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

hi everybody Jade here today's video is about one of the most famous paradoxes in mathematical history it's called Russell's paradox named after Bertrand Russell super cool dude very influential philosopher / mathematician / logician / social activist / Nobel Prize winner so yeah we'll get to know all about him but this is probably one of my favorite paradoxes and it's really one of those ones that just makes you really question everything you know question everything you know so yeah I'm super excited for this video just before we get into it I just want to say that

this is a probably the most to neuen studio I've ever done there's a lot of subtleties and there's a lot of philosophy so if you find that you're getting a bit lost or you need to like rewind or watch the video again that's totally normal and it actually be pretty surprised if you've got everything the first time so yeah let's get right into it so first I'm going to introduce you to a somewhat simpler version of the paradox just to soften some of the concepts that we're going to talk about later on so this one

is called the barber paradox and it goes like this there is a town with one barber and his task is to shave all and only those who do not shave themselves now the question is does the barber shave himself think about it if the barber does not shave himself then he should and if he does shave himself then he shouldn't so so keep this little riddle at the back of your mind throughout the rest of the video but now let's get right into the mat so Russell's paradox is ultimately about the foundation of mathematics but

some of you might be unfamiliar with that concept what is a foundation of mathematics well let's draw an analogy with the other sciences the sciences are kind of like a tree biology is the study of living organisms which are the result of millions and millions of chemical interactions so one could conclude that all branches of biology stem from chemistry or that chemistry is in some way foundational to biology likewise chemical reactions are ultimately physical interaction so one could conclude that old branches of chemistry stem from physics if we keep going down in this way we'll

eventually come to the elementary particles that make up our universe electrons and quarks the laws that govern these particles can be considered the foundations of science in that the rest of the sciences stem from them if we were to discover something completely new about them it would somewhat affect our understanding of the other Sciences even if only at the theoretical level in the 19th century each branch of math was pretty disconnected from the other branches and there was no unifying trunk mathematicians at the time wanted to unify the branches of mathematics and therefore a foundational

theory was needed but what made it tricky is that the approach to figure out the foundations of math is very different to that of the other sciences we learn about the other sciences by observing the outside world whether it be watching cells under a microscope measuring orbits through a telescope or smashing particles together at insanely high speeds but we can't really learn more about mathematics by observing the outside world math seems to be something that just is we can imagine a universe where the fundamental building blocks are different or where time goes backwards or where

everything was made of antimatter but we can't really imagine a universe where two plus two equals five math is different in that it seems to be true by definition and even though it's very good at describing our universe it doesn't seem to be about how universe why that is the question at the heart of this paradox now it's here that we steer away from the harder Sciences and head more toward the study of what reality really is philosophy we come to the first character in our story the question why is math the way it is

captured one of the greatest philosophical minds of all time Plato Plato thought that mathematical objects numbers shapes and the relationships between them were objective truths that is objects that existed independently of us and our worlds he thought they existed in their own world which he called the world of forms but one of Plato's pupils Aristotle had his own views unlike Plato he thought that numbers themselves weren't objects but properties of objects for example if there were four cows in a field it wasn't that cows we're an object and four was also an object but the

number four was a property of the collection of cows he also thought they didn't exist in their own world of forms independent of our world but they described features of our world and so belonged to our world another colorful philosopher Immanuel Kant soon came along and he didn't like the view that mathematical objects were objective truths and believed that to understand the basic principles of math you needed some kind of intuition but yes math described our world but we also brought it to the world from our experience now this is where our quiet hero of

the story comes along Gottlob Frager yes interestingly Bertrand Russell is not the main player in Russell's paradox but a German mathematician named Frager the story of Freya is somewhat tragic as he was largely ignored during his life and twenty years of his life's work was unjustly washed down the drain but here he will be remembered as a God among us Frager disagreed with Aristotle's view that numbers were properties of objects because of this argument if numbers are properties of object then only one number should belong to any object and it shouldn't be influenced by matter

of opinion but then imagine a pair of shoes is it one pair of shoes or is it to shoes depending on how we conceptualize an object the number belonging to it changes again imagine a deck of cards depending on how we conceptualize it it could be one deck of cards or 52 cards so Frago concluded that numbers don't apply to objects but took concepts he also disagreed with kent that to understand math you needed intuition and experience stating that the laws of arithmetic can be known from the basis of reason alone I sought to make

it plausible that arithmetic is a branch of logic and need not borrow any ground of proof whatever for me the experience or intuition frege's main goal was to reduce mathematics to logic and show that logic is in fact the foundation of math this idea is known as logis ISM now the notion of logic that Frager had in mind is pretty similar to how we would use it in everyday language but just to make it super clear logic is a tool for reasoning about how different statements affect each other through nothing more than two duction and

inference for example if we take it as a fact that all dogs have a good sense of smell and Tifa is a dog deductive reasoning will tell you that tiffer has a good sense of smell I don't know if this is the best example though because she's pretty old I don't know if he can smell much anymore well you get the idea Freya started his quest of reducing math to logic by coming up with a definition of what number was now this might seem like a somewhat trivial question to some of you what is a

number but remember at the heart of frege's paradox was the idea that you don't need any kind of intuition or experience to understand mathematics and when you think about it have you ever actually tried to describe what a number is without using the word number or words derived from the word number I've spent some time trying as I was writing this video and I found it I always ended up using the word number again or a word that was directly related to number like amount or quantity give it a go right now and let me

know what you come up with in the comments Frager defined numbers by using this idea of concepts and extensions a concept is pretty much any idea you can think of the color red shape bill and Bob Shakespeare's plays hipsters elephants with no ears literally any idea and an extension is the set of all things that fall under that concept for the concept the color red its extension would be the set of all red things past present and future the concept bill and Bob would be made of the extension of the bill and Bob you are

referring to if we have a concept like square circles which doesn't make sense their extension would simply be the empty set as set with nothing in it numbers frege's stated are extensions of concepts for example the number four is the extension of the concept of all things made up of a collection of this many objects the number seven is the extension of the concept of all things made up of a collection of this many objects Frager built his foundation of mathematics from the axiom that all concepts we can possibly think of have a corresponding extension

and therefore there are as many extensions as there are concepts he named this the general comprehension principle and it sounds reasonable right can you think of a concept that doesn't have a corresponding set to go with it I don't think I could but Bertrand Russell did Frago was just about to have his work printed when he received a letter from Bertrand Russell in June 1901 which read along the lines of Dear Colleague I find myself in complete agreement with you in all essentials in regard to many particular questions I find in your work discussion distinctions

and definitions that one seeks in vain in the works of other logicians there is just one point where I have encountered a difficulty consider the set of all sets that are not members of themselves is that set a member of itself this simple question shattered Freitas Foundation and caused him a mental breakdown so severe he ended up in hospital and later caused him to write my efforts to throw light on the question surrounding the word number seems to have ended in complete failure let's break it down first what about sets sets are a collection of

things you can even have sets within sets take for example the set of all sets of bird species there exists the set of all penguins the set of all seagulls the set of all pigeons and the set of all other bird species and they themselves would make up the set of all sets of bird species most sets are not members of themselves that is they don't fall under themselves for example the set of all teapots is not a teapot the set of all turtles is not a turtle but what about the set of all things

that are not Turtles because it's a set and not a turtle it is a member of the set of all things that are not Turtles therefore the set of all things that are not Turtles is a member of itself a lot of sets of members of themselves and there's nothing too extraordinary about that but the question that Russell asked was is the set of all sets that are not members of themselves a member of itself think about it if it is a member of itself then it is not and if it's not a member of

itself it is sound familiar I'll say it again if it is a member of itself then it's not and if it's not a member of itself it is now the reason that this was so catastrophic is because the last thing you want in the neat systematized acts EMA ties description of anything is a contradiction imagine if one of the foundational laws of physics predicted that gravity both always pulls and always pushes it would be a pretty useless law when your system is able to derive two opposite theorems this puts into question the entire theory numerous

attempts were made to correct this hiccup like Russell's theory of types which tried to put sets into some kind of hierarchy but many thought that this fix was too artificial and ad-hoc Frager eventually felt forced to abandon many of his views about logic and math even so as Russell points out Franco met the news of the paradox with remarkable fortitude as I think about acts of integrity and grace I realized that there is nothing in my knowledge to compare with frege's dedication to truth his entire life's work was on the verge of completion much of

his work had been ignored to the benefit of men infinitely less capable his second volume was about to be published and upon finding that his fundamental assumption was an error he responded with intellectual pleasure clearly submerging any feelings of personal disappointment it was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known apparently he didn't know about the breakdown anyway after a series of fortunate events the foundations of mathematics has now known to be a

system called Zermelo Fraenkel set theory which has a lot of frege's original ideas incorporated into it but although this is currently the most widely accepted theory the search still isn't over with candidates like category theory and homotopy type theory becoming serious contenders for that much desired position at the bottom of the math tree but that is for another video [Music]