 If you treat those, that means something different from that. But yeah, model theory is kind of hard to talk about like this. So often there is lots of background definitions. So with lots of definitions in this talk, we try to make a lot of examples as well. And go slowly, but just stop reading it on your stock or just pick it up and try to do my best to make clear what. What is really going on. So I will start slowly. I will define the graph with axioms of the examples. And this is one of my creating examples. So to talk about the graph, you need a binary relation. So I'm sad to be on this relation. The axioms will just tell you where it's irreflexive and it's symmetric. So for every x, not x. For every x, it will be y, x is y. So these are kind of very simple axioms because you're only quantifying or an element. You're not saying something about every set. Something happens. And you only have a very finite three part here. You don't have the infinitesimal junctions. You create these junctions. So these are called first order axioms called elementary. So that's where the elementary part is. So a lot of theories have been said that the class of graphs is elementable just because it can be axiomatized with an elementary first order of axioms. In fact, it's also finitely axiomatizable. And you wrote finitely many axioms. There are classes that are elementary, but you need infinitely many axioms to use them. So let me give you an example. The axioms of algebraic closed fields. For example, the complex. So just remember, algebraic closed field just means a field where every polynomial equation has a solution. So you will have the axioms of fields. I'm not going to write them. I'm also always going to specify the language what you're using. So here you have the binary relation. There you have binary operations. So you have plus times, and also some fixed constants, 0 and 1. And a thing that's what we need for the field. So you have the axiom of the field. It's just finitely many axioms in this fashion. So for example, you would have one that says where every x to resist y, x plus y to 0. Importantly, just a little more. And then, OK, you have to say something about characterizing 0, so that it would be that 1 is not 0. 1 plus 1 is not 0. 1 plus 1 is 1. It's not 0. So you have infinitely many axioms to say that. And you could also write this with an infinite disjunction. This would not make it elementary anymore. This is still elementary. Each of these axioms is first-order. And I'm just listing infinitely many of them. And finally, I have to say every polynomial has a solution. So no matter for each degree of the polynomial, I had an axiom that says, for every x0 to make sense with that. So I just have to say that the degree of the polynomial is 0. So you want to add 0 all the way to x into the sport? Oh, yeah. You have to add x. Yeah. I'm just making it a polynomial with 1, but it is 1. So that's just the axiom scale. So for each hand, I have an axiom like this. And to fix that, this axiom is elementary. Authentication over a few elements. So this is not financially axiomatizable, but it's still elementary. So this is what model theorists classically study. There's nothing so elementary about it. I mean, it's not trivial. I mean, those classes are interesting model theory. But they're pretty nice. Like, they have some nice property. One of the nice properties they have is their compactness. It says that if you have a list of infinitely many axioms, then you know that each finite part of this list can be realized by some objects. So for example, you have the axioms of groups plus some more axioms. And we'll get some graphs plus some more axioms, say. And you know that for every finite part, there is some object realizing it. Then there is a graph realizing the axioms of the first order axioms. So I'm keeping this name exactly what I mean. I can't even realize how clearly it's clear. There is some kind of set with the right operations on it that satisfy the axioms. Then just see itself can be realized. Remember, this means I don't have everything anywhere. But the first order means that there is quantification only over elements in the sets. So if you have a set of axioms like this, then you have the compactness deals. So this can give you some nice non-trivial results. So for example, you could look at, say, the theory of natural numbers, write about the theory of natural numbers. So let me write some, I'm keeping it vague, but this would say, you know, natural numbers for the addition is commutative on them and the induction principle, those kind of things. Some kind of induction. It doesn't have, you don't need induction. Just to say the operation of addition and times of natural numbers behave like you want. And then you could add something about some special constant x, or some special constant c, that's bigger than one. So in your language, just like I had some constant for, some special constant for zero and one in the language of fields, I'm adding this constant c and I'm missing some of the axioms of natural numbers, whatever you want. And then I have this set of axioms on set that c is bigger than every single natural number. And so every finite subset of t is realized and then let's say, for some appropriates, and because if I take a finite subset of that, I only have finitely many of these axioms, so I can just take c to be the beginning of number so that all the finite many equalities are satisfied. But so this means that by the compactness theorem, I realize my something. So it's some object, some set that we actually contain the original natural numbers, plus some weird, called non-standard natural numbers that's above everything, which would still satisfy one of the properties, or the first of the properties. So it's a very nice theorem, it has many nice consequences. So that's why I'm giving this example. But we know in math that not every topological space is compact and not everything has this finite, the finitary property, so some objects are not elementary at all. They cannot be axiomatized in this nice way. So let's look at a few examples. Okay, so one kind of natural example is what's called a locally finite groups. So there are groups, so that every set that's finitely generated is finite. Yeah, so for example, if you take z, it's not going to be finitely generated because take one, it's going to generate a new group. Plus one, plus one, plus one, plus one, plus one. But there are examples of such things that are infinite, but still every sub, finitely generated subgroup is finite. It could take some kind of direct sum of small, instead of finite groups. Okay, so there are kind of natural mathematical objects that study the group theory. They cannot behave almost like finite groups. We can generalize a lot of the theory of finite groups to them, but they're not elementary. There is no possible set of first order axioms that could axiomatize that. And you know, it's not easy to see, it's not easy to prove. The way you do it is actually you assume that there is some set of axioms that you cannot use to combat this theorem to construct a non-locally finite group that's still satisfied with the axioms. So there is the intuition wise that there is no way to express a set is finite using just, you know, quantification of our elements and finite disjunctions and conjunctions because what to say a set is finite, you have to say it has some size n, but then n is kind of part of your formula and you have infinitely many n. However, it's expressible in the metric object, so let's see how we could still axiomatize it using something like first order axioms. It's expressible in the first order plus infinite conjunctions and conjunctions. I'm just gonna sketch one line. It's not too bad, it's kind of long for good. Okay, so you have to say that every subgroup is, every finite generator subgroup is finite. So you're gonna say for every a zero, for every a n, so for each n you have an axiom like this, and this axiom, and you want to say the group generated a zero again, it's finite. Okay, and how are you gonna say that? Well, you're gonna say either it has size zero or size one or size two or size three and this is an infinite conjunction. So you're gonna do the big conjunction over all the national number and it's gonna say group generated by this. And how do you say this one? Well, this one for a fixed n, I think should be an advantage. So you just write all the possible combination of a zero and a n, oh, but there are many conditions here. So it's not quite an advantage. So what you say when you say, you just say, well, it has size n, so there must exist n many different elements that are equal to the full group. So x zero should be equal to, if you look at, for example, a zero times a one should be equal to a zero to a zero or should be equal to x one or should be equal to something more. And then you do this for all possible words and write any possible combination of that. Continue with it, there might be a pretty many combination, but in the end, we will have to be equal to this. And this is just a bunch of infinite conjunctions. Okay, so somehow it's not completely hopeless. Here is an example that's actually pretty hopeless. Class of well-ordered rinks. Well, it's hopeless from the kind of perspective I want to say. So what are well-ordered rinks? They're just linear orders that have, for each every subset, one empty subset has a list of elements. So it turns out it's impossible to say for every set S, so you're quantifying over sets, so if you're not allowed to do that in first order, S doesn't actually implies S has a list of elements. So this part would be first order, just saying S has a list of elements, you just say there exists x, so that for every y, x less than y, but this part is not because you're quantifying over every single set. So it's not first order, and actually you can show it's not even possible to extrematize it with just a finite conjunction of these junctions. You just need a quantification of your sets. Note also that, so this kind of gives you some mathematically interesting examples that are not elementary, so this motivation for just studying them because not everything is compact, so why not study things that are not compact? However, if you do that, you'll be able to compact this theorem. So compactness, let's say for even an entry. And the example for that would be you can, if you take the previous example, you can write a sentence that says, every x equals to one, or x equals to one, to small. Small, so this whole thing is one axiom because you're not going to finish the junction. And then, let's go to another one. And then you write another axiom which says that here this constant c is the c's bigger than one, it's really not an axiom for me. So this thing is not, it's going to be a shorter example to compactness because the entire set of axioms is not consistent because you're saying that every single element of your structure should be one, one plus one, or one plus one plus one and so on, but you have this special element c that doesn't satisfy this bigger thing. But every finite subset of this is consistent. So this shows that you cannot expect it to stay in there. So it's not as nice as you want to be. So I'm going to tell you a framework where it's possible to study this because at that point it started being a bit at half, like, okay, you looked at those first order axioms, fine, at least this is how you expect compactness, which is nice. But then you're saying, oh, what if I add infinite conjunction with junctions? Well, I could make up some other addition to my logic and could invent another quantifier. Let's quantify, let's not, let's choose something as informal, let's add a quantifier that says for ors, but uncountably, well, for uncountably many things, something happens or for finitely many things. You could make up your own logic and it's kind of ad hoc why you should really consider logic if you have conjunctions and conjunctions. You have some other kind of logic. Also, I'm told regular mathematicians don't really like logic. Like, you know, it's annoying to talk about sentences and make precise what is a formula, what is a language, what's time. You know, regular mathematicians don't really care about how exactly you express the axioms of groups in the first order or not. Who knows? It doesn't really matter, you just want to study the object, the class of objects that are groups, the category of groups. So, we're going to look at some more semantic framework that don't really depend on formulas or to make sense of them. And so, there is one I mentioned abstract for abstract elementary classes, which I'm kidding you about, abstracts, abstract, abstract, but okay. I'm not going to talk about them right now because the definition is kind of complicated and I want to introduce something that's going to be later. I want to talk about a simpler framework, universal classes. So, both were introduced, well, universal classes are the longest, but these ones were introduced by inspiration, which is a famous magician. Okay, universal classes. So, what are universal classes? There are classes of structures satisfying some properties. So, the definition. I'm going to take you to the definition first, and don't worry if you don't understand it, I'm going to explain universal class. If you close on there is more physics, some structures, let me explain this one thing at a time. So, what does it mean? So, first of all, what is a structure? Structure is something like a group or something like a vector space or something like any of the object is moving. It's a set, hey, we need operations and relations. Operations or functions, relations. So, for example, a graph consists of a set, a vertices and then a relation. The group is a set, things and then an operation, time limit. It's filled with a set of times and plus and also zero and one, which you can see either as operations, just spitting out the same thing on the time or as just special elements. That's what structures are. So, does the operations be finitary? Yeah, it's finitary functions and finitary relations. So, the erities are always finitary. So, for example, you could say take the reals with summation, if it is summation only, just in finitary relations. Finitary or energy. Okay, so I've told you what the structure is, what is the message here, the same vocabulary. It just means I don't want to mix up graphs and groups. I might say I don't want to mix up well-ordering and fields. I want to have the same type of relations, each structure, so same, so the vocabulary. I'm not gonna make this precise because it's more worse than it was worth. What is the structure in the class looks like this. All of the other structures also have to look like this. They have to get set with some binary relation. They cannot get set with a binary relation or set with, I don't know, some tree area. Everything has to look the same from the point of view of just the vocabulary. I'm not saying anything about what the relations are. Satisfying, just has to look the same. Same type of relations. Same energy and same number of relations. Okay, so far I haven't said anything I've pretty much all day of that. I think I'll look like this close on the isomorphism or you can define a general notion of isomorphism for graphs, isomorphism, if I add my path, normally it's a bi-direction, can define this for any number of relations and functions. It just means that the isomorphism plays nice with the structure. So this is just saying if you have a group and then you take an isomorphic copy of it, it also has to be much less. So again, it's kind of a norm of rigidity. Okay, substructures, maybe more interesting. So if you have substructure A with a bunch of operations, say relations one, two, so on. It's gonna be a substructure of B. Let's just do it with A and sub-sub-sub-B and then you have to add the operation, much more. So it corresponds to sub-groups, sub-order, sub-field, so for example, 9 through 6, 7, 8, 10, everything. So just to respect it today, you should be able to, you cannot just change completely what a multiplication is on the original group or the order of graph, or whatever it is. And for functions, yeah. Okay, so this is saying if I have a class of group and I take a group in my class and I take a sub-group of it, it's a sub-group. And union of chains, well, let's say a actual number or any order you require the union. And there's a way to make sense of the union. Basically, by this coherence condition, it just means what you think so if the underlying set would be the union of the, the relation of the sub-groups would also be the union. If you think of it in terms of the groups, this just means that if you want to go you should A times B in your union, you just go down to the beginning of pi so that A and B are in the group GI and then you compute the equation there and it gives you the result in the union, okay? So let's see some examples. That's what it means to be a universal class. That satisfies those three axioms. So it's your example of the non-examples. So this looks pretty general, right? I mean, many class of objects would only look like this. Class of graphs, class of all graphs. Any sub-graph of a graph is a graph. It's a little plausible for everything, it's fine. Let's figure out the locally finite groups. So even though I told you that this is not an elementary class, it's still an universal class because it's closer to a sub-structure. If you have a group for which every finite is generated sub-group is finite, it's not gonna change if you look at the sub-structure. It's closer to the union also because any finite set of a subset of a union is contained in a previous element so it's very finite. So anything that's very finite like this would be universal class. There is another example that's got to vary. If you look at the class of algebraically closed fields because it's not closed under being under sub-structure. If you take, say, the complex number, C would be in the class. It's an algebraically closed field. But then you take Q, where it's a sub-field of C, but it's not that you're bringing it closed. So it's not closed under a sub-structure. It's closer to the union though. The union of algebraically closed, algebraically closed with a finite area. So this one is elementary, not a universal. So what's gonna happen? The universal, not an elementary, or an elementary, which is universal. And then well-ordering is not. Because, in this case, there are close under sub-structure. Like if you take a subset of a well-ordering, it's still gonna be a well-ordering because every non-ordering subset is still gonna have a least element. And it's not gonna be closed under a union because you could have something like this going on. You could take the set minus, just look at the integers starting at minus 10. This is a well-ordering because it cannot look like a natural number. So for every fixed time it's a well-ordering and then you take the union. This is gonna give you the full integers and it's not gonna be a well-ordering anymore. For this there's a problem with this ordering. It doesn't play nicely with the faggants. You want it to well-ordering to be well-ordering. You might want to say, well, the nice notion of sub-order would be being an initial segment or something nice. Extends in the right way, doesn't add an element to the back of the ordering. That's not allowed by its framework. Okay, so that's nice. That still manages to encompass some non-elementary examples, but still, right? Elementary examples done obviously are passed by this. If I draw a picture, I mean, you can define what it means to universal add an elementary. Elementary classes are bigger than this and you have mostly universal classes. You need not be an elementary, you want to do so. And we see that local abstract elementary classes stick up here, generalized both. You might ask also why it's called a universal class. So it's, this one goes back to logic. If you remember, it has something to do with logic. The point being that it's a result of the universal class. There is a logic characterization of it. So even though we never use anything about formulas or any concepts like this, we can bring it back. Okay, so that's our side. Our side is quantifier quiz. We don't have any quantifier. We have infinite projections. So it's an exercise to do this with the localifier groups that's got some kind of derivative. I think, yeah, there's smartization, I guess. I gave you the idea of where you were. You can change it so that you can get something. And it's true in general, there's a correspondence. So the universal comes from the universal quantifier. So you also pass closed boundary desertion. Yes, so that's a very good point. Yeah, you have, it's so, in fact, if you give me any subset of the structure of M, that's in your class, you can consider the intersection of all the structures of K containing the A. And this would be in your class because it's in the whole M, in K, it's in K because it's equal to the closure of A under the functions of M. If you look at M, it's like maybe a group. So it has a time operation. So maybe A is not in the class because it's not closed under multiplication. But if you close it under multiplication, every single other thing, well, it's gonna give you a substructure of M. So since K is closed under substructure, you get something that's in the class. And you can argue that it's the same thing as the intersection of everything, closure. Yeah, it's a good point. It works. But algebraically closed field also are closed under intersection, but they don't fit there. So there's kind of a bigger class of things that are closed under intersections, but still might not be universal. So I'm kind of giving you a lot of definitions. You might wonder why here, what are the nice results about it? At least for elementary classes, you have the compactness theorem. Here, there's some kind of compactness we don't want to present it, but I will state results later. I want to give kind of the frameworks first and then I will talk about the model theory. Okay, so now I will make an attempt to define what those abstracted adventure classes are. So the one example you can think have in mind is the algebraically closed fields. They are not universal because they are not closed under substructure and there was a problem. So one thing you want to require is that, well maybe it's not closed under substructure, but there is still some kind, if you give me a substructure, maybe it's not in your class, but you can kind of still close it to a thing that's still small enough, it's pretty small and is in your class. So every subset can be closed to a structure of the class that has reasonable size. So that would be part of the answer. Also I mentioned a problem with the ordering, for example, for what orderings, it was not really the right notion to look at substructure. There are other examples like this, so an AC, we actually have a different orderings. It's going to be a k, right? K with an ordering on it. And then this is satisfied. So k, again, it's a class of structures in the same vocabulary, but I don't want to require it to be closed under substructure or under a union. Yeah. Before that, that's kind of a problem with partial ordering on the class. These k subscripts, that's here. k, that is a k substructure of a, then it would be a substructure in the sense, but it might be something, it might be things that are substructures, but not k subscripts. It's kind of a stronger relation. I would give examples of ordering like this. And also, it's kind of a trivial thing, but you've only isomorphism, like I said, the ordering. If f, and you can take an isomorphism image, an isomorphic image. So this ordering just has monotheorism, but it's not just a random thing that doesn't care about how the isomorphism type is. When it doesn't care, that's what it has to respect, isomorphisms. And then there are three axioms. Coherence, chain axiom. Well, with a please first, I will write it full. This type, the woven line, spoon and torsky axiom. These things have long names, because people thought about these. Then chain axioms. How do you list these? So far I just have a class with a reasonable ordering. I have a set so much. Any of the classes we've seen. Okay, coherence. I'm not gonna explain much about why you might care about such a thing, but it says that the abstract ordering on the class plays pretty nice with the substructure relations too. So if you have this situation, so you have three models, and M1 is a case of structural M2, and M0 is a case of structural M1, M2, then M0 is a case of structural M2. So on this one, I'm not gonna explain why we care about this kind. Comes out from some generalization of some statement. Okay, forget about it, I'm just visiting it for completeness. Okay, the LST axiom is what I was trying to tell you at the beginning about this algebraic closed field business. You want something like close on your substructure, but it's too strong. But you want some closure, so this would say there exists clearly the lambda. Forget about the lambda for now. I'm saying if I take M, okay, subset on the universe of M, then I can close A to kind of a nice case substructure of M that will not be too big, such that first of all, A is containing the universe of M0. M0 is a nice substructure of M, and M0 is not too big. You could take M0 to be equal to M, and then this would not be interesting. So I want to require that the size of the universe of M0. Okay, so, example. I'll give you examples later, but for now, I'll just write the axiom here, then we'll check that some examples satisfy it. The idea is you can close any sets to a substructure. And then two axioms is a replacement for being close on the direction of substructure. I think the universal classes for natural numbers that I really wanted to put in the order I care. So this is increasing with respect to this trans substructure combination. Then I can take the union. You could at least the problem of the chain. So this one is kind of technical again, extant on the MIs, just extant between this. Okay, so long definition. I'm not expecting to just remember every single fine technical point. I just wanted to give you an example, so you have an idea of what kind of thing we're looking at here. Let's look at examples. Okay, so like I said, this in a nutshell means every chain. This says every set can be closed. Example, substructure. And then, okay, so let me explain why, maybe not completely, but unfortunately, all these axioms, coherence is trivial because this thing is exactly substructure, so it's just able to transitivity of substructure. It's actually interesting. Okay, why is this axiom true for universal classes? Well, you can take the closure under the functions of M. Take M0 to be put in the closure. Just think of M as a group, it has just one function. So just close the group under this. And if you think about it, you cannot get too many more elements if you close something under functions. You will get at most the size of M0, will be at most the size of the functions you have times I have not something like this. I'm not explaining why, but basically, you're only plugging finitely many things in your functions. So you only get if you do one application of your functions, you will only get the size of the vocabulary. So let's say, hey, it's finite. If you do one application of your function, you only get size of function, many things. You have another application after a lot of many applications you've closed already. So this is where it comes from. So this means you can take that value to be the size of the vocabulary. So I'm working with infinite coordinates here. Probably some of you are not familiar, so familiar with that, but just think that I'm fine. Just think, anyways, I got the size of the real small, but it does. And there's a big fact about it. If you take another use, that's the only one I would use, you should take lambda plus mu, and the sum is the same as the product and the same as taking this. So that's why I wrote casually, time to change it to the plus, because it doesn't matter. Oh, and so this is for the nsdx here and why does it satisfy the chain? Just because, this becomes very easy once you have substructure itself. So okay, so at least if we analyze universal classes, I'm not gonna explain why, but it also generalizes elementary classes. So if n is elementary, you can ask, what is the order you can put on things? Substructure doesn't work, again, the reasons I'm not gonna explain, but then you can put, see, an ordering that's from your school, an elementary substructure for some order, I'm not gonna explain why. But let's work out, so algebraically closed fields, then algebraically closed fields work, okay? So it's elementary, so I'm going to do this with words, but you also get an explicit description. You can order it with substructure, and this should be an AC, because algebraically closed fields are closed on their union of chains, and the only action that was problematic was this closure on their substructure thing. But for any, if you can take its algebraic closure, you can just close it on their solutions of polynomials. And again, the algebraic closure is not gonna be so big, that it's gonna be the same size as the set, because I don't know, so this is a calculation. What else, oh, non-examples, this cannot be formed in AC. I'm not fully gonna explain why, because you would have to check that no ordering can possibly work for them. But let's see, if I order them with substructure, subordering, we show, we saw already, it's not even a universal class, because you have this behavior with the minus n. This was an example of showing it's not even universal class. So if you order it with substructure, you're not gonna get an AC even, I know, but in two segments, this is gonna fail, too, because now it's closed on their union of chains, because, to argue something, but as long as you don't grow on this part, and you only grow the upper part, it's gonna be fine. But it's not gonna satisfy this, the Lovenheim-Skorna-Darsky business, because, let's see, why? Can you just pick an infinite decrease in chain at first, and it's no, well, it's gonna... Yeah, you can pick some kind of final, you know, you can do more of that. Yeah, like for example, if you have, this is gonna use some theory of the order range that may be something that I'm here with, but basically, here is your order range, let's say, oh, this one, and if in your statement of the, here is it, with a statement of this, on word Lovenheim-Skorna, this Lovenheim-Skorna-Darsky thing, you take A to be equal to singleton A, and A sat here. Well, you want to close it on your substructure, but substructure means initial segment, so you need to pick up all the things that are before A, and there might be much too many of them. There's no fixed bound on this number of things, and you can always pick up, you know, ordering as many, many things. So that's why here A is just a singleton, so the size of A is not gonna contribute as much, but contributes to the weird definition of the order range. So it cannot be an AC. Okay, so I'm not defining this, and now I can try to tell you a little bit about model theory and what questions we're interested in. Working with those. Just take AC equal to nice glass of structures with a nice ordering. I should say I need more examples. You might wonder what is, I haven't given you a real example of the AC where the ordering is not a sub-structure. So you can look at the class of groups. Let's say the median groups, just X, Y, Y, X. Class of the median groups, okay. If you order it on your substructure, it's gonna be an AC, so why? But there are other order range that people that study groups consider. So for example, there is this notion of pure subgroup, pure subgroup of H. Every, if there is an X root in H, it's already in G, so for every G, G, there exists H prime in G. So whenever you have an X root in H, it's already in G. So it's some kind of nice closure property of the group, and this is studied by people doing a median group theory. You might want your group H, you might not want to consider groups that are not nice, not nicely closed, that's why this can be interesting. And this is an AC also, whether it could be a subgroup. It's kind of a financial property, so we can go through the argument a little bit more. Okay, so now in the second part of this talk, I want to talk about this particular AC. The basic observation for all this is, the basic observation is that if you take a vector space, if you take two vector spaces, you might ask, well, are they asomorphic? And the what conditions are they the same? So they must have the same size for a start, but you can show that if they have the same begin of size, then they are asomorphic, so. But you need to take one, two, and vector spaces. I haven't shown this, but vector spaces are also examples of ACs, actually, that's right. So think vector spaces are rational, and assume they have size like that. Okay, so maybe they have the size of the wheels. You can actually see the wheels as a vector space. So what I'm saying is that, and you want to be asomorphic, there's only one subject of space, I think. So let me prove why. It's kind of, it's a very nice argument. Part of the basic fact we're using is that if you want, it'd be a story kind. Well, the basis has to generate your entire space. So the size of B, B is going to be bounded by whatever B generates, the size of what's going to be generated. So any span of the basis will consist of a bunch of rational numbers, and then a bunch of elements of B, any of them, say, ranges over. So what I'm saying is that besides any element of B can be written as a span. So the ideal combination of elements of the basis and the ideal combination of basis is just this. So the size of B is bounded by the sum of the whole add of the size of B to the power add times the size of the rational, so forth. But what is this? What do you think the facts I wrote about? It's just the correlation arithmetic being very easy on the infinite sets. This is the sum. The sum of the rational is at f naught. The power add is still at f naught because you take the max and times. And the size of B, where it could be finite, I guess, you put one more at f naught. And then we take the power add, it doesn't matter. And at f naught times, at f naught, of course, is at f naught. And then you sum this at f times, so you're basically just taking at f naught times this again. So this would just be, and of course, this is bounded by the sum of the total. Because I'm assuming the vector space has infinite sets. So at f naught is bigger than the sum. So this must be an equality, and therefore, the size of B must be the same. So the size of B is lambda. So lambda is uncountable B must be the same. This is just how you show it. You just basically, if B is too small, then it cannot generate enough elements to cover the space. Okay, but notice I have this done because B1 and B2 be basis for B1 and B2, respectively. The B1 and B2, respectively, they must have size lambda. It exists because the size of B1, and once you have a projection between two bases, you can just extend it. So it's nice argument because it shows that this theory of linear independence can help you figure out that there's a unique object in a class, and whenever a unique object is done, because you understand it, you know what to study. You just do this one object. And so that's called categoricity. So a class is categorical in a polynomial lambda if it has only one object of that size. Okay, can be a B, C, a B must be positive. So in this language, vector spaces are categorical in every encounter. You already know the same argument, but it's a view that categorically close fields are categorical, it's categorical, it's zero. So there's this notion of a transcendence basis in the theory of field, and then you have a replacement for the span as well. This is kind of a nice generalization of this. You also have this. And so people notice this in the field. I mean, they notice this much earlier in the 50s, but some logician looked at it in the 50s and said, oh, maybe it's true. So I just made it close to that in elementary classes, vector spaces are also elementary classes. Maybe it's true for all elementary classes. If you're categorical in some size, then you kind of have a uniform reason for it, some kind of basis theory, and then you categorical every word. So it's a projection. If K is elementary, it's traumatized by, countably many axioms. If you have too many axioms, you might be able to read the statement of this. So if you have an elementary class, and K is categorical somewhere, some uncountable cardinal. So the idea is if you have some class that has a single object of a big size, then it must be for some uniform, very nice reason, a simple uniform reason, like the existence of a basis, and then you have a nice theory of independence, and you can understand the class very well. So this looks like a random question, but actually somebody could probably prove it in the 1960s. The proof was very nice because it introduced lots of ideas that allowed you to study the model theory of basically any category called elementary class, any nice enough elementary class, and develop a theory of inner independence and basis that expended that effect of basis to many, many other examples. So this has had a lot of big impact on model theory and also outside of model theory, and a few of them have a few of them. And so of course, you might wonder what's so special about elementary classes? Now I have introduced some sort of elementary classes, and I'm going to prove it to that. So shala, conjecture, it's conjecturing 70s in open memory, since that the same is true for ACs, so every AC category called some high enough part no, must be category called no high enough part. So we're going about the high enough part, here it meant a countable, but no, it's AC is more complicated, so it's just high enough. So there is some threshold of which things emerge. So this is open, and there are thousands of pages of approximation, many by shala, many by others. And so I've also just stated one of the main reasons of my thesis, which is that this is true for universal classes. The proof is very long and complex, and again, it did get a lot of machinery from generalizing independent vector spaces and all that stuff. So in fact, this is kind of funny, so I will let you say it, you can ask what is the high enough threshold in this case? So more precisely, if it's category called, I will write some string of symbols, two to the power of vocabulary, okay, so what is this crazy number here? It's a very big cardinal actually, and you can show it's close to MCNL actually. So for any cardinal, you can define two to the power of something, this is just the claimant of the power set, it's bigger than the set of the set. Oh, maybe I should write the clause because I have to explain it, so it's this way, it's easier to explain it. Okay, and what is the best thing of lambda cardinal, the best of lambda? It's a chain of shoes, chain of expensive shoes, like lambda. So it's kind of hard to explain exactly how you do it. You can do it for finite size, and for infinite size, you cannot take the supremum of these things, and then you keep going lambda many times, okay? So, my tower of two is bigger than the tower of two. So then you take both of this two to the power, which is pretty big, this is the size of the vocabulary, so all the classes we go to have a comfortable one, which you just think inside of felt. This is like the size of every subset of the real, the set of subset of all the reals, this thing on the brackets, this is countable, and then, okay, take a tower of two, is that long? Then you take a tower of two, that long, and then you get this thing, and the joke is that it's actually pretty small compared to other cardinals that set very similar to this thing, so. This is a small, okay, compared to what's called large cardinals, essentially. So for example, if you have a large cardinal in Kappa, Kappa will be bigger than that lambda for every lambda that's in Kappa, so it just dominates all the situation. And usually actually, there are many approximations, this conjecture in the field, and many approximations use actually those large cardinals. The threshold was the next large cardinal, whatever that means, about some size. So this is actually nice, just because you managed to get this thing, you don't do it, it's just ridiculous. Four, very, very good. But these are casual, yes. They pop up pretty often, actually. Okay, so, okay, I had other things to say, but it's later, ready, and it's a good place to stop, so, let's stop. Other questions? Any questions? Yeah, one question. Yeah. So what was the tower of, I don't know. The tower of, that's a bifnode, like this thing. Yeah, so this is two to the two to the two, not many times, and you might wonder, how do you make a bifnode many twos? Well, this is just defined to be the supremum bifnode of, yeah, I think, typically this is defined to be a bifnode to start with, so maybe I should say, I can be any bifnode. It doesn't really matter what's to go with, if I do this with the line, it doesn't matter. So, you take your continuum, like the size of the wheels, and then you add to that the size of the, or the subsets of the wheels, the size of the set, the most subsets of the wheel, you have to get the bigger size, so on, so they are all, this is an increasing chain of cardinals, right? And for any increasing chain of cardinals, the set size, you can take the supremum, and this is something, so. This is like the size of any set that normal mathematicians might care about, is bounded by this, and maybe you can look at functions from wheels to wheels, or the particular spaces of those functions, or something that, if I'm never gonna get past two to the power of two, past a fine string of two, but if you do the Torah, but I know that you go past it, but then you can take two to the power of that, and continue, and define this as always. That's no question or harm, that's the answer question.