condition 2, the "sufficient amount of arithmetic" condition which is the big roadblock to applying Gödel's theorems outside of mathematics. There are multiple examples of formal axiomatizations which are effectively generated, complete, and consistent. They simply cannot represent sufficiently enough arithmetic. One example is Presburger arithmetic, which is an axiomatization of the natural numbers capable of representing natural number addition, but incapable of representing natural number multiplication. (You can still, indirectly, do multiplication as repeated addition, however you cannot encode multiplication as its own operation within Presburger arithmetic.) It turns out that Presburger arithmetic is effectively generated, complete, and consistent.
There are also things called self-verifying theories in mathematical logic. Dan Willard has done some work on them. He has essentially created axiomatizations of the natural numbers which are too weak to fully encode addition and multiplication, but still describe a lot of addition and multiplication, and are effectively generated, consistent, and _capable of proving their own consistency._
All that being said, there are some limitations to where Gödel's incompleteness theorems apply, and I would be wary of people saying they have strong implications outside of mathematics. If you want to apply Gödel's incompleteness theorems outside of mathematics, you should check that you have an effectively generated axiomatization within first-order logic which is capable of representing natural number addition and multiplication. And as Presburger arithmetic shows, being able to _do_ multiplication (in a tedious way) is insufficient. There's also the question of infinities popping up. There are infinitely many natural numbers, and Gödel's proofs rely on this fact. So if you have an axiomatization for the "universe" (whatever that means) and it only allows for a finite amount of space, then you cannot represent the full natural number arithmetic within such a system.
One place where it does have a sort of philosophical implication is that it counteracted a strong form of mathematical formalism within mathematical logicism. Mathematical logicism is a philosophical belief that mathematics can be reduced to logic. And formalism is a particular viewpoint within that, where mathematics can be reduced to formal systems. In other words, you _could_ (not that you _would_ but you could) do mathematics purely with symbol pushing where the symbols don't have any meaning. Because that's what a formal system is. It involves a bunch of meaningless symbols, and a couple logical symbols with logical meaning. And you can prove statements within your formal language from a set of axioms without even giving those symbols meaning, just by following strict formal rules of deduction. This may seem silly to you at first because no one actually does mathematics by pushing around meaningless symbols (for the most part). However, the belief that all of mathematics could be reduced to a complete, consistent, effectively axiomatized formal system would bring a strong sense of objectivity and reliability to mathematics. Because once you figured out how to prove something, you could formalize that proof (by converting it to meaningless symbols), and then have something like a computer check your formal proof to guarantee that you really did prove what you set out to prove.
Gödel's theorems essentially proved, from a philosophical view, that mathematics cannot be fully reduced to formal logic in this way. After all, if you want to axiomatize mathematics within first-order formal logic, you surely will want to have an effective axiomatization and you will want it to be able to represent natural number arithmetic. And Gödel's theorems then, essentially, show that there will always be statements that this system cannot prove but, nevertheless, are _true_ when meaning is given to the symbols. So, no matter what consistent, effectively generated axiomatization of "mathematics" you fix, there will always be facts about "mathematics" which you can prove informally but which you will never be able to formalize into a checkable proof within that system.
This doesn't mean that using formal systems is worthless. It led us to widespread acceptance of axiomatic frameworks (but perhaps not as formal) in mathematics, and tons of subdisciplines (e.g., abstract algebra and category theory) probably wouldn't exist in their current form without Hilbert's formalism having been a widespread philosophy for some time.
I will never do anything as cool as Godel lol
You just did. LOL
As godel, or exactly like godel, lol lol lol lolooo, what are you like jealous of godel, have some rage against him , was a mathematical paper of yours, close to your heart, rejected from being published...,
Is there someone WHOSE ATTENTION AND APPROVAL MEANS THE WORLD TO YOU , UNCONSCIOUSLY, BUT YOU DON'T WANNA ACCEPT IT CONSCIOUSLY, AND YOU THINK GODEL HAS THAT ACCEPTANCE, SO YOU GIVE YOURSELF, BUT HONESTLY TO THAT PERSON OR PEOPLE, A GOOD ENOUGH - PRETENTIOUSLY INTELLECTUAL REASON ...., haha gotcha
I mean that's a weird and kinda dumb comment.
And don't think about why i wrote this comment, i enjoy judging people, who are unconsciously craving attention and approval.... like me, cuz i am a normal human being
Would like to know which Encyclopedia of Philosophy she referring to?
Standford
"It's always NOW", QM-TIME Completeness Actuality of pure-math relative-timing reciprocation-recirculation ratio-rates of Bose-Einsteinian Condensation Quantum-fields Mechanism, aka Superposition Totality of instantaneous log-antilog time-timing here-now-forever,all-ways all-at-once sync-duration holography, Eternity-now.
It is a scenario that every child must recognize and remember, teachers lead out a personal resonant complexity into a comprehensible Fluxion-Integral Temporal Calculus Condensation of intellectual alignment with the functional Universal standing wave-packaging phenomenon of e-Pi-i Superposition-point Entanglement in Time Duration Timing, aka Resonance.
The task of demonstrating the possibility of Zero Kelvin positioned in the Universal flash-fractal exponentiation-ness superposition here-now-forever of nothing floating in No-thing Relativity is an aspect-version of Eternity-now pure-math musical format for mathematical reciprocation-recirculation, instantaneous Interval.., Disproof Methodology is essential knowledge of all Mathematics and by default => Disproof Methodology.. Actual Mathematicians.
Disproof inclusion-exclusion Methodology Philosophy in the hands of Theoretical Physics demonstrates Einsteinian Relativity in Susskind's reasoning format that positioning Singularity-point Condensation Conception, Black Hole Singularity in/of Black-body Holographic Principle, ..is Lensing orientation-observation resonance bonding, a concept in/of QM-TIME Conception Totality of instantaneous pure-math absolute zero-infinity sync-duration vanishing-into-no-thing potential possibilities motion, of sense-in-common at trancendental center of relative-timing cause-effect observations.
Wow, so helpful. Would meta-mathematical theoriticians be answering the same questions as empiricism ?
No. :-)
Questioner: "Everything is incomplete, so why does it matter?" My take: This kind of things mattered to great western intellectuals because they were fighting collectively against the established Christianity in late 19th century. God is complete and consistent, so people believe in God. Those intellectuals aimed to replace God with science or the broader rationality. To compete with God, they rushed to show science or rationality or math is complete and consistent. Chinese, Japanese, Indians do not have the Christian God concept, so they never bother themselves, collectively as peoples, with this kind of questions, even though they are very smart, with very high IQs. Nowadays, western intellectuals have largely succeeded in driving God out of their circles, so they start to feel "Why ask such questions like completeness or consistency?" God is the final driving force behind the western thinking. By throwing away God, western intellectuals are losing the fundamental drive to ask deep questions. As in the old saying, "Be careful, you may get what you ask for."
So your point is that you don't understand math hence god? Dude... that one is getting long in the tooth. :-)
Mir's conjuncture.
If and when taking any positive integers sequence such as a,b,c, And the consecutive part is excluded, there cannot be an equal, meaning there cannot be no mathematics.
You can only say blahhhhhh and that would be as good as a, any and all possible combinations of equals, compiling all of mathematics as equaling to blahhhhhh.
Now let us presuppose there can be such mathematics, and there is such mathematics, where everything is blahhh, we can even call it blahhhh mathematics. I believe there are people which believe there is such beliefs as blahhhh mathematics, and such a presupposition is not, and should no be strange at all.
If and when such be the case, if a, b, c is taken and by all being blahhh, then we only and always have, a (blah), b (blah), and c (blah)..! Even though each might appear and behave sequential, and positively integrating, nonetheless by all being blah the results never change, cannot change.
Making it is true that there cannot be satisfaction of the equation of................................. a (blah) + b (blah) = c (blah) for any integer higher than blah. This is always skipping the fact, any possible (blah) does satisfy the equation simply because it's equal, and consequential results to such equal, are all and always (blah), no matter.
Whereas Mir's conjuncture stands, the one being.. (There is no equal if and when the consecutive part is excluded.)
Another Mir's conjuncture, with a slight high difference from the other conjuncture, meaning in, as an opposite to that being true, and this being also true, is this....
(All prime numbers are not random, all prime numbers are consecutive, in being such, consecutive and not random, all prime numbers equal, and in doing so all prime numbers equal to triangle.
I believe (grouping) to be probably the easiest and best method to achieve and reach a, any and all possible combinations of mathematical proof's, in the fastest and most probably, the best understood way.
You can only group that which exists, if and when grouping that which does not, it has to be achieved in a highly direct regards to that which exists.
Example: One exist, (a) group would easily and can directly show that. One group and one group, can be grouped as two groups, and as a group of two's also, that should directly and very easy, show one plus one (equals) two.
Two halves or two of any equal or unequal measure do not exist, can do so only if and when grouped, making one as a none existing group itself and one as an existing group. Equal, multiplication, subtraction, minus etc do not exist, can do so only if and when grouped making a, any, and all possible combinations of corresponding results.
This is also the difficulty of the simplicity. X Y Z together with functions, halves, one, two or more equal or unequal measures, do not exist, can only do so with and within a highly direct regards to that which does.
The full correlation, knowledge and direct existing correspondence to that which exists, enables for fluid and beautiful mathematics which are a proof simultaneously.
© Mir
March/2024
I enjoyed watching her hands.
"Their status is independent of experience" this is ironically also dependent on how experienced you are with the principle itself.
creativiety of electronic maschien is a list served by orakel and that is what is so worrying with being a computer
I think I got it but... I still have a problem on getting the legitimacy of a statement that refers to its own true/false value (that is autoreferential in that sense). Shouldn't statements that refer to true/false values belong to a meta-language. Hadn't Russell created set theory to avoid such self-refering statements mess with theories?
There is such a language, it is called the Lambda Calculus. You can't model computation without recursion and self reference.
Bravo.
You lost me at G...
It's the essence of the hard problem (at least that's what always comes to my mind). Materialism vs idealism: either mind gives rise to matter, matter gives rise to mind, or something else gives rise to both mind and matter. You can't have mind giving rise to mind or matter giving rise to matter. Any system that is used to describe/model other systems can not effectively describe/model itself. You could never build a computer capable of perfectly simulating itself; it's from a completely different substrate. That's why they are all "incomplete". That's all man. The concept is already kinda programmed into the human mind, we just lose a lot of it as we get older and incorporate more and more symbols and ideas. Most kids have it on lockdown, even if they can't express it - they show it. Godel's huge thing was that he was actually able to spell it out with maths.
it is just that paradoxes cannot be mathematically solved
Just to give me peace of mind : Doesn't Godels Theorem simply boils down to the statement , written on a card , saying :
"EVERYTHING I SAY IS A LIE" ?
"This statement is false."
_______________________
So, if it is true it's false and if it is false it's true..
@@MoiLiberty Thanks for your compressed form of my sentence.
Of course the statement makes no sense , but so does it counterpart :
"This statement is true".
It doesn't add any information .. does it ?
I think no, godel's statement was about its provability and not its truthness. It was a clever way of using self referentiality avoiding paradoxes. That's why we say there's a difference between truth and proof
Well, what he showed was that, by simply using all symbols and rules of basic arithmetics, you can build a sentence which, translated to English, roughly says: "I'm not provable".
And of course you can ask: Is that true?
Here comes the problem.
Suppose that
A) it is true. So now your system contains a sentence which you know to be true but you cannot prove to be true ( because that's exactly what the sentence says about itself )
B) it's false. So, for what the sentence says, it IS provable and so you have now a sentence which can be proven by your system despite being false.
So the conclusion is: either your system is incomplete because it cannot prove all truths that can be legitimately expressed in its language OR it is inconsistent because it can prove a theorem which is known to be false.
In a sense, it is an extension of the famous lyar's paradox but it's more about demonstrability than it is about truth.
That statement might be unprovable in a system of formal logic. However, English is a remarkably imprecise language.
Lee Mary Thompson Brenda Williams Gary
Gödel is really not hard to pronounce.
Very good, but to wordy.
The book is way much better than this lecture
21:26 That's an obvious error. Assuming that philosophy handles meta-questions is a common but incorrect mistake. In reality, philosophy fails to answer these questions and they revert back to science. This includes questions about philosophy itself which cannot be answered philosophically.
51:20 The mention of the unreliability of intuitions is good. These are almost always incorrect.
55:00 Russell's paradox. This isn't a real paradox.
1:03:00 After listening to Goldstein fawning over this proof, it is quite disappointing to see a non-proof presented. The liar's paradox is not a real paradox.
1:05:00 It's hard to believe that someone sees this as a problem. It only exists in an informal system like philosophy where structure is lax, allowing equivocation.
1:07:00 Except the system presented isn't formal.
1:24:00 This is partly true and partly false. It is true that brains are not computers and this is provable. However, it does not seem to be the case that this is absent a formal system. Rather the formal system is not limited to math and logic (both of which are computable).
1:35:36 I can't really criticize Goldstein for being uncomfortable with Penrose-Lucas. She has nothing in her background that would allow her to formally analyze it. We are not delusional computers.
Kurt Godel the type to publish a proof that millennials can, in fact, pull themselves up by their bootstraps.
"um"
ok im a typical man here...... who wouldn't invite this amazing lady for a coffee?
She needs a lot of words to come to the point, a very chaotic mind repeating her words a lot
That's a common attribute of Women. Surface details (about stuff), not depth to the point.
@@godofgodseyesshe’s talking to a lay audience. You expect her to only write out the incredibly dense and complex predicate logic? What a joy all those audience members would have!
I got a feeling of a women quota here.
I can see you don't read much.
Or, you don't get around. Geniuses are eccentric
liars paradox and russell's paradox get at the same thing