27. Gödel and the Black Hole of Mathematics | THUNK 

Gödel was 25 when he published his Incompleteness Theorem! What are we all doing with our lives.

I could be wrong, but after reading Godels work and similar problems that occured with Euclid parallel postulate was that Godel found that in any system logical system (axiomatic systems) which uses arithemetic you can ask a question that cannot be answered "within" the system but with the important caveat that it may be possible to add "more" axioms and expanding the system so that such questions could be answered. In many but perhaps not all cases, this is possible. Your youtube post seems to say we should give up and take a nihilistic approach completely, but I don't think that is the exact meaning we should take away from Godel's work. It could be very natural to assume that axiomatic systems always need to be enlarged in order to answer more questions. In any case, I would not regard this as an "incurable plague" but rather a new insight into how the universe may work. It may be speculative, but I think that this is the way quantum mechanics and various measurement paradoxes therein work as well, although for now there is not a clear connection. It would be very interesting if there was a real connection between the two. Then we might consider that which we do not yet know (that which is unmeasured in both our past and future) as undecideable until a new measurement (or axiom) is added to the system. Automata are similarly restricted and either find a closed loop or we need to add new rules so they can explore a larger space of possibilities. When this is not possible, then the automata recurs (cannot change) and is for all purposes dead to new experience (and time). In a way black holes are similarly sinks of information and limited time in what we believe to be the physical world. If the universe is a type of computational machine (or more likely we as observers are the computational machines) then perhaps mathematical undecideabiltiy where there is not possibility for axiomatic expansion, biological death, and black holes have some deep connection.

Great Thunk Episode! I always walk away from your programs a little better informed and knowledgeable Thanks !

Really enjoyed the video! Subbed. When I first discovered Godel's work it blew my mind since, like most people, I believed the entirety of the universe could be explained mathematically. Godel showed that this wasn't even possible through logic since it is a system based on axioms as well. To me this is exciting since it opens up the possibility of whole worlds or knowledge that we may have overlooked.

I have read about Goedel's incompleteness theorem. Being able to add is not suffficient to trigger it. Being able to add *and* multiply allows creation of an undecidable sentence.

This is maddening,I've always thought that math was the most fundmental thing about science but Gödel just shook it up.

Your joke about Godel refered to itself. Well done.

Russel knew there was a hole even when it was published but he doubted anyone would dig into the books sufficiently. Few people in the world at the time were capable of reading it and understanding it. So because of that he worked for years to resolve it and could not so they published it anyway. One day at a conference a young man approached him and told him about the hole he discovered. Russel was both delighted and disappointed. That young man was Gödel.

THANK YOU SO MUCH FOR THIS VIDEO. The wiki page for Godel’s Incompleteness Theorem went way over my head, and your video really helped me understand some stuff.

This was a terrific inroad for me in understanding the concept of Godel's Universe, thank you!

The most fundamental fallacy in Curt Godel's Incompleteness Assertions is that it proved that every formal system is incomplete. The heart of the fallacy is that a "Godelization" is impossible to formulate for any self-reference. Just try to construct a "free variable" for the assertedly undecidable sentence. You will never be able to construct the "Godelization". It is because of the failure that no proof can be even attempted.

Godel sentence does not say of itself that it is undecidable but that it is unprovable in the system. Undecidability implies that its negation is also unprovable; so it's a bit more complicated. However, it seems to me that a sentence claiming itself undecidable (which exists indeed) would have done the job all the same.

There is a fascinating application of Gödels theorem and the Turing machine during WWII, called "the bomb". Since contradictions makes anything in a system provable, Turing designed his machine to sift through the encrypted messages of the enigma machine to find where there were no contradictions (that the letters "ABC" were transcribed to "BDF" once and "HYT" another time, this with a lot of other related information about the settings of the enigma machine proved to be of value). A lack of contradiction (meaning that the key might be the one to decrypt the message) would complete a circuit and a current could start flowing. Fascinating stuff, I think this kind of "self-reflection" of logic will prove to be really useful, perhaps will it be the center of future AI technology? I don't understand myself, so why should a computer, let's program this into the heart of how it understands the world!

Because no "Godelization" is possible for self-referencing, no Turing machine will run for the impossible self-reference. Self-referential programs are impossible to "Turing Compute" on the same processor where the program resides. A different processor is required to execute the program because a larger, second processor might permit referencing of a truly "free variable".

How do you define something using itself? It becomes an issue of relativity. If you use a part of the same system to describe itself, the part that is describing itself is also part of that system. And therefore is incomplete. You would have to manage for a system to be able to be described itself, without the act of describing itself. Or the entire system is only a function of self describing. Which creates an eternal loop on itself. Hence, if you have a 'set', that contains 'all sets'. That cannot be true because it would have to also contain itself.

What we know; knowledge, is only relative to what we don't know. It is like a text on a paper. Remove the paper, background, contrast, and we end up with nothing which is the backside of everything. So to have a mathematical theory that is both complete and consistent, is the same as removing the contrast, the relationship which I believe is the observer; ourselves. ☯️

The haltingproblem is not a big deal for everyday programming. Most of the time the programmer can see the constellations that would cause and endless loop. And then it is just simple: Put an if-statement somewhere and force the program to end.

It appears you made a mistake in understanding the halting problem. It's not that all programs cannot be proven to complete. In fact there are plenty of programming languages, in which any program written ( no matter what it is ) is known to complete. Since, for the most part we do want a particular program to run indefinitely under certain conditions, those languages are not as useful for general-purpose programming. Additionally, in almost all programming languages, you can write a program that can be proven to complete (eventually). Your statement that that is always some input that will leave it processing forever is wrong. Your conclusion, that no matter what computer you have, there is some number that will just "break" it, is absolutely wrong, and quite silly.

I just got a chance to see this video of yours, and skimmed through a couple more . . . excellent work bro keep up it up and you shall break through . . . the other guys with similar channels got nothing on you (except maybe a little more production value)

incompleteness theorem only face formalism but there are a lot of mathematical philosophy consistent and complete it is not a problem for math its a problem for those who wanted to prove that math is just a languageen.wikipedia.org/wiki/Philosophy_of_mathematics

The best, at any rate most creative, proof of Godel's theorem can be found in a Star Trek TOS episode, Titled: "I, Mudd" (1967) [Captain Kirk: trying to confuse an android] Captain Kirk: Everything Harry tells you is a lie. Remember that. Everything Harry tells you is a lie. Harcourt Fenton Mudd: Now listen to this carefully, Norman. I am... lying. Norman: You say you are lying, but if everything you say is a lie, then you are telling the truth, but you cannot tell the truth because everything you say is a lie, but you lie... You tell the truth but you cannot for you lie... illogical! Illogical! Please explain! You are human. Only humans can explain their behavior! Please explain! Captain Kirk: [giving him the same statement the androids have repeatedly given him several times before] I am not programmed to respond in that area.

Well this is just nifty. :-) It's always seemed to me that mathematics floats in mid-air. Not so much from using to prove or disprove math itself by using math, but rather from the fact that numbers don't exist. You can't eat a 2, or kick a 7. Numbers are metaphysical, not physical. They exist only in the abstract. But fortunately I've managed to keep myself reassured by thinking of mathematics like a ruler. It just measures quantities instead of distances. But the king's foot length is in the mix, nonetheless.

Yes, Godel's incompleteness theorems are a Three Stooges face slapping routine; with Godel doing the honors to every positivist that claimed to have the ultimate answer. Godel, being a mathematician, instinctively knew that the answer to the ultimate question was 42. Stephen Hawking wrote an interesting (very short) paper in 2002 titled: "Godel and the End of the Universe". It can be found here: hawking.org.uk/godel-and-the-end-of-physics.html

Gödel more or less proved determinism impossible. That's impressive.

Spinoza in his writings said that the Idea of the body is just the ideas of the affectations to the body, and the idea of those ideas are just the conscience or the soul, so to put it in a second order logic problem idea(idea(aff)) .--->. idea(idea) only when the affectation (causality) halts but not the idea of the idea now how undecidable is that, so the idea of the ideas are the free or "halting" variables

No Math is not broken, Peano Arithmetic is incomplete, as is ZFC Set Theory. In practice, this means we can not create an algorithm to solve all theorems (true or false) in higher level systems of logic. The main outcome of this limitation is that we can not algorithmically prove that software is free of defects.

It’s only undecidable because scientists are looking for objectivity in a subjective world. This statement is false is up to the discretion of the speaker and the observer

The Continuum Hypothesis is a good example of how the Incompleteness Theorem can blow your mind. There are an infinite number of integers. For every integer, there are an infinite number of irrational numbers. For every irrational number, there are an infinite number of subsets of irrational numbers. We can assign an integer value to each type of infinity (0,1,2,etc...) The Continuum Hypothesis is the idea that there are infinities with non-integer values (0.5,1.5,etc) Would you like to see this idea proven true? Tough. Would you like to see this idea proven false? Tough. In a certain system of mathematics known as Zermelo-Frankel set theory, this idea is *completely unproveable*. Also, Godel found a solution to Einstein's equations of General Relativity that would hypothetically allow for time travel. Just throwing that out there.

So, what's most interesting about this, is what it might say about our universe, to the extent we might believe that our universe seems to follow rules which are mathematical in nature. If we would like to feel that the universe is entirely mathematical at its core, this presents an interesting problem, as Gödel seems to indicate that it should be either intrinsically incomprehensible, or contradictory. It seems also reasonable to expect that a self-contradictory universe ought not to be able to successfully manifest itself, so that option ought to be a non-starter, and so we're left with ..what? What is the exact physical analogue to this mathematical condition? An undecideable physical state, or just an aspect of the laws of nature which we can never hope to get a handle on? Does that make this a proof of the existence of the ineffable? I tend to think it is saying something about the nature of paradox. A paradox arises when an axiomatic system runs into its Gödelian contradiction, and must resolve it by being subsumed by a hierarchically greater, richer system, wherein the paradox is resolved by being revealed to be a doorway to a more complex axiomatic landscape. But this new system will also have a contradiction, which must likewise be resolved by appeal to an even richer system, and this then goes on to form an infinite regression. And a mathematically based universe which exists does so by having embodied that whole infinite regression. ...At least, that how it feels to me...

This video raises some great points, thank you.

I forgot I was on RU-vid. A clip from “I Mudd” (Star Trek - Liar Paradox) is at the following link: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-EzVxsYzXI_Y.html Gene Roddenberry’s take on the Godel sentence.

Can godel incompleteness be fully defined and evaluated? Wouldn't it be avoidable then? You've given an identifiable example that should be avoidable. Only if we don't know if a problem is godel incomplete is it really a problem. I wonder if this line of reasoning is useful. This all seems very distant from being empirically useful.

(abstract) Lets say you have "Tautology" and "Contradiction" mixed with a "anti-Tautology" and "anti-contradiction" at the same time. What will be the outcome?

So, this means the incompleteness theorem is also incomplete?

That quote from Dick stroke me with surprise: I guess I've seen that situation before. Actually, I've LIVED that situation before. I've had a strange thing happen to my mind 4 or 5 times in my life. It happened following a thought or a concept that got to my mind at the time. The first time it happened it was definitely a self-referencing thought. It was the strangest thing I've ever experienced. That thought put my mind into a infinite regress and I actually thought my brain was about to implode the Universe. It scared the hell out of me. This, I figured, might be related to epilepsy, though I never see anyone describing it like this. Well, if anyone wants to know more about it, I'll be glad to talk. Well, good work. Keep up!

Thank you for this very lucid presentation. I'm interested also in how an incompleteness principle might apply to physical inquiry as well. In other words, can any universe provide the means for explaining its own existence? To me, it seems that it can't. Something outside the "axioms" of physics must always be introduced--such as a violation of the conservation of energy or a decrease in total entropy. These violations would seem analogous to what Godel's incompleteness theorems describe for mathematics. I don't know what formal research has been done into this (or if I'd even begin to understand it), but I'd love to hear a brief discussion of it for the layperson.

Thank you

I really think that inherent uncertainty is a fundamental principle of everything. But also not relevant as it doesn't stop macro functionality. This is true of quantum phenomenon and weird math quirks like that, the planets stay good little boys and do what Newton told them too.

Would the system needed to describe the entire universe be bigger than the universe itself? How would that work?

Super bad description of Godel. The undecidability of the completeness of arithmetic doesn't undermine the FACT of the completeness of arithmetic; only it's knowability

Here's a thing not to put on that list : "not this"

Is this the kid from the wonder years?

You've got a step there that amounts to 0=0. But nothing doesn't exist in reality, so trying to do math on it is impossible. Math is descriptive of the relationship between idealised unary entities (that also don't exist) .

OK, so a shared this on my facebook page and my very intelligent cousin-law responded with this " I wish he had further defined the difference between Mathematics and Arithmetic." I told him to send you a comment about it and ou would answer, but he didn't and I would to know your answer :) Hope you are having a great weekend!

My friend. And for all of you that wish the incompleteness theorem is just a special case. You could not me more fool. Sorry for my bad english, but... See Church-Turing thesis first... Okay, I will assume that you know what a Turing Machine is. It receives a input, and raises a output. There are some languages a Turing machine can decide very quickly, but there are others that can not be decided so quickly. There is a class of language called R. This class R contains all the languages that can be decided within a FINITE AMOUT OF TIME. If you know about reducibility then you know that it means that the class R contains all the problems on UNIVERSE that can be solve in a finite amount of time by a turing machine. Okay.. nothing really amazing yet. But, the complement of class R, that is, the class U that contains all the problems that are not on R, or if you prefer U = {x | x is not in R}. Turns Out that U is a LOT BIGGER that R. U is alef_1, while R is alef_0. R is as big as the natural numbers (same size as intergers and racionals), but U is as big as the real numbers. For these problems on U: We are never gonna solve or We need a Machine more general than that turing... Here are more bad news: Turing Machine are so powerful as Lambda-Calculus. So powerful as Primitive Recursion Functions. So powerful as multiplayer-alternation-games, in fact, turing machine ARE our computers. And for the last 100 years, nobody could think of a possible and more general machine. Quantum Computers?? No. Quantum computers aren't more powerful than non-deterministic turing machines, and by that, they aren't more powerful than determinist turing machines. Quantum computers only adds efficiency. Think of this way: Computer Virus, and the problem of recognize if a certain program is a computer virus. They are undecidable. And that is example of a nice problem that would be veeeery good to solve.

excellent

cool stuff!!.....seems to me that a godel sentence is pretty decisive though!...it's decided to be both true and false, the problem is that human logic can't cope with this but nature (is my gutfeeling) can!.....

Who needs the 5 finger death palm money strike to stop the heart of their opponent after 5 steps. I've got a sentence that will freeze your brain, fool!

In the end, you say "maybe Gödel's theorem is just a special case", but I wouldn't say so. In the end, if you think about what it means clearly, you see that his theorem is intuitive, and is way more general that you might think. Let me explain what I mean: Gödel's theorem applies to most (well not most, but most useful ones) first order theories, and it feels totally normal to say : here's my first order theory ,these are my axioms : ax1, ax2, ... axn; these axioms, I checked, are necessary -by that I mean that I could not get the same theory if I removed one-; what happens if I get rid of ax1 ? What can I say about ax1 in the theory ax2,...,axn ? Well if I could prove that it was true, then it's not "necessary", so what I said earlier wasn't correct. If I could prove that it's wrong, well then my first theory is inconsistant, contradictory. That's not cool either, so let's suppose my first theory isn't contradictory. Then I can't prove ax1 to be true, nor to be false. It is therefore undecidable (that's basically the idea). So when Gödel says "all theories that match such and such conditions have undecidable propositions", it shouldn't surprise us: it simply means that we need to "add" some axioms (but of course, we can never add enough). Those undecidable propositions are therefore more than simple gödelites, such as "I am unprovable". But it was an interesting video, and I think it was a good approach to vulgarizing this beautiful piece of maths, well done !

xkcd.com/435/ Well, well, well... how the mighty have fallen. Seriously though, does this have any monumental effect on the way we understand our universe? Hyperbolically, is quantum mechanics flawed because we had to *assume* that 1+1=2?

+THUNK The "ö" in "Gödel" is pronounced more like the oo in "took" or "look" fyi.

I‘d say „This sentence is undecitable within the current system“ is perfectly decitable: It‘s just false

yes... russel sets!

Your explanation is ok, but it makes it sound like the only statements like this (the ones that "break" mathematics) are weird mathy sentences that have no relation to real life, variations of "This statement is false." However, there are important theorems (see the Continuum hypothesis) that are undecided like this. Also, your prononciation of Gödel makes me laugh.

Great

My brain is broken 🤣

And here I am telling my 8 yr old Zoe that there is always one right answer to math and she should always know and not guess it. In second grade that is....

There's a Goedel "Dilema" for every single machine or carbon based form. Nothing can escape the flaw. Not one thing,... 'except maybe in the spiritual world.

It is quite easy to see that the "Godel Fallacy" is, itself, a fallacy. The sentence "This sentence is false" i often cited as proof of incompleteness. However, the sentence "That sentence is false" is never cited as proof of incompleteness. The two sentences differ by more than the pronouns. In the first sentence, the target of the the "This" pronoun resides on the same processor, meaning within the same memory space as the program. The second pronoun, "That", has an unresolved target free variable location that can not be resolved prior to runtime. Therefore, neither of the two sentences is Turing Executable' on one processor.

There's a lot wrong things in this video. First at 2:49, that sentence is not undecidable, it is called the "Liar paradox", it just shows that any system which contains this sentence is inconsistent. Russel came later to give a similar sentence known as "Russel' paradox" to prove the inconsistency of a system given later by someone else. It's after the foundation of set theory which is founded in order to avoid these inconsistencies, that Godel came and gave an example of a sentence which is true but improvable if the system is consistent. This is not a problem for mathematicians. That sentence can be formulated using some mathematics to this sentence: A="This sentence is not provable". In a consistent system, a human mind can see that this sentence is true, but a computer or another algorithmic process cannot decide whether it's true or false. This shows a fundamental difference between minds and machines.

Kurt Gödel 's proof is about as watertight as Zeno's paradox is true. Its not, and its plain in nature that its not. Its a no-brainer that not all sentences can be categorised into the boolean true or false. It makes no sense to ask if "Think about this." is true, any more than it makes sense to categorise any sentence as on or off. We predefine the meaning of true and false (or on and off) as pertaining to the types of things of interest, and we define the elements and methods that are inside and outside of our context. The incompleteness theorem and other similar illegitimate exercises in circular logic are the plaything of insane genius. We find the same invalid type-casting in Zeno's paradox, where Zeno applies an invalid counting method predetermined to fail, to assert a patent nonsense. The riddle is to discover the flaw in his reasoning, and yet we choose to disregard the patently obvious predefined rule that Zeno has broken - namely, Zeno cant count (because counting is done with with fixed intervals, not infinitely decreasing ones). Zeno was an idiot. Russel was a complete nutter, and Gödel was a lonely troll looking for attention who found it in pretending to understand the cryptological latin lunacy of "Principia Mathematica". These guys were INSANE, ok?

stephen cury

Dear Thunk! Please react to my attempt. I have a foundation in math. The proof of Gödel contains the not-discussed and therefore implicit assumption that a sentence containing an empty self-reference is acceptable as a “statement” or “theorem” (a proven statement). In the great tradition of Dutch intuitionistic mathematicians, like Brouwer, I reject this assumption. I claim a sentence containing an empty self-referential construct should NOT be accepted as a statement, and therefore NOT be accepted as a theorem, because a sentence must be a statement before it can become a proven statement. Therefore a sentence like “This statement is false.” should NOT be accepted as a statement, because I will now proceed to show why that sentence is an empty construct that doesn’t claim anything, and when we accept that sentence is NOT a statement, it stops constituting a paradox. You should not just go ahead and claim the sentence is either true or not, because that is a property of sentences that are statements. Statements actually have to claim something other than being a self-referential empty construct. Let us analyze the sentence “This statement is false.” It actually makes THREE claims. The first (explicit) claim is that that sentence constitutes a statement. The second (implicit) somewhat hidden claim is that statements are either true or false. The third (explicit) claim is that the sentence constitutes a statement that is false. Let us now proceed to examine the sentence that claims to be a statement. The sentence contains a reference to the whole sentence. Let us substitute what the reference stands for. We get the sentence: “ “This statement is false.” is false.” However, this stil contains a reference,so let’s substitute again: “ “ “This statement is false.” is false.” is false.” Going to more and more substitutions we notice that inside all double quotes nothing is being claimed, it is an empty construct. In my opinion it therefore fails to be a statement and therefore cannot be concluded to be either true or false. Gödel’s sentence is more complex but contains the same fallacy that every sentence constructed by his numbering mechanism is a valid statement that therefore can be said to be either true or false or rather be provable or not (“Beweisbar”). I suspect that Gödel’s theorems are true, however, his own proof is flawed as I have just shown.

There is an amazing amount of misinformation in this five minutes video. The whole feeling you get is that this guy's language is meaningless, rhetoric and creative... he is not someone you would expect to remember a lot of stuff.

This comment is grammatically incorrect.

Godel also believed that the human brain could not have been created by evolution. Sorry, Darwin.

First, Goedel’s inspiration of the Quine version of the liars paradox, “this statement is false” should drive anyone studying him to suspicion. “This statement is false” is devoid of any semantic value. It is piffle and I cannot fathom how it could be presented in this video as a statement of any consequence in philosophy. The term “statement” in this statement is the subject and a noun. It is also a set definition but without members. Were it to have any, it or they would be previous utterances to which “this statement is false” referred, but then that would not give rise to the paradox. It can only be self-referencing. Therefore, the only means to such a statement is sophistry, a manipulation and abuse of the logical structure of the language. Moving on, Goedel claimed that if the statement “this statement cannot be proved” were true, then it would have to have a proof (if true then it can be proved yet claims it cannot, thus the paradox). Proof of what? Can one say that “there is a proof of that” when “that” has not been identified? The rule of the necessity of a proof in such a case is then as meaningless as the statement which is its object. There is no content in the latter to which the former refers. We are conveying gibberish. If one were to trace the formulation of “this statement cannot be proved” back to the material of its origin, what would be conveyed in that? I would have thought that propositional calculus alone would have been a sufficient means of reformulating mathematical statements into those semantic, but it appears that more translation was required, thus the Goedel numbers. So I believe that if we extrapolate back from “this statement cannot be proved”, a self-referencing statement which is by that alone, illegitimate and conveys no information about anything, to generate such a contradiction which seemed to be his intention, it only follows that a (designed) means permitting contradiction would be required. Consider, a self-referencing statement cannot be formulated unless the subject content is eliminated that the paradox might be manifest. In the case of these paradoxes, the object of the statement in question is made to break the law of non-contradiction by being both true and false at the same time (only possible in the absence of the subject content). Additionally, it is required to be at once the cause and the effect of the paradoxical function. Could there be any more blatant distortion of the logic and the architecture of the language? This is sophistry and not science. If Quines paradox was Goedel’s inspiration then there is something very wrong. How can such a purported feat of mathematics arise from the implementation of such a fraud as self-referencing statements? How can it be accepted that the exercise of a valid proposition could culminate in a contradictory conclusion? What does that suggest of the means of its formulation? Logic would advise us, if its employment is still permitted in the context of discussions of Goedel and his work, that if its conclusion were a conceptual contradiction and the only means to it were prerequisite omissions of the strictures of logic in the process of its formulation, that mathematics, said to be incapable of self-referencing statements on its own does not and could not contain the unprovable statements as claimed.

Godels Incompleteness Theorem is not correct at all. Its all about language and not 'axioms'. 1) This sentence is false - is meaningless because 'this sentence' has no content which can be falsified. Its a word salad and not computing or mathematics 2) We cannot prove that an even number is always the product of two primes although we believe that its true. If we believe it to be true then language dictates that is *provable* because that is the meaning of *true* so 2) should read: We cannot prove that an even number is always the product of two primes although we believe that its provable. Its a language contradiction. Godel never proves his own theorem - its a little money-earning joke he made in order to buy a house :)

yes... russel sets!