 So, I guess, I would say that. Yeah, but, you know, therefore, yeah, I would use this as a speaker. So, welcome back to the Graduate Space Center seminar. And, you know, I haven't seen you for two weeks probably. So, kind of miss you and also, and also the seminar. Anyway, today we have Sebastian, who is going to give us an explanatory talk on that seminar. So, this talk will be to give an explanation of it. So, this is all, our result in this talk are all due to some, an appearing, a book appeared in 1990. But it did the work before in the 1980s. So, what we want to do is look at certain classes of structures and understand them, or say, why we cannot, which are at least in the beginning of the talk. And structure, I mean things, algebraic things, like groups, about these kind of structures. We want to know when we can understand them and when we cannot. So, I'm sure most of you have seen at least one structure theorem in your undergraduate classes. So, let me give you two examples. So, structure theorems are, if in the class that you understood, they can be decomposed into simple objects or even assigned dimensions. So, one well-known example is the classification of finite, a billion groups. So, a billion groups is one where multiplication is commutative. It's finitely generated if there is a finite set from which you can get all the other elements. So, all the other elements are products of this finite set. I can just think of finite at the end groups. First, so the structure theorem here says every finitely generated at the end group, G is isomorphic to a product of Z to the power minus 0 and then some simply groups. So, Z to the power 1. So, this is the simply group with G1 elements. So, here you do arithmetic for G1, take N1 copies of it. So, this gives you a very simple decomposition of each such group. It's not obvious, but you can prove that. Even more well-known example is vector space. So, if you look at the vector space, every vector space, the zero-dimensional, 0.1-dimensional, so line, two-dimensional and so on, and then you can go into the ethyl. So, you could have chanceably many infinitely, infinite countable theorems. Dimension and even more dimension than that. But in any case, maybe only one can be more than that. So, in those two examples, we have kind of dimensions implicitly entering a fiction. So, here, every vector space has one dimension, this lambda, and this dimension just tells you everything you need to know about the vector space. And here, it has what you need more than that in a finite sequence of integers, those N0, N1, N2, Nm, and also those G1 between Pm, which means a small sequence of coordinates that determines the object. Not so hopelessly, but you can say something in a very general framework that says, okay, even in your favorite class of objects, can you assign a dimension function like this? Can you understand it completely with a big composition theorem? Or can you prove that it's impossible? Okay, you've got a class of structures. But when we cannot do it, so it's kind of the interesting point of the result I'm going to discuss. You can say some things for both sides. In theory, for when it is what you have, in this case, you can prove the dimension theorem. But if the criteria is not satisfied, you can also prove that it's impossible. It's not just saying, oh, if my class is the class of finitely generated a bigger group, then I can say something. Because, well, if your class is not this thing, then you can also say something. There are many other examples. So it's not as simple as saying, you're giving an easy criteria. So what exactly do I mean by dimension? Requirement is that you need to respect yourself to simple objects. And you could say, well, dimension is a single coordinate, but this example doesn't really matter. For example, if you look at algebraically closed fields, you need a characteristic plus another dimension. So you need two things. In general, you need a small number of things, but sometimes it's not just a small sequence. So it's somewhat the only issue. So I would just say it should be some small, should be or would be if you see what the dimension would do if you have a nice notion of dimension what it would do to your class. Something you observe here is that it kind of restricts the number of models. So if you have only one dimension, you'll have very few objects because there are very few possible dimensions. So you cannot have too many objects. So for example, I was trying to give an intuition for why the result I was getting next is a bit of that question. If you have a nice dimension function, whatever that is, what would you expect? Well, for example, the case of vector spaces with a small sequence of cardinals, there should be few non-isomorphisms that besides the same number as there are cardinals of vector spaces. So really you want to look at vector spaces of a single size. And either you do that you have some dimension or you don't have it. So let's look at an example. Yes. I'm just wondering what's the precise definition of size. If we don't have a notion of that dimension. Yeah. So size, what's the size? Well, it's cardinality. So having done this, you would guess maybe it's good to do it now. You can define a notion of sizes on sets and also on infinite sets. So let's say sets X and Y have the same size or cardinality but those sizes are nearly ordered. Actually, they are nearly ordered. So when I say some object of size, Linda, I mean look at this universe which will have some ejection. So you know what's the size of? It's the same as the size of the integer. It's the same as the size of the rational numbers. So they are all in the ejection and this size is called that if not. It's the first infinite cardinal. And then you can look at the sizes of the reals. For example, you can show it's bigger than that if not. And it's called the same size of the power set. Three of sizes of infinite objects which are not all infinite objects have the same size but still there is like this nice linear ordering on sizes. So I don't know if this answers your question. This is what I mean when I say some object that says Linda is some infinite number could be finite too actually but most of the time it's just saying infinite objects should be quite the size. So just look at this universe and count it. Let me give you an example. Maybe it will be clear. So here you will just say on the last page everything that has the same cardinality of the structure. So yeah, I want to just look at structures of the same fixed cardinality, Linda. So I'm fixing Linda and I want to say how many objects of this particular size are there? So let me give you an example. So that lambda is not the same as that. Oh, just Linda here? Oh, I guess not. Might not be, but in that actually it would be the same. So let me just, so for vector spaces. So I want to know how many vector spaces over the rational order have only n many vectors. So, well if n is different than 1, there's no, right? So it's like 0, n is not 1. I know there is no 0 vector space. So we can look at finite objects but in that case it's not very interesting. So this one is huge upon 0. So you're right, so this is the standard of the same change. So more interesting is what happens when you look at countable sizes. So how many vector spaces where the rational have the same size as the rational, the natural numbers? So let's try to list them. We have told you that they all have this form and we started with the cube to the bar 0 here. So we can list them. So this is cube to the bar 1, which has the same size as cube. Then you have cube to the bar 2, which you know it's just a bunch of pairs, each of which is a rational number. And you can show actually that, so it's also a type that's underused, I guess. Infinite sizes behave very well in terms of when you hide them and when you multiply them. So if you take two infinite numbers like that, you just get the max. In addition and multiplication are the same. So here, the size of cube squared is the same as the size of cube, is the same as the size of cube, times the size of cube, which is just at f naught times at f naught, which is just that. So you have at least two of them. But there's no reason to stop here. We have cube 3 and so on. And you even have a less than omega. You're taking direct sums. Yeah, so it's a bit of a key issue, but you take direct sums, so every vector is just finite. So we raise kind of this counter. And everyone has a countable size. So in the end, you just add a graph many. This size. You have 1, 2, 3 and so on. And then this one. So this is for the countable case. And what's about the uncountable case? So in that case, I'm sure it has only one. So I don't know. It's slightly confusing notation, but what it says is that the basis has size of lambda. So every vector is a finite combination of things in the basis. So it's basically lambda of the power of finite, which is just the size of this. So last time, in the previous talk, I showed very precisely why this was true. But basically, the idea is the size of q to the power of lambda is just lambda. So then you're just using this result. You can't do something wrong. But this is what the, this is called a spectrum function. This is what it looks like. So we see that it's very small. Like it's 1 eventually. So this makes some sense because we've seen that there is some dimension. So dimension has reduced the number of models. That was great. So now, let's look at an example on the other side. So I've given you many nice examples. I've given you vector spaces. So let me give you an example where this spectrum function is not going to be. If you look at a billion groups, finitely generated a billion groups, they only count to be many of them just because they are finitely generated. And then each one of these can product with these finite sets. So you can show there that only count to be many of them. So again, you're just restricting the number of the bunch. So let's look at k and the class of plans we know. So the two examples that are easy to give are... So let's look at this function there. So first of all, what's about finals? You know what I mean, right? If there were only two elements, if you find one, it would be 3, so for the same reasons, it cannot be 8. State two results, and we'll prove them. For countable ones, what I gave you one example is the rationales. And there's only one. Of course, you can rename the rationales, but you'll always get attention. When lambda is countable, you have two to power lambda. For definition, it's just the size of the power set of lambda. So I'll show actually that this is the maximum number of possible models. Yeah, I guess so. I don't know what to deal with this. We got many models. I mean, you just go get every possible subset of the universe of size lambda. And the best you can hope for is that you cannot put any projection that preserves the structure. So you can always show that. So this is the maximum number you can expect. So this result tells you in a strong sense that there cannot be any reasonable dimension for linear orders. Doesn't mean you cannot understand that in some other ways, but in this very strong dimension kind of sense, it's going to be very possible. However, you should note that here there is only one countable model, but there they were, I have not made it. So that's why I wrote somewhere that you wanted to just look at the asymptotic behavior. Like in countable things and finite things, in an incident where we don't really know what's going on, you just get some noise. Not exactly noise, but you get something that's not too random. So really you want to focus on what happens eventually. So any questions on this thing? So let's see why. So let's look first at the second. So I will show something weaker. It's not that easy. So we just show the idea for something weaker. I will show that and it's hard to deal with countable sizes in general. So I just look at 2 to the i if not. So the size of the reels. And we just show it's much bigger than one. So I'll give some idea on how to be the main models. I'll be the next two. So what's one example I just want to point here is the reels. So we just look at the property of the reels that holds and try to kind of facify it. So in the reels just as an ordering we look at 2 points. There aren't countably many points in between. The same size as the reels, so it's countable. So now let's try to build something that doesn't have this property. And what I know of some linear order that doesn't have this property is the rational because it's countable. So I can just have happened the rational to these reels and see what happens. So this means take the ordering of the reels and then add a copy of the rational and make sure that every rational is above the reels. So we have 2 points between them is countable. So this cannot be as a more thing. The size of this is 2 to the i if not plus i if not. So I've just built 2 different ones. How would I build the third one? Well I can show you what's about q in terms of r in terms of q. Well you can show that it cannot be as a more thing for this one because there are the weaknesses before this copy of r and the after. So also this one is another. All possible sequences of those r and q are calculated in various ways and prove that they are more things. This is the part where it's sketched about. It's not that easy to do actually. You can make 2 to the lambda just of this form of r's and q's. It's a co-stationary set. Yeah, it's not an easy argument. I'm sweeping something under the rug. So anyway, there are at least 3 before those I can build countably many. This number is very big to take lambda to be the reals. This means you have to build as many structures as the number of functions from the reals to the reals. This is the idea. So what happens at the countable stage? Why cannot I do this? So let's see. This is actually due to cantors. This result here is much easier. So I want to show there is only one. I know there is at least one, the rational. So I will just take 2 of them. That's the one. I want to show that they are reasonable. And I will use something on the back and forth. So I want to build an isomorphism. So it's a bijection. Taking elements here, taking elements here, taking elements here. So let's see. I take that amount. It looks the same everywhere since it's countable. So I will just pick some points. Now I will switch my focus. I will start here. I will pick what has not been colored yet. I know that this order has no maximum element. So I can just send it somewhere above. It doesn't matter. So let's take an element here. For example, between the two. And I know that this is a dance in the ordering. So there is always a point between the two. So I can just select one. Now you should get the picture. I would pick a point here, here, there. It doesn't matter. And I can always find the point there. But I'll tell you, at every step I only find the main things. Which means I can always find a point that is relevant. And once I have done this iteration I would have to, you know, if I make sure I exhaust my domains. I've only counted many of them so I can enumerate them. At the end, I would have covered everything. So just continue counting. You can ask, why does this fail when these things are uncountable? Well, you won't be able to exhaust everything until you've counted the main steps. So once you have an infinity of things there, you no longer know that you can find a corresponding element of the other side. So there was this example I just gave. The reals have always uncountably making points between these things. So they are pretty wide, but the rational is only uncountably high. So maybe you've exhausted this uncountable infinity on one side and not on the other. So this is what it is. So this one, there is only one. There is only one. So I've given you two examples. So this one has vector spaces which has some remnants there. And the vector spaces have a nice dimension function that everyone behaved, so their number of models is very small. There's, you know, orders. Our example where it's very impossible to have a nice dimension function because there are many models you can have uncountably covered. So what does the main gap say? Say that it's the only thing that can happen. Either you have very few models before you have many, and it's impossible to have a nice dimension function. And that's of course not for all classes we can speak in. So let me write down the statements and then explain what classes we can use. So it's okay. Maximatized. Or this thing is actually very small. This works, but I guess it can be this way. So, maybe I'll state an interesting corollary of this. That was dominated by Vaskovsky. This function exactly, you can say what it is. You can give all the possible functions. Moreover, I can add for each one you can give. This kind of interesting corollary of this. Since you know exactly what those functions are, you can prove that this function is actually monotonic. So, I think it's easy because you just have some number of models and some small kind of level. You just expand that and you get more above. But it's not that easy to formulate this proof. It doesn't work. If it worked, you wouldn't get the counter examples of countable sizes, right? I've given you an example of vector spaces that has counter-beginning structures at countable levels, this is called Morley's spectrum. I think 1960's. That was proved of Vaskovsky and Vaskovsky's and like the... No, no, so what was it? When was it proven? Oh, the 1960's. This was projected. Yeah. And you might think, what is this ridiculous, the obscure function and I've just managed to compute it. What's the point? Actually, you can... All the machinery that you develop doing that there's actually many other applications to algebra and combinatorial set theory, number theory and all those things. So, there are, for example, those things more than bank conjecture and number theory. Lots of lights, I think, set of them. So, these are kind of test problems or test problems or many other problems just by the machinery that you developed at this talk. I will tell you a little bit about for you how to help you prove these things because if I haven't told you anything about the proofing, I've just given you examples and then the statement, but it's not obvious why this should be true. Yes, you can change the top right. Oh, yeah. Sure. Maybe I will say what I mean by the nice structure of these things. So, what is this? It tells you if you have... So, remember this structure theorem for a billion groups to write everything as a product. Well, it's really not going to be able to write everything as a product, but you can write everything as a tree of those things. So, I will draw a picture. You have a tree, a bunch of bottle, zero, but all those models are kind of free from each other. They don't each write. It has finite size. Every branch is finite. And M is, you know, minimal over these things, so it's generated by this nice tree. And all those models are small. Sides of M, beta is bounded back. So, you can decompose every model as a nice tree. Nice. So, the picture maybe doesn't convey it very well, but every model can be decomposed into a nice video product of things. So, the width and the height? So, the height are finite. And the width is... So, already by this analysis we can kind of bound the cardinal function, because you know, there can be at most two to the height of many models of this size, but since trees are nice, decomposition will be more or less unique. So, you can bound the number of models in this cardinal just by counting the number of trees. And concretely, what does this slow-growing mean? I'll just write it. Yes, slow-growing. You would think that it's not slow-growing at all, because there will be lots of explanations at this table. This is strictly less than Bf1. Alpha is the alpha cardinal. So, why does it slow-growing if you have so many explanations? Well, here I'm taking the alpha cardinal, so this could be much bigger than alpha. And I'm only taking an exponential of this of alpha, not alpha. This exponential is bounded by alpha. So, this is... It doesn't make much sense why it's slow-growing. But, you can show that for certain values of alpha, assuming reasonable set of reactions is not the most true, this kind of potential is strictly less than 2G. So, asymptotically, yeah, so if you look at my hypothesis, this is fine. I mean, do you need to come to your analysis? You don't need it, but it's easy. There are models where this thing is all... Oh, so it can be equal to 2G. Yeah, we can talk about that. Okay. This is why I'm saying it's making more important to remember there's a nice structure here to remember the exact problem of the cardinal. So, how do you prove this? So, this is where the dividing line part of the title is going to come in. Oh, before that, I said I'm working somewhere that is related, I'm working in a countable first-order set of axes of what does it mean. It's something next to resist y, next time y. So, you allow quantification, but only over elements. This is why it's first. So, you're not allowed to do something like for every sub-set of wheels, or something like this. You're not allowed to quantify over sets. You can only quantify over smaller elements. So, you're allowed to have independent things like this. So, for example, groups, you can express all the actions of groups in terms of distance. But, you cannot express the actions of our community of field, for example. So, to express, there exists a natural number n, such that x is less than 1. So, you cannot you can only quantify over elements in your thing. So, this would be in your universe. But, you cannot quantify over natural numbers. So, you cannot write something like this. So, the bottom line is for groups and in the orders, a simple object would work. Otherwise, it would not work. There are counter examples, but still, it encompasses a very wide variety. If it's, you could ask what's about accountability, many actions there is. So, what does it look like? What you just consider. So, you have many possible classes. For each class, I can want to prove something. So, this is the set of all classes. Structures that are first-order and splatizable in this sense. And, I will look at the dividing line. I will look at the property for which I can say something on both sides. So, I can say something if the property fails, if the class is bad, and I can say something if the class is good. So, of course the trick is to find such properties. But, here is one. It's called being stable. So, this has some definition that we give it. The bottom line is, you can show if the class is stable, it's pretty nice. You can analyze it further. If it's not stable, we directly have a method to make sure there are a number of models, and you get the maximum number of the other models. So, it's not possible to do something with all of them. But, you can do something more, and in advance another dividing line to continue the analysis. This is what we call super stable. So, again, if it's not super stable, it's still stable. The time that it's stable will give you extra information as we allow you to prove. Of course, stable is, again, what's better before you can continue. You can find more dividing lines until you've exhausted everything. So, actually five of them are pretty nice. What? Yeah, they are pretty nice. This is a meeting type order property. This is a conventional order property. This is just a word. It's digital. The opposite is shadow. So, if you don't have the auto, then if you're on the bad side of these things, you get many models. If you're on the good side of all these things, then, you know, it's so nice that you can just count everything. This is the idea. And I will just explain what stable is and what's anything about the other properties. But this is, I think, a very nice philosophical concept from the point of view. It's not often the case in mathematics that you do that. For example, if you look at a billion groups, with the property of being a billion, a billion groups are very nice when things commute. Even if you're not a billion, you can say a lot about the group. There's just no structure saying that it's impossible to be anything with it. This is the difference between dividing lines and regular property that one wants. That's right. Of course, it's harder to find them, but even finding them is an interesting question in themselves. Let me search more about stable than I understand. So essentially, what those dividing lines say is that the class has some nice object that can be defined into it. So stable would say that what unstable, to the bad side of stable would say that there is an ordering in it. There is actually a density of ordering inside, and this will allow us to imitate the proof that there are many models in this type of orders. And if there's no ordering, you know a lot about the class. So not super stable would say there is a tree, a weird tree that can be defined. So it's not quite but if there is no tree then you can continue further. Let me define what stable is. So the shell has this for naming properties such that the negation of the property is the good thing. So it's the interesting thing. So not stable is easier to define in stable in some sense. It's not stable if inside it means you can define it somehow. So by the way there is a formula phi xy and this meets the sequence so it's actually allowing me the formula defines an ordering on this sequence. Phi of ai a j holds if and only if i is less than j. So phi defines when ai is less than a j. So let's say k is orders then phi xy is just going to be x0 is less than y0. So you just look at sequences of one element and the order is just defined by the state. It can be m to be the rational for example ai to be and then this is just saying that the natural numbers are ordered by this so this means that it's not stable so it's bad, it's unstable so it has many models but I should say that there are no real examples of that so another example plus any graph that looks like a random graph a 90 sum so what are the axioms for these things? So the elements of your class are going to be graphs and there are going to be graphs that satisfy some opportunity meaning that no matter what planetary factor you have there are going to be graphs that contain every possible finite graph and everywhere which are useful so the axiom is going to say when you have your axioms are just allowed to say vertices x and y are adjacent and we write this by x0 and the axiom will say finite subsets that one axiom so I'm not quantifying over sets in my axioms I'm not counting in my axioms I'm counting in my sets like this so I had one axiom so I fixed s and I would say that no matter what vertices I choose there is a y that's connected to x0, xn-1 exactly like that specified so I have x i connected to y every i in s is y so it's saying I have a bunch of x i's in my graph I want to say oh there is an element that's connected to x0 and x1 but not x2 and for every possible pattern like this there is an element so any model of this theory is going to be like this and this is a class I'm going to study so it's a weird class and you might think it's a bunch of graphs but in order to add relation how could it have the order properties and how could it be unstable but it is so this is where you're going to use that you're allowed to have sequences of several elements x0 is at g0 and now I just want a picture to explain why this works so let's say you're in a very random graph how do you define an ordering with this formula you find some quadruple of vertices 0, x1 I'm just going to say specify there is an edge here there's no edge there so according to this relation x0, x1 is less than y0, y1 and it's a random graph so I can continue searching for this pattern I can find look at some other square like this I find a sub graph that looks like this and in this sub graph well I still have this relation but also y1, y2 will be less than c1, c2 but not the other way around if you reverse, if you swap these two things you can try that because there's no edge here I think I'm not going to work out so it's possible to define an order even in a random graph provided you allow multiple sequences so it's necessary to have multiple elements okay so I hope there wasn't too much so yeah I just find this result very strong and interesting I guess the smallest square in classifying all these spectrum functions what you're talking about what's the what's the n square I'm just curious about what kind of property what kind of dividing lines that are used to get the exact classification oh well actually you don't need that much more so once you have all those five dividing lines you can develop a lot of you know already yes if you are on the bad side of any of these dividing lines you have many models exactly it's just too to learn that but if you have all those properties everything is so nice you can develop a lot of theory you don't have to define any other dividing line you just have to analyze the examples so there's nothing more to do you just have to know the tedious work I mean I'm kind of lowering the work you should have the most of the work you just have to realize you know it's a hard one it's not there's no dividing lines I guess that's what I forget