Russell's Paradox - A Ripple in the Foundations of Mathematics
Bertrand Russell's set theory paradox on the foundations of mathematics, axiomatic set theory and the laws of logic. A celebration of Gottlob Frege.
Thank you to Professor Joel David Hamkins for your help with this video.
Hi! I'm Jade. Subscribe to Up and Atom for physics, math and computer science videos!
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What is a number? (no using the word number)
Up and Atom The existence of an element. whether it exists or not or repeated
An object that characterizes (represents) an equivalence class of sets that are in a one to one correspondence Note: cheated, I already did some set theory
A element that represents position on a particular manifold, such as the space of real numbers.
I'll bite: The quality of a category that describes the observed occurrence of discrete instances within that category. "Instances" is kind of number related, but strictly speaking the concept only requires understanding of the idea of individual things existing, distinct from others. It's enabled by counting, but it doesn't depend on it.
You want to know what a number is? Here you go: kzhead.info/sun/iNGQncqMq2aMdYk/bejne.html It's the first of a 8-part series of 4-min videos. Next video has a tag (G1). Here are all the videos: kzhead.info/tools/RUATK39-y_dSwmN59_-aNQ.htmlvideos
What I learned from this is if you get a letter from Bertrand Russell, don't read it.
Honestly, if you get a letter from Bertrand Russell today, bring that shizz right to James Randi and get yourself a million bucks.
@@DampeS8N There's nothing supernatural in that. They call it 'snail mail' for a reason.
Joe, letter from Bertrand Russell: can you go fetch my teapot, please? (Read as: maybe make a video about interesting thought experiments?)
Russell: Dear Joe, I wholeheartedly enjoy your channel and would not change anything.... Except this small thing that will ruin your confidence for eternity. Joe: Wanna collab?
but it could have money inside!
I enjoyed this video so much. The animation and the explanation are so good!
WoW! , youre here... yay!
Hey dude!!!
You chad LEGEND
@@dakshdheer1681 whaaaaa?
super cool dude? what the hell!
I'm three years late to the party, but I really enjoyed this video and wanted to offer an answer to the important question you asked, "What Is A Number?" The most perfect definition of what a number is that I've ever come across was over 25 years ago when I first read a book called "Mister God This Is Anna." Anna was a truly remarkable 5 year old girl who asked the same question and shared her incredible answer. Anna knew that 1 planet and 1 ant were in no way equal, but wanted to find how and why the number 1 made them equally countable as "1" mathematically. She discovered her answer through a light and shadow experiment. She had an adult set up an overhead projector so a blank square of light shined on a wall. She then placed an apple on the overhead projector screen which made a 2D shadow of the apple on the wall. She then taped a piece of paper on the wall, traced the outline of the apple's shadow and cut it with scissors. She then placed the paper cutout of the apple's shadow in front of the projector holding it at a 90 degree angle, which created the 1D shadow of a line on the wall. She put another piece of paper on the wall, traced the line and cut it out. Then she took the paper cutout of the line and held it over the projector at a 90 degree angle...and was left with a zero dimensional dot on the wall. Then she pointed in excitement and said. "That's what a number is!" No matter what the size, weight or shape of the object was that she conducted this experiment with, she was always left with the exact same dot. She then realized that if there was a projector and a wall big enough, her experiment would get the same dot putting a planet in front of it as an ant. And so Anna concluded that in our three dimensional universe, a number is light's shadow of a shadow of a shadow. I've never found a more beautiful or perfect definition that doesn't use the word "number" and is fully supported by experiment with completely repeatable results.
Underrated
Thanks for this!
@@carlosraventosprieto2065 You're most welcome!
Bravo!!
Dammit! I just waisted 5 minutes of my life because I read really slow.
This is brilliant. I was trained as a physicist and last night - over a bottle of wine - tried to explain the Russel paradox to my baffled adolescent daughters 😃. I now sent them this link 😂
yeah i try to explain too ,but most of them have no idea what am i talking
You just put the barber into a superposition with himself. You do this by putting him in an isolation box as in Schrodinger's box and you use an electron gun pointed at a spin detector. The detector will reveal if the electron is spin up or spin down. Tell the barber to shave himself if the spin is up and not to shave himself if the spin is down. Then start the gun up and close the box. Inside the box the Barber will be in a state of having shaved himself and not having shaved himself at the same time. You can also solve Russell's paradox using this method and any self referral paradox. You have to use the real world which is quantum mechanics and stop living in Newtons Classical world. Let's face it zero and infinity can't exist. Mathematicians completely ignore the uncertainty principle when they do their thought process to develop math. You can't create math that is impossible. That is what they have done. For Russell's paradox just create two sets R1 is the set of all sets that don't contain themselves and include R1 in the set. R2 is the set of all sets that don't contain themselves and exclude R2 from the set. Put these two sets in writing on two papers in a box and have a random quantum event burn one of the papers. Close the box and inside the box will be a superposition of R1 and R2. The superimposed set is labelled R3 and it contains itself and doesn't contain itself at the same time.
This was fantastic. Please don't worry about being overly-nuanced or complex--there is already plenty of dumbed-down content available elsewhere, and you have a skill at presenting complex concepts in a straightforward, understandable manner. Thanks.
I second this message. This is why this channel is great.
I on the contrary think that this video did not explain anything substantial or get into sufficient detail. The talk was mostly about history, and introducing various concepts. For me it only managed to define two paradoxes.
Ivko Stanilov agreed- I absolutely enjoyed the video but was waiting to jump into the abstractions and into the weeds a bit after the intro warning. Hope for a part two going deeper!
Brilliant point! I think if the delivery was simplified, a different audience would be attracted. But I'm grateful that these videos were made in a way that appeals to me.
Well said, sir. I completely agree.
The barber was pulling his hair out trying to solve this problem, which ironically did solve the problem.
until it grew back
@@golfgod1017 when the hair grew back, the problem also came back. So, he found himself once more pulling his hair out again trying to solve it.
@@post1305 unless he learned from the experience and chose a different approach.
@@golfgod1017 There is no evidence to suggest that happened.
@@post1305 or you just didn't see the evidence
The clarity you bring to these difficult to articulate and comprehend topics is exceptional.
Brilliant. There’s nothing else like this. I’ve been struggling with this for too long to mention and this graphic presentation is the clearest I’ve encountered.
"So Baldrick, if I have some beans and add one more bean, what does that make?" "A very small casserole m'lord."
"Three beans and that one."
Great video, but my favorite moment is when you said "Tifa is a dog" and she looks at you as if saying "Wait?? I'm a DOG???"
xacharon insulted that we assumed she couldnt smell cause she’s old. She’s a dog, not a smoker, dammit!
@@alex0589 old people lose the acuity of their senses as they become old, even when they really, really take care of them selves... I think it's just how genetics works, man. i suppose we are probably eventually going to figure out how to prolong this, but i doubt out doggie friend has been genetically modified.
This does bring up a shortcoming of building things out of simple logic: given “dogs have a good sense of smell,” if Tifa does not have a good sense of smell then “Tifa is not a dog” is a logical conclusion, but we can all see she a good gurl
xacharon ohhhhhh shit
This is the best and clearest explanation of Russell's paradox that I've ever heard/seen. Thank you so much. I think I actually get it now :)
Yeah..
omg i've seen videos on this paradox so often but this is the first time i actually got it!! Thank you sm🙏🏼
Consider a sets of all sets that have never been considered. Oh wait, they’re all gone now, never mind.
They haven't been considered, just the set now no longer contains itself :)
I see the joke you did there
so underrated lol
Nice one. I think paradoxes should be hunted and taken as gateways toward unpacking primitives and axioms.
@@muhaimin244 lookup Vsauce
Gottlob Frege: * makes a definition of number* Bertrand Russell : I'm about to end this man's whole career
Foundation is not the right word. These sciences existed for thousands of years before their 'foundations' were even known to exist.
@@Arigator2 It doesn't matter if the word came before the sciences, it can still be considered a foundation. Just as foundations for houses were foundations long before the name "foundation" was invented. This is actually the case with most things. Think of it as "common source" or "common basis".
lol
@@Arigator2 Right, And the same thing is true for mathematics. Science, and math can begin anywhere you like. Whatever you happen to discover, observe, experiment with first. And then it can grow from there in any direction. I think a problem arises when we try to put knowledge into a "tree" format. We make an unfounded assumption that there is a "base" or "foundation" or "root" of the tree, and the rest of science/math grows upward from that. Logic does not have to be at the very "bottom" of the math "tree". Arithmetic works correctly and consistently anyway. We can start with that, and then explore "upwards" or "downwards" as far as we like.
@@fredrikekholm3718 - No, it's not a foundation because they aren't actually built on them. Math and science existed for thousands of years before someone decided to try to come up with a 'foundation'. Math is not based on the definition of a number.
Best visual representation of plato I've ever seen. You are also an amazing communicator
Frege: here's the neat systematic set theory I made. Russell: *I'm about to end this man's whole career*
And hospitalize him
Not only did Russell live a long life (he died aged 97), make huge contributions to logic and win the Nobel Prize for Literature. He also wrote A History of Western Philosophy, a book which remains the standard text for anyone interested in the subject. In short, Bertrand Russell was a truly remarkable guy. This was a great video. Thx for sharing.
It's a great book (so far), working my through it currently.
Bertrand Russell’s provocative _History of Western Philosophy_ is an entertaining account of his biases. Frederick Copleston’s _A History of Philosophy_ is still the place to start for anyone interested in following man’s speculations about himself and his world.
@@jonathansturm4163 I will check it out
Yeah, the dude was bomb!
Dumbass alert, Russell’s text is fascinating but nowhere close to standard. Very subjective
This was great. I have heard Russell's Paradox before and my response was usually, "Ok, but so what?" What you did here was put a seemingly uninteresting paradox into both the historical and mathematical context to help me see _why_ this paradox is so important and interesting. Thank you.
Thank you
I mean the paradox is another way of saying that an axiom cannot prove itself. That happens in logic therefore in math. If you want to go more in depth you can check out the incompleteness theorem of goedel.
Bertrand Russell has been one of my heroes since I first heard about mathematical philosophy
It was actually first discovered by German mathematicians before him, but he was the first to publish it.
nice video, but OMG i feel so bad for frege. imagine being so determined that you would solve all of math and then your years of hard work is just crushed. i understand math is like that because theres paradoxes and all, but i feel like me and lots of other people can relate to the poor man mentally
I love how Immanuel Kant "soon came along" after Aristotle. I once had to teach a Phil 101 course, and our textbook jumped from Aristotle to (I think) Descartes. In the final exam one of my students wrote, "Descartes was a student of Plato, but you'd never know it from the things he wrote."
mathematics students
mmanuel Kant was a real pissant Who was very rarely stable Heidegger, Heidegger was a boozy beggar Who could think you under the table David Hume could out-consume Wilhelm Freidrich Hegel And Wittgenstein was a beery swine Who was just as schloshed as Schlegel There's nothing Nietzsche couldn't teach ya 'bout the raising of the wrist Socrates, himself, was permanently pissed John Stuart Mill, of his own free will On half a pint of shandy was particularly ill Plato, they say, could stick it away Half a crate of whiskey every day Aristotle, Aristotle was a bugger for the bottle Hobbes was fond of his dram And Rene Descartes was a drunken fart "I drink, therefore I am."
Is that because the middle ages philosophy only had to do with religion and Plato and Aristotle's Organon, until renaissance humanism came along?
@@captainzork6109 Not exactly. Even the so-called "churchmen" looked at what we would call philosophical questions about epistemology and ontology and the philosophy of language. There were also philosophers in the Caliphate that I know very little about. More modern thinkers have created theories that people nowadays take more seriously than the medieval ideas, so the medieval philosophers tend to get overlooked and forgotten. It is, however, a deep vein, and I think philosophy is as much about the thought processes as about the end result. Journey vs. destination.
@@sourisvoleur4854 I'm a psychology graduate, and although my Master is in Theory and History of Psychology, it has only been since a year or so I've started learning philosophy and history more generally. But thus far it seems like their epistemological questions have been very broad: What is the world, and how can we know of it? And, as Nietzsche pointed out, even until Schopenhauer the hinterwelt had always been part of the most prominent thinker's philosophies. That is to say, scholars in the past put so much emphasis on some 'more perfect world', getting lost in a convoluted mythos of heaven and hell, that they failed to make any sense of the here and now. As far as Francis Bacon was concerned, those scholars were all just armchair scientists, who come to the wildest conclusions based on singular experiments Except, of course, when it came to practical things, such as geometry and algebra, which presumably was also helpful for engineering This is all to say: People's worldview used to be wild and stupid, and we are much more sensible nowadays But despite the sources I've come across, I can't help but wonder if it's really all that true there really weren't any unsung heroes from those middle ages. After all, the ancient Greeks had people like Ptolemy, Socrates, Plato, Aristotle, and Galen, and though they believed in the gods, they still came to great thoughts and discoveries I wish there'd be such nice examples of the medieval times, who were influential, but were just overlooked by those in the 14-15th century, who called themselves renaissance humanists
5:05 ‘Another philosopher, Immanuel Kant, soon came along’ Soon, as in 2000 years!
hey, like jesus!
relative to the history of people, 2000 years is soon. relative to how long this problem has been around, probably not so soon.
Soon is a relative term
Thanks for this really superbly explained and illustrated video. I have no idea how to define a number; but I think it has to encompass the concept you described whereby different numbers can apply depending on context (one pair of shoes vs two shoes). The context can go all the way up to everything: One universe vs however many points in space; even one multiverse vs however many universes it contains. All the way down to one proton vs three quarks. As I see it, for there to be any numbers other than 1 or 0, there has to be some mechanism of differentiation or segmentation. Consider an infinite, completely empty space. Number would have no meaning as far as I can tell. Even if one proposes an imaginary grid or coordinate system, one is imposing differentiation or segmentation onto that empty space. I sometimes have pondered that I think we can generate our whole number system from just the digit 1 and a set of operations (which I think may implicitly assume the existence of differentiation or segmentation): 1 - 1 = 0, 1 + 1 = 2, 1 + 1 + 1 = 3, .... So I suspect the essence of numbers somehow has to do with 1 (unity/emptiness/etc) and ways of breaking up that unity. Anyway, I will stop my rambling now lol.
Thank you very much for this video, it was just what I've been looking for in weeks!
Just discovered this channel and spent most of the day just watching a bunch of your videos. Seriously some of the best and most accessible, entertaining science content I've ever come across.
In the UK, we use the Brexit method to solve the Barber Paradox: the barber keeps saying he's going to shave himself, but he never does :-)
usvalve If you can only define what you don’t want but not what the heck you do want instead, that’s what you end up with. Just like in mathematics: it’s much easier to debunk than to confirm something.
he can Vax himself and shave others 😂😂
Please don't refer to the Europeans as armpit hair, they are sensitive about that
You mean, it's much easier to bunk than to debunk? I'd say that's true... @@yaff1851
Barberxit... Barbrexit... Barbarella?
Yep, there we go. New channel that I must watch every single video. Great content Mrs. Jade!
You are an extraordinary communicator of a difficult subject, well done!!
What a wonderful comment about Frege by Russel! Frege, one who put the search of truth above all other matters. You know, as an older retired person, who used to be in the get-ahead-game -- though not particullary dedicated to that -- it is heartwarming to thnk that you can dedicate your last years to Fregel's ideals, and not be penalized for it.
"my nose will now grow" said pinnochio
Pinnochio's nose would disappear.
Ah...I see
lol
In this case, isn't Pinnochio making a promise and not lying? Pinnochio's nose doesn't grow when he breaks a promise.
@@argumengenichyperloquaciou4115 Let us put it in this way, "My nose is growing now"
Upstanding presentation.., I found it very insightful w/thought provoking explanations and great animation!
Quick point: There isn't actually any paradox with Frege's theory of concepts and extensions at all (as it was presented in this video at least); that idea is used in ZFC set theory all the time (every well-defined property φ induces a class of sets satisfying φ). The reason this isn't a contradiction is that there is no notion of a class containing another class - so you can't have a class of classes that do not contain themself. The contradiction seems to just be in the way he defined "set". If you swap it for the modern idea of a set, then you get a perfectly good model for set theory.
And then Gödel wrote a letter to Russel.
I was going to say: Why are people still looking for a foundation post-Godel?
@@GrantDexter Exactly. Every meta-system that could provide a foundation is itself subject to incompleteness, infinite regress.
@@GrantDexter Because they are no longer looking for a complete coherent foundational framework. They are just looking for a list coherent list of axioms that lines up with what we commonly picture as a set. Not complete, just coherent and with the least possible amount of vagueness.
Then Schrödinger asked him if the barber was observed, since he obviously was both shaved and unshaved
@@crackedcandy7958 That's interesting. If we consider a foundational theory such as ZFS to be equivalent to quantum states in physics, is it possible for a theory to be superpositional? If so, Russel's paradox becomes a superposition, not a contradiction.
I have a question observation. We routinely define math such that we exclude certain conditions because there isn’t a clean definition. We cannot divide by zero. We used to not be able to take the square root of negative numbers. And we used to insist on only rational numbers. We have determined a means to work around these issues, except we still say that dividing by zero is undefined. The other place I think we see the rules change is when we talk about sets of infinite size. We have limitations on what we can compare with these sets. Hence we exclude properties because of the paradoxes that arise. The Russel paradox looks like the divide by zero concern. He’s just pointing out that there are these cases that tend to act like dividing by zero. These cases are self referral cases. Any set that refers to itself can create this paradox. In fact, all of the paradoxes I’ve seen here have this same property that the rule because it applies to itself changes the state of the object and so self referral creates the same type of condition as dividing by zero. Hence, for the same reasons we exclude divide by zero; can’t we also just exclude cases of self referral that create the paradox? If it works for dividing by zero, it appears that it works here as well?
That's basically what happened in the future. Some dude's (Zermelo and Fränkel) developed a new axiomatic set theory (Zermelo-Fränkel set theory) specifically to exclude paradoxes like this.
@@epicmarschmallow5049 Thx!
This is also very applicable to computers. This is all about sequential calculations. It’s about multiple calculations and the results depend on sequence. This is the heart of computing. It is also the heart of observation.
Thank you so much for the material and reminding me you can do anything with functional notation. Great video. Intersecting multi dimensional sets.
I'm so glad Up and Atom is a channel on youtube. Keep up the good work. I can't wait to binge on all your videos.
13:23 "Apparently he didn´t know about the breakdown." 😂😂😂 I think this says something about us all; happiness lies in not trying to belong to the set of all sets because this action alone just excludes ourselves 😉.
Hello, I have to say I discover your channel today. I’ve been watching some of your videos and I really really like it keep going and this is amazing!
That was absolutely beautiful! Explained at a level that even I could understand, and with great animation. Yes, it did bring me out in a chuckle with the sets within sets. Proves what a mess we all are! 🤭
Wow, Russell literally broke Frege with that meta-set question. Physically and mentally.
Did he really have a breakdown? I looked around briefly to find something about this, and the closest I found was this quote from him: "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion." -- seems level-headed, if emotionally charged, to me. Curious about whether this was just hyperbole for storytelling purposes (which I'm mostly fine with, though it kinda undermines the "I guess he hadn't heard?" line, to me), or if there's more to the story than what I managed to find (in an admittedly not-extensive search).
@@DavidLindes The german wikipedia article about Frege is covering his breakdown, but attributes to it to the death of his wife in 1904, two years after Russels letter.
Krmpfpks: ah, cool. Danke!
Does it also work in order to make a robot (or AI) crash? :D
I'm definitely not a math person but I believe you hit upon a key component which could have saved thid logician his methodology. She describes concepts and the extensions which derive from them, but the paradox revolves around a superset which in and of itself is not exactly a set, therefore could not be included within itself as a set since it is a superset. Further I believe the paradox is embedded within the definition that these sets can innately even be included together. How can a concept extended to things which are not themselves be comprehensively extrapolated into a group? That super set would essentially be a collection of all things. To put it simply, if one set was a list of all people who were not you, and the other set was a list of people who were not me, then your list would include me and my list would include you, therefore our super list would include everyone, therefore there is no rational way to innately and properly categorize a super list of every individual that is not an individual, that is without merging the definitions of the sets themselves, as in a list of people who were neither you nor me. Anyways I also think that Plato seem to have a proper by excluding numbers into their own realm. I'll probably get raked over coals for this because I don't know it very well at all, but I would presume that this paradox came prior to the concept of imaginary numbers, and somehow I innately think that quantum physics and it's possible underlying foundations have undermined numbers directly being able to describe reality directly and rather reverting to statistics to become a catch-all for all of the inconsistencies, hence the revolutionary qubit, which is now somehow at the foundation of both physics and mathematics subverting what appeared to be logic with something new entirely, where in our super set of individuals who were not individuals might include a matrix of possibilities [ just you, just me, you & me, everybody, nobody, & every interative factorial between nobody & everybody, even duplicates through infinity given the 'probability' of there being a finite limit of particle configurations in and infinite expansive universe beyond our observable one ] Simultaneously! ~ B) Yea Logic !
A great little video so well scripted and cut and a testament to the ability of its creator. I got to the end without needing to rewind but I can call on a degree in Philosophy to help me. I have never seen set theory explained so well. Thanks and well done.
Im new to your channel. Thanks for making these videos. I watched the one in Russel and Whiteheads book, which i learned about while studying philosophy. Its nice to learn more about it. Thanks for doing that. I really liked it
Thanks, i really wasn't understanding it, but now i understand (Also: loved the drawings and the sound effect on the breakdown of Frege, astonished me! )
12:50 WOW, did Russel clearly know his way about elegantly rubbing salt in the wound ^^!!
Russell s question was plain stupid n irrelevant. Idiotic man, overrated as f
I have subscribed. This video was as clear and concise of a description of Russell's paradox as I have seen. It was enjoyable. Good work, Jade.
Jade, you have a gift in presenting complex concepts. The only thing more fascinating is you!
Thank you for doing these videos. I really enjoy these.
I'm Russell and I approve this paradox.
If you're Russell you cannot approve your own paradox, if you approve your paradox you're not Russell. :-)
@@sumeshrajurkar5922 First of all, I'm not going to publish something that I don't approve of. Secondly, I'm the other Russell.
@@sumeshrajurkar5922 IT WAS A JOKE YOU ABSOLUTE DUMBASS. HOW DID YOU NOT GET THAT?
Plus one.
@@derylpetersonnnnnnnnn Yes, how dare he reply to a joke! Please be more mean to him.
Frege's breakdown almost made me cry. I can't even imagine how it must have felt to have his life's work be disproved by a single sentence. Great video, you've earned a subscriber!
I literally laughed out loud when she said that.... Time to find the Ted Talk about why we laugh at other people's pain...
@@broffutt Well it is funny from a dark comedy point of view and also we are all different I guess. So I think there is nothing wrong with you 😄
He wouldn't be the only one of these clowns with a screw loose. Pondering different flavors of infinity defies all intuition, until you're deep enough into it to develop a new kind of intuition.
Same...
There is no evidence Frege had a breakdown due to Russel's letter.
I love this. I would say, not having watched further in the video yet then the preposition to define a number without using number, that a number is: a mathematical object defined by a relation to the empty set and non-empty sets using logical operators.
Wow. This is an excellent video. The visuals are great and the explanations excellent. Before now, I had found it difficult to fully grasp Russell's paradox. But, while watching this video, I found myself understanding the concept while laughing. Well done!
keep up the good work i really love your videos. I'm an enthusiast about physics and maths and it's hard to find people speak about actual interesting things like this.
I love your videos :) great explanation for complicated topics, and the animation is amazing and creative, thank you jade
You're such an intelegent and articulate math communicator. I subscribed and looking forward to endulgr more of your videos.
You've reminded me of binding loops in QML (a programming language for user interfaces). Binding loops highlight a bug in the code: you've crafted something that can't work / doesn't make sense. Much like the paradoxes you mention in the video.
This was a really good overview of some of the philosophical problems surrounding math, and I would love to see more on this subject.
The Buttersotch Paradox - It tastes neither like butter or scotch. This Butterscotch Ripple is more upsetting to the foundation of life than Russell’s Paradox ever could be.
J J if you throw butterscotch hard enough, it tears space-time so you can step out of this reality and can taste thoughts and concepts instead. Try it.
@@alex0589 we've got a synesthete!
My butterscotch paradox is that my butter is usually messy while my scotch is always neat
Amazing video! You are so cool and explain things so well. If you could please do a video on Category Theory, that would be so helpful, I've been trying to learn it for a while now and most of the videos are pretty abstract. Thanks for what you do!
The best answer you can possibly give to Russel's question is to modify the question a bit, because every question along the same lines and similar, creates a paradox. "Any unifying set, such as this, cannot be part of itself, as this means it has not unified everything"
11:27 -- I love that Jade's gives the camera that same look you'd give anybody when you're pretty sure you've said something that's gone over their head.
I am currently reading Logicomix and this video really helped me understand the novel. Thanks!!👍
I bought it thanks to you…
Paradoxes drive you nuts. But this supremely charming teacher makes you love them, paradoxes and nuts.
experiencing the segments of memory and our individual relationship to those fragmented segments, objects, as we recall we apply number to parts of the memory and to the complete group of fragmented memories this is done with hopes to organize our experience to the extent that we can give identification of these experiences.
As for the barber paradox, a similar solution (to my solution for Russell's paradox) can be applied. Once again, the trick is to divide what the barber is into two reciprocal aspects. Instead of sets and elements, we must divide the barber into that part of himself that is an actual barber and that which is just an ordinary person. Now, the definition of a barber is someone who shaves or cuts the hair of another person for money. Now, these two aspects of the barber must be kept separate because (like sets and elements) they have certain characteristics that are incompatible with each other. For instance, the [person aspect] is a necessary characteristic while the [barber aspect] is optional since he could choose to be something other than a barber in a way that he cannot choose to be a different person. Now the barber aspect is the aspect that shaves people. This is true whether he's shaving other people or himself. Thus, if his barber aspect shaves his person aspect, then the person aspect is NOT shaving himself. Now, there are two possibilities. Since the barber aspect isn't charging his self aspect any money to shave himself, then the barber isn't functioning as a barber, since that requires the acceptance of money. Thus, if the person aspect shaves himself, the barber aspect is not involved in the shaving. And the situation is not paradoxical. On the other hand, if the person did not shave himself, he would have to pay someone else to do it, and thus, he is receiving value (the absence of need to pay someone else) by shaving himself... but if we acknowledge that value, then we must also admit that the barber aspect kicks in and it is the barber who is shaving his person aspect, not the person, and so once again the barber is shaving an aspect that is not shaving itself. Either way, there is no paradoxical confusion.
Or he could just be barber on the weekdays and plays with ZZ Top on the weekends and doesn’t cut his beard, just grooms it. Boom, paradox solved.
Excellent description of the paradox. Any similar insights on the essence of the Russell paradox or Godel incompleteness
@@mikedougherty1011 Thanks for asking and yes, I do, although a detailed look at Godel is probably beyond the capabilities of this format. Russell's Paradox... can be resolved (I think) by distinguishing between the nature of an element and that of a set. A set is that which contains elements, an element is that which is contained by a set. It's like the relation between a father and a son. The same person can be both a father and a son, but he can't be both of these things to the same person. Similarly, a set is like a [container], while an element is like [that which is contained]. You can but a small box (that contains something into a larger box) but the relation between the boxes is such that only one contains the other. Thus, since R is the set of all sets that do not contain themselves, R is necessarily the [set of all set], since no set contains itself. The opposite of R is the empty set. We can think of this distinction as the [name of the set] vs [what the set actually is]. Like the single word "English" versus the set that contains all the English words, [English]. The set [English] contains a the name of itself, which is that single word "English" but it does not "contain" the set [English] it simply is that set. In the same way, we can create a set R that contains all the sets that do not contain their own name. But since a [name] is not the same thing as the [thing named], there is no paradox. A Quick look at Godel's Incompleteness. Without getting into the weeds, G can be essentially understood as a set that makes a self-reference to itself, as follows: (G) [G is false] Again, the error is to assume that (G) and [G is false] are the same thing and that they are interchangeable. In reality, the G in [G is false] is only a name. It is not the whole set [G is false]. We could try to substitute the whole set for the name, in order to get rid of the name aspect, but this only produces [G is false is false] We can substitute as many times as we want, but it will never get rid of the name aspect. And this creates a necessary vicious circle that is identical to the way two mirrors partially reflecting each other create an "infinite" series of mirrors in mirrors. We see the same thing with a camera records it's own monitor. We see an infinite series of smaller monitors. Again, with sound feedback, etc. Every time we encounter this same structure, we always see an infinite regression. Godel's error was to treat the [name] and the [thing named] as if they are the same thing, when clearly they cannot possibly be the same. His trick of using astronomically large numbers to represent the name and the thing named, however, makes it very difficult to see what is actually happening, since it is literally impossible to actualize either the [name] or the [thing being named] in his proof. This makes it very easy to ignore the infinite regression that must occur. However, since the infinite regression is unavoidable, the construction of the proof is invalid and thus it does not show what it claims to show. If you're interested in a more detailed analysis, still using layman's language, but definitely much more precise and closer to Godel's original language, let me know your email, or some other place where we can discuss more and I'll be happy to expand.
Your take on the paradox is intriguing, you have divided the barber into two personalities, one who is a barber and one who is just another random guy who doesn't shave himself. You are suggesting that the barber shave his non barber self from what i understand. However that does not actually solve the problem, in fact the problem remains. You are just proposing he has schizophrenia, which might solve the problem from his point of view, but what if we change the frame of reference and set it as an observer? The paradox would be deemed solved only if everybody agrees. To other people he is still the barber who shaves himself.
My solution to the barber paradox is that the barber is a woman. Easy.
“Life” is immeasurably and incomprehensibly complex. Words are at best rough approximations of anything resembling “life” or “reality”.
Intuitively, numbers are variables: “One” : l “Two” : ll “Three” : lll … Those variables point to a collection of objects (not quite…) 1) A number can be considered a scalar, is a tensor, is a sequence, is a class, is an object. Given a sequence, we can apply the system from Tensor Theory, and define numbers like that… 2) A number can be considered 1 more than the previous number, where 0 represents nothing. 3) A number can be considered as the *magma* of generic collections mapping to the explicitly defined property: “quantity”.
Your explanation is very clear and helpful. Thanks a lot
Frege: I'm finally done with my work! Russel: I'm about to end this whole man career.
Lmao love this meme
Story of PhDs
Frege's work got known thanks to Russel though. And although the problem he found was at the base of the theory, most of the work still was very important for the future develpment of formal logic.
@@Deguiko So he destroyed his mind in order to build him back up? I've heard of tough love, but savage love? Damn.
Because you didn't ask me to subscribe and hit the notification bell, I did. How's that for a paradox?
Bruno Bronosky that is not a paradox. It was an exploitation of your nature. The fact that it simultaneously is and was, that’s a paradox.
@@rolyf100 is and is not*
Try subscribing to the channels of all KZheadrs who don't subscribe to their own channels.
My real analysis professor gave us a definition of a set as "any mathematical object" which in retrospect punts the definition down the road, as it leaves open the question of what is "mathematical". He also raised the question of when a set can itself be a member of the set and the example he gave was the "set of ideas".
I would claim sets are groups, specifically groups of images. But first I will show that numbers are built from images Example , 4 always represents 4 images, like 4 squares for instance. To be specific numbers are "labels" for groups of images 1. The main idea here is that maths is built from images (a) example , geometry is clearly made of images b) example 2, We claim numbers are built from images too, as say 4 , always represents 4 images, like 4 squares for instance. C) imaginary numbers are connected to images too , which is why they have applications in physics D) In general any mathematical symbol that comes to mind is connected to images too.
Attempt at definition of number - It is property of a group having common category but different object instances. The property is such that each new instance in the group changes the property by same value and each instance destroyed from the group also changes the property by same value.
People: Imagine if everything was absurd!? Quantum Mechanics: Well hello there :)
@7:50 oh, gotta try defining number, because I love trying to do this kind of stuff. So, I thought through stuff like she said, quantity, amount. Then I went on to stuff like sets of things (the things you would count). Then I thought of a series, the series of numbers, and a correspondence between each number and the item in the set being counted. So, removing the word "number" and such to make the definition non-circular: A label selected from a series of unique labels that are in a fixed order, with a beginning. For a given set of items, each item receives a unique label given out in the same order as the list of labels is defined, beginning with the beginning label. A set itself can receive a label that is the same as the final label applied to the items in the set. Then you just have to invent names for those labels. The beginning we call "one", and so on. And the above also shows how "quantity" and stuff comes from this. Anyways, rough sketch, now on to watch the rest of the video!
@9:10 Huh, looks like Gottlob Frege got extensions and concepts reversed when he talks about numbers... Clearly, 4 is the concept, and the extension is every set of objects that have that amount.
Also, I slightly cheated because I know Richard Carrier said that set theory is the foundation of mathematics or something, so I knew "sets" had to be important, that helped me! Yes, Richard Carrier is the cheat-sheet for philosophy.
And, to handle two dimensional numbers (so-called "complex numbers") can generalize from a list in one dimension to naming locations on another dimension as well.
When I was a little lad of five I started school, the teacher started talking about the ' numbers ' one, two, three, four ..... We were encouraged to count these on our fingers. It was several decades later that for me the big intellectual leap was made that from ' one ' to ' two ' is to accept the fiction that two objects are the same, so the concept of ' two ' is a DOUBLING of the original object. It is a matter of convenience [ a first level of abstraction ] that the second one is the same in the present concept. So that a right shoe is for the moment treated as no different from a left shoe, or a red sock is equivalent to a blue sock to serve as working assumptions for some a priori result. The question of whether or not such a priori results are useful in some real-world application seems to be an important consideration for most of us. So if you are selling oranges it's an important abstraction that each individual orange has the same properties as every other orange on the stall. This is an essential abstraction necessary to facilitate the sale of oranges.
A set of sets is like a search query of prior search queries. A search for prior searches will never show up in its own set of results. But it will show up in all such searches in the future. Thus this paradox is easily resolved by requiring its formal resolution at a precise time. I think most most paradoxes might be resolved by requiring all infinite concepts or parameters be defined or constrained.
There are many of us who are nuanced beyond adequate descriptions. I enjoyed your presentation. This comment is what came to mind at the problem description. I thought that according to Kant, the physical world does not imply the existence of mathematics. Therefore, mathematics is a synthetic construct that may or may not have descriptive use. This can only be determined posteriori or after the fact through analysis.
Up and Atom Kurt Gödel shed a lot of light on self referential statements with his work. You should consider a follow up video covering his work on meta-mathematics and consistency vs completeness. I really enjoyed this video btw! 😁
She's done Gödel I think, and the related Halting Problem. But yes, a follow-up video on how one led to the other would be well warranted.
Really interesting, what I like about this paradox is that in a way it’s the same as the problem with quantizing gravity. One of the problems with quantizing gravity is that it’s not a quantum field on top of space time, it is space time, I see here similarity to this paradox, the set of the sets that are not members of themselves is sort of different than other sets in the same way that gravity is from the other fundamental forces.
Yes! I was thinking the same thing
A "number" is a collective grouping noun. It can also be used as a verb to assign titles to objects.
Frege didn't coin the term "General Comprehension Principle". It was simply his Basic Law #5 (a property of concepts). W. V. Quine pointed out in his 1955 paper that Frege's Basic Law 5 was implied by the Principle of Abstraction/Comprehension, but Frege wasn't directly concerned about the properties of sets per se.
"Apparently he didn't know about the breakdown?!" 😂
Jade, you should get a Nobel prize for your teachings to a large audience on KZhead. A growing audience at that!
Reminded me of this idea I had which I sometimes enjoy exploring and it goes as such : Does nonexistence exist ? If it does then it doesn't, and if it doesn't then it does.. I keep thinking about it (Enjoying my brain going on this loop at that), because it almost feels as if I am touching one side of the the human brain capabilities, and what really intrigues me on the subject are the other limits that I don't know of, and how exhilarating it would be if ever could experience the limits on a different perspective.
I have no idea how to describe number, all I know is your explanation is awesome and I love Tifa.
Deep philosophical question that comes to my mind from watching this video: Who shaves the turtle?
Achilles
Turtles all the way down to the Mock.
"Who shaves the turtle?" ==> Mitch McConnel's wife.
One thing I am sure of, turtles don’t get electrolysis. That would leave them shell shocked.
@@bobbimke82 yeah, he definitely is the turtle.
"You're all individuals." "I'm not."
Tom van Leeuwen but you are in the set of all individuals
No, no. It goes: "You're all individuals." "Yes, we're all individuals!" "You're all different." "Yes, we're all different!" "I'm not..."
@@AvanToor I know, just tried to simplify so that it was easily read. :-) The ones who know the scene understand.
@@fakkmorradi I am the set of all individuals.
@@AvanToor The very best movie quote of all time.
A number is an item or items with a label depicting how many items are existing, being referred to, or are non-existent. This works for zero and integers. Negatives, decimals, etc. are expressions of those labeled items.
Thank you!!! Very helpful as I’m trying to make my way through an advanced logic course in university 😅
Great Video! Apropos, I happen to be in a reading group studying Homotopy Type Theory. I look forward to your take on it.
This is great. You speak so clearly about this. :) Does this translate or connect with Goedels theorem as well?
My definition of number: An individual member of a fully abstract series. For example: If I create a uni-dimensional series, I get all real numbers. If I create a uni-dimentional space made out of individual member with a set distance between them, I get natural numbers. The names "one", "two", "1/3", etc. are just that, names. Sets like even numbers are only defined based on another set. If you take a fully abstract series that is only described with the most basic concepts like dimension or space you get the numbers.
Russel’s paradox was always that quirky thing I was taught half way through a Discrete Math course. I didn’t know it basically ruined a dude’s life lol.
The femininity of her is remarkable. Who cares about mathematics?
@@seanleith5312 Dude, go away.
@@seanleith5312 holy shit dude. touch grass
@@icefire6622 I came here for science, apparently, but ended up giving up on science. I am happy.
I refuse to join any club that would have me as a member - Groucho Marx
And this leads us to a different paradox. Imagine a town where every possible set of citizens forms a club. Would it be possible to name all clubs after a citizen, in such a way that every club is named and no two clubs have the same name? Of course, this can't be done with a finite town; it would have more clubs for than citizens. (For example, with just 10 people there would be 2^10=1024 clubs.) But could it be done in an infinite town? Turns out, the answer is: no, it can't be done either. Take any naming scheme (where no two clubs have the same name), and ask: does it cover all clubs? If every set is a club, then so is the set of all citizens who are not a member of their own clubs. But this club can't be named; otherwise, can the citizen who the club is named after be its member? It can be seen that he's a member of our club if and only if he isn't a member. This is an impossibility, so the club can't have a name.
@@MikeRosoftJH do these clubs come with membership benefits? otherwise I must decline your offer.
@@MikeRosoftJH Anyways, you seem to be running out of letter combinations, so here I propose an infinite alphabet to go along with the naming.
@@MikeRosoftJH The number of clubs is summation(n choose r) for 0
I thought that was Woody Allen
Would love to see an episode on Kurt Godels Inconsistency and Incompleteness Theorems. A lot of people who think of math as a perfect form of logic need to understand that Kurt Godel removed that assertion as a possibility with his proofs that no system of logic can be proved internally consistent or complete (for essentially the same reason we don't use a word in its own definition). You have a great show here.
As far as the "Barber Paradox" goes, which is also called the "rational loop-line", all numbers can be determined as either rational or irrational, since I can split all numbers by a solid, definable line. The question you're asking is "is the set rational?" And I would reply that adding the barber as a member will make the set both rational and irrational, in different states. So, now you have a liquid-defining set, and it will evaporate and should be recondensed to consider the computation again. Fun use of set-theory, but not a paradox! Hope to see more videos too!
Frege: I have the most fundamental theory about maths Russell: I'm about to end this man's whole career
Thank you, great explanation! ALso loved the graphics, especially the birds :)
The mapping of the Concept to the Set is an operation. No-one calls "0" a paradox in operation "divide by". It's totally acceptable to have an exception in your theory, as long as you know about it.
"Common denominator approach": Patterns (including patterns of influence). Concepts are patterns of neural activity, and application of patterns influences other patterns. A number seems to be a mental pattern where rule patterns are set up, including associations to other patterns. Anyway, you know where I'm driving in this...this may help understand where some patterns cannot have influence over targeted patterns (the rule patterns impede association or influence).