Відео

Thanks! I wish I had a good reference. For what it's worth, whenever I have talked to an actual economist or person in finance, they react as if my interpretation is insane. But they do not point to exactly where I got any of it wrong. As far as I can tell, they are operating in a reality that is built upon a very nested self-referential jargon, and their view makes complete sense after you have adopted that jargon.

@@anuprao11 Thanks, appreciate the response. Mirrors my difficulties in finding a good source :)

By true they just mean true in a larger system that we are using to talk about Meta Mathematics. Just ignore the "true" part as it is meaningless. Gödel's incompleteness theorem simply says that there are statements that cannot be proven even when I just used logical reasoning within the system I am talking about (i.e. without having to invoke metamathematical arguments to do it).

@@zemm9003 First, thank you for responding. I love these discussions and I really want to understand fully the topic but I have no math background due to having a sufficiently debilitating case of dyslexia so I could never study it as was typical when I was in school (a long time ago). Let me clarify my position….IF one formulates a proposition he can only do so by logic or it couldn’t possibly make any sense. The strictures of the structure of language would impose and make that clear. The logic by which a proposition is formulated, by which the conclusions are drawn and by which it is presented is that by which it should be critiqued. Is that not so? I would think that if this is not the case, the structure would not exist to have this discussion. Anyway, I have never seen a video on this topic that didn’t present it as a translation of a sort from mathematical language to what I think is the system of propositional calculus of B. Russell and Whitehead. Is that the case? If I am correct in this then I would think that my critique applies. When you make a statement in which a part is contingent upon another, i.e., (“this statement is true) but cannot be proved”, the latter contingent upon the former, the logic by which the dependent statement was defined has to also define the former. As stated in my post, the statement “this statement is true but cannot be proved” or “this statement cannot be proved” (as I have sometimes seen it) can only be either a self-reference, that statement referring only to itself, which I don’t think can be the case for as I understand it, Goedell translated the mathematical statement through his number system to the semantic we are discussing, or as I mentioned immediately above, it is a statement referring to some previous statement in mathematical language then translated. If the former, it makes no sense and why would he have required the translation scheme at all? If the latter, then the essence of my critique I would venture, still applies. If referring a previous statement in mathematical language, the claim of the dependent statement “…but cannot be proved” is dependent upon the former for its very existence. We are back then to consider, how and why is it asserted or assumed that the statement translated is true? Once again, if that is known or defined as such by the translation scheme then it can only be that the how and why it is true is known and again, that would be the proof. So I don’t get how this makes any sense at all and of what possible use it could be. Materiality does not permit contradictions, e.g., a rock cannot be both here and there at once. This applies to the realm of the abstract as well, itself a reflection of the material realm. All abstractions/concepts are a product of material contextual referents by which contradictions are also not permitted. Again, one cannot “appeal to truths to formulate a position which denies the existence of truth, i.e., one cannot claim “I think I am not thinking” and expect that it be considered valid. I don’t think that Goedell found a hole in math or logic but rather introduced one by the same sophistry employed in the paradox of Quine, whom Goedell apparently admired, something I cannot understand. Quine’s liar paradox, “this statement is false” was pure piffle. What do you think?

@@jamestagge3429 Gödel's Theorem is purely about syntax. So that he avoids pitfalls by purely computing what he is saying. Gödel's Theorem just means that there are some end strings (Theorems) that cannot be obtained from the starting strings (Axioms) by using inference rules to calculate what are the valid and "true" strings of the language and inside the Theory.

@@zemm9003 Thanks again for taking the time to respond. I guess what I don’t understand is the context from which all this arises. 1 + 1 = 2 is logical. Why? because 1, an arbitrary name applied to a concept which is unequivocally as to what it is, e.g., 1 is simply, logically a “thing by itself”. By virtue of the nature of our minds, whatever they are, when another “thing by itself” is observed, it is understood by the very structure of the mind as “another single thing by itself”, the second distinct from the first. With two “things by themselves, we apply the name 2 if we wish to refer to the two together so the + sign simply is a proxy name for “and” and = a proxy name for the act of their placement together. If this kind of thinking is extended out into more and more complex formulae, I don’t see the problem in there being ways to prove each and every such formula. What am I missing? That Goedell chose to translate these kinds of things into semantic analogies, certainly the means of the critique the conclusions of those translations would be by means of the brand of logic by which the semantic constructions were formulated, yes? Are these not reflective (as analogies) of the mathematics from which they were derived/translated? If not, what was the point of his scheme? If I am correct in this relationship between the original math and the consequent semantics, I would venture that to successfully critique (show the faults in) the conclusions of the theorem as expressed in language, the mathematics problem would be dispensed with by definition. How is it that this might not be true?

@@jamestagge3429 deducting valid formulas is not the same as deducting theorems. Valid sentences depend only on the definition of the language you want to use. Provable sentences depend on the language and on the axioms you want to use as a starting point (and inference rules). Not all valid sentences are theorems (even if they are good enough to be provable outside the system in a bigger system) This is what Gödel has shown. That you can always find new sentences that look like they should be Theorems (or their negation) but are actually completely independent from the Axioms and are thus undefined.

During the argument, we showed that there is a long list of statements such that most of the statements are true, and none of them can be proved. This established that there is a true statement that cannot be proved. So, we managed to give a proof that most of the statements are true. But we did not find a proof of a specific true statement in the list.

@@anuprao11 No, it doesn’t help to resolve the contradiction in Goedel’s reasoning (if the videos such as this one are to be considered correct), but I do thank you for your response. It is very instructive and makes this discussion all the more pleasurable. “If” you make the claim that there are these or those statements which are true then we are required to assume that these statements are in fact true (or there is little reason to have this discussion) and if so, we know that logically, you cannot make that claim unless “you know how and why they are true” that you might state them as such. To argue on the side of Goedel, i.e., this argument, you deny the very logic by which he formulated his proposition, he basically “appealing to truths to define a position which denies the existence of truth”. I don’t care how smart he was supposed to be, this is illogical, makes no sense and certainly does “not” show that there is a “hole in mathematics or logic”. Again, to claim that “there is a long list of statements such that most of the statements are true, and some of them can’t be proved”, one must explain how he could make the claim that they are in fact true. If he can explain how he knew them to be, THAT IS THE PROOF! This is piffle. Goedel employed structurally sound logic to formulate his propositions/theorems without which he could not have done so, i.e., if he could not have understood exactly how it was that they could be determined to be true (which is again the proof). One cannot make the claim that there are statements that cannot be proved (or “this statement cannot be proved) in the context of that above. The statement, “this statement cannot be proved” in terms of its meaning (even as a self-referencing statement) is in a sense, contingent upon the assumption that there is a logical structural necessity for the determination for certain statements to be considered true and could not then determine the lack of some statements, i.e., the logic by which the statements would be demonstrated to be unproveable would have to encompass the means to their truth. So this logical, structural necessity is the means to the proof of any true statement. I don’t see how Goedel could be considered having found a hole in logic, etc. What do you think?

If you do not find the explanation in this video convincing, then please point to the exact step of the argument made in this video that you do not find convincing, and we can talk about that.

@@anuprao11 Sounds good. I will watch in its entirety today. My comments are a result of many other videos on the same topic, none of which make sense to me. But let me leave you with this and tonight I will watch the video and comment, to which you can respond. In Quine’s liar paradox, “this statement is false”, which is obviously self-referencing, he is employing the worst kind of sophistry. I indict Goedel for he did the same. How could a man considered so brilliant think well of the liar paradox which is pure piffle and want to employ a similar self-referencing statement in his discipline? In Quine’s paradox, the term “statement” is not a subject noun like rabbit or rock, the two self-defined in a sense, but rather it is a set definition but in the paradox, one without any members, i.e., no reference object by which it violates the law of non-contradiction, it being required to be considered true and false at once. (The same violation of logic was employed in the Nelson-Gelling paradox). This is sophomoric in my view. The statement that they sky is green is false would be a valid statement formulated by uncorrupted logic. The paradox is not. So too with “this statement cannot be proved”. Also, I believe that both of these challenge the context in which set theory is discussed so the damage is twofold. In order to formulate a proposition as Goedel did, there must be a structure of logic within which this is accomplished. That structure imposes certain strictures and realities which cannot be ignored or the proposition cannot be defined initially (and it was, or we could not be discussing it). As I mentioned in my previous post, to make the claim that [there are true statements(which cannot be proved)] one would have to know why those statements which are claimed to exist but which cannot be proved are true and again, that is the proof. The logic by which that is understood as such, is the same by which the claim that “they cannot be proved” is made. The knowledge that they cannot be proved is contingent upon that by which they are known to be true, or vice versa. How can one know that this or that statement is true “and” that it cannot be proved, the latter of which would be dependent upon why and how he knows it is a true statement? One cannot have it both ways.

@@anuprao11 no response to my last post?

Glad you like them!

You can find notes here: kam.mff.cuni.cz/~matousek/akt.html Follow the link to chapter 2 for several proofs. There is a geometric proof, which is the first one that Matousek describes.

@@anuprao11 Thank you! Now I have read the proof you linked. I have spent a nice chunk of time trying to find the sketches of these other proofs but it was a torture for many reasons, anyway... I managed to access just enough proofs and I think my proof actually coincedes with yours or the one that uses Tucker's lemma. I didn't really follow any proof together with all details, but I feel like yours has the same spirit as the one via Tucker's lemma but is more direct maybe. Still, both proofs are presented relatively rigorously and are imemdiately jumping on the utilization of simplices (Borsuk's still does). My proof also runs through "odd mapping has odd degree" part, but all utilization of simplices is already spent on defining the degree (popular and well understood), so my intuitive proof can focus only on geometric ideas and geometric properties of the degree (geometry = smooth surfaces exclusively, no PL). I like to think that such approach is easier to grasp and understand compared to those which enjoy arbitrary constructions with simplices, but it only makes sense if we assume homological algebra or simplices are hard to understand in the first place and that the mapping degree is still easy to conceptualize. All "different" proofs of this theorem describe the same phenomenon anyway, only the language (algebra, combinatorics, geometry) differs. It is not some complicated and hard theorem after all... Maybe I will proceed with my wish to make a UA-cam video on my proof (or rather my perspective on the proof), regardless of the facts I mentioned earlier. Hopefully some other people share that geometric intuition with me and will appreciate it. I shared all this here even after I lost the reason to do it, so don't mind me please, I'm sorry for taking your time.

Fantastic! Yes, the proof is presented here is identical to the second proof of Tuckers lemma in the notes. I myself am not completely comfortable with simplicial homology, and I assumed most people who watch a video like this are not. That’s why I did not define degree or any of those concepts, in order to get to the proof as quickly as possible. If you wrote up another presentation of a proof, I at least would be interested in seeing it!

@@anuprao11 That sure is nice to hear, thank you! If I really make it later this year, I will let you know.

Yes, I meant that it is not a subset!

Great question! What is the length of the shortest word in the free group with this property? If you want a short word you should double the number of nails in each step (rather than adding one nail at a time). That would give a word of length O(n^2) for n nails. For two nails I think it is clear that the commutator is optimal. That suggest some kind of inductive argument to prove that this is optimal, but I don’t see how to do it at the moment.

Here’s another question: what if you want the property that any k nails will hold the picture up, but k-1 nails will fail to do so. Can you design a solution?

@@anuprao11 Hmm so if we stick to concatenating the commuter it would grow exponentially, something like 4^n for n nails i would guess (maybe add a constant c to be safe). For 2 nails I agree the commuter is optimal. With some friends we thought of a way to (as you said) for n nails bound the growth to O(n^3) by rewriting the n in binary and isolating the 1's as examples of 2nails (perhaps this what you already thought of). So for example for 4 nails A,B,C,D and for loops a,b,c,d it would not be [[[[a,b],c],d] but [[a,b],[c,d]] which is smaller than the first word. For 5 = 101 this would be [[[a,b],[c,d]],e]. For your other question, wouldn't the concatenating commuters work, since if any loop turns trivial (by removing its associated nail) the whole expression disappears.

Nice! Yes that works. I guess that gives a solution of size (n choose k)^2 which is again exponentially big for k linear in n. Is there a polynomial sized solution?

@@anuprao11 Right so the idea I mentioned should indeed be in O(n^3) unless there is some faulty reasoning somewhere. I have a video where i recorded solving the above problem, its rather long but with the timestamps you should find an exact explanation of the idea above. I thought of typing it up on latex but classes just started again :X and we're still wondering if there's any better way than quadratic growth. If you have any ideas that would be wonderful ! edit 1:grammar edit 2: the bound is actually in O(n^3) not O(n^2) there was a missing term in the simplification.

You’re right! I guess it’s good to have some errors in the videos to make sure the audience is paying attention :).

@@anuprao11 It was indeed a good test of my understanding :)

There are many generalizations (try googling it!). For example here’s one higher dimension version: www.cs.princeton.edu/~zdvir/papers/DvirHu14.pdf

Shapes A_i where boundary(A_j) is within boundary(A_k) for j<k for i in naturals

A nice feature is that the properties we are asking about are invariant under linear transformations. So, since the integers points work, the points of any lattice also work!

Thanks for the comments! You're right about both errors!

Because every line has at least 3 points, there are always at least 3 red points! So at least two of them will be on the same side of the normal as the black point. The closer one to to the normal is always available to use as the green point.

It’s not so complicated! The statement I presented here is self-contained, crystal clear, unambiguous and does not require any further interpretation. There is no gesturing or pointing required.

What Gödel proved is that there is a sentence "G iff no proof of G exists (within the system)". He deliberately shifted away from semantics and made it into a syntactic problem. Precisely because he wanted to avoid the problem with involving semantics. His argument can easily be translated into computer functions. It is equivalent to saying that not all valid strings can be built using certain rules from the axiom strings and this can be simulated in a computer, it is equivalent to saying that given a sentence S I can try to create a valid sequence of strings that starts from the axiom and ends at S and Gödel's Theorem is roughly equivalent to saying that there is at least an S for which your search will never end.

​@@zemm9003 Thanks for that. I wil make two devils advocate arguments against it. First, look at "G iff no proof of G exists (within the system)": irrespective of proof, we know what G is nonetheless. A proof of G, in any form, could not possibly be better or more certain than our knowing G. But didn't Godel suggest that there was a problem with knowing G? In which case, G would be an incoherence, whether presented as a proof, as a proposal, or in a proposal (such as a sentence). For example, it is an incoherence to say "we cannot know X", or "only I know X". X falls away. Also, I cannot envision any computer function that could be built around this observationn, an incoherence is an incoherence after all. Second, "Gödel's Theorem is roughly equivalent to saying that there is at least an S for which your search will never end." But isn't this a truism, plainly evident without proof? Because I note that Godel, in order to make such a claim, invokes an infinite, he assumes the necessity of the very thing that is an element of his conclusion. Let Godel make a proof without invoking an infinite.

Thank you Thatchaphol! And thanks for your great work on these topics :).

Yes, maybe. But if there is a lesson that can help the human checker learn, then you could call that lesson the first part of the proof, and the theorem again applies. So if you think of the checker as the code of our brains when we are born, then there is a truth that cannot be proved no matter how much we are taught during our lives. Now the reality is that our brains are not a closed system with input and output… and our brains themselves change during our lives. So maybe we can think of the "program" as a simulator for the molecules inside our brain, and the initial configuration of that system is hardcoded in the program. Then during our entire lives, the program takes as input all interactions that our brain has with the outside world, and simulates the changes to the structure of the brain until the actual proof shows up, and then it simulates that too. The proof discussed in the video applies to this situation if the program is of finite length and all the input to it is of finite length. So, unless some step of these simulations cannot be carried out by programs, you have that there is a truth that we will not be able to verify!

Isn't the statement a proof, and a traditional proof is simply a duplicate of S in another form? Doesn't the statement prove the method of the proof? Which of them, statement or proof, has the honour or right to be considered as THE Proof?

The statement is not the proof of the method.Rather the rigorous method is the proof of the statement.

If there are two unprovable truths such that if you update the checker to be able to prove wither of them, the other becomes false, then the theorem is bad news again.

It was bad news for science because Godel said that it proved that the universe was not determined - against science which claimed it was determined. It's all crap of course, just a mathematician succumbing to a paradox and lessening his defeat by redefining it, as Godel did.

I defined exactly what I meant by "cannot be proved" in the video! The result is much stronger than saying that a truth is not provable in ZFC.

@@anuprao11 There may well be truths the can't be proven. The statement sounds similar to Fitch's Paradox of Knowability. However, Gödel's Incompleteness Theorem doesn't say there are general "Truths" that can't be proven. He only says that higher order systems of logic (such has Peano Arithmetic), are incomplete. There is nothing GIT that says the sentence G can not be proven (or disproven) in some other system of logic (or even Math).

Sounds like you are talking about an older proof of GIT. The proof I explained is due to Chaitin. It is stronger.

​​@@anuprao11yeah, is more deep what you are saying

This is because in the algorithm, we first use the ideas from the previous step to fix shift the potential function so that it works for all of the edges going in one direction (outside to inside). Then we are only left with negative edges in the other direction.

First there is a typo on this slide. The case is supposed to be that there are no incoming edges to v. Then we are worried only about the outgoing edges, and if phi(v) is set to be -min over u of zeta(v,u) - phi(u), that would be good enough. But there are no outgoing edges either, then set phi(v)=0 and that works, because v is an isolated vertex. In general, edges in one of the directions between two sets can be handled using a max or min. The most general bad case is case 3. The point is that if all the edges go in one direction, then shifting the potentials of one side far enough in the appropriate direction does the trick!

@@anuprao11 1) So if there's no incoming edge to v, phi(v) = -min(zeta(v,u) - phi(u)) over all u works 2) If there's no outgoing edge from v, phi(v) = min(zeta(u,v)+phi(u)) over all u works 3) In case there are partitions V1 and V2 with potentials phi1 and phi2 respectively with no edge from V1 to V2, following works: F(u) = phi2(u) if u is in V2 F(u) = phi1(u) +min(zeta(u2,u1)+phi2(u2)-phi1(u1)) for all u2 in V2 and u1 in V1 where (u2,u1) is an edge from V2 to V1 so that F(b)+zeta(b,a)-F(a)>=0 where b is in V2 and a in V1. So, in slide at 41:25 there's a typo in sign of min in phi1 prime ? There should be plus in front of min instead of minus ? Also, (2) is a special case of (3) where V1 is just one node and phi1 = 0 ? Is my understanding correct ?

I used Omnigraffle to draw most of the pictures!

@@anuprao11 Thank you so much. You and your lectures are awesome

I’m glad you like them and I appreciate the support!

f(x) is O(x)… this means that f(x) is bounded by a linear function in x, as you yourself explained is true!

@@anuprao11 Oh, yes, sorry. I mixed it with little o notation. I also noticed some typos in the presentation, for example, on 29:00 in the second formula you are integrating with respect to t, however in one integral dz is written.

You're suggesting that I should steer *away* from controversy? Interesting approach...

That is one of the lessons I learnt from talking to economists when I was doing this more actively. At first, I came up with more concrete metrics that I thought are reasonable, but over time I realized that the most significant idea here is that trade networks can be used to do "something". Determining what that "something" is requires significant additional work.

There are some things stronger than proof or knowledge. For example, pain. We don't "know" we are in pain, and our certainty is weak in the face of the experience itself.

@@JohnJones-tx6rt Can you elaborate please?

@@gokutarzan We can't know we are in pain. We can't check our certainty. We simply are in pain, stronger than certainty or doubt.

That’s not an accurate quote. What I said was that we do not know of any physical phenomenon that cannot be simulated by programs. If you disagree with that, I’d love to hear about a physical process that’s impossible to simulate with programs.

@@anuprao11 Then why did you speak it?

@@James-ll3jb it’s taken out of context.

@@anuprao11 Whatvis the context if not your own stated one of quote unquote THE WHOLE WORLD!

@@anuprao11plenty of black hole stuff , quantum physics cannot be accurately simulated. Only approximately guessed. 🤯🤦‍♂️🤦‍♂️🤦‍♂️🤦‍♂️🤦‍♂️

Yes: you can show that the bound is tight for the convex body that is the convex hull of the points {+1,-1}^n.

My pleasure!

There is no such assumption made here. The theorem says that there is a true statement that cannot be proved, not that there is a paradox that cannot be proved. The statement that we found is either true or false: it does not reference itself.

A statement cannot be said true unless it is proved.If you trust in the truthfulness then it is provable.

@@anuprao11 the theorem in the video is a proof by contradiction which demonstrates that C cannot have finite length. Its an uncomputable function, and so it has infinite kolmogorov complexity. There are no true, unprovable statements. Thats a gross misunderstanding of godel incompleteness perpetuated by model theorists. The godel statement is not true in every model, and so its not true in the theory. Its just unprovable.

@@samb443 The problem is that our brains only have finite size. All of the computational objects we can hope to build seem to have finite size. So, the theorem is very much relevant.

@@anuprao11 But the "proof" that "for all n, there is an x making S true" is non-constructive, and so non-computable. So the "truth" of the statement is ethereal at best in this computational context. This is where the extra assumption that cmilkau was talking about comes in. That proof uses LEM, "its not false therefore its true". The theorem is only relevant if you accept LEM, and if C(S,p) has finite length.

We are not claiming that program A is shorter than L. We are claiming that program A is shorter than n. Also, the length of the program has nothing to do with x or L. The length of the program is the length of the code that describes the for loops etc. x and L are variables in the program, and their size is not relevant to the length of the program. Here n is fixed during the execution of the program, and n is hardcoded into the program using its digits. Hope that makes sense.

Oh I get it. You want to show if the integral converges then the thm 2 is true. I get that.