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Oh yeah, I'd be looking much forward to the Goedel video.

all smurgles are flumpy, but some are more flumpy than others

there is no such thing like "mathematical logic", logic is a mathematical object, the only axiom that really exist in today math is Voevosky's Univalance Axiom that enable you to go rigorously into intuitionistic logic without the trouble third middle excluded introduced in first order logic formalized with ZF set theory all math is completely being redefined in HoTT

The algorithm has thrown a golden video my way! Thank you so much.

13:54 that's just wrong, Kurt proved you can have a complete system!!! Why does nobody get that!!!

"in the human language like english"brightaining explaining lord of ring,nothing personely,just ..this young expiranto made for global control is under press ion.greetings*sonickeymaker.datanti IUOrg.

They aren't holes. Call them inconsistencies. A small price to pay for completeness. Even Kurt created a 3-valued logic. He would have found RM3, too

Okay, great videos. Maybe I had alteady asked, but what tool do you use to make the animations in your videos?

10:47 As the joke goes, 2 + 2 DOES equal 5, but only for exceptionally large values of 2.

5:11 This expression begs for parentheses

Not entirely true that only one formula exists, its best expressed in glsl vec2 cmul(vec2 a, vec2 b) { vec2 u = a.xy*conjugate(b).xy; // ai*bi is conjugate vec2 v = a.xy*b.yx; return vec2(u.x+u.y, v.x+v.y); }

0:34 me: "Huh, this will be a vid about sets?"

What nonsense 😂

It is worth noting that the semantics of classical logic that you mentioned is a choice, not intrinsic; it is 'a semantics for logic' not 'the semantics of logic'. The same syntax can have a number of different semantics in different settings, and it is a statement about the choice of semantics that it is sound and complete. There are other semantics of classical (and non-classical) logics which can be sound, complete. For example classical logic has semantics in boolean first order hyperdoctrines.

Id love to see some category theory after introducing monoids

Truth is ALL inclusive! False indicates the perceiver's inability to find it!

The soundness and completeness of logic ~~~ Logic is 'sound' only on it's on home field, within the stricture of the rules by which it is bound! Logic, alone, can never 'completely' define Reality! Beliefs are not bound by such limitations; all that triggers a belief is a perceived threat! The First Law of Soul Dynamics; "For every Perspective, there is an equal and opposite Perspective!" "The complete Universe/Mind/Reality/Truth/God... or any feature herein, can only be completely defined/described as the 'synchronous sum-total of all Perspectives'!"

How dare you. I know countless of flumpless smurgles, and they are all shniving their best fleebs.

I recommend the paper "The Provability of Consistency" by Sergei Artemov. Gödel's Second Incompleteness Theorem says PA can't prove its consistency. It takes as given that consistency is represented as a single formula. This is kind of a silly choice, since it requires PA to prove that nonstandard numbers don't encode proofs of contradiction, even though (IIUC) we normally only consider proofs encoded by standard numbers. This paper represents consistency as a scheme instead of a formula. (I don't know what a scheme is.) This allows them to formalize a proof of PA's consistency in PA. Pretty cool!

i see some inaccuracies have been pointed out, maybe this one as well but i haven't seen it yet. it's seemingly by the same conflation of two different concepts, you say, by the end, that at least we can rest assured everything we CAN prove, is true. but that's actually unknowable, which Gödel also proved. all systems of axioms (with basic arithmetic) are either incomplete, or inconsistent, but you can't ever prove consistency (which would imply incompleteness) from within the system itself, only can you prove inconsistency by eventually finding a contradiction. it is similar to the halting problem being unsolvable, in that we could design an algorithm that would eventually prove something halts, by just running for long enough, but before you're there you don't know there will be an end, in general. i love the topic though, and i think the presentation is great, and i could learn a lot from a series about this. i have to say i don't fully understand the notation in the introduction rule of the implication, but that could very well be due to a lack of familiarity. i think you are explaining things very well, just the content you're explaining needs some touch-ups but i'm excited about the different directions you're exploring in your content. I would like to share a verse of a song i wrote: De mysteries van mijn leven Soms ben ik het even kwijt En dan denk ik het te vinden Maar dan blijkt het tot mijn spijt Een comfortabele illusie Beslisbaar en bekend Maar alles is uiteindelijk Onvolledig Of inconsistent

I regargus this kleb.

"If your system can classify everything as true or false, it can't resolve every contradiction. You can't know if axioms themselves are true, false, or contradictory. This applies to God too: If God can't explain contradictions, then God's knowledge is incomplete. If God can, then God is acting illogically. Either way, there's always something unexplainable within axioms or truths. In simple terms, we can't know everything. Contradictions just shift around, never fully resolved. In math, every axiom spawns systems with contradictory paths. We extend systems to handle these contradictions, but new ones always emerge. It’s about usefulness, not absolute completeness."

3:10 you mean the group axioms are too complicated? We haven’t even gotten to rings or fields yet 😂

where are the next videos

That's because form and formless are 2 sides of the same coin. See my paper How Self-Reference Builds the World, author Cosmin Visan.

You misinterpreted Gödel's incompleteness theorem, for you confused the semantic completeness for the syntactic or negation completness ; you see, Godel proved that if a system S is sufficiently complicated , then S is syntacticaly incomplete ie there are sentences φ of the language of S such that neither φ nor ¬φ can be proven in S ie φ is Undecidable by the axioms of S or φ is independent of S This is different from semantic completeness, for φ is not a logic consequence of S ie S⊭φ The gödel sentence is not a logical consequence of the axioms; the relation ⊨ is not total over the wffs of S

1:08 Axioms for Smurgles and Flumpiness Existence of Smurgles: There exists at least one smurgle. ∃s:s is a smurgle. Flumpiness is an Inherent Property: Every smurgle is flumpy. ∀s (s is a smurgle  ⟹  s is flumpy.) Flumpiness is Transmissible: If a smurgle interacts with a non-flumpy object, that object becomes flumpy. ∀s,x (s is a smurgle∧x is not flumpy  ⟹  x becomes flumpy after interaction.) Smurgle Union: When two smurgles interact, they form a new smurgle whose flumpiness is the sum of the original smurgles' flumpiness levels. s1+s2=s3s1​+s2​=s3​, where s3 is a smurgle, and flump(s3)=flump(s1)+flump(s2). Neutral Flumpiness: There exists a smurgle s0 with neutral flumpiness that does not change the flumpiness level of another smurgle during interaction. ∃s0:∀s (s+s0=s). Inverse Smurgle: For every smurgle ss, there exists an anti-smurgle s−1 such that their interaction results in neutral flumpiness. ∀s ∃s−1:s+s−1=s0. Flumpy Symmetry: If a smurgle has flumpiness F, then its reverse smurgle has −F, and their combined flumpiness is neutral. flump(s)+flump(s−1)=0. Flumpiness Conservation: In any closed system of smurgles, the total flumpiness is conserved. ∑flump(si)=constant.

wow very good

Too bad model theory wasn't mentioned in the video although a hashtag reference is made in the description. A quotation repeated in the Wikipedia article for it says: "if proof theory is about the sacred, then model theory is about the profane" and "proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature." There is an obvious duality between proof theory (how) and model theory (what) which would be nice to explore more in a future video. Related to model theory is universal algebra, which is the generalization of abstract algebra. It helped me realize that symbolic math (in contrast to the "natural" notions of geometry and topology we experience in the physical world) mainly boils down to things (the underlying sets), one or more actions that permute those things (n-ary operations from the n-fold Cartesian product to the set), and one or more properties that each of those actions always have (associativity, identity, commutativity, etc). Abstracting over the underlying set results in a "variety" of sets with the same behaviour. Relaxing the "closure" property of being a homogenous total operation to a heterogenous partial function results in a (monoidal) category, since category theory is basically just an associative, unital algebra of the composition operation between various functions. And if all those functions in the category have inverses in a way that each morphism is an isomorphism, then it results in a groupoid, which is like if a group and finite state machine had a baby.

Interestingly related to what can be proven--it seems that all paradoxes have both self-reference and negation: Russel's paradox, "this sentence is false", etc. This revelation led me down a rabbit hole with associated concepts like the open vs closed world assumptions, negation as failure, and intuitionistic logic without the law of excluded middle. There's an amazing Strange Loop programming talk by Peter Alvaro called "I See What You Mean" which discusses this source of paradoxes. Now I'm always cautious when I'm dealing with something recursive when it's not fully constructive (falsehood and contradiction are not just related to the void unconstructable initial type). Also, type theory is interesting: it seems common that things are either defined recursively like a cycle with the collection of things containing its own definitions inductively, or else it's like a chain with the things being defined in terms of a collection of higher-order things. Set theory takes the first approach with being recursively defined but that leads to issues like Russel's paradox. Type theory is like the second approach in that there is always a higher order type which can define the lower order types ad infinitum. But dependent typing allows some backwards reference from higher order referring to lower order types, which gives a best of both worlds approach: it can be circularly defined when the goal is consistency but the ability to jump to a higher universe of types allows it to avoid paradoxes.

Actually all planets are oblique spheroids.🤓

Bravo!

Brilliant! Yes, we need a video on decidability and the connection between Tarski, Gödel, and Turing 🙏

Amazing video.

prove to me that P->Q is true when P is false and Q is true

This is a fantastic introduction. Your explanation of semantics of logic reminds me quite a bit of the concept of Differance by the "non-math" philosopher Derrida! When you get to the foundations of mathematics/logic, there is quite a bit of overlap with continental philosophy haha.

The first one is also false technically. i don't think any planets are spheres, I think the whole thing is that, in space, nothing is spherical. In space, there's no such thing as a straight line. All planets bulge, a little bit at their poles, and you would need a really specific set up to, like counteract that I i just don't think it's possible like you'd never end up with a system that the planet is spherical, even if they were made of only one element.

During my logic course, it was actually quite satisfying to prove that First Order Logic is both complete and sound.

Kinda weird to see all the commas and "hence" operators instead of actual conjution/disjunctions. At first i though it was an enumeration of arguments (A,A) not sequence A & A=> B

I think you should be a bit more precise in the completeness part. You cannot prove it *within* the system. You can switch to a higher order system and prove it there (assuming there is consistency).

A particularly interesting issue with logic and inference models is the 'raven paradox' by Carl Hempel. Nice video, keep them coming. Have a great 2025!

You seem to have confused "incompleteness" in the sense of Goedel's Incompleteness Theorems with "incompleteness" in the sense of not having syntactic proofs of everything that is true semantically. These are two unrelated ideas that confusingly have the same name. This is why the English Wikipedia page for the incompleteness theorems has a warning at the top: "for...the correspondence between truth and provability, see Goedel's completeness theorem". His completeness theorem actually says that in a first-order system (like the standard set theory axioms ZF or ZFC), the thing you're worried about towards the end of this video *can't happen* ! Every statement that's semantically true has a "blind" syntactic proof. (So unless you're a logician working with "second-order logic" or similar, all the math you do can be done on a foundation of set theory and be complete in this sense.) Goedel's *incompleteness* theorems are about something else entirely: the question of whether every statement is provable or has its negation provable (syntactically or semantically, since those are the same by completeness). For reasonably complex first-order systems, there are unsettled questions like that, but completeness still holds, so that *if* a question is settled by semantics, it's settled by syntax, too.

Brilliant

I think, for expressions like the one appearing at 4:28, parentheses would help a lot. Apparently, the expression is supposed to mean (A, A ==> B) |- B. But my first interpretation would have been more something like (A,A) ==> (B |- B). This, of course, doesn't make any sense (...or does it? ...it could be a tautology, I guess - "Given A and A, it follows that we can conclude B from B"? ....yeahhh...that sounds like it should be true, regardless of A and B 🤣 )) and was therefore confusing. Seems like one really has to know operator precedence rules here - the typographical spacing not only doesn't help but actually leads one on to the wrong way. I also often notice this effect with expressions involving the wedge product. I often get it wrong, what operand it "binds to".

It is distressing to me that this video here before us today tells us we must simply accept a thing as true (Goedel's incompleteness theorem, in this case) without this video providing any pretence whatsoever of justification for this claim. We should accept a thing as true merely by fiat, just because a slick video is telling us? That sets a regrettable precedent for educational videos. That would be unremarkable if this were, for instance, a gossipy news video. But the title of this video makes it clear that it intends to be telling us something about logic. Isn't logic supposed to refrain from making claims without attempting to justify them? In fairness, the video was extremely well organized and produced. You can tell a lot of thought, work, and care went into constructing and producing it. I just wish if a video about logic isn't going to give any justification for a statement, leave it out. If you're going to make a video in the future explaining the incompleteness theorem, fine. Talk about it there. Making categorical pronouncements concerning the truth or falsity of claims with no justification is the opposite of constructing and extending the network of human knowledge: it fosters and encourages the growth of human ignorance and error. Please don't do it. Please refrain from digressing into claims about a subtopic unless you intend to offer support for the claim. Thank you for taking the time to read this.

“There will always be unprovable truths” is not what incompleteness actually is. This is a common misconception. Rather, any sufficiently powerful logic system (such as first-order logic plus peano’s axioms) are possessed of a degree of expressive power sufficient enough to construct ONE single self-referential statement (the liars paradox) whose inability to be evaluated by that system imbues it with a trivial aspect of being true, but unprovable. That it no way implies other statements share this property, or that the rest of mathematics is “full of holes”, anymore than the liars paradox in colloquial speak implies our grammar is “full of holes”. Rather it simply shows we have the power of self-reference in such a language.

Spheroids*

As an expert on smurgles, I can confirm that all smurgles are flumpy.

Crystalline. Thank you!

Thank you for introducing Gerhard Genzen to me