 In this video, we're gonna include lecture two about sets. We've introduced sets and talked about how does one prove whether an object belongs to a set or not. And I wanna come back to something I promised I would do in the first part of this lecture and dig deeper into this idea of set membership and what does it mean to be well-defined? Determining membership for a set comes down to checking whether a fixed object satisfies the defining property of a set. Unfortunately, there are times where this can be very difficult to check, like in the previous video when we talked about perfect numbers. That's a hard problem, even though it's an easy property. And so there's times where it can be impossible to actually determine whether an element belongs to a set or not. How could it be impossible? The following example is gonna give us that. And actually it's a story. So imagine the following story here. In a small town, this is a fictitious town, but it could be real, right? In a small town, there exists a single barber shop owned and operated by a single barber. Now this barber is very proud of his shop and his workmanship. He proudly displays a plaque on the barber shop wall that says the following, I will cut every man's hair so long as they don't cut their own. I mean, it does make sense. I mean, if they cut their own hair, why would they come to a barber shop, right? He cuts other people's hair. So yeah, I mean, it's kind of an interesting thing to put on the wall, our barber takes it very seriously. He proudly displays it on the wall. And I mean, this barber is so proud of his mantra here that he takes it literally. He cuts every man's hair in town who does not cut his own hair, each and every one of them. So let us consider the set of men in the town for which this barber cuts their hair. We're gonna let that be set A, right? So set A is the set of men in town who the barber, I should say whom, he's got some proper grammar here, whom the barber, the barber cuts his hair. All right, that seems like a fairly simple, well-defined set. If you take a man in town, I should be able to decide does that person belong to A or not, especially by this plaque right here. This barber cuts every man's hair in town who doesn't cut his own hair. So for example, there's a fellow in our town, his name is Thomas. Now, what do we know about Thomas? Well, Thomas, he actually gets his hair cut by the barber every two weeks at the shop. So Thomas belongs to set A. Okay, on the other hand, we have another fellow in our town, his name's William. William, on the other hand, cuts his own hair at home. So he does not belong to the set A. Again, on the surface, this seems like a fairly simple determination here, but there is one issue we have to come down to. What about the barber himself, all right? Does the barber belong to the set? He is a man, right? Does the barber belong to this set? Well, let's consider the possibilities. Well, I mean, the barber either belongs to A or it doesn't belong to A, aren't those the only possibilities here? So if the barber belongs to the set A, that means the barber cuts the barber's hair, but the barber is himself. So if the barber cuts the barber's hair, then he cuts his own hair. And by his motto, he only cuts the hair of people who don't cut their own hair. So actually, if the barber belongs to A, then that actually implies that the barber doesn't cut his own hair. It's kind of silly, but that would be the logical conclusion you get from such a statement there. The barber only cuts the hair of people who don't cut their own hair. So if he cuts his own hair, then that's a man who cuts his own hair and he doesn't cut the hair of someone who cuts their own hair. I know, it might seem silly, but play with it for a little bit longer, right? So it turns out that the barber can't belong to set A because if it belonged to set A, it would have to not belong to set A, all right? But what if we consider another possibility? What if the barber doesn't belong to A? I mean, because we put barber in the set, we got a contradiction. So it's gotta be the case that he doesn't belong to A, but wait, if the barber doesn't belong to set A, that means the barber doesn't cut the barber's hair, but the barber is himself. So if the barber doesn't cut the barber's hair, that means the barber doesn't cut his own hair. He's a man in town who doesn't cut his own hair and therefore the barber cuts the hair of every man in town who doesn't cut his own hair. So the barber cuts his own hair. So the barber should belong to A, but wait, how can the barber belong to A and not belong to A? So notice the situation we're in right now. If barber belongs to A, then it means barber doesn't belong to A. And but if barber doesn't belong to A, then it means barber does belong to A. We get this contradiction in both situations. This is an example of what we refer to as Russell's Paradox. Russell's Paradox is a situation where it is undecidable on whether an object of a fixed universe belongs to a collection or not. So our fixed universe were the men in town. We can't decide whether the barber belongs to the collection of people who the barber cuts their hair. Because of his literal interpretation of his motto, he will cut every man's hair who doesn't cut himself. We can't actually decide whether the barber should belong to it or not. The barber would not be able to know whether he should cut his own hair because of this contradiction. Now it turns out the discovery of Russell's Paradox was a great controversy to early set theory because Russell, who it's named after, shows that sets may not always be well-defined. A term that they didn't quite have in our modern understanding. Early set theory, what's often called naive set theory, didn't specify what a set was. And so it was just collections of stuff. And then you got into problems like Russell's Paradox where it couldn't be decided whether the barber belongs to A or not because the set wasn't well-defined. So this discovery of Russell's Paradox eventually led to the axiomization of set theory. And today this is referred to as ZFC set theory. Set theory, for which it's an acronym. We're not gonna delve too much into what that means right now. But it turns out that there are rules to set theory that determine exactly what a set is. Now for this moment as we're transitioned to advanced mathematics, we're not ready to delve into the axioms of set theory. They're a little bit more complicated than we're ready for. So as we approach this class, we very much are gonna live in naive set theory. But because we're a naive set theory, we have to be cautious of things like Russell's Paradox. We have to make sure that our set is well-defined, that there is an unambiguous rule that decides whether an element belongs to a set or not belongs to a set. Now, how did we resolve Russell's Paradox? Using ZFC set theory? Well, without going to all the details of it, the question basically comes down to, is it possible for a set to belong to itself? Because after all, there's nothing that forbids a set to belong as a member of another set. I mean, for example, we could come up with a set for which the three elements of the set are the set of real numbers, the set of natural numbers, the set of integers, something like that, right? We could take a set like this and these could be like, this is the set of all numbers I like to do math problems for, right? For some reason, maybe I don't like to do stuff with rational numbers or complex numbers. I don't know, but this could be a set. And so the elements of a set themselves could be sets. So the idea of a set being an element of a set is not a controversy, there's no issue there. The question comes down to can a set, since sets can be elements of sets, can a set be an element of itself? And fundamentally, that's where Russell's paradox comes from and the ZFC axioms of set theory forbid this from being a thing. You cannot have a set which is an element of its set. That's one of the axioms that forbids it. And by doing that, that resolves the Russell's paradox in a ZFC set theory here. So let's go back to our barber example and see how does A be an element of A have anything to do with our barber here? So to see how the barber example has consequences to set theory, consider the set A defined in the following manner, okay? A is the set of sets. So the elements of A are themselves sets. And A is gonna be the set of sets such that the sets are not members of themselves. So we're gonna take all of the sets in the entire universe, every possible set so that the set is not a member of itself, okay? And so you put all those sets together into a set and that set we call A. And so A is the set of all sets who do not contain themselves. But then we get into a problem. The exact same Russell's paradox shows up again. Does A belong to itself, okay? Does A belong to itself? So because if A belongs to A, then that means A is a set that contains itself. So then by definition, A is not inside of A. Well, it's a problem. But on the other hand, if A doesn't belong to A, then A is a set that doesn't contain itself so it belongs inside of A, so you get A inside of A. This is the exact same thing with a barber. Now I'm not talking about barbers anymore but it's the exact same issue. If you don't forbid this, then such a thing is constructible and this would be considered a set. Then this set would, you couldn't decide membership. So the thing is how you resolve this is that the ZFC axioms say that this is not a set. It is a collection, but it's not a set because it's not well-defined. And because a good definition of a set doesn't allow for a set to be a member of itself. So that's how we try to resolve it. And so I'll let you kind of take that here. Russell's Paradox is a very fun problem to think about. There's a lot of deep stuff here that we're not gonna worry so much about but the issue I want you to get from Russell's Paradox is one needs to be careful when we define a set. If we're careless, it turns out what we defined is not actually a set such as this right here. The set of all sets that don't contain themselves is not a set. Another problem is if you take X to equal the set of all sets, something like that, this is also not a set. And this of course leads to another Russell's type paradox. These type of paradox will occur. So the set of all sets actually is not a set. It's not a set. And so I'll let you explore that a little bit more because we'll be ending our video at this moment. But before closing, I didn't wanna thank you all for watching. If you've learned anything in these videos about sets, please like the videos. If you wanna see more videos like this in the future, subscribe to the channel. And as always, if you have any questions about anything that you've seen in any of these videos, feel free to post your questions in the comments below and I'll be glad to answer them as soon as I can.