Russell's Paradox - A Ripple in the Foundations of Mathematics

[@upandatom]

What is a number? (no using the word number)

[@essam352]

Up and Atom The existence of an element. whether it exists or not or repeated

[@mohamedbelgoumri6729]

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

[@ARBB1]

A element that represents position on a particular manifold, such as the space of real numbers.

[@autumndidact6148]

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.

[@saarangsahasrabudhe8634]

You want to know what a number is? Here you go: kzbin.info/www/bejne/i566npmKqZmJeLM It's the first of a 8-part series of 4-min videos. Next video has a tag (G1). Here are all the videos: kzbin.info/door/RUATK39-y_dSwmN59_-aNQvideos

[@joescott]

What I learned from this is if you get a letter from Bertrand Russell, don't read it.

[@DampeS8N]

Honestly, if you get a letter from Bertrand Russell today, bring that shizz right to James Randi and get yourself a million bucks.

[@michaelsommers2356]

@@DampeS8N There's nothing supernatural in that. They call it 'snail mail' for a reason.

[@NowanIlfideme]

Joe, letter from Bertrand Russell: can you go fetch my teapot, please? (Read as: maybe make a video about interesting thought experiments?)

[@devvynully]

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?

[@Blox117]

but it could have money inside!

[@johnathancorgan3994]

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.

[@xway2]

I second this message. This is why this channel is great.

[@ivkost]

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.

[@romwil]

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!

[@harrypehkonen]

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.

[@Amateur0Visionary]

Well said, sir. I completely agree.

[@blackpenredpen]

I enjoyed this video so much. The animation and the explanation are so good!

[@xcalibur6482]

WoW! , youre here... yay!

[@darthvader-oz2om]

Hey dude!!!

[@dakshdheer1681]

You chad LEGEND

[@ccole1255]

@@dakshdheer1681 whaaaaa?

[@seanleith5312]

super cool dude? what the hell!

[@davidsapir3764]

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.

[@muzukashiinamae]

Underrated

[@carlosraventosprieto2065]

Thanks for this!

[@davidsapir3764]

@@carlosraventosprieto2065 You're most welcome!

[@ananyagupta1409]

Bravo!!

[@daithi1966]

Dammit! I just waisted 5 minutes of my life because I read really slow.

[@garyhughes1664]

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.

[@bensadowyj1974]

It's a great book (so far), working my through it currently.

[@jonathansturm4163]

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.

[@bensadowyj1974]

@@jonathansturm4163 I will check it out

[@anonymike8280]

Yeah, the dude was bomb!

[@spinZ186]

Dumbass alert, Russell’s text is fascinating but nowhere close to standard. Very subjective

[@post1305]

The barber was pulling his hair out trying to solve this problem, which ironically did solve the problem.

[@golfgod1017]

until it grew back

[@post1305]

@@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.

[@golfgod1017]

@@post1305 unless he learned from the experience and chose a different approach.

[@post1305]

@@golfgod1017 There is no evidence to suggest that happened.

[@golfgod1017]

@@post1305 or you just didn't see the evidence

[@DanHoke]

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.

[@saxonalexander3888]

Thank you

[@lucashuerga1368]

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.

[@jeffbguarino]

@@lucashuerga1368 A paradox means using classical logic that there is a basic mistake in the foundations of math. Quantum math would fix this but no one seems to be working on this. You can have things that are true and false at the same time if you create a superposition. So put the barber in Schrodinger's box and have a random quantum event like measuring whether an electron is spin up or down. Close the box and if the spin is up , then the barber shaves himself and if it is spin down he doesn't. So you have a box and in the box is the barber in a state of having shaved himself and not having shaved himself at the same time. This cures Russell's paradox. You can do the same set up with the set of all sets that don't contain themselves.

[@prototypeinheritance515]

​@@jeffbguarinoRussel's paradox has been resolved for over a hundred years. It is a problem of naive set theory, but modern set theory gets around it by simply restricting what is considered a set. Can you please eloborate what "quantum" math is? I doubt such a thing would be very meaningful as quantum physics is already easily described by classical mathematics. What you are describing sounds very similar to so called "fuzzy" sets. These can easily be modeled in set theory like this: If we have a superset S of all things we want consider, a universe of discussion so to say, all relevant sets are subsets of S which can be uniquely identified with a function f : S -> {0, 1}, where every element of S is assigned the value 1 if it is considered to be in the subset. A fuzzy set is defined in a similar way as f : S -> [0, 1], where [0, 1] is the interval of real numbers from 0 to 1. So every element is assigned a probability of being in a particular subset. The point is that these kinds of set are easily modeled in classical mathematics.

[@robertferraro236]

The clarity you bring to these difficult to articulate and comprehend topics is exceptional.

[@xacharon]

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???"

[@alex0589]

xacharon insulted that we assumed she couldnt smell cause she’s old. She’s a dog, not a smoker, dammit!

[@Inertia888]

@@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.

[@AnkhAnanku]

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

[@archprep131]

xacharon ohhhhhh shit

[@vamsikrishnan9714]

Gottlob Frege: * makes a definition of number* Bertrand Russell : I'm about to end this man's whole career

[@KenjaTimu]

Foundation is not the right word. These sciences existed for thousands of years before their 'foundations' were even known to exist.

[@fredrikekholm3718]

@@KenjaTimu 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".

[@fahmiyassin2517]

lol

[@PhilBagels]

@@KenjaTimu 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.

[@KenjaTimu]

@@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.

[@hvhvgitaar]

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 😂

[@Blankpaper_openbook]

yeah i try to explain too ,but most of them have no idea what am i talking

[@jeffbguarino]

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.

[@malfunction5448]

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 :)

[@EM-qr4kz]

Yeah..

[@VexLimenOfficial]

"So Baldrick, if I have some beans and add one more bean, what does that make?" "A very small casserole m'lord."

[@jason-paulwells6696]

"Three beans and that one."

[@denniskeefe1979]

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.

[@chrisrichards7063]

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.

[@robertmontgomery6256]

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.

[@rotflmaopmpqxyz]

Consider a sets of all sets that have never been considered. Oh wait, they’re all gone now, never mind.

[@Hailfire08]

They haven't been considered, just the set now no longer contains itself :)

[@brianjoelbasualdo7436]

I see the joke you did there

[@talhabedir3812]

so underrated lol

[@muhaimin244]

Nice one. I think paradoxes should be hunted and taken as gateways toward unpacking primitives and axioms.

[@NewtonMD]

@@muhaimin244 lookup Vsauce

[@sourisvoleur4854]

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."

[@anymaths]

mathematics students

[@hetrodoxlysonov-wh9oo]

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."

[@captainzork6109]

Is that because the middle ages philosophy only had to do with religion and Plato and Aristotle's Organon, until renaissance humanism came along?

[@sourisvoleur4854]

@@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.

[@captainzork6109]

@@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

[@usvalve]

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 :-)

[@yaff1851]

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.

[@anshulbhardwaj4038]

he can Vax himself and shave others 😂😂

[@pigeonlove]

Please don't refer to the Europeans as armpit hair, they are sensitive about that

[@j3ffn4v4rr0]

You mean, it's much easier to bunk than to debunk? I'd say that's true... @@yaff1851

[@Sunastar4D]

Barberxit... Barbrexit... Barbarella?

[@xletix69]

omg i've seen videos on this paradox so often but this is the first time i actually got it!! Thank you sm🙏🏼

[@Ygyhhhhhhhh]

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.

[@SeriousNERD1]

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!

[@broffutt]

I literally laughed out loud when she said that.... Time to find the Ted Talk about why we laugh at other people's pain...

[@SeriousNERD1]

@@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 😄

[@harrymills2770]

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.

[@moadot720]

Same...

[@retry4z]

There is no evidence Frege had a breakdown due to Russel's letter.

[@RussellSubedi]

I'm Russell and I approve this paradox.

[@sumeshrajurkar5922]

If you're Russell you cannot approve your own paradox, if you approve your paradox you're not Russell. :-)

[@RussellSubedi]

@@sumeshrajurkar5922 First of all, I'm not going to publish something that I don't approve of. Secondly, I'm the other Russell.

[@derylpetersonnnnnnnnn]

@@sumeshrajurkar5922 IT WAS A JOKE YOU ABSOLUTE DUMBASS. HOW DID YOU NOT GET THAT?

[@russellnavin7882]

Plus one.

[@RussellSubedi]

@@derylpetersonnnnnnnnn Yes, how dare he reply to a joke! Please be more mean to him.

[@kayleym8947]

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

[@sclarkaz]

And then Gödel wrote a letter to Russel.

[@GrantDexter]

I was going to say: Why are people still looking for a foundation post-Godel?

[@sclarkaz]

@@GrantDexter Exactly. Every meta-system that could provide a foundation is itself subject to incompleteness, infinite regress.

[@franciscofernandez8183]

@@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.

[@crackedcandy7958]

Then Schrödinger asked him if the barber was observed, since he obviously was both shaved and unshaved

[@mikekelley9125]

@@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.

[@hunterterrell9930]

"my nose will now grow" said pinnochio

[@zfloyd1627]

Pinnochio's nose would disappear.

[@nene_san]

Ah...I see

[@fahmiyassin2517]

lol

[@argumengenichyperloquaciou4115]

In this case, isn't Pinnochio making a promise and not lying? Pinnochio's nose doesn't grow when he breaks a promise.

[@priteshsoni3891]

@@argumengenichyperloquaciou4115 Let us put it in this way, "My nose is growing now"

[@deandeann1541]

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.

[@hughbarton5743]

A wonderfully clear analysis and presentation. Brava, Ms. Jade!

[@monjitchetia6768]

I am currently reading Logicomix and this video really helped me understand the novel. Thanks!!👍

[@10418]

I bought it thanks to you…

[@garybrisebois2667]

People: Imagine if everything was absurd!? Quantum Mechanics: Well hello there :)

[@sean..L]

Wow, Russell literally broke Frege with that meta-set question. Physically and mentally.

[@DavidLindes]

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).

[@Krmpfpks]

@@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.

[@DavidLindes]

Krmpfpks: ah, cool. Danke!

[@SedatKPunkt]

Does it also work in order to make a robot (or AI) crash? :D

[@maniacalbarbarian]

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 !

[@retry4z]

The claim that Frege had a breakdown due to Russel's letter is a fiction added for dramatic purposes. Frege was going through a combination of poor health, the early loss of his wife in 1904, and disappointment over the continued poor reception of his work. There is no evidence that frustration with his failure to find an adequate solution to Russell’s paradox was the primary reason for his hospitalization.

[@JJ-kl7eq]

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.

[@alex0589]

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.

[@Hydrastic-bz5qm]

@@alex0589 we've got a synesthete!

[@aznkingdom12345]

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.

[@stevesomers7366]

Jade, you have a gift in presenting complex concepts. The only thing more fascinating is you!

[@magnusjonsson7303]

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 😉.

[@Ivan_1791]

Frege: I'm finally done with my work! Russel: I'm about to end this whole man career.

[@shlovaski8393]

Lmao love this meme

[@hamman_samuel]

Story of PhDs

[@Deguiko]

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.

[@salaciousBastard]

@@Deguiko So he destroyed his mind in order to build him back up? I've heard of tough love, but savage love? Damn.

[@briancotton]

I refuse to join any club that would have me as a member - Groucho Marx

[@MikeRosoftJH]

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.

[@Blox117]

@@MikeRosoftJH do these clubs come with membership benefits? otherwise I must decline your offer.

[@milanstevic8424]

@@MikeRosoftJH Anyways, you seem to be running out of letter combinations, so here I propose an infinite alphabet to go along with the naming.

[@WBoettchStevens]

​@@MikeRosoftJH The number of clubs is summation(n choose r) for 0

[@nicolaiveliki1409]

I thought that was Woody Allen

[@yuridesideri7144]

Thank you very much for this video, it was just what I've been looking for in weeks!

[@thecompanioncube4211]

Frege: I have the most fundamental theory about maths Russell: I'm about to end this man's whole career

[@RyanVJones]

Jade, you should get a Nobel prize for your teachings to a large audience on KZbin. A growing audience at that!

[@jameshoffman552]

5:05 ‘Another philosopher, Immanuel Kant, soon came along’ Soon, as in 2000 years!

[@fukpoeslaw3613]

hey, like jesus!

[@joeyd1734]

relative to the history of people, 2000 years is soon. relative to how long this problem has been around, probably not so soon.

[@stonerdave]

Soon is a relative term

[@TXLogic]

The Barber paradox is not a “simpler” version of Russell’s paradox. Logically, It is *exactly* Russell’s paradox. Substitute “set” for “barber” and “contains” (i.e., the converse of “is a member of”) for “shaves”. It follows as a matter of pure predicate logic that there is no barber/set that shaves/contains all and only those things that don’t shave/contain themselves. What gives rise to the puzzle in the case of sets is that it seems that, for any description, there *should* be a set containing exactly the things satisfying the description (i.e., the principle of unrestricted comprehension) - I mean, intuitively, however many things satisfy a description, there should be the *set* of them, right? - whereas no one is inclined a priori to think that, for any description (in particular, “barber who shaves all and only those that don’t shave themselves”), there should be something satisfying the description.

[@ywenp]

12:50 WOW, did Russel clearly know his way about elegantly rubbing salt in the wound ^^!!

[@email4ady]

Russell s question was plain stupid n irrelevant. Idiotic man, overrated as f

[@chuckm1961]

“Life” is immeasurably and incomprehensibly complex. Words are at best rough approximations of anything resembling “life” or “reality”.

[@nielsen425]

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?

[@epicmarschmallow5049]

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.

[@nielsen425]

@@epicmarschmallow5049 Thx!

[@JJ-kl7eq]

Deep philosophical question that comes to my mind from watching this video: Who shaves the turtle?

[@videofudge]

Achilles

[@EddieVBlueIsland]

Turtles all the way down to the Mock.

[@bobbimke82]

"Who shaves the turtle?" ==> Mitch McConnel's wife.

[@JJ-kl7eq]

One thing I am sure of, turtles don’t get electrolysis. That would leave them shell shocked.

[@kandysman86]

@@bobbimke82 yeah, he definitely is the turtle.

[@CHOCOLATIONZ]

Frege: here's the neat systematic set theory I made. Russell: *I'm about to end this man's whole career*

[@colinjava8447]

And hospitalize him

[@MrEmrys24]

"Apparently he didn't know about the breakdown?!" 😂

[@adamad1958]

Thanks!

[@RichardBronosky]

Because you didn't ask me to subscribe and hit the notification bell, I did. How's that for a paradox?

[@rolyf100]

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.

[@force3413]

@@rolyf100 is and is not*

[@rodschmidt8952]

Try subscribing to the channels of all KZbinrs who don't subscribe to their own channels.

[@rcb3921]

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.

[@djtomleeuwen]

"You're all individuals." "I'm not."

[@fakkmorradi]

Tom van Leeuwen but you are in the set of all individuals

[@AvanToor]

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..."

[@djtomleeuwen]

@@AvanToor I know, just tried to simplify so that it was easily read. :-) The ones who know the scene understand.

[@cosmojg]

@@fakkmorradi I am the set of all individuals.

[@happinesstan]

@@AvanToor The very best movie quote of all time.

[@scottsabey2114]

Upstanding presentation.., I found it very insightful w/thought provoking explanations and great animation!

[@vmp916]

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.

[@seanleith5312]

The femininity of her is remarkable. Who cares about mathematics?

[@Red-Brick-Dream]

@@seanleith5312 Dude, go away.

[@icefire6622]

@@seanleith5312 holy shit dude. touch grass

[@seanleith5312]

@@icefire6622 I came here for science, apparently, but ended up giving up on science. I am happy.

[@antoneogzewalla2040]

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.

[@mikedougherty1011]

Excellent description of the paradox. Any similar insights on the essence of the Russell paradox or Godel incompleteness

[@antoneogzewalla2040]

@@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.

[@preetanwitasarkar3794]

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.

[@abigailcooling6604]

My solution to the barber paradox is that the barber is a woman. Easy.

[@PukeSkinwalker]

@@abigailcooling6604 She has legs. The concept is that a set cannot contain itself or can it. The description of it being like a mirror is intriguing. I think the solution is that a set cannot contain itself. Just as much as you cannot divide by 0 and get anything that exists. If you divide something into nothing then you get undefined due to the limit of 1/.X as n approaches 0. It leads to infinity. A set that contains itself would divide by nothing because it technically wouldn't contain itself. Therefore, it would create an infinite loop like dividing 0 does. It is the same. If you divide a set that contains nothing but itself, it would be 0. Dividing by 0 leads to an infinite loop which creates the paradox. I hope that makes sense as to why the paradox exists and why a set cannot contain itself if it is the only thing within the set. It is because there is nothing but the description of the set which means that the set contains nothing but itself so that it is 0 and you cannot divide by 0 as the limit leads to infinity. I have to go now.

[@duggydo]

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! 😁

[@AgentOccam]

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.

[@rambysophistry1220]

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.

[@zeinmahmoud9931]

I love your videos :) great explanation for complicated topics, and the animation is amazing and creative, thank you jade

[@michaelmj1964]

This was a really good overview of some of the philosophical problems surrounding math, and I would love to see more on this subject.

[@nafrost2787]

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.

[@aparnaga1182]

Yes! I was thinking the same thing

[@13rolf]

I have no idea how to describe number, all I know is your explanation is awesome and I love Tifa.

[@erwinveenhoven]

"Immanuel Kant soon came along." Well that took some 'human' time but on the scale of the universe I'll let that one stand :)

[@wierdalien1]

I mean, what does soon even mean?

[@YambamYambam2]

This thread xD

[@Briaaanz]

Bertrand Russell has been one of my heroes since I first heard about mathematical philosophy

[@pimpom1345]

It was actually first discovered by German mathematicians before him, but he was the first to publish it.

[@mohamedbelgoumri6729]

It's worth mentioning that Russell's aim also was destroyed by the great Kurt Gödel, guess karma is a thing?

[@upandatom]

haha i know. poor logicians... such great men too

[@JoshuaHillerup]

But hey, we got computer science out of all of it.

[@monad_tcp]

​@@JoshuaHillerup I'm waiting for the Wolfram Alpha idea (A new kind of Mathematics) that Computing is the basis of Mathematics. I'm all for HoTT.

[@Krmpfpks]

Gödel, like others working on the foundation of logic, ended up mentally unstable. Guess this stuff is very hard.

[@PeterJavi]

@@Krmpfpks That raises an interesting question, because it means that people will lose their senses when trying to make sense of logic, which in itself should be logical, but apparently we as humans can't deal with this logic?

[@knowledgeckr786]

Thanks

[@phaysaal]

Q- What is mathematics? A- Something cool Q- Show me a proof A- This video

[@marcperez2598]

"Show me proof" isnt a question

[@worldhola5090]

@@marcperez2598 Q is not a question. Q is just a name of someone.

[@marcperez2598]

@@lewis3774 absolutely

[@namefinder]

That bottom quark will haunt my dreams...

[@jeffreymilitante4597]

Oh no XD

[@vitabricksnailslime8273]

I too, have nightmares about Michael Jackson.

[@bpansky]

@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!

[@bpansky]

@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.

[@bpansky]

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.

[@bpansky]

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.

[@crustyoldfart]

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.

[@PresentFocus]

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.

[@boletarianbread7349]

It's one of those divide by zeros situations where you just have to axiomatically declare the set of all things which are not sets an undefined set. It doesn't have an answer. There's no answer to some number divided by zero. There's no answer to the whether the barber shaves himself as presented, so you add an axiom that the barber shaves people who don't shave themselves *and aren't barbers* and himself.

[@WonderzStevey]

@@tthung8668 i do believe this human nature at fault, because we try categories nature from sentience. Nature true particle physics does not bind to mathematics, albeit a great measuring stick does not give you the full scope of what is at play.

[@Curt_Sampson]

@@tthung8668 Yes, it's a very different situation. The so-called "imaginary" numbers (as derisively named by Descartes) are no more imaginary than zero or negative numbers, both of which were in the past equally derided. It's just that in our current state of mathematical sophistication many non-mathematicians have a comfortable intuition for zero and negative numbers that was not previously widespread. In the same way that if you're comfortable with extending a number line infinitely in both directions, giving you negative numbers, you can get comfortable with having two perpendicular number lines extending infinitely in both directions, giving you a plane on which each complex number is a point, at which point you have an easy way of developing an intuition about how _i_ and the like work. The issue with division by zero is that (in a _very_ arm-wavy sense here) division itself tends to be intuited as an inverse of multiplication in the same way that subtraction is an inverse of addition. Thus, "if we can subtract any numbers we can add, we should be able to divide any numbers we can multiply." But division is not that at all, which is why it doesn't work as I just described, and there _are_ numbers that can be multiplied but not divided.

[@therealjezzyc6209]

@@Curt_Sampson friendly reminder that KZbin use Markdown not LaTex. * for *bold* and _ for italics

[@Curt_Sampson]

@@therealjezzyc6209 Actually, my attempt (which I should have checked after posting) was HTML markup, not LaTeX. But thanks for the reminder; I can never keep straight which commenting systems use which markup, but I've made a note to myself about what KZbin uses and fixed the comment.

[@hansijawns]

So, the barber was given an impossible task, and you have to make an exeption for him to complete his task, makes sense. However, if you are trying to create a theory that is supposed to be a completely logical foundations of all of maths, creating an arbitrary exeption like that would completely defeat the purpose of what you are doing

[@deslomator]

Your graphics are so cute that I sometimes have to rewind and focus my attention on what you are saying. Totally worth it, though, nice explanation.

[@safwanshahriar4108]

Thank the animators lol

[@SciPunk215]

"A set is a collection of things." Great, now all we have to do is figure out exactly what a collection is.

[@ast453000]

and what "things" are

[@deponensvogel7261]

You see, collections are a number of- ah!

[@mrporcupine4140]

A collection is a set of things, duh. And a set is a collection of things, therefore a collection is a set of things, and...

[@happinesstan]

Alternatively, we could stop trying to define everything, and simply experience it for what it actually is.

[@mrporcupine4140]

@@happinesstan While that may work for day to day life, you can't do logic without definitions, imo.

[@ericvondell5157]

I've seen the Jeffrey Kaplan tribute to the Russel's Paradox, but, I got Way More value from your approach. Those adorable animated Thingies remind me a lot of those Delightful Saturday Morning Educational Interstitials. "Interplanet Janet", "Conjunction Junction ", "Bill ", Twelve Toes", "Hero Zero"... There Were So Many!🙀💕 You'd have Loved My Century!😻 THANK YOU FOR WORKING SO HARD ON THIS!💖 Tifa might not be able to smell things as well as when she was younger, but, we could Use Logicism to infer, correctly that "All Dogs Smell!" And, This is accurate to At Least, ONE POV concerning Dogs. How to describe a Number w/o using the Word "Number" Might be expressed as :the value and/or *"Quantity" (I know 🤪) of a something denotes how it may be combined with another Value and/or Quantity. Ie: I have a quantity of Shoes, but, only enough for One person at a time to Wear." I think that THIS is How Most animals determine if everyone in the family is present! NOT with artificial inventions, but, by direct Observation and memory. The Linnaean system of classification operates similarly by determining What items Belong together based on intensity and plurality of similarities. The Cow clearly doesn't Belong with Humans and Apes. But, Could belong with Cats and Dogs, but, moreso with Sheep and Goats. Phylogenetic classification deals with ancestral relationships between organisms. 1+1=3! Is True because there are Three value symbols shown. But, we know that This is Incorrect. The Quantity of All symbols in the equation would Be 5! Which is how an animal would perceive this equation's Value. 1+1=2 is logically correct to the purpose of combining integers. My Dyscalclia operates like this giving me an instinctive value of the symbols First and then resulting in everything getting all mixed up in my mind disrupting computation. *But, like you said: Words Such as "Quantity" are related to the Word "Numbers"! But, what if I use a Made Up term such as "Accumulation", but, here, again, we've got a number related word. Humans Are the ONLY Indigenous Species Currently Extant on Earth to Utilize Complex Structured Utterances to convey information. It's completely artificial and unnatural. That's Why It has to be taught and learned! But, a mother Duck determining if everyone in Her brood Are present, is Not Likely accomplished with artificiality. Even words Such as Concepts and Sets are, precisely The same sort of thing As Quantity and Value! And, again, we're back to the Linnaean system! Try conveying Any Information w/o employing Artificiality! Like That Party Game "Taboo"! And, that's At the Root of this type of Paradox: Just like with The Raven Paradox! Although, you might be able to Break it down with Sets, Subsets, Ifrasets, Quantum Sets. You can say "All Ravens in Subset A are black. But, NOT All Ravens in Set 1 Are Black. And, only some of the Ravens in Collection 3 10:06 are Black!" Er .........🤪😵‍💫 Does the Concept of "Nothing" have an Extension?!🙀😱💫 Getting Off Tangent... Sorry 😹

[@tiborkoos188]

Thank you, great explanation! ALso loved the graphics, especially the birds :)

[@UnkleRiceYo]

I’d argue the most shocking and interesting part of this story comes after, when Gödel throws a spanner in the works

[@zephon13]

Number - A manmade reference, used for logical analysis of objects and the interactions with their environments.

[@IrvingNestorRandom]

Are you implying that logic is man made? We cannot use a man made concept to conclude something absolutely given in a logical proof...

[@performancetesting1]

Yes, you can use manmade objects as tools to simplify and organize your thoughts in the prime concept of logic. That’s all numbers are are placeholders for our minds so we can keep track of what we know and draw implications from what we know, especially when we can’t focus on too much knowledge at once using our brains at their current evolutionary state.

[@AriaNight]

10 month later .... This definition contains many things that aren't numbers. like words, language, philosophy and logic itself.

[@elliem7339]

@@AriaNight What if you add "to a quantity" after "A manmade reference" and keep the rest the same? Yes, it uses the word "quantity" which is a word that's a direct reference to the word "number," but it seems like a pretty good definition at least.

[@AriaNight]

@@elliem7339 you answered yourself, it's recursive definition, ofcourse sounds good. But it's not gonna be useful

[@tirr1]

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!

[@littlemikey46]

Please include more of Tifa! Dogs are so lovely and older dogs deserve love just as much as puppies!

[@AbhijeetSaxenaIN]

While the paradox holds a folklorish status, there's some less reported work too. Frege talked of sense and reference, not just extension. Russell worked on theory of description, not just theory of types. Wittgenstein came up with ideas that led to Russell's logical atomism. Before you proceed to Godel, a shoutout to these shall be appreciated much 😊. All in the spirit of giving a 360° overview!

[@paoloazzini9003]

Psychologist: hey mr. Frege, Gödel doesn't exists, it can't hurt you...

[@frankmccann29]

Thank you so much for the material and reminding me you can do anything with functional notation. Great video. Intersecting multi dimensional sets.

[@atsmyles]

Great Video! Apropos, I happen to be in a reading group studying Homotopy Type Theory. I look forward to your take on it.

[@zembalu]

A brilliant video, thank you! I have sipped it in small doses, and then tried to explain it to a fictitious friend. Now, I am eager to learn about Zermelo-Fraenkel :-).

[@erictaylor5462]

Barber paradox solution: The Barber doesn't need to shave, because she is a woman.

[@erictaylor5462]

@E Mathematics has nothing to do with it. Gender bias to assume the barber must be a man.

[@isadoreladuca1112]

I had a grandmother who could have used a barber...

[@erictaylor5462]

@@isadoreladuca1112 When you know her. Women can get more hirsute after menopause.

[@AllanSavolainen]

Other options, barber is a kid, so he doesn't have a beard and adults are not allowed to be barbers. Or barbers are required to have bears by law.

[@christopherdrake54]

According to Russell - not a paradox because the barber simply does not exist. This does not work with class theory however, as he points out.

[@ifechukwuejiofor8479]

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!

[@mangalvnam2010]

And soon after, Kurt Godëvil stroke, with his infernal indecidilities: math would never be the same... Poor Hilbert could not even dream for long of his perfect formalistic math-topia, ruined forever...

[@steevemartial4084]

Very interesting video. Now my interpretation of them might be wrong but I believe Gödel's theorems prove that there can't be a "fundation of maths". Since no system of axioms can be both coherent and complete, and we can't even use a given system of axiom to determine whether it is incoherent or incomplete.

[@redvel5042]

No system of axioms advanced enough for arithemtic, i.e. second order logic. First order logic still can be both coherent and complete.

[@steevemartial4084]

@@redvel5042 You're right! That's still a problem for any candidate to the title of "fundation of maths"

[@redvel5042]

As long as it's first order logic, which I'd say sounds fairly fitting for the *foundation* of mathematics, there isn't much problem. Of course, you'd still have to get a good candidate and prove its coherence and completeness, and indeed, it won't be capable of even arithmetic, but that may still be sufficient for the foundation of mathematics. But if you want the foundation to be second order logic [or above, I guess], then yes, there's the probelm of the necessity of either incoherence or incompleteness. Thankfully, though, not all of them would be on the same level, so I'd say you could perhaps scream pragmatism once you're tired and pick the one with least problems and most benefits, lol.

[@steevemartial4084]

@@redvel5042 We may not be using "fundation" in the same way here. I think about it like I think about the laws of physics in relation to chemistry and biology. You only have to accept those axioms and the rest will sort of appear. If a system of axiom does not allow for arithmetic I don't understand how it could be considered fundamental. Perhaps you could explain?

[@redvel5042]

Oh, on, it's not like the system of axioms doesn't allow for arithmetic. Instead, it simply doesn't go that far. Since you brought up physics as an example for a foundation for chemistry and biology, I guess it'd be kinda like how physics isn't exactly chemistry. It's got the building blocks for chemistry, but it's not so much concerned with chemical reactions and as far as I know [not a physicist, so feel free to correct me if I'm wrong], doesn't actually deal with such reactions or even say anything about it in particular. So while first order logic is too simple for arithmetic, it's not like it outright doesn't allow it. You have to get to second order logic to get arithmetic. The good thing about first order logic is that it can be both coherent and complete. Only because it doesn't go as far as arithmetic, which does indeed render it mostly useless, trivial, but that's as far as you can go with coherent and complete systems of axioms.

[@suryashivaprasad73]

Um the question I'm struggling to get a grip on is whether mathematics just exists and we "unearth"/"discover" it or if we "invent" it. If we do just discover mathematics using our intellectual capabilities as they develop, then is it even right to look for a "foundation" of mathematics in the first place? If we're really just discovering numbers woven into the fabric of our reality then where does the logic idea fit in?

[@Adraria8]

They’re not just embedded in our reality, they’re embedded outside our reality too! They are embedded in truth itself. The idea of using different foundations these days is less about finding one that is “correct” (internally consistent), since there can be more than one correct foundation, but to find one that is both the simplest and most descriptive. All a foundation is is something that other math concepts can be described in terms of, so for example if the idea of a triangle can be described as either a set of points or a set of lines, and in both cases you would still be able to prove all of the properties of a triangle. Math concepts don’t require a foundation to exist, foundations are just useful for describing them in simpler and more unifying terms

[@suryashivaprasad73]

@@Adraria8 So it's mostly just us trying to find the simplest set of axioms to construct the rest of mathematics with? And it doesn't clash with the idea of us discovering mathematics rather than inventing it?

[@Adraria8]

Surya Shivaprasad Exactly. And no it doesn’t because you could think of looking for foundations as either discovering them or inventing them.

[@jasonslade6259]

Personally I don't think math is a fundamental property of the universe. Instead it is a fundamental property of the human mind.

[@takashitamagawa5881]

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".

[@raheem2845]

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.

[@chrisnelson6991]

Note to self, next time make sure you are sufficiently caffeinated before stumbling round KZbin and clicking on a random mathematics paradox video. I liked and subscribed after my coffee. Thank you.

[@christianheichel]

Coffee and math 2 thumbs way up! Lol

[@sofia.eris.bauhaus]

mathematics is the study of computation. unsolved problems are programs that have not yet terminated. unsolvable/undecidable problems are programs with an infinite loop.

[@Biglover29]

"Tifa is a dog" Tifa's like, "I am?!" lol

[@VeteranVandal]

"Always knew I liked balls way too much."

[@kreiderdonny]

I gotta say I have long been a "fan" of Russell(he conceived and signed the "Russell-Einstein Manifesto"[right before the wonderful Einstein died in 1955]) and of course Godel.However I was lucky enough to stumble on Frege long ago in a library and was impressed and inspired.Excellent video and looking up the definition of "number" ( 1 ) in my Principia Mathematica was a chore haha.(I'm serious).Thank you for the information.

[@KatBaumgarten]

Jade, you're like a scientist disney princess, love you!!

[@anymaths]

mathematician

[@madhatterhillbilly4267]

I wonder if the barber also had a breakdown.. "To shave or not to shave?" He later became Sweeney Todd.

[@pratikt5702]

The demon barber of fleet street

[@williamcasey1927]

fifteen yard penalty for "alternative" use of a razor. and another thirty yard penalty for undermining the innocent trust a Briton deserves to keep, concerning meat pies and sausages. and yes, I'm using American football terms. Britons also deserve a better sport than continental football.

[@Suman-zp8gs]

Wow that's awesome!👌 Thanks👍

[@alannetherclift678]

A "number" is a collective grouping noun. It can also be used as a verb to assign titles to objects.

[@MrMZaccone]

This is a demonstration that paradox seem to be inherently semantic.

[@irrelevant_noob]

But you need semantics to define stuff, right? Otherwise, there's nothing to study...

[@MrMZaccone]

@@irrelevant_noob Yes, but definitions are descriptive not prescriptive.

[@irrelevant_noob]

Edmond Dantez Not all definitions though... Defining the series of Fibonacci numbers using recursion is quite prescriptive IMO. ^^

[@MrMZaccone]

@@irrelevant_noob Um... no. It's still merely descriptive. You're not really "defining" the Fibonacci series using recursion, you're just applying an already established description (definition) to the case of solving the larger problem, now based on the original result. That essentially IS the definition (description) of a Fibonacci series.

[@irrelevant_noob]

Edmond Dantez uhh, wth are you on about?! What "larger problem"? What "original result"?! And if not-even-that is an example of a prescriptive definition, then what was your point again? Why would it matter that "definitions are descriptive not prescriptive"? o.O