There's some confusion in the comments about implications of the theorem. I've studied Gödel quite a bit. An analogous finding was Turing's "undecidability", which proved that every program has a problem which it fundamentally cannot solve even if you gave it an infinite amount of time. The two are analogous to everything: if we build some kind of thinking process out of rules, that system will ALWAYS be flawed. I'm attempting to write a book on expanding the logical implications and some more commonsense analogies.
All systems are necessarily finite and therefore flawed to that degree. Mind, if properly cultivated, can intuit the Infinite and realize It eventually.
@@imasciencegeek I agree. I did not say our mind could grasp an uncountable infinity, only that we can realize that we are part of that infinity, which is our Source.
I'm absolutely sure that not everybody should learn math continually all their lives. Who'd grow the food and who'd pick up the garbage if everybody was obsessed with math? For most people math is mental torture and not satisfying at all.
Never was very good at math anything past basic shop- type math. That’s why I’m not an engineer. One thing I learned about all that hard math I couldn’t do. Unless I was and engineer or something like that I would never use it in life
Agreed! Blind faith in religion has simply been replaced by blind faith in Science. But i guess the flip side is that the educational system mostly stifles creativity and curiosity, and children are "made" to study rather than them "wanting" to. This state of affairs will never lead them to ask questions which are on the edge, which question the results, which stretch the science beyond the banal examples which are given in the classroom and which lead to incremental learning...
What Godel shows with incompleteness theorem, is that the human mind has a capacity for intuition and creativity, which ultimately lay at the foundations of reasoning. This is in direct opposition to logicistic attitudes, which suggest that the formal axiomization of mathematics could lead to all truths. Godel understood that truth (including logic) relies on the foundations of axioms, not all of which have been discovered, and our infinite.
I suggest readers look into Godel's Ontological proof on why God exists. Ultimate, it is what keeps humanity free... free from other men who purport to be gods. It's a beautiful system, but the public is not supposed to understand this, because it would render those seeking "godship" powerless in everyones eyes.
I first heard this presentation as a podcast and was impressed at what a good communicator Mark Colyvan is, and what an interesting talk he gave here. I still think it's one of the best offerings on Godel out there.
The alarming way to clear a crowded room is to yell “Fire!!” However, the safest way to clear such a room is to tell the crowd that you are going to talk about complex problems in math. You will soon be enjoying the empty space.
I am amazed that people are still talking about the mind/machine issue. The resolution is simple. An analogy goes like this: Ask this question: "In general, is it possible to trisect an angle?" The knee-jerk mathematical response is: "Of course not! Everybody knows that!" But the better response is: "Of course you can! Just not with a compass and straightedge." The mind/machine issue similarly hinges on the technical definition of the word "algorithm": a finite set of instructions that is guaranteed to produce a correct result in a finite time. Human minds are not an algorithm. Neither do they need to be anything more than a computer to do what they do. A human mind is a gigantic (but finite) non-terminating (except by death) Monte Carlo process, some parts of which run in a deductive (mathematical) mode. It has been possible to build something mind-like for decades now. The trick is to recognize what a mind is by recognizing what it accomplishes, and then noting how it does it. It's like evolution: once you see the trick, there is nothing hard about it. What turns out to be hard is encompassing all of the consequences of the little trick.
+Steven Abell Well, it is possible to trisect an angle using origami. There are some problems that are impossible by ANY finite construction, like, for instance, squaring the circle. So this means the mind/machine issue is solved? Um, ok. So how did you reach the conclusion that the mind is a giant Monte Carlo process? What exactly do you mean by a Monte Carlo process? The Monte Carlo process refers to the generation of random numbers in a computer to solve some problem. So, if the mind is a Monte Carlo simulation, does that mean the human mind is a computer generating random numbers or... I'm not clear what your point is.
@prof5string: The sentence "this sentence has five words" is NOT self-referential. It refers to a numbering system that defines the number of words in the sentence, and that numbering system is outside the language of the sentence.
@Dent Niggemeyer: "Uncertainty" is an enormous threat to the established system and those who control it and benefit from it. If the public were to become uncertain, then they would become less vested, and perhaps turn to alternatives, or turn to themselves, or perhaps turn to direct relationship with the spiritual. All of these trajectories disintermediate the current power structure. Hence, Goedel's findings are extremely dangerous to the status quo.
Starts out as a bit of a dry treatment for those not intellectually inclined. But for those who can hang in for a few minutes there's a lot to enjoy, and even a few good laughs.
I don't really know how anyone could NOT believe in some higher power after understanding Godel's First Proof, the Incompleteness Theorem. Man simply does not have command over nature, and that is obvious no matter how many elitist academics, media personalities or the like state to the contrary.
No one said set theory is inconsistent. It is not a fabulous question. Being unable to prove consistency does not demonstrate that something is in fact inconsistent. The only way to do that is to find an inconsistency and that has not been done.
I have 1 question. If Godel proved that we cannot guarantee that the mathematical axiom's are consistent, doesn't that ironically undermine his own proof? He used a framework to prove that there is no framework wich is fale safe, or at least we can't determine it.
You are right about that....We cannot prove any property of any system inside the system itself. But Godel proved the INconsistency of arithmetic outside of its system
I'm not sure what you mean. Completeness has to do more with formulas in a system which are also theorems in that system. So a system is considered incomplete if there is a formula in a system that cannot be proven. I haven't taken enough number theory, though I would imagine it wouldn't be trivial to attempt to prove this rigorously enough for mathematicians. If there is something I'm missing, I would like to know.
"For 2000 years, mathematics has been the model-the subject-that convinces us that certainty is possible. Yet Now there's no certainty anywhere-not even in mathematics." Are you certain?
Consistency allows some things to be true and others false. Inconsistency makes everything and its opposite true. You really have to expect the answer to the consistency question to be “yes”. In an inconsistent world, you can answer the Consistency Question and any other question “yes”. In a consistent world, “yes” is the obvious answer to the Consistency Question. So that’s two choices. One is “yes” and the other is “yes”. Gödel"s Proof showed the answer to the Consistency Question was “no”.
@Pooja Deshpande Set theory is not inconsistent. In order for a system to be inconsistent, it must be the case that both a formula of the system and that very formula's negation can be proven within the system (I.E. you can prove some formula P and you can also prove NOT P). Set theory is a consistent system. It is impossible to prove both a formula of set theory and that very formula's negation within set theory. Godel's incompleteness theorem does not demonstrate that set theory is inconsistent (this cannot be demonstrated, because set theory is consistent). Godel proved that set theory is incomplete. A system is incomplete if there is some formula of the system which is true, but cannot be proven within the system. Godel's theorem demonstrates that there is at least one formula of set theory which is true, but cannot be proven according to the deductive rules of set theory (it follows that there are actually infinitely many formulae of set theory which cannot be proven). Now, to answer your question: you asked why set theory is still taught in schools even though it is inconsistent. If set theory were inconsistent, it would hardly make sense to teach it at all. For example, if basic arithmetic were inconsistent (it is, in fact, not), then we would be able to prove both that 1 + 1 = 2 and that 1 + 1 =/= 2. In an inconsistent system, you can prove anything, no matter how crazy-sounding! So teaching it would be a breeze because every formula you demonstrate can be proven. The only downside would be that you could come in the next day and teach the exact opposite of what you had taught the previous day without breaking the rules of the system. That is why it is important to teach consistent systems. However, there is a much more interesting question of why it is that we still teach incomplete systems (like set theory). I find this question much more open to discussion, as there are many different arguments in favor of teaching incomplete systems. One point to be made is that systems which are both complete and consistent are often not considered "interesting" as fields of study. For example, first order predicate logic is a system which is both consistent and complete, but the complexity of provable statements within first order logic comes nowhere near the complexity of some of the results provable within set theory. Another argument one could make in favor of teaching incomplete systems is to appeal to the results provable within the system themselves. Much of our discoveries in mathematics rest on set theory as a foundation for demonstrating our results, and giving up set theory might also mean giving up on those discoveries. Again, my answer to this second question is much more speculative than the first, but I hope I was able to clear up the confusion about the consistency of set theory and explain why we still teach set theory in schools even after the discovery of Godel's incompleteness theorem.
What about when Nature seems to conform to mathematical laws? I am not advocating a sort of Platonic mathematical ideality, but I still believe that mathematics is something objective and independent of the human mind. I am not altogether sure I understand my current conception of what mathematics -is-.
(con't).....I realized that there were as many branches of math as there are types of literature & in fact maths is literature of numbers, relationships, etc. So the equivalence is that maths & literature begin as one, then diverge developing into their recognizable forms based on how people decide to develop their characters.
@Pooja Deshpande: What a fabulous question! What's preposterous is that children are taught these tools without being told of their limitations, especially when compared to the human mind or the real world. But I guess if you want to create an elite supersystem you must convince all of the people within that system that science is a god that its subjects must religiously follow. And when science takes over, humanity is marginalized, as numbers, charts & algorthms drive all decision making.
because it's a good approximation that works in enough cases to be useful. that's the answer for all theories - we'll probably never find a perfectly consistent system that perfectly describes the real world, partly because of the limits of language, our brains, logic, etc.
The speaker makes an error by saying "set theory was shown to be inconsistent". What he means to say that the Naive Set Theory of Cantor where sets could be described without types or classes or additional axioms restricting what defines a set. In this case, he is evoking Russell's paradox, namely "is the set R which is the set of all sets which are not members of themselves contained in itself?"
I'm not an expert in any sense. But from the description i have seen here. Incompleteness is an observation if anything. It doesn't need to have a mathematical framework justifying it.
No we are NOT, and Godel helps to show this. John von Neumann took Godel Numbering and used it to help create binary numbering systems, which can be "gamed" to create a Complete Formal System where there is none, via a computer controlled virtual "reality".
It's only inconsistent if you're too loose with your definition of "set". That problem was eventually solved. The concept of cardinality (or "size"- two sets are the same size if their elements can be put in 1-1 correspondence) is, for finite sets, the underlying concept that the "counting numbers" 1,2,3... are based on, so sets are indispensable in math.
"A word is the skin of a living idea". We are talking about mathematical certainty here, not human certainty. The importance of the proof is not that when you measure the length of a building's shadow and its angle with the ground that you should distrust the height you calculate. The importance is that a software developer who is writing a software program that determines the correctness of other software programs can give up the impossible and create a video game instead.
Wrong. It refers to, splinters off into several ideas- sentences, words, numbers, etc. If you think that is the same as "The apple is false." You just don't get it, sorry. Very different. Thank you very much for proving my point. Next? QED.
Why did Bertrand Russell need advanced set theory for his part of this? That's just an internally inconsistent definition of a set; you could do the same thing with a variable in beginning algebra: let a = a+1 If a is 1, then a must be 2. If it's 2, then it must be 3. Etc. This variable is just as self-contradictory as Russell's set.
does "truth" exist outside of the human experience, or, does the fact that human consciousness is capable of defining "truth" isolate man made logical functionality from so called natural systems and turn it in to a tautology?
I am thinking that that is either inconsistent or incomplete...
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Heidegger posited the theorem of oppositional logic in which he says (paraphrasing) that anyone who purports to know, with certitude, that there is no God is guilty of the same sin they accuse the ecclesiastes of committing: that is, they purport to KNOW. He constructed the proposition as an epistemological function of time and empircism: since we can only believe information gleamed from the senses, and we materialize as humans after the creation of all that we see, how can we know its genesis?
It was not Russell, but Cantor that showed that *naive* set theory was inconsistent. The axiomatic set theory that mathematics is based on is *consistent*
Precisely, that was Russell's conclusion, your reasoning, is a shallow view of Russell paradox, indeed, if you read the first conclusions on that, you would notice that Russell refer that paradox to explain why a set may not be self referential. Off course, if you are very rigorous, you would notice that this is not a paradox a posteriori since ZF system exclude this kind of rare sets thanks to Russell's ideas. If you talk about language, you need to set a framework.
For incompleteness theorem...isnt there just one true infinite set that all other 'infinite sets' are contained in. the totality of all infinite sets is true infinity.and also,the only set. That is to say, when sets are made, it is us that are breaking up the true infinity into these categories, when really, there are no sets or categories,real numbers, irrational numbers, all the numbers exist in their infinite ways coincidingly.and that existence of every possible number,is the only true set.
@QuantumBunk: To be more specific, the "problem" is that the public does not understand the limits of Mathmetics. Further, I believe that these limits are deliberately hidden from the general population, so that science can be sold to the masses as its new god.
What is the highest level of math that you have used in your everyday lives? Me? In the days before supermarkets listed the price/volume, I would do a simple ratio to determine if the economy size was really a better deal than the other. I weigh stuff to make sure that I am not being cheated, too. I'm not quite sure why I shared this...
I once when working in retail used the integration of a disc to prove the length of a coil of wire was accurate...now I don't work retail anymore... I do that same thing for best pricing at the market! I'm glad I'm not alone on that
Godel and Feynman's gods reminds me of Q from Star Trek. I can't seem to find any connections though. Does anyone know if the Q is an exploration of this idea of Godel's or Feynman's gods?
Def 1 is an arbitrary biconditional which relies on undefined terms like 'positive'. Def 3 concerning necessary existence replies on these ontological terms (essence) that are not well-defined and furthermore this word 'exemplified', which isn't well-defined. It seems to be one big 'begging the question' fallacy, wherein we invent some terms, use those terms to enforce our system, and then say, 'look at that, I've proved it.'
Can someone correct me if I am wrong; according to the first incompleteness theorem in every axiomatic system related to arithmetic there will be statements that are true but not provable within that system, second incompleteness theorem is special case of the first one - each axiomatic system related to arithmetic cannot prove its consistency. If true, what is relation between first and second theorem?
You're account of gödels incompletness theorems is roughly correct, depending on what you mean with "related to arithmetic" and "provable". The english wikipedia article on the theorem is probably an ok read. It is wrong though, to say that the "second incompleteness theorem is special case of the first one". I see no connection between them. It is however very hard to explain that, as the whole conext of argument is extremely technical and it feels impossible to give an appropiate account without laying out the formalisms. A few things I can say though: - beware that the 1. and 2 . theorem do not exactly speak about the same sets of axiom systems. - the first incompletness theorem does not exactly give you "statements that are true but not provable", instead it gives you statements whose truth cannot be determined, within an appropiate axiomatic system. it is not clear wether these statements are true or not in an intuitive sense, for all I know, it could be that alle the statements whose truth cannot be determined are false. (the concept of truth has to be treated very carefully here, as the only way we can communicate truth are axiomatic systems, so its weird to say that something is true even though all the seemingly good axiomatic systems do not tell us it is true) - the second gives you statement where one cannot infer that it is true. that is different from saying you cannot infer its truth value, because "one cannot infer that it is true" still entails the possibility of you infering that the statement is wrong. "you cannot infer its truth value" does not leave that possibility open. the first theorem gives you such statements. - the wikipedia article says that the proof of the second theorem can be obtained by formalizing the proof of the first one. I don't know about that. We didn't do it that way in my class, never read about that either, not even in philosophy books. seems like a cumbersome way of doing things.
I am a man. Self-referential, but not "stupid and fake". The problem with those self-referential paradoxes is that the object of those sentences are not complete enough to ascribe "truth" or "falseness" to. "This sentence" is neither true nor false, by itself. It does not assert or deny anything. Same thing for "time", or "space", or "apple", or "man", or anything like that.
Of course not. You can't prove/disprove a negative. Why should something that doesn't exist (like the hundredthousands gods of the humans) be part of a scrutiny in the first place?
@Nukutawiti:Right, one has to rely upon an EXTERNAL system to prove consistence of the target system. That's incompleteness. If the mathematical system were complete, then it would be able to prove it's own consistency. But it can't.
You must get ahead of inconsistencies and find involvement with true conjecture a future which satisfies a dream yet unattainable in present circumstances of inner desertion.
Is mathematics merely the natural language of physics, or is it also indispensable at the core of its conceptual development, as well? In other words, is the conceptual foundation of physics ultimately mathematical in nature?
"For something to be either true of false it cannot be self referential, it must refer to something outside itself." Really? "This sentence has five words" is self-referential and true.
+anushka trivedi Let S be the set of all sets not members of themselves. Since all elements of S are not elements of themselves, it follows that S is not an element of S. Well, then the statement 'either S is not an element of itself OR 2 + 2 = 5' is a true statement, since the first statement of the disjunction is true. But then, since S is not an element of itself, it is by definition an element of itself. Therefore, '2 + 2 = 5' is true. By the way, this works for any statement. For instance, the existence of this paradoxical set S implies that squares are round.
Thanks for giving such an enthusiastic talk about Gödel. Maybe you know my booklet (written together with Casti) "Gödel: A Life of Logic" (Casti+DePauli, Perseus Books, Cambridge MA, 2000.) Also: "Wahrheit und Beweisbarkeit" htp+bv (Hölder-Pichler-Tempski + Bundes-Verlag, Wien 2002 Volume 1 and 2). Or: "EUROPOLIS5 Kurt Gödel, ein Mathematischer Mythos" NOVUM_Verlag, Horitschon, Austria 2003). There exists also a film with the same title, produced by the ORF= Austrian Television network, 1986, you can buy from the ORF-shop (But the film is also in German).
@QuantumBunk: One could say paradoxes are "stupid & fake" if they are KNOWN and ACKNOWLEDGED. One of the biggest problems with mathemetatics, science and computer technology is that they are being sold to the public as techniques and systems that are infallable. The general population does not understand the weaknesses and faults inherent in these system, and therefore they place them on a pedastal that is undeserved.
Meh. What happens if we allow 'logical' statements (the Russel Paradox) in set theory to violate the law of non-contradiction? Set theory becomes inconsistent. What a surprising result. I still think such (self-contradictory) statements should be held to be undefined (much like dividing by zero), unable to take a truth value because in a very real sense they are neither true nor false, but meaningless. So much for 2,000 years of controversy.
@HebaruSan If you start out with pure logic you cannot do things like a=a+1. Pure logic doesn't admit the concept of infinite procedural truths. This goes back to something called the paradox of analysis. Either it is true or it tells you something useful, not both. a is a a is a+1 then a+1 is a If a is 1 then a+1 is 1 Unless you want to claim that 1=2 you cannot hold the proposition a+1=2. This is prohibited by the law of the excluded middle.
@Bloke Poppy: My point in the earlier post is to illustrate that a system where the public believes in God-over-men creates more freedom for the people living in that system. When God is disbelieved, men can fill that role and exert godly powers over the public, resulting in massive suppression.
The labyrinth finally destroyed Goedel as it did Newton, Weierstrass, Cantor, Frege, Ramanujan, von Neumann et al. They used bad infinities & not the transfinite fractions of Non-Cantorian set theory. I painted a large triptych called The Madness of Mathematics.
That is of course the question which haunted him , near the end of his life . I simply find it hard to believe the great Kurt Godel actually believed in the Judeo/Christian personal god . I would side with those who say he believed in Spinozas god , which I do . Honest people can disagree on which one he believed in , no ?
Point of clarification. When I was 22 I had my first profound realization about mathematics. Up to that point I had continuously struggled to find the relationship between maths & literature. Is there a difference? Which is most fundamental, etc. Well I discovered that the field of mathematics is = to literature: some fiction, some non-fiction, but just hadn't been labeled as such.
Please do not forget, that people like Einsten and Gödel lived in times, when being an atheist was not an option. If anyone would have taken the position of an atheist, he wouldn't have the chance to study or the chance for a job. At this time, churches have been overcrowded on sundays. If you're not there taking part in praying silly crap, you'd have a very good excuse or become an unadapted outlaw. Like in the bible belt today.
There's some confusion in the comments about implications of the theorem. I've studied Gödel quite a bit. An analogous finding was Turing's "undecidability", which proved that every program has a problem which it fundamentally cannot solve even if you gave it an infinite amount of time. The two are analogous to everything: if we build some kind of thinking process out of rules, that system will ALWAYS be flawed. I'm attempting to write a book on expanding the logical implications and some more commonsense analogies.
COVID got him maybe. Wish we could see his work
All systems are necessarily finite and therefore flawed to that degree.
Mind, if properly cultivated, can intuit the Infinite and realize It eventually.
@manuelfrn still writing, not dead
@@thenowchurch6419 humans can recognize a loop and intuit that it has no end, but we cannot grasp an uncountable infinity.
@@imasciencegeek I agree.
I did not say our mind could grasp an uncountable infinity, only that we can realize that we are part of that infinity, which is our Source.
I love maths, and I think that everybody should learn math continually all their lives.
It is very satisfying.
I'm absolutely sure that not everybody should learn math continually all their lives. Who'd grow the food and who'd pick up the garbage if everybody was obsessed with math? For most people math is mental torture and not satisfying at all.
dlwatib
They could fake it.
I like it very much but its to hard for me.
Never was very good at math anything past basic shop- type math. That’s why I’m not an engineer. One thing I learned about all that hard math I couldn’t do. Unless I was and engineer or something like that I would never use it in life
Agreed!
Blind faith in religion has simply been replaced by blind faith in Science.
But i guess the flip side is that the educational system mostly stifles creativity and curiosity, and children are "made" to study rather than them "wanting" to. This state of affairs will never lead them to ask questions which are on the edge, which question the results, which stretch the science beyond the banal examples which are given in the classroom and which lead to incremental learning...
What Godel shows with incompleteness theorem, is that the human mind has a capacity for intuition and creativity, which ultimately lay at the foundations of reasoning. This is in direct opposition to logicistic attitudes, which suggest that the formal axiomization of mathematics could lead to all truths. Godel understood that truth (including logic) relies on the foundations of axioms, not all of which have been discovered, and our infinite.
great stuff. i've been watching too much garbage on youtube lately. i need to get back to this stuff. it's very satisfyingly interesting.
calabiyou Watch "Gödel, Escher & Bach" instead. :)
I suggest readers look into Godel's Ontological proof on why God exists. Ultimate, it is what keeps humanity free... free from other men who purport to be gods. It's a beautiful system, but the public is not supposed to understand this, because it would render those seeking "godship" powerless in everyones eyes.
I first heard this presentation as a podcast and was impressed at what a good communicator Mark Colyvan is, and what an interesting talk he gave here. I still think it's one of the best offerings on Godel out there.
The alarming way to clear a crowded room is to yell “Fire!!” However, the safest way to clear such a room is to tell the crowd that you are going to talk about complex problems in math. You will soon be enjoying the empty space.
*Tips fedora
or was that enjoying the empty "set"
Or bring a baby to a lecture
I would not leave the room based on someone's claim alone. I would wait until the proof, e.g. smoke.
@@artoffugue333 smoke on its own isn't a rigorous proof, it's just a hint you can build a conjecture on.
I am amazed that people are still talking about the mind/machine issue. The resolution is simple. An analogy goes like this: Ask this question: "In general, is it possible to trisect an angle?" The knee-jerk mathematical response is: "Of course not! Everybody knows that!" But the better response is: "Of course you can! Just not with a compass and straightedge." The mind/machine issue similarly hinges on the technical definition of the word "algorithm": a finite set of instructions that is guaranteed to produce a correct result in a finite time. Human minds are not an algorithm. Neither do they need to be anything more than a computer to do what they do. A human mind is a gigantic (but finite) non-terminating (except by death) Monte Carlo process, some parts of which run in a deductive (mathematical) mode. It has been possible to build something mind-like for decades now. The trick is to recognize what a mind is by recognizing what it accomplishes, and then noting how it does it. It's like evolution: once you see the trick, there is nothing hard about it. What turns out to be hard is encompassing all of the consequences of the little trick.
The best trick I learned in this life was how to tell shit from shinola. :D
My father know how to trisec the angle but you need a different tool for it.
Nice. I like what you say sir, and I appreciate you saying it.
+Steven Abell
Well, it is possible to trisect an angle using origami. There are some problems that are impossible by ANY finite construction, like, for instance, squaring the circle. So this means the mind/machine issue is solved? Um, ok. So how did you reach the conclusion that the mind is a giant Monte Carlo process? What exactly do you mean by a Monte Carlo process? The Monte Carlo process refers to the generation of random numbers in a computer to solve some problem. So, if the mind is a Monte Carlo simulation, does that mean the human mind is a computer generating random numbers or...
I'm not clear what your point is.
Agreed. I think this opinion is similar to that of AI pioneer Marvin Minsky.
@prof5string: The sentence "this sentence has five words" is NOT self-referential. It refers to a numbering system that defines the number of words in the sentence, and that numbering system is outside the language of the sentence.
@Dent Niggemeyer: "Uncertainty" is an enormous threat to the established system and those who control it and benefit from it. If the public were to become uncertain, then they would become less vested, and perhaps turn to alternatives, or turn to themselves, or perhaps turn to direct relationship with the spiritual. All of these trajectories disintermediate the current power structure. Hence, Goedel's findings are extremely dangerous to the status quo.
No mention of Euler?
Starts out as a bit of a dry treatment for those not intellectually inclined. But for those who can hang in for a few minutes there's a lot to enjoy, and even a few good laughs.
I don't really know how anyone could NOT believe in some higher power after understanding Godel's First Proof, the Incompleteness Theorem. Man simply does not have command over nature, and that is obvious no matter how many elitist academics, media personalities or the like state to the contrary.
No one said set theory is inconsistent. It is not a fabulous question. Being unable to prove consistency does not demonstrate that something is in fact inconsistent. The only way to do that is to find an inconsistency and that has not been done.
He paces back and forth with a rhythmic tempo. The acoustics change in this distance. It sounds like he's being recorded with a slight flanger.
I have 1 question. If Godel proved that we cannot guarantee that the mathematical axiom's are consistent, doesn't that ironically undermine his own proof? He used a framework to prove that there is no framework wich is fale safe, or at least we can't determine it.
You are right about that....We cannot prove any property of any system inside the system itself. But Godel proved the INconsistency of arithmetic outside of its system
Hi- who is the speaker here? I'm sorry if it is mentioned somewhere, I just couldn't see it myself... Thanks for putting it up.
I'm not sure what you mean. Completeness has to do more with formulas in a system which are also theorems in that system. So a system is considered incomplete if there is a formula in a system that cannot be proven. I haven't taken enough number theory, though I would imagine it wouldn't be trivial to attempt to prove this rigorously enough for mathematicians. If there is something I'm missing, I would like to know.
"For 2000 years, mathematics has been the model-the subject-that convinces us that certainty is possible. Yet Now there's no certainty anywhere-not even in mathematics."
Are you certain?
I'm tempted to say that "this sentence has five words" is a true self referential statement but i dont really know.
Consistency allows some things to be true and others false. Inconsistency makes everything and its opposite true. You really have to expect the answer to the consistency question to be “yes”.
In an inconsistent world, you can answer the Consistency Question and any other question “yes”. In a consistent world, “yes” is the obvious answer to the Consistency Question.
So that’s two choices. One is “yes” and the other is “yes”. Gödel"s Proof showed the answer to the Consistency Question was “no”.
A power set has a property {a,b} = {b,a}
but, where {a,b} *= {b,a} the set is defined by a linear algorithm
@Pooja Deshpande Set theory is not inconsistent.
In order for a system to be inconsistent, it must be the case that both a formula of the system and that very formula's negation can be proven within the system (I.E. you can prove some formula P and you can also prove NOT P).
Set theory is a consistent system. It is impossible to prove both a formula of set theory and that very formula's negation within set theory.
Godel's incompleteness theorem does not demonstrate that set theory is inconsistent (this cannot be demonstrated, because set theory is consistent). Godel proved that set theory is incomplete. A system is incomplete if there is some formula of the system which is true, but cannot be proven within the system. Godel's theorem demonstrates that there is at least one formula of set theory which is true, but cannot be proven according to the deductive rules of set theory (it follows that there are actually infinitely many formulae of set theory which cannot be proven).
Now, to answer your question: you asked why set theory is still taught in schools even though it is inconsistent. If set theory were inconsistent, it would hardly make sense to teach it at all. For example, if basic arithmetic were inconsistent (it is, in fact, not), then we would be able to prove both that 1 + 1 = 2 and that 1 + 1 =/= 2. In an inconsistent system, you can prove anything, no matter how crazy-sounding! So teaching it would be a breeze because every formula you demonstrate can be proven. The only downside would be that you could come in the next day and teach the exact opposite of what you had taught the previous day without breaking the rules of the system. That is why it is important to teach consistent systems.
However, there is a much more interesting question of why it is that we still teach incomplete systems (like set theory). I find this question much more open to discussion, as there are many different arguments in favor of teaching incomplete systems. One point to be made is that systems which are both complete and consistent are often not considered "interesting" as fields of study. For example, first order predicate logic is a system which is both consistent and complete, but the complexity of provable statements within first order logic comes nowhere near the complexity of some of the results provable within set theory.
Another argument one could make in favor of teaching incomplete systems is to appeal to the results provable within the system themselves. Much of our discoveries in mathematics rest on set theory as a foundation for demonstrating our results, and giving up set theory might also mean giving up on those discoveries.
Again, my answer to this second question is much more speculative than the first, but I hope I was able to clear up the confusion about the consistency of set theory and explain why we still teach set theory in schools even after the discovery of Godel's incompleteness theorem.
It's not set theory, but Arithmetic that Godel examined, and he showed not that it is inconsistent, but that it is consistent only if incomplete.
We need more solicitation to improve college learning facilities and expand our horizons into the next generation of well wishers.
paradox lies at the heart of reality.
Just to be clear, Russell's letter to Frege was written in 1902. Zermelo's work on set theory was published in 1904-1908.
What is the name of the lecturer?
What about when Nature seems to conform to mathematical laws? I am not advocating a sort of Platonic mathematical ideality, but I still believe that mathematics is something objective and independent of the human mind. I am not altogether sure I understand my current conception of what mathematics -is-.
(con't).....I realized that there were as many branches of math as there are types of literature & in fact maths is literature of numbers, relationships, etc. So the equivalence is that maths & literature begin as one, then diverge developing into their recognizable forms based on how people decide to develop their characters.
A few seconds in and I'm already getting seasick. Dude stop moving back and forth! 🤣
@Pooja Deshpande: What a fabulous question!
What's preposterous is that children are taught these tools without being told of their limitations, especially when compared to the human mind or the real world. But I guess if you want to create an elite supersystem you must convince all of the people within that system that science is a god that its subjects must religiously follow. And when science takes over, humanity is marginalized, as numbers, charts & algorthms drive all decision making.
because it's a good approximation that works in enough cases to be useful. that's the answer for all theories - we'll probably never find a perfectly consistent system that perfectly describes the real world, partly because of the limits of language, our brains, logic, etc.
Video grapher should have zoomed out long ago. Great information but film can make your eyes tired.
The speaker makes an error by saying "set theory was shown to be inconsistent". What he means to say that the Naive Set Theory of Cantor where sets could be described without types or classes or additional axioms restricting what defines a set. In this case, he is evoking Russell's paradox, namely "is the set R which is the set of all sets which are not members of themselves contained in itself?"
i have a question that if set theory is inconsistent, then why are children still taught that at school?
I'm not an expert in any sense. But from the description i have seen here. Incompleteness is an observation if anything. It doesn't need to have a mathematical framework justifying it.
WHO is the speaker,please?
No we are NOT, and Godel helps to show this. John von Neumann took Godel Numbering and used it to help create binary numbering systems, which can be "gamed" to create a Complete Formal System where there is none, via a computer controlled virtual "reality".
It's only inconsistent if you're too loose with your definition of "set". That problem was eventually solved.
The concept of cardinality (or "size"- two sets are the same size if their elements can be put in 1-1 correspondence) is, for finite sets, the underlying concept that the "counting numbers" 1,2,3... are based on, so sets are indispensable in math.
"A word is the skin of a living idea". We are talking about mathematical certainty here, not human certainty. The importance of the proof is not that when you measure the length of a building's shadow and its angle with the ground that you should distrust the height you calculate. The importance is that a software developer who is writing a software program that determines the correctness of other software programs can give up the impossible and create a video game instead.
This guy is a very gifted lecturer. What is the setting of his presentation? Thanks.
Wrong. It refers to, splinters off into several ideas- sentences, words, numbers, etc. If you think that is the same as "The apple is false." You just don't get it, sorry. Very different. Thank you very much for proving my point. Next? QED.
Why did Bertrand Russell need advanced set theory for his part of this? That's just an internally inconsistent definition of a set; you could do the same thing with a variable in beginning algebra:
let a = a+1
If a is 1, then a must be 2. If it's 2, then it must be 3. Etc. This variable is just as self-contradictory as Russell's set.
What Is The Point Of His Marcus Elieser Bloch ?
does "truth" exist outside of the human experience, or, does the fact that human consciousness is capable of defining "truth" isolate man made logical functionality from so called natural systems and turn it in to a tautology?
Who is the speaker?
The reason why not many good books are written is that people that know stuff don't know how to write.
The movement, pacing back and forth, induces nausea.
I doubt the correctness of the subtitles
I am thinking that that is either inconsistent or incomplete...
Heidegger posited the theorem of oppositional logic in which he says (paraphrasing) that anyone who purports to know, with certitude, that there is no God is guilty of the same sin they accuse the ecclesiastes of committing: that is, they purport to KNOW. He constructed the proposition as an epistemological function of time and empircism: since we can only believe information gleamed from the senses, and we materialize as humans after the creation of all that we see, how can we know its genesis?
Incredible videos. Who is the speaker?
@HebaruSan but what if a = infinity?
It was not Russell, but Cantor that showed that *naive* set theory was inconsistent. The axiomatic set theory that mathematics is based on is *consistent*
Precisely, that was Russell's conclusion, your reasoning, is a shallow view of Russell paradox, indeed, if you read the first conclusions on that, you would notice that Russell refer that paradox to explain why a set may not be self referential. Off course, if you are very rigorous, you would notice that this is not a paradox a posteriori since ZF system exclude this kind of rare sets thanks to Russell's ideas. If you talk about language, you need to set a framework.
For incompleteness theorem...isnt there just one true infinite set that all other 'infinite sets' are contained in. the totality of all infinite sets is true infinity.and also,the only set. That is to say, when sets are made, it is us that are breaking up the true infinity into these categories, when really, there are no sets or categories,real numbers, irrational numbers, all the numbers exist in their infinite ways coincidingly.and that existence of every possible number,is the only true set.
who is the lecturer?
@QuantumBunk: To be more specific, the "problem" is that the public does not understand the limits of Mathmetics. Further, I believe that these limits are deliberately hidden from the general population, so that science can be sold to the masses as its new god.
What is the highest level of math that you have used in your everyday lives? Me? In the days before supermarkets listed the price/volume, I would do a simple ratio to determine if the economy size was really a better deal than the other. I weigh stuff to make sure that I am not being cheated, too.
I'm not quite sure why I shared this...
I once when working in retail used the integration of a disc to prove the length of a coil of wire was accurate...now I don't work retail anymore...
I do that same thing for best pricing at the market! I'm glad I'm not alone on that
Godel and Feynman's gods reminds me of Q from Star Trek. I can't seem to find any connections though. Does anyone know if the Q is an exploration of this idea of Godel's or Feynman's gods?
Very good general explanation of Godel incompleteness theorem and it's implication;
I hoped it was longer :(
Def 1 is an arbitrary biconditional which relies on undefined terms like 'positive'. Def 3 concerning necessary existence replies on these ontological terms (essence) that are not well-defined and furthermore this word 'exemplified', which isn't well-defined. It seems to be one big 'begging the question' fallacy, wherein we invent some terms, use those terms to enforce our system, and then say, 'look at that, I've proved it.'
Can someone correct me if I am wrong; according to the first incompleteness theorem in every axiomatic system related to arithmetic there will be statements that are true but not provable within that system, second incompleteness theorem is special case of the first one - each axiomatic system related to arithmetic cannot prove its consistency.
If true, what is relation between first and second theorem?
You're account of gödels incompletness theorems is roughly correct, depending on what you mean with "related to arithmetic" and "provable". The english wikipedia article on the theorem is probably an ok read.
It is wrong though, to say that the "second incompleteness theorem is special case of the first one". I see no connection between them. It is however very hard to explain that, as the whole conext of argument is extremely technical and it feels impossible to give an appropiate account without laying out the formalisms.
A few things I can say though:
- beware that the 1. and 2 . theorem do not exactly speak about the same sets of axiom systems.
- the first incompletness theorem does not exactly give you "statements that are true but not provable", instead it gives you statements whose truth cannot be determined, within an appropiate axiomatic system. it is not clear wether these statements are true or not in an intuitive sense, for all I know, it could be that alle the statements whose truth cannot be determined are false. (the concept of truth has to be treated very carefully here, as the only way we can communicate truth are axiomatic systems, so its weird to say that something is true even though all the seemingly good axiomatic systems do not tell us it is true)
- the second gives you statement where one cannot infer that it is true. that is different from saying you cannot infer its truth value, because "one cannot infer that it is true" still entails the possibility of you infering that the statement is wrong. "you cannot infer its truth value" does not leave that possibility open. the first theorem gives you such statements.
- the wikipedia article says that the proof of the second theorem can be obtained by formalizing the proof of the first one. I don't know about that. We didn't do it that way in my class, never read about that either, not even in philosophy books. seems like a cumbersome way of doing things.
@dekippiesip: I too, am struggling with this same question. If anyone has anything to add here, I'd appreciate.
ZFC set theory is not inconsistent, but the standard set theory at the time was. Modern mathematicians use a set of axioms to avoid paradoxes.
Very interesting and worthwhile video.
I am a man. Self-referential, but not "stupid and fake". The problem with those self-referential paradoxes is that the object of those sentences are not complete enough to ascribe "truth" or "falseness" to. "This sentence" is neither true nor false, by itself. It does not assert or deny anything. Same thing for "time", or "space", or "apple", or "man", or anything like that.
Everyone holds all truths of life in their heart, me and many others have found all truths needed. I feel blessed. God bless you all on earth!
Of course not. You can't prove/disprove a negative. Why should something that doesn't exist (like the hundredthousands gods of the humans) be part of a scrutiny in the first place?
I mentioned it because the lecturer mentioned some of Godel's personal beliefs.
@Nukutawiti:Right, one has to rely upon an EXTERNAL system to prove consistence of the target system. That's incompleteness. If the mathematical system were complete, then it would be able to prove it's own consistency. But it can't.
You must get ahead of inconsistencies and find involvement with true conjecture a future which satisfies a dream yet unattainable in present circumstances of inner desertion.
Is mathematics merely the natural language of physics, or is it also indispensable at the core of its conceptual development, as well? In other words, is the conceptual foundation of physics ultimately mathematical in nature?
"For something to be either true of false it cannot be self referential, it must refer to something outside itself."
Really? "This sentence has five words" is self-referential and true.
Very good Global!! Very good.
Great talk but the constant pauses and lip smacking were incredibly annoying.
how does an inconsistency in the set theory prove that 2+2 = 5?
+anushka trivedi
Let S be the set of all sets not members of themselves. Since all elements of S are not elements of themselves, it follows that S is not an element of S. Well, then the statement 'either S is not an element of itself OR 2 + 2 = 5' is a true statement, since the first statement of the disjunction is true. But then, since S is not an element of itself, it is by definition an element of itself. Therefore, '2 + 2 = 5' is true.
By the way, this works for any statement. For instance, the existence of this paradoxical set S implies that squares are round.
Thanks for giving such an enthusiastic talk about Gödel. Maybe you know my booklet (written together with Casti) "Gödel: A Life of Logic" (Casti+DePauli, Perseus Books, Cambridge MA, 2000.) Also: "Wahrheit und Beweisbarkeit" htp+bv (Hölder-Pichler-Tempski + Bundes-Verlag, Wien 2002 Volume 1 and 2). Or: "EUROPOLIS5 Kurt Gödel, ein Mathematischer Mythos" NOVUM_Verlag, Horitschon, Austria 2003). There exists also a film with the same title, produced by the ORF= Austrian Television network, 1986, you can buy from the ORF-shop (But the film is also in German).
@QuantumBunk: One could say paradoxes are "stupid & fake" if they are KNOWN and ACKNOWLEDGED. One of the biggest problems with mathemetatics, science and computer technology is that they are being sold to the public as techniques and systems that are infallable. The general population does not understand the weaknesses and faults inherent in these system, and therefore they place them on a pedastal that is undeserved.
No one compares Aristotle with Plato. In the same way Kurt Goedel can not be compared with Aristotle since he is a Platonist.
Set theory has not been shown to be inconsistent. It is taught because it is useful.
oh Mark, still waiting for Godel.
Meh. What happens if we allow 'logical' statements (the Russel Paradox) in set theory to violate the law of non-contradiction? Set theory becomes inconsistent.
What a surprising result.
I still think such (self-contradictory) statements should be held to be undefined (much like dividing by zero), unable to take a truth value because in a very real sense they are neither true nor false, but meaningless.
So much for 2,000 years of controversy.
Thanks for this! Want to do a Comp Sci degree, really enjoy this.
@HebaruSan If you start out with pure logic you cannot do things like a=a+1. Pure logic doesn't admit the concept of infinite procedural truths.
This goes back to something called the paradox of analysis. Either it is true or it tells you something useful, not both.
a is a
a is a+1
then a+1 is a
If a is 1 then a+1 is 1
Unless you want to claim that 1=2 you cannot hold the proposition a+1=2. This is prohibited by the law of the excluded middle.
@MagisterPridgen: yes, the "anchors" are outside the mathematical system
the mind which rules the dream is not within the dream, the brain waves are not part of that system. Holographic 3d universe is ruled from 2d reality.
@Bloke Poppy: My point in the earlier post is to illustrate that a system where the public believes in God-over-men creates more freedom for the people living in that system. When God is disbelieved, men can fill that role and exert godly powers over the public, resulting in massive suppression.
The labyrinth finally destroyed Goedel as it did Newton, Weierstrass, Cantor, Frege, Ramanujan, von Neumann et al. They used bad infinities & not the transfinite fractions of Non-Cantorian set theory. I painted a large triptych called The Madness of Mathematics.
What's your game?
That is of course the question which haunted him , near the end of his life . I simply find it hard to believe the great Kurt Godel actually believed in the Judeo/Christian personal god . I would side with those who say he believed in Spinozas god , which I do . Honest people can disagree on which one he believed in , no ?
Point of clarification. When I was 22 I had my first profound realization about mathematics. Up to that point I had continuously struggled to find the relationship between maths & literature. Is there a difference? Which is most fundamental, etc. Well I discovered that the field of mathematics is = to literature: some fiction, some non-fiction, but just hadn't been labeled as such.
Anyone know the name of the guy giving the talk?
+Sholto Bateson 0:07
Kurt Gidel
Yes, my point was that the inconsistencies have been resolved.
did he try?.......can you proof, what you say?
Can one actually give an example of any set that is not a member of itself? Seems like nonsensical playing with words
Why is it wrong?
Please do not forget, that people like Einsten and Gödel lived in times, when being an atheist was not an option. If anyone would have taken the position of an atheist, he wouldn't have the chance to study or the chance for a job. At this time, churches have been overcrowded on sundays. If you're not there taking part in praying silly crap, you'd have a very good excuse or become an unadapted outlaw. Like in the bible belt today.