Gödel's Incompleteness Theorems: An Informal Introduction to Formal Logic

[@nHans]

Great video. Of course, it'll get compared to Veritasium's 34-minute video on the same subject-there's no avoiding that. At 21M views, it's the most popular one on the subject till date. But I believe yours will hold its own-not least because you conclude on an optimistic note, while he calls it _Math's Fundamental Flaw._ I will keep coming back to this video-it's a very good introduction to Formal Logic and Peano Axioms. I have a couple of suggestions/requests: - Could you list the books that you've recommended in the description, aka dooblydoo? You could even include affiliate links to online booksellers and earn small commissions. - A table of contents with timestamps would be most helpful. KZbin has built-in support for this.

[@mathpunk6493]

Excellent points! I'd intended to demarcate the chapters when I uploaded, but I got distracted and never followed through. I'll do that now. And I'll also add a bibliography to the description.

[@santiagocalvo]

this video is perfection, it makes me so sad to go on your channel and see there are no more videos!! Please keep em coming!! This is an absolute treasure!

[@luiz00estilo]

This must be the best explanation of Godel's Incompleteness Theorems that I've ever seen! Glad to have found this gem.

[@adsoyad2607]

I cannot believe I didn't come across this video last summer. It randomly popped into my recommended today and I just watched the entire thing at one sitting. I absolutely loved every little split second detail/gimmick/joke you've put in an essentially movie-long video. I'm also studying maths and have taken a course in introductory first-order logic and set theory this semester, and I would love to see you talk more about these subjects. This might literally be the most underrated video I've come across on KZbin. Fantastic stuff

[@thomasokeeffe5050]

Simply amazing video.

[@caalher]

This is just a master piece and I'm not gonna say more.

[@dbharat9576]

I am yet to encounter a more succinct and comprehensive overview of formal logic on the internet. I am very impressed and looking forward to your next video. Thank you.

[@larianton1008]

I'm in for a treat. I absolutely love incompleteness, and I have many philosophical ideas upon it.

[@larianton1008]

This video was way too advanced for me. I could follow until we introduced gödel numbering, which is the point in which my own study of this subject has reached.

[@derelbenkoenig]

Something I think people also don't realize often is that we do in fact use inconsistent systems all the time for useful things, namely almost all programming languages. By the Curry-Howard correspondence, a Type corresponds to a proposition, and a term in a lambda calculus or programming language (e.g. an expression or a function) serves as a proof of that proposition. The validity of this is checked by the programming language's type checker. But if these programming languages are complete, then they can compute any computable function, which more or less entails being able to perform arbitrary recursion. This amounts to admitting an axiom like 'forall a. (a -> a) -> a', which is the type of the Fixed-Point combinator or Y combinator. An expression which inhabits this type is a function like `y f = f (y f)` or `y f = let x = f x in x`. Or to take an even shorter shortcut to proving-everything-including-false-things the type 'forall a. a' being inhabited by a term defined by simply referring to itself ('a = a'). So clearly these systems are _complete_ and _inconsistent_ and totally unsound yet it's what everyone uses for computation almost everywhere all the time, and we simply rely on other systems to filter out programs which are incorrect. We could instead use only programming languages with type systems known to be sound but this is so inconvenient for programmers that it's not even considered an option in most cases. For now, at least. They're working on making programming languages with soundness but enough convenience to use but there will likely still be a higher learning curve

[@mathpunk6493]

You have glimpsed into the far future, friend. The Curry-Howard Isomorphism is something which is on the cards for a future video, but preferably one for when I have a much deeper understanding of lambda calculus and computability theory.

[@crus_crus]

There wasn't enough papyrus or comic sans for me to understand this. Please fix 🙏

[@tolkienfan1972]

If true why don't programming languages suffer from explosion?

[@棗こころ-z1x]

They do suffer from explosion. For example in Typescript you can get a value of type “never” (corresponding to we can proof False), and you can use that variable as any type (corresponding to the rule of explosion). So it means we can’t use that type system to build proof assistant on top of it, however in practice that type system is still useful for general programming.

[@tolkienfan1972]

@@棗こころ-z1x I'm not sure I take your word that "never" in Typescript corresponds to proving false. But either way, whether the corresponding system is irrelevant. We don't use it as a mathematical system. We don't prove things in the corresponding system. Explosion in the dual system is irrelevant unless it proves something about programs written in the language itself. If the corresponding system is inconsistent then clearly that inconsistency says nothing interesting about the language or programs written in it. Also, while I haven't looked at this correspondence in any detail, it looks like it derives a system only from the types. If true, then it leaves most of the language on the table. I'm pretty sure there is no useful inconsistent system, because no proof is valid in an inconsistent system.

[@MCredstoningnstuff]

I am going to disagree with one of those opinions: you said at the end that wasn't the best video essay out there on this topic but I absolutely loved this and I do honestly think that this is one of if not the best intro to this topic out there, especially for people with a math/logic background. I appreciate that you didn't shy away from the formal logic and showing how some of the informal proofs work while casually setting a proper foundation so it wasn't overwhelming. 10/10 will share with all of my math friends and maybe a professor or two!

[@christophermiller3031]

I just watched a horribly cringe inducing video about logic that was KILLING me!!! I searched to find a person discussing logic that was NOT actually the ramblings of an idiot... I've watched enough of this video to know... I AGREE... This video is the reality... Math is the logic I respect most. Sometimes I forget how rarely people even consider how AMAZING MATH IS. Oh.. and this video. I literally needed this... Cuz if my search yielded all results about the logic of god, and how logic doesn't care about your feelings, I would literally have gotten madder and madder like a teapot. Lol

[@samuelstermer6437]

watching this a year late and your channel is right up my alley, dreading checking the rest of the channel and finding you havent posted since. hope you're doing well

[@samuelstermer6437]

I am the big sad, please post more if you can and want to

[@mathpunk6493]

@@samuelstermer6437 You haven't seen the last of me. I'm working up to something big, but it's tough going as I'm having to do a lot of independent study alongside my degree. Watch this space, I might try and put out a few smaller projects in the meantime just to get my editing skills back up to scratch.

[@spacebuddy5339]

​@@mathpunk6493Can't wait!

[@amarasa2567]

Okay, while this has not yet been successful, I really hope your work becomes more appreciated over time: it is a wonderful video, and you even managed to teach me a few things (I have a quite heavy pure maths background, although I stopped at Master's degree level) :) The fact that you went into such detail while staying accessible to the general SoME public shows quite a bit of mastery in both the topic and the art of explaining what you know. I hope to see more in-depth videos like this in the future :)

[@PowerhouseCell]

I can't believe I went through the full 90 mins, it felt so fast! As an educational video creator myself, I understand how much effort must have been put into this. Liked and subscribed, always enjoy supporting fellow small creators :)

[@meccamiles7816]

I sat here and watched this end-to-end in one sitting and was enthralled. Very, very well done.

[@emiainscough2945]

This is wonderful. You are a rockstar.

[@vrsgng]

thanks! this is the best and most detailed introductory video i've seen so far. please keep them coming.

[@BrooklynRoseLudlow]

This video brings a level of depth and rigor that's missing from a lot of math KZbin content. You've got a great concept here. I'm a video editor and if you make more of these, I can show you some things to make them more punchy and algorithm-friendly.

[@ff_helge4107]

High quality content

[@derelbenkoenig]

What an excellent video, I am certainly looking forward to seeing more. I had already a low-resolution understanding of a lot of the stuff but this really helped see more of the formal details for me. I'll likely rewatch again and actually pause and read all the proofs in the middle as well but even without doing that, very informative

[@meccamiles7816]

You appear rather young to be so well-acquainted with these topics and at such a depth. Kudos to you.

[@pedrogabrielbueno9151]

Wonderful video, congratulations for exposing it so well and clear

[@nanamacapagal8342]

Holy shit this is fucking cinema in a math exposition. You somehow made math even more thrilling than I've ever believed could be possible for math, and you did a good job with this 90 minute movie. You and your team deserve more than a good job, you deserve a congratulations.

[@samuelstermer6437]

The opening is by far the best explaination of Godels incompleteness lol i love it so much

[@sarchet3149]

I'm 10 minutes in I'm understanding nothing but I'm loving everything

[@AB-gf4ue]

This is such a comfort video for me. I feel like I have learnt so much from it.

[@pl412]

What a wonderful video. Loved the humour throughout and found the exposition very approachable. Fantastic work!

[@AssemblyWizard]

At 1:19:52 you introduced Proves(x, not y) as an abuse of notation, yet at 1:23:24 you used PR(not x, z) to mean "z proves not x(x)", but by the definition of PR it should mean "z proves not x(not x)". The first abuse is using the "not" operator on a Godel number to get the Godel number of its negation, which is understandable, but is the second case a different abuse of notation which wasn't explicitly mentioned? Do you define PR(not x, z) as "z is the Godel number of a proof of not x(x)"?

[@mathpunk6493]

Yes, your interpretation is correct. PR(not x,z) means "z is a proof of not X(x)", and I should have stated that unambiguously in the video.

[@matthewmorgan7103]

Absolutely fantastic. I have been trying to find a way to express logic statements to my students.

[@tanchienhao]

Can certainly tell that a lot of effort was put into this video! Some2 has blessed us again, great job!!

[@christopherluikart9656]

MathPunk rocked my world! What a blast !

[@dominvs935]

hey you weirdo, listen to me. you need to create a whole playlist on formal logic from the very start because this is one of the best videos i've seen in a long time.

[@mathpunk6493]

Be careful what you wish for.

[@MathCuriousity]

Heyy I love your channel and may I ask a question: If in set theory, I can create a relation which takes a set of elements which are propositions (like set a is a subset of set b) and map it to a set of elements containing “true” and “false”, then why is it said that set theory itself can’t make truth valuations? I ask this because somebody told me recently that “set theory cannot make true valuations” Is this because I cannot do what I say above? Or because truth valuations happen via deductive systems and not by say first order set theory ?

[@mathpunk6493]

Hi, sorry for the delay. Set theory is able to make assertions about collections of objects, but it cannot interpret what those assertions might mean. For instance, taking your example, if a function maps 1 to T and all other natural numbers to F, that means set theory is "able to prove that" f(1) = T and f(n) = F for all n!=1. But T and F don't MEAN anything, and this is the critical point - just because we give the numbers these values, doesn't imply anything about the meaning of that assignment. In order to permit truth valuations, some system of interpretation is needed, which is usually the domain of model theory. It's only under these interpretations that functions like the one you mention come to gain meaning. I'm currently learning about how to apply models to set theory, which I plan to share with the channel at some point in the not-too-distant future.

[@lucifermorningstar1902]

Not a mathematician by any means (I study jazz, and also Bach, I’m a musical nerd haha), but I’ve always been fascinated by pure mathematics and theoretical physics, it’s a huge interest of mine. And as someone who isn’t at all a professional I have to say this is one of the best and also most fun explanations I’ve seen. It is so approachable and makes such an abstract concept easier to comprehend. That’s all I can say as an outsider haha

[@willowjavery4652]

This is pretty brilliant work and I hope you end up making more stuff for this channel

[@lilmoesk899]

great job! and funny too. You should keep making these kinds of videos

[@carloselfrancos7205]

Incredibly powerful incipit

[@compegord07]

Was worth it just for the introduction. 😂. Ty.

[@MrMeltdown]

Watched the whole thing. You lost me very early on. It's clear you know what you are on about. I've been taught by many people who mentioned the Incompleteness theorem and possible problems. This gives a really good place to start from when trying to understand the whole thing. Great Work.

[@qqii]

Really enjoying the video so far, just finished 2.2. This is a topic I really enjoy and I'm looking forward to the next part. As a minor bit of constructive criticism, I think introducing predicate logic via definitions becomes a word salad very quickly. I really liked the storytelling of the introduction and up to 2.1 you did a really nice job weaving in history and motivation, something that a lot of introductions lack. I would suggest presenting intuition first, as this would also follow nicely from the end of 2.1. "We'd really like to be able to say ", and then present the formalisation. Anyway, I'm going back to the video!

[@mathpunk6493]

Yeah, that's a good point. The problem with any video of this kind is that in order to go from 0 to Gödel in 90 minutes, you need to introduce a lot of concepts very quickly. I had intended to include a glossary of terms in the middle, reminding viewers of the meanings of the terms I'm using before launching into the theorems, but I simply didn't have time to include it. Were I to do this again, I'd put a lot more time into predicate logic, because I'm not hugely happy with how that section turned out.

[@qqii]

I'm in the middle of the peano arithmetic section with the same comment - I think you even acknowledge this by trying to "make it interesting" with a tier list. Once again I think it would have been better to actually show numbers 0 = 0 1 = S 0 2 = S S 0 etc, which everyone has intuition for before formalising them. After all this is the order of history too.

[@qqii]

Ah, I've found your SoME_Topics github issue: > explaining why or how the Godel sentence is able to assert that the theorem with its own Godel number is unproveable. A noble goal, and explains why you were as rigourous as you have been.

[@Rtwbjb24]

great motivation for me to formally look into formal logic

[@UnathiGX]

I struggle with multiplication...but here we go!

[@nomanzafar4938]

LMAO

[@RecursionIs]

Great work, I espescially loved how you summarized multiple of my university courses in a couple hours 😁

[@hedegard100]

Lovely video. What is the music in the first chapter? Mahler?

[@mathpunk6493]

Brahms' German Requiem, movement 2! One of my favourite pieces of classical music, and the 6th movement (to which I set the proof) is just as great. The opening was initially to be set to Rachmaninoff's second piano concerto, but when I came up with the conceit for the sequence I settled on something longer but no less dramatic,

[@abcde_fz]

. 🙂 Thanx for pointing out that the term "formal" is similar to the term "spherical" when used in the context you're discussing here. . Now I fully understand the term "formal logic".

[@MB-ue2rf]

entirely enjoyed, thank you

[@identityelement7729]

This is harder and easier than I thought.

[@rickeichmann7272]

Well done.

[@thephysicistcuber175]

24:06 don't you need to require the set to be ordered and conditions on psi_k to be deduced only from psi_i,psi_j with i,j

[@mathpunk6493]

You're quite correct, but since it's a relatively minor point in the grand scheme of the video I thought it better to just skate on over it rather than adding in unnecessary detail. Plus, that section was filmed in the middle of a heatwave, in a room with no AC, and the line was hard enough to say as is.

[@thephysicistcuber175]

@@mathpunk6493 Fair enough.

[@raydencreed1524]

At 35:26, you mention that one can quantify over variables not present in a formula, producing things like ∀y∃xPx and ∀xP. Since that’s the case and one can consider functions in algebra which do not actually depend on their inputs (such as f(x) = 1), I wonder if there’s anything preventing one from expressing a proposition (zero-place predicate) as a one-place predicate which does not actually depend on the variable it admits, and then quantifying over it.

[@juancristi376]

Nice video. Thanks for making it!

[@yairraz6067]

Great video , clear and funny thank you

[@nahometesfay1112]

18:53 "I've tried so hard to rub this off, but it's not coming"

[@larianton1008]

Sorry for asking something so broad, but I would be very interested in hearing your personal thoughts about what intuition is? I believe this facet of our mind is what Penrose uses in his book "the emperors new mind", if I'm not mistaken (haven't gotten into reading it yet). Do you think there is anything to be said about intuition in regards to Gödels theories, or for any other reason?

[@tastypie2276]

Great video! Thank you!

[@_thenyounoticeyourethinking]

Captivating!

[@kheammachart]

This better not be the one and only of your video in this channel!!!

[@mathpunk6493]

More are in the works, but I am overstretched enough already so have patience!

[@vovinlonshin3708]

"the word is not the thing, the map is not the territory"

[@blankboy-ww7jt]

Very impressive

[@ShadowKestrel]

bit of an objection to the C vs Python comparison - even once you understand C, Python is still a far more powerful and expressive tool for getting programs written (even if it leaves a little to be desired in the speed of running them). Perhaps a more natural comparison would be C vs a functional language, like Haskell - in C, you give a series of instructions, from a fairly small and mostly fixed set of instructions of what to do (like individual inferences of natural deduction) whereas in functional languages you start with a small set of built-in functions and define new functions by modifying existing functions (with these modifications themselves being functions). In fact, the Curry-Howard isomorphism states that function application follows the same rule "A, A->B becomes B" as modus ponens (albeit with no analogue to negation)

[@mathpunk6493]

Thanks for the comment! You're right, and doubly so for bringing Curry-Howard into things (watch this space for more on that, it's on the list!). I could have gone into significantly more detail had I had the time and expertise, but for a throwaway line I thought focusing on speed of computation would suffice.

[@ciceron-6366]

very good vidéo, worth a comment for SEO. Wasn't quite sure about the duration of this video that may be a little scarying at first, but in fact I went through the whole video very naturally (at the beginning, because I gotta admit that I had to pause for loooong periods of time during the last Proofs...). It was exceptionnally accessible for my level (highschool in your country?, corresponds to 18 yo), and I think I even understood the proofs at the end,which is something I wouldn't have bet. it was also very entertaining, with a moderate (but sufficient) dose of humour. Hope you'll be rewarded for your work, goodnight sry for english, ima frog

[@bean2046]

First of all I love this video and the editing is amazing, absolutely wonderful. However, I'm not a mathematician and this might be a very stupid question, but why do we wonder whether the universe is maths and we found it or if it's a model and we made it? Because if I found a wooden house and built a wooden model of it, I found the house and built the model, but I wouldn't suggest that wood is either a house or a model. The house would give me some insights for my model and my model helps understand the house better, but wood is just the material. I'm so confused

[@mathpunk6493]

Not even remotely a stupid question. If I understand your metaphor correctly, you're suggesting that mathematics is like the wood, in that it's the tool we use to understand the universe by building models. Mathematical philosophers are then the people who ask where the wood came from, because if we're building our house out of wood (or even models of houses) then we want to make sure it's good wood. Nobody wants a model built out of moldy, termite-ridden wood; and it's the same with mathematics. If mathematics proves to be like rotten wood for model-building, it's then VERY bad if the universe also turns out to be mathematical in nature (read: you build a model out of moldy wood, then discover your house is made of that same moldy wood). I hope this helps, but if it doesn't then please feel free to ask for more clarification (it boosts my engagement, and also this is a really fun metaphor!)

[@bean2046]

@@mathpunk6493 Thank you so much for your response! Even better, a response I actually understand, while it does not quite clear my confusion as to why there is a "science vs philosphy" debate. And I have to preface this by admitting to the fact that I am very biased and basically filled with hatred towards the thought of "only science is really smart and arts are stupid". I hope that's correct in English, in German it's "Naturwissenschaft" (science of nature) vs "Geisteswissenschaft" (science of the mind), with the latter always seen as inferior. That bothers me, especially with mathematics because the debate always sounds like "we really want to be seen as only science, not that stupid thinking book stuff, we're REAL scientists", however - and that's why I really like the metaphor - no matter how much we look at the house and the model, the wood comes from a tree. Same with mathematics, for me at least, it's just our language for understanding. Yes, mathematics as a language and a tree, just like Tolkien intended. Because what is that question other than "maths has to be physics or else it's wrong". If nature is mathematical and the way we use maths to describe physics is incorrect, then that's incorrect, but I can still calculate the area of a square and apply statistics to a linguistic study. So if this is an important question for mathematicians, what is even the point of maths for you? Is it still the expression of understandment or just wanting to prove equality to physics or something entirely different? I hope that makes sense, I'm oddly passionate about the purpose and characteristics of mathematics for someone who can barely count to twenty, let alone calculate anything with numbers over three

[@mathpunk6493]

​@@bean2046 If only everyone in the world had the same mindset as you. I'm hoping to spend the rest of my life studying the answers to the questions you raise - not just about the nature of mathematics, but also about the point of studying maths in the first place. I originally typed a 400-word response about the role of mathematics education in society, but things spiralled out of control and I've shelved it for a future video. All I'll say on the issue is that, whilst I don't think the importance of mathematics is unduly inflated, I do believe that the importance of the Geisteswissenschaft is massively underestimated by pretty much everyone. I entirely agree with you that there's a certain stigma about it in the education system, which must be banished for liberal education to be as holistic as it wishes to me. As to your second point, I'll try and keep it brief. I believe that we should study mathematics for the same reason Mallory tried to climb Everest: because it's there. Quite aside from its utility (though that's still a very good reason), I myself am a firm believer in knowledge for knowledge's sake, and since mathematics arises very naturally in many different contexts we must interrogate it. I don't seriously believe that the wood is rotten, but the more interesting question is: could it be? Could there be something wrong with mathematics? If yes, IS there something wrong with it? If not, why not? What makes mathematics so infallible? These are questions I don't ever expect to answer, but they're too interesting to leave alone.

[@shaunlanighan813]

More please.

[@luserdroog]

The Chinese Room is Plato's Cave.

[@nickk6386]

This video is brilliant

[@redapplefour6223]

got very lucky this one was recommended to me, while clearly needing a decent amount of background knowledge (that which i only have some of), i think it’s interesting how it fits into the class of (good) meandering leftist essays about stuff. hope grant gives some kind of shoutout to this one!

[@biblebot3947]

10:40 that’s a containment breach right there.

You explain almost everything so clearly, but then near the end, the most important steps are not commented at all, you just dump symbols onto the screen with music and hope that people understand it. It would have been nice to have explanations of what each step means.

[@mathpunk6493]

You're absolutely right. The reason the ending is so rushed is twofold: I was working to a pretty tight deadline, and I didn't understand all of the intricacies of the proof myself. Were I to come back to this, I'd definitely spend more time in the latter sections, and possibly include some kind of recap of the terminology established earlier in the video (which was part of the original plan but got cut for time). Overall, I'm reasonably happy with this video, but I agree that there are many things to improve in future.

[@IkedaHakubi]

Pure Math is good for itself!

[@NikolajKuntner]

1:31:37 oh noes, not my Shades of Gray. I'm not sure what you mean thereafter, when you say arithmetic has proven difficult to formalize. Especially as you contrast it to "very good" ZF in the same sentence. If anything, second order arithmetic is categorical and we barely stumble over unprovable PA statements, while set theory has a lot of loose ends, is open ended upwards, with undecidable statements that people research into and it's not all too helpful for natural number arithmetic.

[@crawkn]

Nail polish remover is a great hoopla solvent.

[@coast-guard-1cargo-spectio552]

Yo dis b lit I'm shaking

[@周品宏-o7w]

My guess of why the proof isn't rigorous is because g is the gödle number of G(x) not G(g). Great video.

[@mathpunk6493]

Not quite! You're correct that g is only the Godel number of G(x), but the Proof Recursion function I define specifically asserts something about the existence of a proof of G(g), i.e. a formula with one free variable fed its own Godel number. The reason it isn't rigorous is because of steps like "If PRPA(g,0), PRPA(g,1), ... then we can deduce that for all n PRPA(g,n) is false". I can't remember if that's exactly how it goes, and the formatting in KZbin comments isn't exactly LaTeX level, but that step isn't entirely justified within the system of logic I'm using. Sure, it's obvious, but that's the thing with rigor: the obvious must still be explained in full.

[@周品宏-o7w]

@@mathpunk6493 thanks

[@meccamiles7816]

That intro though.

[@bongo50_]

Good video. Please make more m

[@ophello]

Rubbing alcohol might get the HOOPLA off.

[@orterves]

sometimes just tracing over the letters with a non-permanent marker will dissolve the ink enough to erase it

[@rageprod]

As a first year applied math major, this is woke. And did not go over my head. I promise.

[@MMarcuzzo]

Where's number 3 peano axiom?

[@Mpire101]

Is that a benry plush

[@mathpunk6493]

Yes, it is!

[@Mpire101]

@@mathpunk6493 this is one of the most queer videos I have ever watched I love it

[@harveysasaki6284]

In studying formal logic and systems, perhaps I should take a look at this Gerhard Gentzen character.... Uh oh, whoops, haha let's move on

[@davidlythgoe809]

very cool!

[@TheMemesofDestruction]

Thank you! ^.^

[@morgengabe1]

I think the nothing/ham sandwich paradox is _at least_ almost economical in nature

[@drako3659]

I didn't realize Exurbia was a mathematician?

[@jamestagge3429]

“This statement cannot be proved using axioms and rules.” Consider as part of the understanding that from which the statement was formulated, i.e., to back-engineer it. This statement mirrors Quine’s liar paradox, “this statement is false”, which in my view was pure sophistry and meaningless. The contradiction arises from the defiance of the very logic also used in its definition/formulation. The term “statement” is a set definition though empty of members so there is no reference object that might present meaning in its use. “False” is the adjective meant to judge it though is kept from that task in that it becomes in its place in the statement, both the cause and effect at once of the paradoxical function. So to introduce this into mathematics is no less a scam than it was for Quine. “This statement cannot be proved by the axioms and rules” was formulated from a statement which either proposed a truth and by the clever structure of the Goedel numbers gave rise to the statement above or was a hodgepodge of mathematical symbols (of no particular meaning in a mathematical context) which when manipulated, meant to result through Goedel’s translation scheme to formulate his statement. If the former, it means that the mathematical statement had to have expressed that which we would be obliged to consider true, but for which there was no proof. How is it that such a mathematical formula could have escaped the notice of those in the profession for centuries? How could they not have seen that there were formulae for which there were not axioms and rules by which to provide their proofs? To have generated such a statement as “this statement cannot be proved by the axioms and rules” from mathematical formulae and axioms, which made sense in the discipline prior to Goedel’s translation seems a bit unlikely. One is driven to conclude that this ridiculous paradox of Quine’s would only act to corrupt mathematics as it did philosophy. How could a mathematical statement which when translated by Goedel’s numbers into a semantic statement make any sense in the context of mathematics? So, if we obey that suggested by the statement “this statement cannot be proved by the axioms and rules”, which is that it is a true statement (or what is the point of the discussion?), we must accept that it is either a statement standing on its own in judgment and thus self-referencing or it references the mathematical statement from which it was derived. The problem remains the same in either case. If it is not self-referencing, the statement from which it was translated had to have actually stated something in mathematical terms which made mathematical sense. I find it hard to believe that this is the case in that there is no structure to mathematics which is mirrored by the structure of language. Creating this statement out of the translation of disconnected mathematical statement fragments proves nothing. Such a mathematical statement would merely define something out of context of any associated mathematics axioms/rules normally employed. If this is not so, then the world of mathematicians would have known readily that such mathematical statements existed and were unprovable and there would have been no need of Goedel. Something seems wrong here. Three points to consider; • If there is a statement that is claimed true then it has to be known as to how and why it is true and that would be the proof. If we are to understand that this Goedel statement references a mathematical statement (from which it was derived) that is true, how is it that we can make such a claim? By what means can we understand it as true? • If there are simply bits of mathematical formulae or axioms, etc., which are manipulated that by the Goedel numbering system formulate this statement, of what consequence is that? It is able to make no claim about any incompleteness of math or logic. • This statement cannot be proved…generated by the translation of bits of math via the numbering system, means nothing if the terminology employed in the conclusion does not have counterparts in the language of mathematics. I seem to always come back to indict Goedel because of his stated admiration of Quine and his paradox which was pure nonsense. What do you think?

[@timdion9527]

I enjoyed the video. But, now I feel incomplete ...

[@crawkn]

Maths and physical reality overlap, but they are not one. But wait . . . what is one?

[@hugofontes5708]

A punk explaining maths and logic? Subscribed My brain is melting but that was a very cool video and that first but if opinion on how it is *just* proof that arithmetics is incomplete was very important

[@mikefocal5770]

great

[@chrstfer2452]

I mostly agree with your ending opinion, but not with your using the chinese room as an example. All the chinese room proves is that the book is a static intelligence of sufficient complexity/capability, and the man in the room is nothing but the computational substrate in the problem. Its basically a red herring that has convinced half or more of the great thinkers in the world.

[@abcde_fz]

. Connectives? Conjunction Junction, what's your function? "And", "But" and "Or" get you pretty far! . -- (School House Rock)

[@nHans]

You know, Gödel's first Incompleteness Theorem has always made intuitive sense to me. (The Consistency one, on the other hand, is counter-intuitive.) See, in any axiomatic system, you can't prove the 2nd axiom from the first. Nor the 3rd from the first two. And so on. Until you reach the 15th (or whatever) axiom. And then you suddenly say that that's enough; all other statements can now be proved from these 15? Never made sense to me. Are you _sure_ you got all the axioms? Couldn't there be one or two more hiding somewhere in the infinity of statements? Or even an infinite number of them, like primes among integers? Take the analogy of prime numbers. There are an infinite number of integers. You cannot factor them all using a finite number of primes; you need an infinite number of primes. Euclid proved it millennia ago; nobody gets worked up about it. But when Gödel says something similar about statements and axioms, everybody is suddenly acting like the world's about to come to an end or something. I don't get all that hoopla! 😜

[@qqii]

Godel incompleteness not only applies to a finite set of axioms, but also an infinite one. I think this is non obvious even in your analogy.

[@nHans]

​@@qqii And that's why I called it an analogy, not an isomorphism! My point was that it's quite intuitive. You have an infinite number of statements. To prove (or disprove) them all, you might need an infinite number of axioms; a finite number of axioms may not be sufficient. That's all. Obviously, analogies don't hold if you stretch them too far. And there are lots of subtleties in Gödel's theorems that can't be covered in the dozen lines of a KZbin comment. So yes, "effective axiomatization." Finite sets of axioms are clearly insufficient. Even certain infinite sets of axioms are insufficient. _This does not contradict what I said earlier!_ In fact, when formulating Peano Arithmetic (PA) in first-order logic, you do end up with an infinite number of axioms-the so-called axiom _schema_ of induction. However, these infinite axioms are not arbitrary. Rather, they're "recursively enumerable." Consequently, PA remains incomplete via Gödel's theorem. BTW, you can stretch the prime number analogy a little bit. See, even if you have an infinite set of prime numbers, you may not be able to factor all integers-particularly if one or more primes are missing. For example, all odd primes form an infinite set of primes. But clearly, you can't factor the even integers that way. Similarly, an infinite set of axioms is no guarantee that you can decide every statement. Of course, Gödel goes even further and states that it's just not possible. Another important point is that the axiomatic system must be "sufficiently powerful." That is, it must be able to define natural numbers with both addition and multiplication operations (or some isomorphism thereof). Peano, ZF, ZFC, Robinson etc. can do that. Consequently, they are incomplete by Gödel's theorem. Weaker systems-like Presburger, Skolem, Algebraically Closed Fields, Real Closed Fields, Tarski's Axioms of Euclidean Geometry, Dan Willard's Self-Verifying Theories etc.-cannot formulate natural numbers with both addition and multiplication. Consequently, they are _not_ subject to Gödel's Incompleteness Theorem. In fact all those systems are provably complete. (Consistent too, but that's a different story.) And then there's this whole semantic confusion about _true_ statements that cannot be proven. I'm not even sure what "true" means in that context, so I just ignore it. For me, within a formal system, either a statement can be decided or not. If it's not decidable, its "truth" is irrelevant.

[@kreynolds1123]

Try some wd40 to get rid of Hoopla.

[@crawkn]

Seeing human reason as superior to, or unachievable by machines ignores the fact that the difference is in our penchant for flawed logic. I think it has already been demonstrated adequately that learning models are amply capable of flawed logic (although we are still and will always remain the masters of that realm). What it may demonstrate is that attempting to eliminate such flaws, while probably a fool's errand anyway, might also not achieve all of our goals for the technology.

[@qqii]

1:11:12 😂

[@qqii]

1:15:00 Oh shit, it all comes together!

[@Gaxi2]

Make a small video nerd

[@Jazzmaster58]

The story you tell in the beginning is either poorly told or makes no sense at all, first, how come that the sword can only kill, "all of his enemies" or "only his enemies" and not both, second, why is the supposedly inherent downfall that if you are his enemy you cannot be killed and if you can be killed you are not his enemy, this is the opposite of the sword's attributes.

[@mathpunk6493]

The story is not meant to be taken literally. It functions as a metaphor for the concepts I outline over the course of the video. I talk about how this story reflects on the Incompleteness Theorems further into the video; it is because of these theorems that the sword (any axiomatic theory of mathematics) cannot kill (prove) every enemy (true statement of mathematics). In order to show this, Gödel constructed a statement of mathematics which, if it can be proven, is false, and if it cannot be proven is true. This is equivalent to saying that some person exists who cannot be killed (proven) by the sword (maths) if they are your enemy (true), and if they can be killed by your sword (proven using maths) then they are not your enemy (false). I hope this clarifies things.

[@Jazzmaster58]

@@mathpunk6493 I need to ask my math expert "friend" to explain me this one. I'm the guy that when asked "if you're in a race and surpass the second placed in which place will I be" ? First. Just kidding, I already know where it goes, but after your video I still need you or a friend to explain how it works. Thanks anyway.

[@frechjo]

@@Jazzmaster58 I already knew a bit about Gödel's Incompleteness Theorems, but I had to rewind that part, because I didn't follow the metaphor too well at first, haha. It does make sense, but it confused me that killing enemies was proving theorems. Maybe because I'd expect theorems to be friends in the metaphor, not foes XD You can have a sword that will kill all of your enemies (= it's complete), but it will also kill friends (= it's not consistent). You can have a sword that will not kill any friends (it's consistent), but it won't kill all of your enemies (it's incomplete). You can't have a sword that's both complete (kill all of your enemies -> prove anything true) and consistent (kill only enemies -> only prove true things) at the same time.

[@mathpunk6493]

@@frechjo I was worried that people would struggle with the metaphor, and I tried to find a different (and less hostile) way to word it, but every other way I tried was needlessly complicated so I stuck with the one I began with. Thanks for not giving up on it, because I think it's quite a good metaphor if you can get over the hurdle of associating theorems with enemies rather than friends.

[@frechjo]

@@mathpunk6493 It's got narrative interest, it's visual, conveys emotion... It's a great metaphor, and the narration nailed it! :) And, one could say, proving a theorem is just like slaying the enemy of uncertainty XD

[@mcol3]

Starts a 3:05

[@AceHack00]

What are your pronouns?

[@marinoceccotti9155]

Logic from ... you??? I flee.

[@mathpunk6493]

😘

[@MMarcuzzo]

Where's number 3 peano axiom?