My teacher told me that "all rules have exceptions" and I told her that that meant that there are rules that don't have exceptions. Because if "all rules have exceptions" is a rule then it must have an exception that contradicts it.

That's a good one.

Very True, However; it is a universal Constant that "He who has the Gold, Makes the Rules". Without Exception!!!

needed a minute to figure out how does that work haha

but if a rule has the exception of not having exceptions, it is still a rule with an exception; right?

​@Harry Mingelickr _Except_ there are societies that have rules but don't have gold, _and_ there are societies that have rules but don't _value_ gold. 😉

I could sit through a 5 hours math class of this guy, he somehow made a math subject entertaining.

go out in the sun, look at a plant - seriously .

@RazaXML some people (just like myself) need that 5th grade comprehension to even begin to understand math, so this is actually really valuable for someone like me and others like me. My math classes were always taught to those who actually understood math, like two people, the ones who didn’t (the rest of the class) were left in the after classes and usually got 1-3/10 grades..

So we always needed to go after class and rewrite the tests, it’s a fucked up system in a lot of Latvian schools, probably a lot more places in the world as well

This isn't maths - where is the practical application? This is a waste of time.

@jbooks888 math has applications that transcend the merely practical.  It’s a playground of logical thought where black holes are discovered and the contents of atoms and nuclei found. More than that, math describes and circumscribes the limits of our understanding of what’s *out there*.

I love the style in which you present your podcasts. You have personality, you connect, you hold our interest ❤

I can't help thinking that on some perverse level Russel was pleased with himself that his ideas had the power to literally blow someones mind.

or it was all pointless rhetoric.

@Diana Prince Sounds like you just prefer to not think about it.

@Somedude Ok Sounds like youre relieved someone replied.

@Diana Prince ??? I replied to you...

​​@Diana Prince Aw gee, my little snickerdoodle, you're too precious for words, darling dumpling. They make Midol for what ails you, bless your itty bitty little old heart, pumpykins. Koosh koosh!

Jeffrey, you are a born teacher, thank you for explaining this ❤

In my opinion, this "predicate paradox" isn't a big deal. A paradox in set theory is important, because it was intended to be a logically sounds system with no flaws or contradictions. But language doesn't work that way, and no one really needs it to work that way. So sure, there's a paradox in language that we can't easily remove, but that's completely fine. The paradox in set theory was the real issue, and it has been resolved by altering set theory to remove it.

Never thought I could have such an enjoyable time watching a 30 min video on advanced mathematical theory. I chuckled and even laughed multiple times. Well done sir

same here

I as well and I'm math phobic.

Wow, it was that long? I didn't even notice.

It's basic set theory.

Advanced?

As someone who is much more linguistic in my thinking than mathematical, this was a great explanation.

Was it though?

I agree! We can easily (or almost) understand the paradox! Great video!

While the style is good... his information is wrong.

@James J how so

@respite To quote my OG comment I made when I first saw this video: Your very first premise is wrong. Even if math itself is a product of the human imagination, that doesn't make mathematical truths subjective. The *_units used_* are subjective, but the truths themselves are objective, as shown by the ability to come to those truths no matter what type of mathematical system you choose to use. The *_system_* is subjective, but the truths discovered by that system are still *_objective_* no different than measuring distance. You can shoose to measure in inches, or centimeters... or even use cubits or any other mesaurement you choose, but the distance remains an objective distance. Only the representation of that distance is subjective. Since your very first premise is demonstrably incorrect, I'm going to assume the rest of your arguement is as well, although I will still watch the rest of the video to be certain. As such my comment may get edited as I see more of this video. Edit: Holy shit, 7 minutes and 21 seconds in and this is downright *_riddled_* with erroneous claims and misinformation about mathematics and sets. This is... fucking hell this is bad.

As a History teacher I must say that all set theory you explained starts to make more sense when you talked about the linguistic! Really interesting!

I love hearing the linguistics of set theory. English major who was good in math here. I want to live again.

Loved your summary, normally it takes much longer to develop all these topics. Amazing work Jeffrey

Loved this. Highly entertaining and instructive. Top marks.

Crazy how well Jeffrey can write backward on the glass! Probably was inspired by DaVinci or such.

Or he is hanging upside down and then they simply mirror and rotate the video.

Great video. I think it perfectly illustrates the fundamental flaws inherent to viewing language as a logical system. Even Wittgenstein, once considered the greatest champion of linguistic logic, decided later in his life to abandon that path. Language is not and never will be logical, because the purpose of language is not description but communication. All language is at its core more concerned with forming connections and being useful than with being accurate to reality

I was sifting through the comments because he never restated is original claim, “math is a human construct”? You put words to my thoughts. Math is objective, and any apparent paradox speaks to the limitations of our tools, namely language and human thought.

I must disagree. The purpose of language is not communication. The purpose of language is deception. There is no reason for language to be as complex as it is if it is designed to communicate ideas. It only needs the complexity we see if its goal is to deceive.

@Benjamin Rainey NO! Math is not objective! Math is completely subject to human minds. If human minds (or some other complex mind) do not exist then math does not exist. Being dependent on a complex mind by definition makes it subjective. You clearly do not understand the definitions of objective and subjective.

@acvarthered Fair enough. Math is a language. Our understanding of the universe, described through math, can be complete or incomplete. The principles of the objective universe are objective.

@acvarthered So did you just communicate or deceive?

Awesome video! your explanation is right on the spot. - Also "predicate" means pre= Anterior/prior, Dicate= made/Known, on the same context "Contradiction" is Contra=opposite/denial, diction= said/mentioned. and they both open the door to "Conditionality", which in on itself is conditional. - :)

Well done! A fantastic video, and an excellent explanation.

From what I have found this paradox has been resolved by the Zermelo-Fraenkel set theory. If a given set z and predicate n exists the subset is {x belongs to z: n(x)} instead of the set {x: n(x)}. This means only subsets can be constructed and that there does not exist a set containing all sets. Interesting stuff! Great video.

Doesn't that negate rule #1 again?

@Pablo UcanI'm not sure actually but seems a well established math concept. I wanted to post the link of the article I found on it from Brilliant but it wouldn't let me post to comment.

Right -- another way to look at it is, if we use Kaplan's linguistic predicates (without the ZF restriction) then we have a paradox. Given we have a paradox, we can construct a proof of anything (including things that are false by definition.) So, we don't use that formal system to try to prove anything. If we accept the ZF restriction, then a proof might actually be useful. ZF does not abolish paradoxes in other formal systems. It just provides a more limited formal system that is free of the paradox. Regardless, Kaplan's analysis is fascinating because it clues us into the fact that in order to have a consistent formal system, we have to abandon how we normally think and talk. We have to construct a language that is a little less like what some might call "objective reality."

You are extremely smart and well spoken, thanks for explaining this concept so thoroughly!!

This problem of self reference, infinite recursion, strange loops, or whatever one chooses to call it comes up again and again. Gödel’s incompleteness theorem is essentially another form of it, Hofstadter has made a career writing about it, and classical philosophers knew all about it and expressed it in many ways that we might boil down to the Liar’s Paradox or the most efficient form, “this statement is false”. They’re all logically-topologically equivalent. Good presentation for lay people, I like your channel and have subscribed. Going to check out your other videos. Cheers.

Honestly, there's a lot beyond my understanding. So it was weirdly reassuring to hear about the genius guy whose brain just straight-up blue screened because of this paradox.

How could it be reassuring? Because if set theory was “beyond his understanding” then something tells me this dude is not gonna be hospitalized over reading a letter he doesn’t understand.

Fun fact: basic set theory was part of Mathematics education in elemetary school in germany of the 1970s (and not a small one) . My estimate is, this forever reduced Germanys BIP by 2-3%.

Your honest comment is genuinely reassuring, because weirdly you solved the paradox. If there are not things beyond understanding, then the concept of understanding itself becomes nonsensical. The narrative construct of reality is a instance of mathematical induction, moving from the known to the unknown. Reality is chaos and the unknown, determinism is an emergent property of the process of understanding. So if you take away hope and possibility (which resides in the unknown) you take the life out of reason and the reason out of life. Fcvk I just blue-screened myself 😂

Your head was designed with paradox-absorbing crumple zones.

@Jacob Wiren it’s like the “designed-to-be-dropped” cartilage system never disappears. It merely adapts to be hit you take.

Very interesting and very well communicated 👏

Interesting discussion which reminds me of 1959 when I was in 7th grade in Junior H.S. (oops, got to be culturally correct - Middle School). In Math class the first book we worked out of was called 'Sets and Sentences' - an introduction to Set theory for 12 year olds. Thank goodness we weren't introduced to Russell's Paradox that year.

Set theory in middle school????

@Derp Yes. Public school education was more rigorous before teachers’ unions

Absolutely mind bending. Loved it.

This paradox is only possible because we're talking about immaterial things that can be inside themselves, which is impossible in the physical world. I can also imagine a character named Fry who is his own grandpa but thats not possible, but it is a paradox. Its just sort of like our brains being capable of thinking of things and comparing them to reality.

How about modeling a universe that was created without intelligent design?

@Joseph Coon How about believing in a god that is so powerful he can do anything and is therefore powerful enough to stop himself from being able to do anything?

If Fry marries a widow with an adult daughter, and Fry's father then marries the widow's daughter, then the widow's daughter becomes Fry's mother, and since Fry's mother's mother is Fry's wife, his wife becomes his grandmother. As the husband of his grandmother, Fry becomes his own grandpa. There.

@Litter Picker That would be a grandfather-in-law. There. LOL!

I was following along really well thanks to the Discrete Math class I took years ago. Rule 11 really tripped me though. I really liked the video and your communication style. Subscribed!

"A set of all sets" seem to cause a problem. Require to call it "a propper class" and you fix the problem for now by preventing self-references.

You’re a brilliant teacher; it’s not an easy feat to deliver such an entertaining intro to set theory in several minutes! Looking forward to checking out your other vids. Thanks!

As a historian of the ancient world, all I want to say is that if you took a learned Roman from the past and made him watch some math and physics videos, he would likely inquire how it is that the peasants from Iudaea seem to be the smartest people in the world. Because of you, I'll say. Thanks for that. It's made everything better.

If you liked this discussion, I recommend checking out Gregory Bateson's essay "A Theory of Play and Fantasy" from his Steps to an Ecology of Mind (1972). He shows that the paradox runs deeper than even predicate-based syntax/human language. In the process, Bateson shows that Russell violates his own rule (a set may not be a member of itself) by even positing his rule. And here's the kicker: Bateson argues that without such violations/paradoxes, communication as we know it (beyond rigid mood-signaling), would not be possible (including that peculiar game we call logic). This includes non-human behavior such as mammalian play, threats, and other metacommunicative interactions from which the metalinguistic rules and thus spoken language evolved. Bateson ends his essay with imagining what the world would be like without such paradoxes: "Life would then be an endless interchange of stylized messages, a game with rigid rules, unrelieved by change or humor."

You just explained this perfect. Expanded my mind. Love your energy. Ty

Brilliant. I already knew about set theory, except for Russell's Paradox. That was more than interesting. I can see that I'll be spending the rest of my free between learning german on DuoLingo and watching your videos. By that's fine be me!

The fact that he is so involved and tells you the story as if it is conspiracy tale is just amazing

Yes truly amassing!

Dude you need to chill 😂

items amassed = set

{x: is a youtube comment, x: is grammatically correct, x: does not contain malapropisms, x: is read by anyone}

@scambammer The set of all things that are amassed ?

Fascinating - I wish I understood more of it than I did, but fascinating nonetheless. One minor technical point, Bertrand Russell was Welsh, not English - even Mark Strong said so 🙂

I truly enjoyed your explanation, it is just down to earth and graspable. Thank you! The way I see it is like this: life is full of contradictions that don't always have an explanation, or also life is paradoxical and we have to accept it. Russell just grasped this via Set Theory :P

This was really good. You're going to be a star. I've been studying logic and set theory intermittently for 6 years. Wish this video was around when I started.

I read about Russell's paradox 30 years ago. You probably missed one of the first lectures.

I remember a phrase in a scientific book that I have used in understand some situations. " Rat is a feature of Rodent, but Rodent is not a feature of Rat." I had no idea it was perhaps part of set theory. It was used to define " Feature" in the a following discusion.

We covered this Paradox in my Intro Mathematical Proofs course when I was an undergrad! Nice seeing a video on it :)

I have loved proofs and now miss them, English teacher that I was.

@Sunny Greenings You should go back to reading some of them! Some are so cool. You should look up some of George Cantor's proofs. Those were my favorite

For a 57 year old man who cannot even recite his times tables (my head just doesn't do maths), I'm stunned I actually followed that, I really did!! That speaks volumes about this guys ability to convey information. I applaud you Sir, especially for the ability to hold my attention for the entire video. I quite enjoyed that!! I've no idea what use it is to me personally, but it was fascinating!

So are you really not good at maths or has it just been explained poorly to you in the past ...

@Grant Day probably a bit of both swaying more toward poor teaching. Ive always been very good at art from being a kid. Don't get me wrong here I'm not saying love me love me I'm thick! I have a BA (Hons) degree in the social sciences, but honestly I've never been able to recite my times tables. If you fire one off I can tell you the answer eg. 7x8 or 9x6 etc. I just can't recite the whole tables they way your taught to more or less sing them if you get my meaning. I got by for a while but when they got to algebra and sticking letters in that was it, I just lost the plot and switched off and had a giggle instead. I enjoyed Pythagoras like working out areas was easy, and the simple letter stuff like 3 × X = 9, but when those equations got a bit serious my head just switched off!! In retrospect, I wish I had a maths head as now theres so much Id like to ask questions on but feel I cannot explain myself because I'll look stupid. I'm absolutely fascinated by Jeremy Strides math on Coral Castle, but I'm lost when he talks about prime sets etc. Anyways, I'm waffling now. But yeah, I just don't have a brain that handles maths, but.... We can't have everything can we!! Gotta work within your limitations so I'll keep on trying lol. Have a good day

What a lovely comment for me to read! You've made my day. Thank you!

I was thinking pretty much the same thing. I've often told people that I am "math stupid", and blunder through anything that involves math. Jeffery's presentation was both captivating and inarguable. At least I think it was - LOL. But you shouldn't take my word for it; I'm math stupid. BTW - I've never watched any of Jeffery's videos before today. He reminds me of a mix of both Sheldon and Leonard from the Big Bang Theory. "Sheldon and Leonard" is a set. Heh. See? I learned something. :D

@Niche Bundles I did took set theory in the university explained by a very good professor and I have to say that the way Jeffrey was able to explain all of this in only half hour, while keeping me focused (maybe helped by the use of LeBron and Garfield) was just flawless! PS. I was also looking at Sheldon's eyes at some point of the video

Fantastic. Thank you. I am very confused so it must be that I understand all of it!

great stuff! thank you. I learned a lot here. And I learned that the set x : x has the name Jeffrey Kaplan is a non-Singleton

This seems like an unavoidable destination for making summaries of summaries. When you make really large abstract groupings, you will eventually try to make a grouping that breaks its own conditions. It seems more like a logical or necessary upper limit on how we can make groupings, but I'm only eyeballing it.

Rule 10 explains your problem. You said, "No one really understands this one, that's okay." Until rule number 10 is understood, Russle's idea is a paradox. So, set theory doesn't exist, but, this explains why everything, everyone, everywhere, at every time, in all of space, are one. We are all one, we are all members of a greater body within which we are all contianed.contained. Nice breakdown of it brotha ❤

Took 18 mins to get there but I definitely enjoyed it! Thanks for sharing.

It's the same as the liar paradox, "this sentence is false". Whenever you allow self-reference in a logical system (where true/false are the expected outcomes), you enable the paradox. "Sentences can refer to themselves", "Sets can contain themselves", "Predicates can refer to themselves" - - these are all equivalent, and all problematic in a T/F logical system. The solution, as Russell and others proposed, is to not allow self-reference (or self-containment), which makes sense because self-reference creates endless loops for the T/F evaluation, as you aptly demonstrated. The solution is to show how self-reference, though feasible semantically, isn't logically valid in subject/object relationships - to get into that is beyond the scope of this comment. Another solution is to allow self-reference but to specifically handle endless loops as "undefined", i.e. have 3 possible outcomes: T, F, Undefined. Great video but I disagree if you have hit on anything new using Predicates.

He wasn't claiming he hit on something new by showing predication is structurally equivalent to set theory. He was claiming that predication is how we naturally think. And that the paradox arises from the way we naturally use predication in language (and maybe in thought), and can't be solved way by saying "self-reference isn't allowed". Because in language, self-reference is allowed. And the rules of language are observations, not rules we can change.

@thenonsequitur Yes, Jeffrey's proposal is that the problem is real because it's noticeable in real language - the use of predicates, which is a new twist on the problem. However the problem with predicates is no more real than the problem with "this sentence is false", which has been around for a long time, and which he didn't bring up in the video, oddly... The video is initially about logic. When you bring up logical problems in language, you need to address the relationship between logic and language... what's valid/acceptable in one system is not necessarily so for the other. "This sentence is false" is a misconstruction (in logic) or a syntactic curiosity (in linguistics).

It is the same as all paradox That’s what a paradox is

@Bryo Jafa yeah this video, while interesting, felt anti climactic for me. He seemed to leave out some important bits to make it as mind blowing as it seems to be for him

@darren collings ... and like a paradox, you can tie up more than one boat

I always was terrible at Math in high school, but I was always good at philosophy.......I had to learn to stop talking with the teachers during the class because it'd often end up in an hour long conversation in front of the rest of the class...which wasn't really cool for them lol. The thing that I realized growing up is that my issues with mathematics and science often came from almost philosophical questions....When I was 14 and my science teacher told me that atoms were separated by ''empty space made of nothing'' and I genuinely asked how "nothing" could be "something", she kicked me out of the class..........seemed like a legitimate question for a 14 year old....I had to retake the class the following year...I learned the answers she wanted to get and pretty much killed any scientific curiosity I had up until that point. Long story short, this video is fascinating and I love it. I don't remember having this kind of interest for anything even remotely related to mathematics in over a decade. It is so interesting and so well explained. I thank you with all my heart. P.S. Sorry if my message is hard to read. English isn't my mother tongue, my right hand is broken in 4 places and the computer that I'm using, which isn't mine, insist on identifying every single world has being misspelled....It makes the whole thing a bit confusing lol. Anyway, I insisted on thanking and congratulating you for this great video. P.P.S. to be super clear, I'm not pretending that watching this video somehow made me less of a terrible mathematics student lol. All I'm saying is that it managed to peak my interest and fascinate me lol.

In situations like this I tend to fall back on the Venkman Axiom: _It's more of a guideline than a rule._ It's not the most scientifically robust of arguments but on the other hand I've so far avoided any letter-based hospitalisations.

just depends on how you define the set. once defined it can be indispensable, depending on if you want it or don't want it dispensable. contains its self or many sets of its self or if you want want it containable then it isn't.

It has really been some time since i ACTUALLY ENJOYED any kind of lecture so thank your for that! Great communication and hooking skills, pleasant voice, tones, flow and extra bonus for me you remind me of tobey maguire so im having spiderman teaching me science which as a thing is contained in the set {x : x is cool}.

Thats a cool set

😍Brilliantly written, presented and edited.

Nicely done; this video reminds me of one of my favorite books, Gödel, Escher, Bach: an Eternal Golden Braid (1979, Douglas Hofstadter). A set of things that are allowed to define themselves will always be incomplete or in other words if a class of something is allowed to define itself, then an instance of "something" can always be constructed that leads to a contradiction. I think Russell's Paradox, Gödel's Incompleteness Theorem, and the Halting Problem are all just different instantiations of this same underlying problem.

I frikken love that book.

I have that book, loved its metaphors with achilles and the turtle, but even with such informative and illustrious metaphors to help me understand the subject... man... it is way beyond me. I could try to to understand it but I fear I would suffer the same fate as when Gotleb Frege when he got Russell's paradox in the mail.

Yeah - this is just another version of 'All Cretans are liars', so it's very old.

@Grant Hearn Also reminds me of sitting in class listening to some rule and then shouting out the paradox to get a laugh. This paradox is everywhere whenever you start curving arrows inwards

yes

Rule #12: Sets cannot be a contradiction. I appreciate how well you write backwards also.

... and wear shirts that button up the opposite way to most! (its almost like we're seeing a mirror image)

@Karl Rodgers hah, indeed. I figured it was mirrored and was looking for a tell and spotted the shirt. men's shirts and jackets button up such that you can draw a weapon (sword or dagger) with the right hand.

Breaks rule no.1

​@TheFlyingEpergne Well yeah, rule 1 is the problem. That's why it's different in other forms of set theory.

@t Yeah. Seems like it should be Rule #1 Unrestricted Composition where not an inherent contradiction. Rule #1 causing the paradox would indicate to me that Rule #1 has to be modified, because it doesn't fit all data. The assumption inherently leads to a contradiction, therefore it's false.

this reminds me a bit of the classic riddle – you encounter one person who always lies and another who always tells the truth, and you can ask them one question to determine which is which.

That thing that goes bump in the night, something that we’ve all heard at one time or other and puzzled over, is actually caused by the collective force of people’s brains figuratively exploding while trying to suss out an understanding of Russell’s Paradox and reoccurs every time we hear a sound that we label as “something that goes bump in the night”. That it’s all figurative is the very thing that allows us to continue existing.

It would be interesting to consider quantum computing with this as a solution as a qubit can represent multiple states at once for the boolean expressions.

That's exactly what I was thinking.

Great video, loved your explanation. One thing I kept thinking when talking about the predication paradox is, once the predication becomes the subject, is it still a predication? This hurt my brain..

This man legit put his own death year in the quote what an enigmatic legend I dig this guy

off the grave yeah

I have long known that *I shall die on 21 April 2052* , aged 89; I am so sure, in fact, that it's been up on a poster (containing my favourite quote) I created and stuck in my office 26 years ago. My hope is that if nothing funny happens on that day, some gentle soul -- knowing of the prophecy -- will be kind enough to do me in. There are, after all, many ways to generate a self-fulfilling prophecy.

@Olaf shom Kirtimukh I will hopefully live tomorrow starting from the day I post this comment. There you go.

@Olaf shom Kirtimukh Olaf, if this is your true desire, I may be able to help

Thank you, I found the explanation captivating.

Absolutely Brilliant! Bravo. Could the paradox serve as an explanation for a subjective nature within mathematics & predication? Meaning, the paradox can only be generated from a particular perspective.... Also, it seems that creating a set of sets of contradictions is permissible under rule #1.

I have a degree in mathematics. So when I took any math class, most professors began the class with an overview of set theory.

As a philosophy student I feel I have to defend Frege and Russell, the intention of “counting” in set theory was to explain the way in which something like the number 4 always picks out 4 entities (like 4 apples) but is never itself those entities (4 is a number, apples are apples). So they thought it must be the set of all possible 4 entities that makes up the number 4.

this is exactly how I feel every day, all the time, weather I'm sleeping and dreaming or I'm awake and living, not that I'm constantly aware of it, but when I am and do get aware of it I only know that I, and I is a curtail statement here, am aware of the emotions that come out of the feelings that are connected to experiences, not that it matters weather I dreamed of them or actually experienced them, but rather the construction of it all, based on the moment, is what it is and it will always be that... So, it does not matter what we think it can be, but rather what it has been, so the "SETES" can only be the past and not he future, here in this dimension, and if you ever go and approach this problem with future in mind you won't be able to solve it, because it is out of your dimension and understanding, at least, but that won't stop us from pushing the boundaries of our dimension and who knows where lies the solution to this problem ""in the future" where we will move the boundaries of all KNOWN EXISTING LAWES !

I've always been fascinated by the other set in Russell's paradox—the set of all sets that do contain themselves. If you ask which of the two sets that one belongs to, you can show that there's no problem with saying it belongs to either set. There's a certain freedom there that mathematical objects are not supposed to have.

Gödel.

That "this statement is false" is undecidable gets all the attention, but "this sentence is true" is just as undecidable.

that's basically what philosophy offered to all the mathematicians, a back door

thinly veiled mysticism, opium for the few. adults speaking in codes according to complete arbitrariness. you might as well expel hot air the other way round!

Well if you want to have that much fun, add into the paradox the consideration of "infinity". Is infinity with boundaries truly infinite? But a can loop of round material can be infinite and finite at the same time? You could chase your tail on that one for a long time. You can go an infinite number of loops around the loop returning to the same point each time whether you go in a loop around the larger loop or the smaller circumference (also a loop) :)

Based on your description, I understand the interpretation (mathematical argument?) of the paradox for rule 11 (26:32) to be: -1X = 1X, where X = -1

Bravo Très intéressant facile à comprendre, très dynamique

Very interesting video, but it confirms my belief that stopping mathematics after calc 2 was the right call for me. It was easy to understand and I followed along but I'm still dumbstruck afterwards.

In object-oriented programming, Sets and Object follow most of the same rules. Different objects must interact with reality while Set theory can complete intellectual processes. You frequently have deal with paradoxes like an object containing itself

Btw, I like the way you explained it very much. Very clear arguments. I wish all my fundmaental and logic math classes were given this clearly.

I haven't read much philosophy and even that was a long time ago, but I remember Plato going on about the skill philosophers need to practice - abstraction, maybe? Well, he didn't use that word, it was something about learning to see the idea behind the things. The point is that it is a skill to be practiced. So maybe professors don't always explain the best way possible, but you gained the skill anyway - which is useful, since it is rare that math comes the most consumable way possible. My logic prof was also quite dry btw. Well, I rambled, sry.

The first thing that I was thinking was: didnt Gödel do the same theory more or less? And what about the halting problem? And to my delight they are indeed connected.

Great video! Never have I heard such a complicated way to explain mumbo-jumbo. Nice!

That video was very, very good and fascinating! 👍👍👏

Aha! I think the best solution is to add a new rule: a criteria (a.k.a. condition) can only apply once per set: i) to prevent infinite recursions ii) to eliminate Russell's 'paradox'. Another formulation is each set must be UNIQUE - if it shares the same criteria as another set, it is not a true set but is the same as the set it duplicates. *per the duplicates-don't-count rule Hmm this basically means take out the rule that says a set can contain itself 😁

Let’s say just as a means of description that a paradox is constantly in-between each other in a endless continuum that’ grades into one another

Wow! You're both, a philosopher and an entertainer with substantive information. It's refreshing to have come across this video purely by chance. It brought me back to the carefree days of early college dilemmas that provided endless hours of conversation over liquid refreshments before stumbling back to class at 7 in the morning still wondering whether anything was relevant to life in general. Kudos!

I believe things like this are even more evidence of God. We can't even begin to comprehend things that we take for granted (like language), yet we want to believe that we figured everything out. God does not abide by our rules, and this is yet another sign.

@Luis Arevalo Belief systems are based on evidence, not blind and arbitrary selection as I feel you're implying (a straw man argument). We have overwhelming evidence in the case of Islam.

@Ziad Hamed as you say, belief systems are based on evidence, but that also means not proof, just evidence. Belief systems also lack a systematic approach to testing and properly validating their own beliefs. They are not dynamic for that reason, they're static. They never evolve or get any better with time. It comes down to the very definition of what the word belief means. If you have to "believe" something it's because you have given up knowing it for a fact and have to rely on someone's better wisdom for it. That's what "believing" means. It's the choice to not pursue knowledge.

Another variation is the word "heterological" which means "does not describe itself." Most words are heterological, while words that do describe themselves are autologicial. However, if you try to define the word heterological as either heterological or autological, you run into the same self-reference paradox: If the word describes itself then by definition it does not describe itself, and vice versa.

Isn't it the case that the only predicates you mention that validate this theory are simply the ones that quote themselves? All the other ones fail. If so wouldn't adding a law (which it still seems to me is an option) that prediactes can't directly reference themselves only, be a solution to this paradox?

Questions: Is a clause which contains a verb still a predicate once it is removed from it's subject ? Might said clause actually be nothing more than a subject in this argument ? This set theory, and by extension the paradox, may be applicable to other areas of logic besides mathematics, but I'm not convinced by the predication examples.

Great video. I kept wondering though why ‘can’ is being interpreted as ‘must’. To me, rule 11 read as “it’s possible, but not mandatory”. So my question is “would accepting that premise remove the paradox?”

Jeffrey, I have solved this paradox - in a way that explains all the points you made in this video. Is there an accepted answer already?

I think the paradox falls apart on a use/reference distinction. The predicate, once taken out of context, is now simply a reference to the actual predicate. The fact that the reference predicate occasionally has a meaning that seems to relate the sentence it is in is just sleight of hand. Once a predicate is removed from its context, it no longer retains any truth value. It simply references something that does. Of course, a reference predicate still attributes, but only trivial ones, like "is written in English".

Exactly what I was saying about “colors”

well reasoned, you perfectly articulated what i more vaguely surmised was the trap

I somewhat agree. Someone wrote in the comments "All rules have exceptions". Does that rule have an exception? I think that's what the video was trying to convey.

@Conversations With Gambit Yup. That's a sticky one. I think that is a paradox. But it's not the sort they are discussing here. I think here they are just engaging in a really complex version of "This sentence is false". "Sets" (like "sentences") are containers that don't have any non-trivial predicates. So you can't create a paradox with one. Fun fact, I still don't think set theory is the right way to "prove" math. I think we need to look at it as an emergent psychological function. Like "what are the mental states required to create a mathematical idea" and how certain concepts are reified and privileged by their utility. I think once we have a better grasp on that, these questions might seem a little more solvable. Possibly.

Patience… not so much a paradox as an error created by the limitations of the mathematics applied.

I enjoyed this talk. I wish I'd had more "philosophical" math teachers in high school. I had to laugh: "The world" is a specific location (earth) in space! All the cats on earth (at a specific time) are thus in "one place" in a sense. Ha ha (Of course he was speaking about a much more limited notion of "place" like a room and not earth versus the moon, etc.) I learned naive set theory in elementary school. We were taught Venn diagrams. It has helped me many ways. {x: (x = all sociological theories pertaining to the general idea of legitimate authority)} is the "set" I am very interested in. I am especially interested in one sub-set: {y: (y = a sub-set of x, where the theory is derived from thought published by Max Weber or his heirs)}. I tried to understand Russell's ideas about sets in his co-authored Principia Mathematica and got what I thought was a pretty good idea of what he was writing/thinking about. In 1901-1902 the idea of a set which does not contain "itself" was indeed paradoxical. (I am a little bit impressed that Russell wrote his letter in German.) The axioms do seem contradictory. Can a set "contain itself"? Does the empty set contain itself? Since "Garfield" does not really exist as an actual physical cat it is false to say: Garfield is a cat. (Think: "This is not a pipe.") The "predication" of a fictional character (e.g., Mickey Mouse, Sherlock Holmes) is only sort of "true." Hence, "Is a cat" is a predicate that can serve as a "subject." All of this is lots of fun (at this level). It does all seem a lot like the liar paradox. I think it can be "escaped" by considering the tricky way in which the predicate of a sentence sort of ceases to be the predicate of a sentence when it is no longer used in that (or a similar) sentence (or equation).

I'm starting to think Russell had a foreign assistant who just kept saying, "But that's a set too." to anything he asked him or said to him. "Can I have some tea and crumpets?" "But that's a set, too." Ahhhh, Genius.

When you were making this video did you experience that phenomenon where you repeat a word so many times that it stops making sense? Because after a while I couldn't even mentally utter "set" without getting confused. Awesome video.

Lol I just had this same experience. Though I have had it with many other words at different times, about halfway through him saying “set” so many times I just kinda forgot what “set” was and it seemed like a foreign, unknown word to me.

This actually has a name, it’s related to Deja Vu, it’s called Jamais Vu

@Ryan Brown thanks I knew there was a name, just had no idea how to even begin to look it up.

Russel summed up his paradox by asking "Who shaves the barber?" In a certain town in Bavaria, the barber shaves everyone who does not shave himself. "Who shaves the barber?" When a liar says, "I am lying," the liar is telling the truth. When a liar says, "I am not lying" the liar is telling a lie. This is a semantic paradox that underlies (pun unintended) inquiry into truth particularly in legal proceedings.

"Mr. Morton is the subject of my sentence, and what the predicate says, he does." As soon as he started talking about predication, I heard Skee-Lo rapping his wonderful interpretation of my young childhood: https://www.youtube.com/watch?v=Dj4H3Ioxs6s Also, really great video!

You have a gift. I actually understood this whole thing. I wish I'd had you teaching all of my philosophy and math classes in college.

I wish he would endeavor to explain why Zenos Paradox isnt really a paradox, because it surely is not. That would make an interesting video

If you understood it, you would realize that you can't really understand it ;)

Reminds me of the riddle of two doors and two guards: One door to freedom, one door to horrible death; a liar guard and an honest guard; you get one question of one guard. What you ask is kind the Russell's Paradox.

No it's not. You ask the left guard "will the right guard tell me this door leads to freedom?" and if he says yes then take the other door. That's a riddle, not a paradox. Asking one guard what the other would say will get you the opposite of the truth.

Brilliant lecture !

1:15 “Four” exists in the same space as “blue.” Both are psychological constructs used to represent something in reality which allows us to virtualize and simulate reality in our minds. 18:30 Also known as the Heisenberg Set: it is in a superposition of containing and not containing itself. 20:00 Wouldn’t “unrestricted comprehension” just mean anything that is comprehensible is allowed, so since the {x:c is the set of all sets that don’t contain themselves} being incomprehensible, becomes excluded based on Rule 1? 22:15 Predicates aren’t necessarily true of their subjects. “You dunk” is an example of a predicate that isn’t true of the subject of what you said earlier is true. So, every pair of contradicting statements can have one example that the predicate is true of the subject and the other predicate is false of the subject. You can even have contradicting comments where bothers predicates are false. 24:45 “Is a predicate” is actually a subject. 25:15 “Taste like chicken” could taste like chicken. It all depends on how your brain is wired. See: synesthesia.

I think like you. We should be friends.

Comprehension just means you can specify a set based on a property that a set may have, so the set of all sets that contain themselves is entirely well defined if you accept comprehension.

@vitalstatistics Well defined implies you comprehend the definition, no?

@Likafoss I got a lot of outrageous ideas…you might want to reserve judgement. 😂 Cheers 🍻

Robin Williams: "Reality, what a concept."

Where do I find an intellectual man to pick my brain like this man does? Great video, very well explained! ❤

I made it to minute 17 as far as even vaguely understanding this! Strangely enjoyable nonetheless.

This sort of paradox has a very deep and sublime influence on all of mathematics; from basic to QFT and GM. For whatever theorems we try to prove in advanced maths or physics, if we are not careful then something like this can ruin whatever experiment or theory we wish to prove. I’ve always felt that one of the hardest things to understand about mathematics is that it can be it’s own meta-language. 0:04

Huh. I cannot remember ever having a problem with that. Of course, that leads directly to Gödel's incompleteness theorem, which is the exact same argument as this video, just changed enough that you can apply it to proofs. Oh, and let's not forget that this kind of proof, while more complicated in the case of Russel and Gödel, is really pretty much the same as Cantor's diagonal proof that the set of all real numbers is larger than the set of all natural numbers. It's basically all the same idea. The question is how you deal with it in each case.

​@Kai Henningsen You don't, and can't, deal with it. And that is the fundamental issue. The entire thing is infinitely recursive - when you derive the meta-language of mathematics, that set/ system/ proof/ etc. itself can't be it's own meta-language once more. There is, and always must be, something outside the set of all sets.

@duane oldsen I can deal with it. It is only a problem if you want every statement to be either true of false... but then it's your problem, not mine.

@Alain Bruguieres In fact, the very idea that a paradox exists at all is entirely made up.

@Kai Henningsen What's the similarity between Russel's paradox and Cantor's diagonal proof?

I very much enjoyed this video. I also very much enjoy that the set of all-time leading scorers of the NBA is a singleton set which contain Lebron James...but the set of greatest of all time is a singleton set which contains Michael Jordan.

The concept you are missing is that of ‘logical type’ - this is not only russel’s solution to the paradox but a very important idea if you are keen to develop some METAlinguistic awareness

A concept you are been attracted from, that now dictates the way you think, is "logical hype". A paradox cannot be solved. A paradox is the symptom of a logical issue, it just says that your theory is wrong, you want to solve or prevent the cause of the issue instead.

You dont know about the history of paradoxes being resolved - paradox inspires refreshed/refined conceptualization: consider the ancient paradoxes of motion, and how they are resolved with the aid of the new concept of *the limit of a series* . . .

@julian Holman If it has been resolved, it wasn't a paradox, by definition.

@Dino Sauro a paradox will always be in relation to a particular conceptualization

I used to write on the back of glass like that to read from the other side. Even writing backwards and with either hand, my handwriting was still recognizable. I was surprised.

Interesting concept. I love a good paradox. What was most interesting to me about this video is how Mr. Kaplan can write on the glass in front of him so fast in mirror image.

He can't. You assume what you see is a moving picture. It's flipped left/right in post production.. still cool.

@andrew cobb He's probably not writing backwards, that doesn't necessarily mean that he can't.

@Mace Thorns You are correct. Anyone who can write can write backwards. It's rare for it to be effortless. His quiff, watch and buttons give away the mirroring in post production - and I bet he's a rightie, not a leftie.

You made this vid really interesting and funny. I enjoyed this a lot

To those who would like to learn about the context of how that theory came to be along with Godel's incompleteness theorem and the birth of set theory in a rather fun and pedagogic way i would recommend the comicbook "logicomix" that centers around the life , with some apocryphal event, of B. Russell in search of mathematical truth . apart from that Great video as always from you Jeffrey

Would the creation of a set for those sets with oscillatory conclusions help? It would contain, for example, the "I am a liar" undetermined statements.

If I had a nickel for every time I've watched a video about Russell's Paradox and Set Theory and learned more about LeBron than set theory and the paradox itself i'd have 5 cents. It's not much money, but it feels weird that it happened.

I think that you can get out of this paradox by making a rule 12. if that itself makes a paradox you make a 13 and so on.

Think of it in terms of summations / series. Some number series goes to infinity, some go to negative infinity, some stabilize to a concrete number, and some oscillate. The "paradox" is simply an infinite series that oscillates. It's easier to understand in terms of "the set of all sets" ... you can try to imagine this set in your mind as the process of building an ever growing set: for example start with putting LeBron in the set. Ok cool, that created the set of LeBron, which is a new set, so then put the set of LeBron into our set of all sets. That is also a new set - a set that contains the set of LeBron, so put that into our set of all sets too. That again is also a new set... and so on and so forth. Thus the set of all set can be seen as an infinite series that grows to infinity [number of members]. There is nothing very strange about infinite series, and thus I argue that there is nothing very strange about an oscillating set either. Oscillation in system of logic is something that very much happens in the real world all the time.