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

refers specifically to a fundamental branch of mathematics called set theory, so in this lecture taught them all of set theory in about eight minutes. and then I show how the paradox arises. Russell himself and many other mathematicians thought they could solve this paradox, but I maintain that they can't and don't, so let's begin. What is a number? Take, for example, the number four. I'm not talking about four potatoes, four tomatoes or four hairs, I'm talking about the number four itself.

We know a lot about the number four, like that it is divisible by two and that it is the square root of sixteen, I think. I am talking about the number itself and I am also not talking about this 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 numbers that represent numbers like This is the Roman numeral four, we use it to count Super Bowls here in the United States. The number itself is something no one has seen or touched, but we seem to know things about it and all other numbers are essential to science.

Technology and all human life Immanuel Kant was a Prussian philosopher who lived in the 18th century. this painting of his was actually done in color, but I changed it to black and white to make it look even older. He thought that mathematics was a construct of the human mind and if that is correct, then mathematical truths are in some sense or rather subjective, but the German and English philosophers Gottlob Fraga and Bertrand Russell did not like this, they thought that the Mathematics had to be objective to refute and counter Kant's view, they developed a view called logicism according to logicism, mathematics is a branch of logic and the most basic type of mathematics, which is arithmetic, could be reduced to basic logic of first order and set theory.

I'll explain what set theory is in a second, they thought that if they could reduce 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 the numbers are sets. Well, what is a set? A set is a collection of objects. The branch of mathematics that you study. Sets or collections of objects were invented by the Russian German mathematician Georg Cantor in the 1870s. Cantor showed that some infinities were larger than other infinities. Yes, that's right, the idea is that you can have an infinite number of something and then an infinite number of something else. but that you would have more of one than the other, yes it's crazy, to do this he had to invent set theory, so now I'm going to teach you set theory very quickly, but then set theory will meet with a

problem

. terrible problem, it's going to seem terrible, they're going to try to solve it but then I'm going to show them that they really didn't solve it.

The version of set theory that I'm going to explain to them in the next few minutes is a naive set theory. It is called naive simply because it is ordinary set theory that we can formulate in ordinary languages ​​like English, and it is contrasted with formal or axiomatic set theory that is formulated in an artificial logical language that you don't have to worry about. That a set is a collection of objects like the set of these markers, here there are three markers and the set of these markers contains three elements, but the objects in a set do not need to be collected together in space or time. 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 the people who watch this video and who are spread throughout the world, perhaps They are distributed over time, furthermore, the objects in a set do not 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 elements anything we can refer to anything we can imagine that could be in a set we could have a set that consists in lebron james the four time nba champion and harry potter the young wizard that does not exist that set contains those two objects one of which has won an nba championship four times at the time of the recording of this video and the other is a non-existent child with magical powers, in fact sets can even include objects that cannot be imagined, we could have the set of all objects that cannot be imagined, that is also a set and contains many, many things, perhaps an infinite number of things that You can't imagine seeing these wavy brackets.

The squiggly brackets are used in set theory to select the set and all the things inside those brackets are the things in the set, so this is Lebron James's set and the number four in there. two things in this set, a number and a basketball player in this form of notation, in order to select a set in writing, we would have to write down all the things that are in that set, but that is too complicated if it is a large set like the set of all cats the set of all cats includes a lot of things, too many cats to list and plus 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 use this notation which is red as follows the set of all x such that 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, just as a set is a set that allows be considered as one. gay or singer never said that someone simply said that he said that but in reality he did not say it a set is a meeting in a hole of distinct distinct objects of our perception or of our thinking that are called elements of the set that he made I really say that, but he said it in German when Cantor invented set theory in the 1870s.

He wasn't just inventing a mathematical game for the sake of it. No, no, we deal with outfits every day. Suppose someone says that that pile of potatoes is huge. They are not talking about the individual potatoes. They are not saying that the individual potatoes are huge. Some of them may be quite large but others are small like fry. I'll try to find a photo of a fingerling potato or something. other types of potatoes anyway when we say that pile of potatoes is huge we are talking about the pile, not the individual potatoes and we do the same with objects that are not spatially gathered the world population of cats is huge the cats are not gathered in space, but we are talking about all cats as a whole and we say that this population is enormous.

We're not saying that individual cats are huge, although some of them can be quite strong, the paradox. The logical problem that Russell discovers in 1901 has to do with one or more rules of set theory, so let's review a few rules quickly before we get to paradox rule number one, unrestricted composition which simply means that we can form any set. that we want What does this mean? What does this mean? a certain set the set that is is just what's inside it it doesn't matter how we label those things it doesn't matter how we label the entire set as a whole the only thing that matters is what's in the set informal axiomatized set theory this is called axiom of extensionality, you don't have to remember that rule number three, the order of the elements in a set does not 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 in which you put the elements in the objects does not matter now you can see this rule this actually comes from the number two the number two says that the identity of the set is determined by the membership the only thing that matters is who is a member or what elements are members of the set, it doesn't matter the rule of order number four is repeated, don't change anything, this set here that contains number one, number two and number two is exactly the same set we were talking about five seconds ago is the set of just the numbers one and two, if you repeat a number or a member of a set it does not change the set because the identity of the set is determined by the membership, that is the rule number four. which actually follows from rule number two, rule number five, the description of the elements of a set does not matter, for example, the set that contains only Lebron James and the set that contains all the x's, all the things of so that thing is the NBA. -The playoffs of the leader in time scoring are included.

I'm including the playoffs, if you're just talking about the regular season, then Kareem Abdul-Jabbar as of this recording is still the all-time points leader, but if you're including the regular season and playoffs, then he's already Lebron James. those two sets are the same set, it doesn't matter if you describe it as lebron james or king james or the nba's all-time scoring leader either way, it's the same set containing the same element and that's that guy who is a basketball. player's rule number six, the union of any two or more sets is itself a set, this rule only means that if you take two sets like the set of all cats and the set of all dogs and combine them, then you have the set Of all the cats and dogs, well that's a set too, this rule number six comes from rule number one because rule number one says you can make any set you want, so if you have two sets and then the together, that's another set you can do. that too rule seven any subset is a set a subset is just a set that contains some of the elements that are contained in another set and that comes from rule one two if you can make any set then any set you have, well any group of those elements there that is a subset that is a set two rule eight a set can have only one member this set contains one element or one element and that is lebron james four time nba champion a set with only one member is called singleton set One important thing to note 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, has never won an nba championship and has never played a basketball game because it is a set, this rule number eight is also derived From rule number one, if we can make any set we want, then we can make a set with just one element, okay, we're getting close. the point at which one of these rules is going to generate the paradox is going to blow everything up, wait for the next rule number nine, a set cannot have members, this set is called an empty set, you can write it like this with just the scribble and nothing intermediate or you can indicate it with this which is like a zero with a line through it, which means nothing this is an empty set the empty set or is called a null set the fact that such a set can exist simply follows from the rule number one, unrestricted composition, you can create any set you want, including a set that contains nothing, but it follows from rule number two, that the identity of the set is determined by membership, that there is only one empty set or only a null set, the set with nothing in it is defined by the fact that it has nothing in it, so if you have two sets and they are both empty, they are the same set, the null set, now things get juicy, rule number 10, you can have sets of sets, this simply also follows from number one.

If you can make an outfit out of anything you can think of, you can also think in outfits. You can not? Sets can have sets, for example the set of all singleton sets, you would write it. like this one that reads the set of all x such that x is a singleton set which is a set of all sets contains the set that only includes lebron james contains the set that only includes the number 17. does not contain lebron james because this is the set of all singleton sets and lebron james is not a singleton set, it is not a set at all, it isfour-time NBA champion or you could have the set of all sets, the set of all x, so that x is a set of this by the way, is how fragile and Russell answered that question before what are the numbers.

They thought, at least they thought until Russell's paradox blew everything up. They thought that number one is simply the set of all unique sets. The set of all sets. with one member and the number two is simply the set of all sets with two members, now you might be thinking I can't understand this, what does it really mean that the number four is just a certain set? that really means, don't worry, no one really understands what that means, not really, anyway, it doesn't matter because the whole thing is going to explode right now with the next rule, which is rule number 11.

Sets can contain alone this It's strange and it's going to give us the paradox, but that only comes from rule number one. Also, if you can, if you can think about it, you can cast it in a set and therefore you can think in sets and therefore you can cast 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 is a set and everything it contains is a cat, so it does not contain itself. What about the set of all sets, the set of all x's? that x is a set, does that set contain itself?

Yes, that set contains all sets and is itself one of those sets, so it contains either itself or the set of all the things I'm thinking about. This set generally does not contain itself, but it does. Now it does because right now I'm thinking about that outfit. Let us follow the pattern of thought that Russell was following in 1901 and 1902 when he discovered this paradox. He was just thinking and developing this idea. Rule number 11 that sets can contain themselves. about all sets that do not contain themselves, does the singleton set with only lebron james contain itself? No, because he only has one thing there and it's Lebron, there is no set and much less himself.

The set of all cats does not contain itself because it is itself a set, not a cat, the set of all singleton sets, does that set contain itself? No, because there are many singleton sets, so the set of all singleton sets has many sets, so it itself is not a singleton. the set has many, many members, many elements, so it does not contain itself and then here are some sets that do contain themselves, the set of all sets that contain themselves, the set of all non-singleton sets, yes, that set contains itself. because there are many non-singleton sets and that set, the set of all non-singleton sets has many members and therefore is a non-singleton set, so that set satisfies its own condition and is within itself or the set of all sets. that have been mentioned in this room, this room that is an endless black abyss from which I am speaking to you right now, until now that group did not include itself, but I just mentioned it, so now yes, here it is the next thought Russell had. in England in 1901.

Okay, let's take all those sets that contain themselves, put them all together and make another set with them, the set of sets that contain themselves, now that we have a set of those, we would write it like this . this is the set of all x such that as it rushes towards a paradox that is going to blow up the entire project to form the basis 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 would write like this the set of all x such that x is a set that does not contain itself and then something happened he realized a problem and wrote this problem in a letter in 1902 on June 16 to fraga and fraga received the letter like this is what the letter really looked like fraga received the letter and physically broke it had a nervous breakdown and had to be hospitalized reading this two page letter and the letter asked this question the set of all sets that do not contain themselves does this set contain itself here is again the set of all x 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 over both possibilities if it does, then this set is here, it is in itself and the only way to be there is to meet this condition is the condition of not containing itself, so if this set contains itself well, then it meets this condition and this condition says that it does not contain itself, if it contains itself then it does not and consider the alternative possibility if it does not. does not contain itself then it satisfies the condition, it is a set that does not contain itself, so if it does not contain itself then it satisfies the condition and then it is there which means it contains itself itself so if this set does not contain itself then it contains itself if it does not then it does and if it does then it does not so this set contains itself and does not contain itself 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 not a problem, set theory was just some made up rules, right, that's math, we just make things up, the axioms are stipulated, that is, made up statements, so we just make up all these rules, including rule 10. and rule 11. and rule 11 is really the source of the paradox, once you let the sets contain themselves well, then you can formulate this set and this set is what leads to the paradox, so let's change the rules. right, that's exactly what Russell tried to do, he tried to just change the rules of set theory, he invented some new rules and according to his rules sets cannot contain themselves and that's what others have done mathematicians who do set theory, such as Zermillo Frankel's set theory. that's a version of set theory with some restrictions, they just remove rule 11. and of course if you remove rule 11 you say that sets cannot be contained well, then you also need to remove rule 1, unrestricted composition or the axiom. of unrestricted understanding or whatever you want to call it, that's what all mathematicians do, but does it work?

Can we just change the rules before, when we were going through all those rules, rule one and rule 2, all the way to the rule? 11. Do we just make up the rules? No, we don't just make up the rules, so we can't just change them. That was me. I just said that I don't know what I'm going to argue next. two minutes is that the rules of set theory are not invented rules, they are real objective rules, not invented, that already exist and that govern perhaps one of the most fundamental practices of human existence and that is predication, now I have to teach you some linguistics this will take me I don't know 60 seconds 90 seconds something like that once I've done it 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 language and the thought itself let's go here is a sentence garfield is a cat contains four words grammatically speaking although it has two essential parts it has this first part which is called the subject of the sentence and it has this second part it is a cat which is called the predicate the subject is what what the sentence is about is about garfield the cat and the predicate says something about garfield it says that garfield has a certain characteristic and that characteristic is being a cat lebron is the subject of this sentence he is the What the sentence is about and dunks here let's go is from the predicate, we can say that the predicates are true for certain subjects, so the predicate is a cat is true for garfield and the predicate dunks is true for lebron and it is not true for me what was the relationship between sets and objects the relationship was one of containment the sets contain objects the objects are in the sets what is the relationship or the relationship between predicates and subjects being true of the predicates are true of or a predicate is true of a subject what am I going to do What I have to What to do now is to take advantage of this similarity to try to regenerate Russell's paradox, but now it is not with some invented rules of set theory, but with some rules of predication that are not invented and, let me remind you, it is just the practice of saying things about things, so the practice of preaching is ubiquitous, it is completely widespread, we do this constantly, we do it linguistically all the time out loud and almost every thought we have preaches something about something, remember rule number one, unrestricted composition for sets theory that the rule was the rule that there is a set for any imaginable collection of a thing or things.

Well, that rule seems to be true for preaching. There is also a predicate for any imaginable characteristic of a thing. Anything you can say about something. There is a predicate for that, of course, of course there is, and those rules of set theory that allowed Russell in 1901 to generate the paradox of him. Those rules of set theory, the relevant ones, are also true for predication. Rule number 10 of preaching, you can preach. predicate things just like you can have sets of sets well, that's true, you can predicate predicate things here's a perfectly grammatical English sentence it's a cat it sounds funny yeah right it sounds funny when you think about it it's a cat it's the cat is your cat sure 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 says something about the subject what is the predicate is a cat sure we can do that, that's a perfectly meaningful thing in any natural language, we can predicate things from predicates, yes we can, can you smell where this is going?

Predicates can be true in themselves, just as sets can contain themselves, is that correct? I think it's something that should be considered. For example, the sentence is a cat is a cat is true not because it is a cat it is a predicate so it does not have a tail it does not have fur it is a cat it is not a cat but what about this is a predicate? a predicate yes, that is true is a predicate is a predicate is and that is a case in which the predicate is a predicate is true by itself says of itself that it is a predicate and is correct so the predicates of the rule number 11 some predicates can be true by themselves, of course, they are not true in themselves, as it is a cat, we already saw that it is a cat, it is not a cat, so that predicate is not true in itself, dunks, dunks, he doesn't dunk, dunks, it's a predicate, so he can't play basketball. t dunk so it's not true to say that dunks dunks is not false tastes like chicken that's a predicate something can taste like chicken tastes like chicken tastes like chicken no, it doesn't taste like chicken it's a predicate so it doesn't taste like anything but then you have a lot of predicates that are true by themselves is a predicate is a predicate so is a predicate is true by itself is a string of words is a string of words yes it is so that predicate is true by itself typically comes at the end of a sentence , normally comes at the end of a sentence, normally comes at the end of a sentence, so the predicate is true in itself.

Now let's try to make a predicate that is true for all predicates that are true for themselves and that predicate is simply true for itself is true for itself is true for all predicates that are true for themselves well, there's nothing wrong with that predicate, but then what would happen if we tried to make a predicate that was true for all the predicates that are not true in themselves, that predicate would be not true in themselves and now we are going to generate the paradox. I can't write this in a letter to Gottlob Fraga because Gottlobfrega is dead, but I'm writing it to you in a video. here's the question is it not true in itself this predicate is true in itself let's review both possibilities if it is true in itself well then what does it say about itself says that it is not true in itself so if this predicate is true in itself then it is not true in itself if it is then it is not what about the alternative if this predicate is not true in itself then it is not true in itself but then it satisfies the stated condition in itself satisfies it has this characteristic specified by by the predicate is not true of itself then if it is not true of itself then it is true of itself if it is true of itself then it is not true of itself and it is nottrue only means false so if it is true of itself then it is false of and if it is false in itself then it is true in itself, so it is both true and false, which is a contradiction and is not a paradox from which we can escape by declaring that rule 11 does not exist.

In the case of set theory, yes. perhaps we could simply declare that sets cannot contain themselves, but in the case of predication, which is simply speaking, in the case of saying things about things, we cannot simply declare that predicates cannot be true on their own. themselves because they can do it. This rule 11 is like that. true and once you give me rule 11 I will generate the paradox. This is a paradox from which we cannot escape.