Logic at its Limit: The Grelling-Nelson Paradox

Whoever said logic was infallible? Certainly not the German logicians who, in 1908, discovered this mind-bending paradox. Learn how they beat reason at its own game, all by asking a single question about two simple words. Don't skimp out or you'll miss the surprise twist at the end! Plus: a very brief, totally not copyright-infringing appearance by The Sorting Hat.

Contents:
00:00 - A Tale of Two Words
02:04 - Semantic Sorting
03:30 - Paradox in a Nutshell
04:41 - Grammatical Formulas
06:53 - Breaking Logic
10:12 - The Naive Objection
11:18 - This was never about words, was it?

[BirdBrain]

This may be a separate paradox, but this sounds exactly like the linguistic equivalent to the yes or no question "Is the answer to this question no?"

[justin.t.mcclung]

How about a follow up video (or videos) which explain Russell’s Paradox, The Incompleteness Theorem and The Halting Problem, and then show the equivalence of each to the others. This would increase the value of this excellent video exponentially by making it just the first step in a much deeper journey

[realbrickbread]

I solved the paradox!




Instead of putting heterological and autological *in* the box, you stick them to the side as labels. No strange sorting needed.

[SolsenMk]

Thank you for explaining the Russell's Paradox using language as a substitute. I've always struggled with maths, and when we did it in Philosophy, I had no idea what was going on XD

[AlexandrePorto]

Adopting Russell's system in "On Denoting" solves the paradox. Basically, the thing is words have no intrinsic meaning, only full propositions do. No words can denote by themselves, so the whole idea of those two words is inconsistent, they don't exist the way they appear to. The propositions "there is X such that X is a word and X describes itself" (or doesn't) are false propositions, there can be no such words. Logic is flawless, we are flawed.

[nedearbwormback]

So this paradox is essentially having two bins, a trash and recycling bin, and all the stuff is sorted into one of the two bins, but then you’re handed the recycling bin itself and asked to throw it away and it's like, "well, you can't throw a bin away into itself, yeah?" Seems like a problem of trying to throw your bin away when you don't got no bin for your bins, I'll tell you hwat.

[fireballferret]

Tenletters is my favourite autological "word"

...it's been living rent free in my brain over a decade and I finally have a word for it now, thanks!

[MinerUser]

I feel like you can easily sort autological into the autological category. There is only a problem when you try heterological.

[Xbob]

Seems kinda like the issue is just that the sorting issue becomes incoherent when trying to sort the sorters. To me it comes across less a problem of logic, strictly speaking, and more an issue of constructing the thought experiment.

[rjstegbauer]

I loved how you pulled together Russell's, Godel's and Turning's paradoxes!

[dimiturtabakov]

I'm glad you mentioned Russel's paradox (more easily digestible in the form of "the barber paradox") and Godel's incompleteness, cause they popped into my head and it occurred to me that a lot of paradoxes are a mere product of our ability to say "A equals not A."

[Alorand]

There are three categories:
1) Autological
2) Heterological
3) One of those self-referential paradoxes
Category 3 can not be grouped with other categories.
If you want to sort between 1 and 2 you don't start from all words, but from all words in the combined bin 1+2 which is first separated from box 3.

[stevemenegaz]

Logic is in the eye of the beholder. You choose your axioms carefully, minimizing the number to the minimum necessary and hopefully admitting the right ones required to resolve paradoxes yet not produce contradictions. When I studied Axiomatic Set Theory, there was a whole list of paradoxes that were problems, but not necessarily problematic towards getting a working theory of Set Theory. For example, Russell's paradox was just a example of admitting too much and was easily fixed with the Axiom Schema of Separation. I am not sure if Grelling Nelson is a hurdle to set theory, but as more of a mathematician than a philosopher, I only pick the battles I have to fight.We have evolved from ZF to ZFC to Von Neumann -Bernays Goedel to Grothendieck Tarski to ? By Goedel Incompleteness, we know we can not close the box. We just hopefully have enough to get er done for our particular application. .I wouldnt call logic broken, That is the way Mother Nature is and we should learn to work with it.

[unclejuju]

I love how you brought together all 3 of these paradoxes. They are like the NP complete set in that if we can solve one of the logical paradoxes we unlock all of them lol. Amazing video!

[gridgaming_]

this channel is amazing, i love the style of narration you use. too unnatural to be described as fully human, but too unique to be generated. really makes the video interesting to listen to. I also love problems like these, it was nice to see this covered as clearly as you did.

[seismicdna]

Wasn’t the set theoretic answer to actually make a new type of object called classes which were basically collections of objects which could not be included in other sets?

So in a way, even though it’s not resolved, the problem is side-stepped by creating a slightly different system which avoids the problematic recursion/self-reference.

I wonder what the linguistic analog to classes are.

[danielvarga_p]

Gödel incompleteness in action?!

Thank you it is a really great work!

[XetXetable]

The examples given at the end aren't really the same; rather, they are all theorems proven via a diagonal argument. This makes them special cases of Lawvere's fixed point theorem, but that's not the same thing as them being "translations" of each other into different domains. Other examples include Cantor's theorem about the uncountability of the reals (and varients there of), the non-definability of satisfiability, Tarski's theorem on the undefinability of a truth predicate, the non-enumerability of computable total functions, Borodin’s Gap Theorem in complexity theory, the Knaster-Tarski theorem in preorder theory, (the existence of) Kripke’s theory of truth, Brouwer’s fixed point theorem and the Ascoli theorem in topology, Helly’s theorem in distribution theory, Montel’s theorem from complex function theory, and Nash’s equilibria theorem from game theory are all, similarly, fixed point theorems proved via a similar scheme. This pattern is pretty common. The first to use it was Cantor in the proof of the theorem bearing his name, in which he remarked (originally in German); "This proof appears remarkable not only because of its great simplicity, but also for the reason that its underlying principle can readily be extended."

Perhapse diagonal arguments are the true topic of this video, and the claim at the end that these theorems are essentially translations of eachother is a rationalization for not naming the thing itself. If you actually go through the task of proving the theorems formally, you'll realize that the bulk of the work is in finding/constructing either suitable epimorphisms for the argument to go through (thus concluding that a fixed point must exist) or finding a suitable endomorphism without a fixed point (thus concluding that an epimorphism doesn't exist). The actual diagonal argument itself is, usually, the easiest part of the proof, however unintuitive a newbie might find it.

[doomtho]

To me, this “paradox” actually just exemplifies the fallacy of attempting to define a concept in terms of itself. In a way it’s a bit like trying to plug a power strip into itself. The thing is, logic can only function within a framework of fundamental rules or axioms (in our power strip analogy, this framework would be akin to a power source of some sort, e.g. a battery). So this “paradox” is essentially just what happens when you try to perform a logical operation on the framework of the system of logic itself.

[julianemery]

I like how clear this is, how you break this problem into smaller parts after first showing the overall idea, and how you showed different ways of approaching the problem.