Russell's Paradox - a simple explanation of a profound problem

in 1901 the english philosopher and mathematician bertrand russell discovered a problem a paradox at the heart of mathematics and all of science the paradox specifically concerns a foundational branch of mathematics called set theory so in this lecture i teach you all of set theory in like eight minutes and then i show how the paradox arises russell himself and many other mathematicians thought that they could solve this paradox but i argue that they can't and they don't so let's get started what is a number take for example the number four i'm not talking about four potatoes or four tomatoes or four hairs i'm talking about the number four itself we know lots of things about the number four like that it is evenly divisible by two and that it's the square root of sixteen i think i'm talking about the number itself and i'm also not talking about this right here this is not the number four this is the arabic numeral written on a piece of glass the arabic numeral that represents the number four we have other numerals that represent numbers like this one that's the roman numeral four we use it for counting super bowls here in the united states the number itself is something that no one has ever seen or touched but we seem to know things about and also it and every other number is essential to science technology and all of human life immanuel kant was a prussian philosopher living in the 1700s this painting of him was actually done in color but i've changed it to black and white to make it seem even older he thought that mathematics was a construction of the human mind and if that's right then mathematical truths are in some sense or rather subjective but the german and english philosophers gottlob fraga and bertrand russell didn't like this they thought that mathematics had to be objective to refute and counteract kant's view they developed a view called logicism according to logicism mathematics is a branch of logic and the most basic type of math which is arithmetic could be reduced to just first order basic logic and set theory i'll explain what set theory is in a second they thought that if they could succeed in reducing arithmetic to logic and set theory then they would be able to answer this question what is a number and the answer would be that numbers are sets okay well what is a set a set is a collection of objects the branch of mathematics that studies sets or collections of objects was invented by the russian german mathematician georg cantor in the 1870s cantor proved that some infinities were larger than other infinities yes that's right the idea is that you could have an infinite number of something and then an infinite number of something else but that you'd have more of one than the other yes it's wild in order to do this he had to invent set theory and so now i'm going to teach you set theory very quickly but then set theory is going to run into a terrible problem it's going to seem terrible they're going to try to solve it but then i'm going to show that they didn't really solve it the version of set theory that i'm going to explain to you in the next few minutes is naive set theory it's called naive just because it's the ordinary set theory that we can formulate in ordinary languages like english and it's contrasted with formal or axiomatic set theory which is formulated in an artificial logical language you don't have to worry about any of that a set is a collection of objects like the set of these markers there are three markers here and the set of these markers contains three items within it but the objects in a set don't need to be collected together in space or time in order to be a set for the purposes of set theory so on the one hand we could have the set of those three markers but we could also have the set of all of the people watching this video who are spread out all over the world maybe they're spread out throughout time moreover the objects in a set don't need to be related to each other in any significant or meaningful way another set could be the set of lebron james the four-time nba champion and the top half of the eiffel tower these things have nothing to do with each other really but we have a set with those two things in them those two members those two items anything that we can refer to anything that we can imagine that can be in a set we could have a set that consists of lebron james the four-time nba champion and harry potter the young wizarding boy who does not exist that set contains those two objects one of which has won an nba championship four times as of the recording of this video and the other one is a non-existent boy with magic powers indeed sets can even include objects that can't be imagined we could have the set of all objects that cannot be imagined that's a set too and it contains many many things perhaps an infinite number of things things that can't be imagined you see these squiggly brackets the squiggly brackets are used in set theory to pick out the set and all the stuff inside those brackets are the stuff in the set so this is the set of lebron james and the number four there are two things in this set a number and a basketball player on this form of notation in order to pick out a set in writing we would have to write out all of the things that are in that set but that's too clumsy if it's a big set like the set of all cats the set of all cats includes many many things too many cats to list out and also we don't even know all the names of all the cats all the cats in the world all the cats in the universe so instead we use this notation which is red as follows the set of all x's such that x is a cat that's called the set builder or intentional notation you don't have to remember that the other crucial thing is the word contains we say that a set contains all of the objects that are members of that set so this set from before contains lebron james and the number four the set of all cats contains garfield and all the other cats as well a set is a many that allows itself to be thought of as a one gay or cantor never said that someone just said that he said that but he didn't really say it a set is a gathering together into a hole of definite distinct objects of our perception or of our thought which are called elements of the set he did really say that but he said it in german when cantor invented set theory in the 1870s he wasn't just making up a mathematical game for the sake of it no no we deal with sets every day suppose that someone says that pile of potatoes is enormous they're not talking about the individual potatoes they're not saying the individual potatoes are enormous some of them may be quite large but some of them are small like fingerling potatoes i'm gonna try to find a photo of a fingerling potato or some other kinds of potatoes anyway when we say that that pile of potatoes is enormous we're talking about the pile not the individual potatoes and we do the same thing with objects that are not collected together spatially the world population of cats is enormous the cats aren't collected together in space but we're talking about all of the cats as a whole and we're saying that that population is enormous we're not saying that the individual cats are enormous although some of them might be quite hefty the paradox the logical problem that russell discovers in 1901 concerns one or more of the rules of set theory so let's go through some rules real quick before we can get to the paradox rule number one unrestricted composition that just means we can make any sets we want any set that you can think of that's a set in the formal axiomatized version of set theory this is called the axiom of unrestricted comprehension you don't have to remember that rule number two set identity is determined by membership what does this mean it means that what makes a certain set the set that it is is just what's inside of it it doesn't matter how we label that stuff it doesn't matter how we label the whole set as a whole all that matters is what's in the set informal axiomatized set theory this is called the axiom of extensionality you do not have to remember that rule number three the order of elements in a set doesn't matter here for example our two sets the set of the number one and the number two and the set of the number two and the number one these two sets are the same set because the order that you put the items in the objects it doesn't matter now you can see this rule this really comes from number two number two says that set identity is determined by membership all that matters is who's a member or what items are a member of the set it doesn't matter the order rule number four repeats don't change anything this set right here which contains the number one the number two and the number two is exactly the same set that we were talking about five seconds ago it's the set of just the numbers one and two if you repeat a number or a member of a set it doesn't change the set because set identity is determined by membership that's rule number four which is really just derived from rule number two rule number five the description of the items in a set doesn't matter for example the set that contains just lebron james and the set that contains all x's all things such that that thing is the nba all-time scoring leader playoffs included i'm including the playoffs if you only talk about the regular season then kareem abdul-jabbar as of this recording is still the all-time points leader but if you include regular season and playoffs then it's already lebron james those two sets are the same set it doesn't matter if you describe him as lebron james or king james or the nba all-time scoring leader either way it's the same set containing the same one item in it and that's that guy who's a basketball player rule number six the union of any two or more sets is itself a set this rule just means that if you take two sets like the set of all cats and the set of all dogs and you combine them and then you've got the set of all cats and dogs well that's a set too this rule number six comes from rule number one because rule number one says you could just make any sets you want so if you got two sets and then you put them together that's another set you can make that one too rule seven any subset is a set a subset is just a set containing some of the items that are contained in another set and that comes from rule one two if you can make any set then any set you've got well any group of those items in there that's a subset that's a set two rule eight a set can have just one member this set contains one item or one element and that is lebron james four-time nba champion a set with just one member is called a singleton set one important thing to notice is that this set this singleton set is not the same as lebron james lebron james is a four-time nba champion this set the singleton set containing lebron james is a zero time nba champion it has never won an nba championship and it has never played a game of basketball because it is a set this rule rule number eight also derives from rule number one if we can make any set that we want well then we can make a set with just one item in it okay we are approaching the point where one of these rules is going to generate the paradox it's going to blow the whole thing up hold on it's coming rule number nine a set can have no members this set is called the empty set you can write it like this with just the squiggles and nothing in between or you can signify it with this which is a like a zero with a line through it which means nothing this is an empty set the empty set or it's called the null set the fact that there can be such a set just derives from rule number one unrestricted composition you can make any sets you want including a set with nothing in it but it is derived from number two that set identity is determined by membership that there's only one empty set or only one null set that one set with nothing in it it's defined by the fact that it has nothing in it so if you have two sets and they're both empty they're the same set the null set now things get juicy rule number 10 you can have sets of sets this just follows from number one as well if you can make a set out of anything that you can think of well you can think of sets too can't you sets can have sets in them for example the set of all singleton sets you'd write it like this which reads the set of all x's such that x is a singleton set that is a set of all sets it contains the set that just includes lebron james it contains the set that just includes the number 17. it does not contain lebron james because this is the set of all singleton sets and lebron james is not a singleton set he's not a set at all he's a four-time nba champion or you could have the set of all sets the set of all x's such that x is a set this by the way is how fragile and russell answered that question from before what are numbers they thought at least they thought until russell's paradox just blew the whole thing up they thought that the number one just is the set of all singleton sets the set of all sets with one member and the number two just is the set of all sets with two members now you might be thinking i can't wrap my head around this what does it really mean for the number four to just be a certain set what does what does that really mean don't worry no one really understands what that means not really anyway it doesn't matter because the whole thing is going to be blown up right now with the next rule which is rule number 11. sets can contain themselves this one's weird and it's going to give us the paradox but that just comes from rule number one also if you can if you can think of it you can throw it in a set and so you can think of sets and so you can throw sets in themselves consider for example the set of all cats does that set contain itself no because that set is not itself a cat it's a set and everything in it is a cat so it does not contain itself what about the set of all sets the set of all x's such that x is a set does that set contain itself yes that set contains all the sets and it is itself one of those sets so it contains itself or the set of all the things that i'm thinking about this set doesn't usually contain itself but right now it does because right now i'm thinking of that set let's follow the pattern of thought that russell was following in 1901 and 1902 when he discovered this paradox he was just thinking about and developing this idea rule number 11 that sets can contain themselves think about all the sets that do not contain themselves the singleton set with just lebron james does it contain itself no because it just has one thing in there and that's lebron not any sets let alone itself the set of all cats doesn't contain itself because it itself is a set not a cat the set of all singleton sets does that set contain itself no because there are a lot of singleton sets and so the set of all singleton sets has a lot of sets in it and so it is itself not a singleton set it's got many many members many many items in it and so it doesn't contain itself and then here are some sets that do contain themselves the set of all sets that contains itself the set of all non-singleton sets yeah that set contains itself because there are many non-singleton sets and so that set the set of all non-singleton sets has many members and so it's a non-singleton set so because that set meets its own condition it is inside of itself or the set of all sets that have been mentioned in this room this room which is an endless black abyss from which i am speaking to you right now up until right now that set did not include itself but i just mentioned it so now it does here's the next thought that russell had in england in 1901.

okay let's take all those sets that do contain themselves let's collect all of them together and make another set out of them the set of sets that contain themselves now we've got a set of those we'd write it like this the set of all x's such that x is a set that contains itself now let's take all of the sets that do not contain themselves collect them together make a set out of those russell thinks to himself in 1901 as he's barreling toward a paradox that's going to blow up the whole project to form the foundation of mathematics and all of science this set that includes all of those is the set of all sets that do not contain themselves and we'd write that like this the set of all x's such that x is a set that does not contain itself and then something happened he realized a problem and he wrote this problem in a letter in 1902 on june 16th to fraga and fraga received the letter this is what the letter really looked like fraga got the letter and it broke him physically he had a breakdown and had to be hospitalized upon reading this two-page letter and the letter asked this question the set of all sets that do not contain themselves this set does it contain itself here it is again the set of all x's the set of all things such that x is a set that does not contain itself does this set contain itself well let's go through both possibilities if it does then this set is in here it's in itself and the only way to be in there is to meet this condition the condition of not containing oneself so if this set does contain itself well then it meets this condition and this condition says that it doesn't contain itself if it does contain itself then it doesn't and consider the alternative possibility if it doesn't contain itself then it meets the condition it's a set that doesn't contain itself so if it doesn't contain itself it meets the condition and then it's in there which means it contains itself so if this set doesn't contain itself then it does contain itself if it doesn't then it does and if it does then it doesn't so this set both contains itself and doesn't contain itself and that's just a contradiction so set theory doesn't work that's the paradox now you might be thinking oh okay well that's no problem set theory was just some made-up rules right that's how math is we just make things up axioms are stipulated which is to say made-up claims so we just made up all these rules including rule 10 and rule 11. and rule 11 is really the source of the paradox once you let sets contain themselves well then you're going to be able to formulate this set and this set is the one that leads to the paradox so let's just change the rules right yeah that's exactly what russell tried to do he tried to just change the rules of set theory he made up some new rules and according to his rules sets cannot contain themselves and that's what other mathematicians doing set theory have done like the zermillo frankel set theory that's a version of set theory with some restrictions they just get rid of rule 11. and of course if you get rid of rule 11 you say that sets cannot contain themselves well then you also have to get rid of rule 1 unrestricted composition or the axiom of unrestricted comprehension or whatever you want to call it so that's what all the mathematicians do but does that work like can we just change the rules back before when we were going through all those rules rule one and rule 2 and all the way up to rule 11.

were we just making up the rules no we didn't just make up the rules and so we can't just change them that was me i just said that what i'm going to argue for the next i don't know two minutes is that the rules of set theory are not made-up rules they're real non-made-up objective rules that already exist and they govern perhaps one of the most fundamental practices of human existence and that is predication now i have to teach you some linguistics this is going to take i don't know 60 seconds 90 seconds something like that once i've done that i'm going to regenerate russell's paradox and show that it was never a paradox just for set theory it's a paradox for all of language and thought itself let's go here is a sentence garfield is a cat it contains four words grammatically speaking though it has two essential parts it has this first part which is called the subject of the sentence and it has this second part is a cat which is called the predicate the subject is what the sentence is about it's about garfield the cat and the predicate says something of garfield it says that garfield has a certain characteristic and that characteristic is being a cat lebron is the subject of this sentence he's the thing that the sentence is about and dunks here we go is the predicate we can say that predicates are true of certain subjects so the predicate is a cat is true of garfield and the predicate dunks is true of lebron and it's not true of me what was the relationship between sets and objects the relation was containment sets contain objects the objects are in the sets what is the relation or the relationship between predicates and subjects being true of predicates are true of or a predicate is true of a subject what i'm going to do now is i'm going to leverage this similarity to try to regenerate russell's paradox but now it's not with some made-up rules of set theory but with some very much not made up rules of predication and predication let me just remind you is just the practice of saying things about things so the practice of predication is ubiquitous it is utterly widespread we do this constantly we do it linguistically all the time out loud and almost every thought we have predicates something of something remember rule number one unrestricted composition for set theory that rule was the rule that there's a set for any imaginable collection of a thing or of things well that rule seems like it just is true of predication also there is a predicate for any imaginable characteristic of a thing anything that you can say about something there's a predicate for that of course of course there is and those rules of set theory that allowed russell in 1901 to generate his paradox those rules of set theory the relevant ones they're just true of predication too rule number 10 of predication you can predicate things of predicates just like you can have sets of sets well that's true you can predicate things of predicates here's a perfectly grammatical english sentence is a cat sounds funny yeah sure kind of it does kind of sound funny when you think about it is a cat is the cat is your cat sure it sounds funny in this sentence the predicate is a cat is functioning as the subject of the sentence sounds funny in this sentence is the predicate and it's saying something about the subject which is the predicate is a cat sure we can do that that's a perfectly meaningful thing to do in any natural language we can predicate things of predicates yes we can can you smell where this is going predicates can be true of themselves just like sets can contain themselves is this right i think it is consider for example the sentence is a cat is a cat is that true no because is a cat is a predicate so it doesn't have a tail it doesn't have fur is a cat is not a cat but what about this one is a predicate is a predicate yes that's true is a predicate is a predicate it is and that's a case where the predicate is a predicate is true of itself it says of itself that it's a predicate and it's right so rule number 11 predicates can be true of themselves some predicates of course are not true of themselves like is a cat we already saw that is a cat is not a cat so that predicate is not true of itself dunks dunks doesn't dunk dunks is a predicate so it can't play basketball it can't dunk so it's not true to say that dunks dunks no false tastes like chicken that's a predicate something can taste like chicken does taste like chicken taste like chicken no it doesn't taste like chicken is a predicate so it doesn't taste like anything but then you've got plenty of predicates that are true of themselves is a predicate is a predicate so is a predicate is true of itself is a string of words is a string of words yes it is so that predicate is true of itself typically comes at the end of a sentence well typically comes at the end of a sentence does typically come at the end of a sentence so that predicate is true of itself now let's try to make a predicate that is true of all of the predicates that are true of themselves and that predicate just is is true of itself is true of itself is true of all of the predicates that are true of themselves fine nothing wrong with that predicate but then what if we tried to make a predicate that's true of all of the predicates that are not true of themselves that predicate would be is not true of itself and now we're gonna generate the paradox i can't write this in a letter to gottlob fraga because gotlobfrega is dead but i'm writing it in a video to you here's the question is not true of itself this predicate is it true of itself let's go through both possibilities if it is true of itself well then what does it say about itself it says that it's not true of itself so if this predicate is true of itself then it's not true of itself if it is then it isn't what about the alternative if this predicate is not true of itself then it's not true of itself but then it meets the condition set out in itself it meets it has this characteristic specified by by the predicate it's not true of itself so if it's not true of itself well then it's true of itself if it is true of itself then it's not true of itself and not true just means false so if it is true of itself then it's false of itself and if it's false of itself then it's true of itself so it's both true and false which is a contradiction and this isn't a paradox that we can just escape by declaring that there is no rule 11.