Afternoon talk

0:00 That's what it's like. So you shouldn't be happy with that. There's no value to it. You should be happy that you have just two answers going all the way. So, that is quite remarkable. But to describe everything... Because you know. No, but you see, you have the scientific sense of our time. That's what you get. You know what you're trying to classify. You want to check that something isn't wrong. But that is another problem. But that's the kind of thing that you deal with when you want to do research. Well, yes, but that is another issue. So it might come to be easier. I mean, it's not a discussion whether there is having the explicit definition of different topologies. It's great to have it. So that's the kind of... But that is, you know, there are intrinsic algebraic... But you know that physics is a part of the local range. Yes. But when you want to compute the world of physics, you want to do it. Yeah, thanks. The fact is that you are able to obtain such an elegant description by pure algebra. I mean, algebra is playing a role. But I mean, really the theory is playing a role, so it's not a kind of general argument, you can easily adopt that, because you, it's something that you achieve only by working in a very sound way.

2:30 Well, you can test the thing at some general fees for the quotients of... This is again very easy because the covering things are exactly those which contain those number of quotients of A going to the quotient of A by the element Ai. The final collection of AI is such that the products of all of them are here. So, you know, it's very easy to say, yes, I'm going to grab the content, so I collected it, for example. So it's not the first time that it's going to be in a way. Yes. So I'm saying that it should be possible to obtain more of those. But, I mean, there is something specific there, which deals with the fact that your category is the category of rings, and that you can construct the portions, and the portions behave in a certain way, so I think that there is something algebraic and ring-corrected that results in the category of rings. But eventually it becomes the finite probability. Finite is just a probability. But really it has nothing to do with the fact that... Some algebraic categories are quick to guess. It has to do with the explicitity of the language. So, and the rings is not the only case for this question. It's a great job, isn't it? It takes NPR to prove it. I don't mean the rings in general. There's also representation theory in terms of machines of localness. They're not local things. There are some things in some parts of my career that should help you solve this kind of problem. But anyway, it's something you have to get, really.

5:00 It's not immediate, but that doesn't mean that I can't solve it. Well, anyway... I mean, you calculated something taking profit off of it. In a general case, you go to the elementary description. But in between, there's a whole space for asking for a number of very stringent conditions that might help you. It's an interesting question to find, you know, an intermediate class such that you have such descriptions, but even to find the common voice, I mean, for example, the thing for integration and for logic, on one hand, they have a similar structure, but... So, on one hand you obtain the quotient by principal ideas. In the other case, you are trying, you ingress an element. Yeah. But that's sort of the particular case that sort of accidentally you managed to do directly. Yes. No, no, no, no, but still. Using your techniques, you could cover that. But my impression is that the techniques should be applicable to similar problems or problems of the same kind. Notice that the problem I'm giving you, in this particular case of rain, is not really of the sort of asking for a mirror. It's just one of those coincidences. My son came to the house just while we were driving. Thank you for your attention.

7:30 They are very nice so far as I was able to get to hear them. They seem to be very personal. It is much more general, your idea of just thinking. Well, you're not sure you could, are you? What's essential for the structure? Is the way it's drawn. Can I prove it? One model of... I mean, what's the...

10:00 I'm just being utterly naive, because if that was the case, then I would see an obvious role for the construction that Bill was talking about last night, of taking co-limits of those which are kind of union of all topos, and that might do something for you, I don't know. Well, you're really not talking about numbers. I think the emitter factor in the infinite system should be topological meaningful with this, but the whole, I'm just thinking of taking, I think just taking us through.

12:30 The original motivation for his own going right back to notes at the time he did this stuff was tried in the early 70s for the definition of the natural number of J. Well, and I can see Mr. Asselin there, so that's why I'm happy to use this. Yes, as you say, discrete-state dynamical systems really structuralize this notion of going on and on. Well, yes, and in fact, that's something that's... No, we know, and the fact that the television number object is so natural in its structure, you know, Thomas probably is already telling you that, but it is too strong. But that's why I was wondering whether this sort of two-category of what I call relativized Thomas, and Thomas is, that Bill was talking about last night, but that's an absolute...

15:00 But do they have to be, I mean, does the top, does the structure of the, or in the case of the metal number object, could one not do some kind of analogy of that, again, in terms of, it would be all sorts of behavior on the coverings. But behavior might make quite a lot of sense. I don't know, that's one of the things I'm not sure about. Here's my answer to all these questions, if my questions will be set back in place. I really at some point wanted to get into the talk more generally about this whole notion

17:30 Across the back, of course, is a big one of my topology ideas, but then there's also this whole theory of equality, very important, you know. And the other is the notion of the way one system has a variation in its logic, which I guess is summarized in the book on training, where the notion of objects is just the absolute general notion of what is logic, and what we ask of what we are asking of what objects are, and that's why it suffices to say that variations are really useful. Well, that really abolishes the intuitive notion of variation on the other hand. And here, of course, that is in terms of quantum physics and its variations, where the quantities are just might not point. The cost of maths may not be one that might be much more general, even the notion, the appropriate notion of point. But I really have to understand that in, in, in, that's already obviously a very, very general topic.

20:00 And there are some things he said about the agency in that same paper which we were talking about a little bit last night on the way back. In fact, there's also a paper where he comes up for the first time with this generalization. It's two categories. It's a category with built-in co-ledgers in which top-I or sub-top-I are the kind of two-top-offs that relate, yeah, I guess that's more the idea of that kind of category. With that society, you just have the discounted co-ledger statement. At the end of that statement, he has the full section on the economy. But he says he's now reinforced his statement. He thinks that he's made his own thoughts about that, I'm not going to go into any of that, but he says all sorts of interesting things about the reasons for all of these domains, which he termed it by, and they can't do until, I'm sure they can't do until it's gone, particularly with some kind of action in them, or it might even be a group of different domains. So the G sets are the instances, but that's just one very, very important thought that's been decided here. There are two topological characters of the main. I think that what I understand is that it can be a space locally, but not necessarily globally, and I see it a bit vice versa, I think, just that the question side of the topic is the case within.

22:30 These do behave like sets, domains of variation, these do behave like sets, but the others don't, and they're precluded from being sets, precisely because of this kind of internal variation. It's got nothing to do with size, but they are precluded for reasons to do with the non-existence of the global... I just always thought that was actually fascinating in terms of understanding what he sees as fitting with geometry and it's something I always wanted to understand in both conceptually firmer and I can see it's not top of the race when he's detailed. Thank you for your attention.

25:00 You said something very interesting, which was that you and John were occasionally a block ahead of the issue of syntax. So, John, can you please let me know if there are some questions? Thank you very much for your time, and I hope to see you again soon. You know, and his paper on the English. Oh, that's it. There's a lot of his papers. I've seen a lot of people publishing his parts in the British Journal of Philosophy of Science. Oh, okay. And there's a big article that you've probably seen two years earlier. It's a piece about that. That's why I find it useful. Right. Let them be white. But it's a good start. Thank you for your attention. You have done over all four, which is one of the lowest conceivable operations of doing mathematics.

27:30 I will say, okay, I'll write something down. I've been doing our archival work, and it's very nice. I've had to exercise to actually have the balls to make that. Really? I would be happy. There you go. You can do very well in this reception. I mean, I just can't stand stuff like this, it's pathetic. I mean, any car I can afford, you can drive around like any old chameleon in a car, right? So, instead, I drive around in a disreputable one. It was that old git, because I've got terrible fits of road rage because of this angry old guy in it. It doesn't matter that they're hard. More people should have scattered with this. If you just carry on, where were you left off? I think it's for you. I'm sorry, but Richard, didn't you have something? What was it that you were saying to Bill in the pub? I'm afraid I wasn't able to capture everything.

30:00 It's time to do something that other people might also jump to. Maybe we should leave that. Because I'm looking to see it tomorrow. There are a set of things to see. Or I could talk to Bill. I don't know, because we have this thing that we have to perform actual calculations, so we would need some time to do that, and it's not something, you know, interesting to show to others, it's just the technical calculations that we have to perform. It involves the incompletion of the natural numbers. It's very big. In the finish. I don't think it's a... It's too much. It's too much. It's too much. It's too much. It's too much. It's too much. It's too much. It's too much. It's too much. It's too much. It's too much. It's too much. It's too much. It's too much. There's a second natural number problem, because the one defined by universal maturity does not have the least number of properties, in the sense that an arbitrary sub-object of it may not be representable by element.

32:30 Isn't this a consequence of the fact that the overall logic is essentially constructed, sort of global logic, the intrinsic global logic? I mean, that's a phenomenon that occurs in intuitionistic logic as well. There's no, there's, there's reduction holes and recursion holes, but there's no snuffer. If you chew over a billion ounces, you also, you have to move. It's just the fact that you have variation, relief, connected with DNA. I think the semi, the semi-continuous, it's a semi-continuous, It's bold. It's convex. It's bulging, trying to accommodate all these ideas you're forming. It has space. It can restore all that. Yep, there he is. Now, there is a topos of sheaves, and if you're given any topos of x that goes through the cells, then you can consider geometric workings from x to sheaves on W.

35:00 So, these deserve to be called continuous tons of gathered functions, even though x is not necessarily a self-spatial function. In general, what happens is that the continuous math values in a locality code course like this, in a general code course, they all factor through the locality reflection, doing a sub-category that has a core operator that's the part that's generated by the truth values. The sum objects in one are the code codes for the truth values. And just take direct elements of those, ignoring all the other objects, and carve out the SOC category, which is the construction generated by some objects that are not in the locality. This is the reaction of the metric worker who extracts the locality part of any object. I'm keen on modeling devices myself, and I think that the co-reflection is formally the right adjoint to the inclusion function. We could really just be called the core, extracts the core, different kind of core of an object. So it's a sort of core of an object that lives on its logic, logical core. Okay, so, you know, particularly, also in any topology, we can make certain constructions.

37:30 So the basic observation is that for things like this, the top sub s, I mean the category of all the x, can be achieved on the ordinary real number, two sided fields as built in sets, but that corresponds to the internal real number. So this is sort of a general picture. Given any particular topological space, then in any golden deep tokos, the picture of that space seems like that tokos. So the same thing for the semecus in these fields. And the same thing for truth value. The simplest case is when you bring the category of all of your mentioned morphisms into the Gs on the Sapinski space, which turns out to be just the morphisms on the sets, because for that three-point space, the sheath condition just tells you that the N-sheath should have value 1 at the empty set. It doesn't tell you anything else. You have two sets.

40:00 The global norm and that favorite open set and restriction now. Let me extract that. Oh, that is the truth value object of the topo. We'll point this again, putting in the variation. The truth value object of the topo is just that representor for the structure of the science. Here's the truth value. So really the internal truth value is strongly related to the external truth value. What about the natural numbers? If you take the order of the natural numbers as a discrete set, well then of course the cheese on that is just a big product of copies of sets. The same sort of thing is true, but it's just given, if you take the ordered set of numbers, and it was not in a topo, it's just an S, and consider there's a topological space. These points are the points of the natural numbers, and the topology is intervals. The topology is derived as the interval topology from the ordering. Again, one-sided ordered topology. So the open sets are co-components. If you take the sheets on that, again, one can compute that exactly. These are sequences, a little omega, as the ordered set, that satisfy a semi-continuity condition.

42:30 What Hobart said is an upper array, but the one with the covers, the strict ordering covers, the actual, the semi-continuity, I mean, again, as I learned in that gradebook, they don't tell us topology. The seven containers just means continuous with respect to the topology and the coding. So this is something extremely specific. Non-autonomous mathematical systems with the speed of time predict what's going to happen next. But the whole history is, they have similar techniques. And so, the thing is, should we internally do any of those and try to calculate what this means? Sorry, just a minute, let me just say, does that mean it's determined by a finite stretch of the mathematical state? You get the value of the function by just looking at arguments. Thank you for your attention. No, but I'm just trying to think. I'm trying to remember the second one. I think the semi-continuity is this. Suppose those who have a future history, no apologies to those I like, those who have a future history, i.e., at hand, for everything that comes after hand.

45:00 We've got to be, we've chosen a point, and what the state will be, they have a string of the dates. They satisfy the condition that under the transition, those states are allowed to choose. Then they're existing in each state at this stage, which can rise to all of them. So it's already determined like that? Yeah. I think that's, is that sort of the thing? I hope I'm not going in the right direction. Does that work? Definitely works. Yeah, but it's a non-trivial condition. I think what you were saying, I think it's reversible. It's not a full appreciative. What is this? This is representing something inside the totals, which is different from the natural numbers. It's not at all. The natural numbers are the ones that satisfy the universal truth. It is embedded in this as the opposite of the order-conveying maps, which are semi-continuous.

47:30 Semi-continuity is a very simple case of a boolean deep psychology. It's a topology where every object in the site has a smallest covering. Well, covering is nothing else but just any sub-object that contains that one sub-object. So that's the way of semi-continuity because the representable function is to every element instead of all elements less than or equal to it, let's say. There's a preferred sub-object of that, sub-object of that, mainly strictly, everything is strictly less. So that's the main thing that's dancing, is the way that it is, of one side of it. So I can impose that condition on an order-preserving map of m to omega. The thing is that the ordinary, the discrete natural number object has an order of m, but it's not, it doesn't have m. On the other hand, this does, and it's the smallest one, particularly as the empty infuses the greatest elements, or attacks the greatest elements, on the top as a bipod, and avoid that, but it's sort of natural to have the top element there for things, in other words, a well-ordered set is one that's complete and, moreover, has the property of those things that are presentable, but elements.

50:00 In terms of numbers, I thought of this realizing you actually need it. A lot of things you do with natural numbers. There's a method in combinatorics where you embed things into real continuous numbers. You've got these counts, but then you view the counts as the special real numbers and form real algebra. That means in particular that you're getting the truth values. The truth values themselves are not embedded here. They're not, not, not embedded. They do, but they aren't embedded here in this construction. Just like the sets are embedded in every piece of the omega that, let's say, the omega part of each cell. So this embedding is the first step of all those common core arguments. There are further events, some of which are real and some of which are not, but you have to get started by filling in between the natural numbers the truth value objects worth, what I remember now, the second natural number. I mean, I motivated it by working with Recep and so forth, but in the end it's a purely internal construction. The semi-construction that appeared in the talk by Davide, which took the rational study, when embedded, the objective really seems to appear quite different from how it might have been written on a computer.

52:30 Yeah, yeah, yeah, right. I mean, the dedicate, you know, it's one side of everything. I'm aligning with the fact that something apparently discrete, like the natural numbers, this is very different. These numbers, yeah, not in here, but here. The nth of a set is the number that represents the set. Of course, it's a further condition that that nth should actually be a member of the set we started with. You mentioned the possible application of certain variable linear algebra over algebraical complex analytic spaces, because dimension of vector space is a semi-continuous function. Or besides, there is a good module in the ring topos, and for each x in the topos there should be a mark, the max to the semi-continuous natural numbers, which is the fiberwise dimension of that. That's another application of the same thing. By a good module, I mean something like a vector bundle and its function. For an electrostate object, then the category, then the combos, and things over x. That is fine. It's a change of step. Fine, I guess, but it's just that each fiber, the electrostate, in a lower sense, it still has a dimension, in a fluid sense, but dimensioning, this is well known in different branches of geometry, the dimensioning of any semi-continuity.

55:00 And so on and so forth. Or maybe up. Whichever. And that will tell you kind of the semi-continuity. I think the two kinds make a difference to the natural one. The third thing, going down, seems to be the opposite of the trivial kind of thing. It's a definite direction, it's relevant. Whereas by contrast to the real, it's both the up and the lower some of the values. But it seems that the integral, the one that's the integral, is what's relevant to, for example, metric space. Typically the distance between the house or distance between subsets of the metric, the infimum of distances between points in the two sets.

57:30 Even in a Riemann and Manifold long path, you take all possible paths, which are geodesic paths between two points, they have a value for smooth reasons. But now you quantify them, and that's the end that pops out of the ordinary wheel. Of course, going down in the natural universe is not entirely true. No, I know it's not entirely true. The complexity of functions going down is connected with various kinds of internally representable trans-finite reductions. The implication is that it permits, unlike the standard natural numbers, the use of standard method of finiteness Not only 1 plus 1 equals 1, but instead in a way in which natural numbers of this result in generalized characteristic functions are added, multiplied in, say, according to a machine as a quadratic analysis and then translated back into the universe of the original final structure. Right, so that's it. The point here is you have this map that's sort of 20 units and all that, but when you study posets and everything, it's 2,000 units in a special plan, including real data. If you think of it in terms of the lowest category, the semi-continuous real plus is the lowest category, and the centric space is down similarly.

1:00:00 Omega is the closed category in the dictionary because it's a co-set object. But then the third line, the K-2, goes out of the natural number. So again, it goes to the closed category. You know, the study of, I always like to look at this in maturoids, and I see maturoids are actually also the new category, and valued in this closed category, more about it being carefully tested and used. Namely, in a Metroid, not in a Metroid, you have between any two elements you have a rank, which roughly tells you how much you have to work to get to one, and that the rank, that's what you have to do. I think you're recalculating the simplest, non-trivial case. Sorry. I don't know. Yeah, oh yeah. But maturoids are usually described by McLean, Stein, and it's interesting how that translates into the properties of this composition for each category.

1:02:30 And in the case of sets, that's different, right? Yeah, right. In the case of sets, there is this... You see, that's the one you have to be careful of the direction. The illusion of the up and the down requires a minus figure. Besides, you don't want to know the minus figure to calculate the initial object. Whereas in the one that's the illusion of the metric space, the wheels are ordered backwards, so the infinity is the initial object. There are examples where you have stuff in between zero and minus, but I'm saying it's just one more element.

1:05:00 There may be an analysis that you can't have minus infinity in the same system, but you can if you realize that if there's a column, a, b, and there's an h-star attention group, which are often equal, the case is that infinity plus minus infinity is definitely one or the other. Whereas infinity minus infinity is a binary object with different values. But in all those examples, semi-continuities, it should have, you know, an example of how many applications you have on dimensions and vectors, and in general, if you're doing two-value matrices, you're going to increase the number, and so on. It strikes me as odd and awful, because the things that are, are fixed and rigid.

1:07:30 There's something really odd about this. I mean, I'm just trying to think why you would call those natural numbers a sort of series, whether that natural number obviously is going to be a big thing, right, at least, depending on the topic, it seems to me. At least on the last part, as I say, in some sense it really only lives on the part that's totally not true, but the true value of an abstract says that it's giving out these new metrics. But yeah, I mean, depending on that, it's like, you know, the Stone-Weir-Strauss type, you recover a suitable kind of topological space from its grave with its new structures. Well, there's a little piece of actual mathematics involved in that, but actually it's derived from something relatively trivial, which is if you rig a semi-continuous space, it's sort of very easy to recover from space. Why? Because open sensors are just things that are additively added to your input. Just within that, you get back the lattice. And so on and on and on and on and on and on and on and on and on and on and on and on and on

1:10:00 Simple semi-continuous topology, but it's already a new language, plus other than natural numbers, it's semi-continuous. So then the thing we were getting there is actually the ring, the rig of semi-continuous natural number values. Many kinds of spaces you can recover from that. That's why it's extremely variable, it's as variable as space, it could be different kinds of spaces. And within each space there's a lot of... Sometimes I maintain philosophically that all this constant stuff, which we think of classically as mathematics, is really derived from things that the real world is really are. Because of the fact that it is so derived, it's actually connected up with it. So you can go back and see the sort of double negation, how the things are varying. It's not that it's just constant and sits there dead for all eternity. I must say that I brought this up here because the whole thread started with the question, Very good. What are the Kwiatkowski's suggestions? It was marked with a smaller subset of its power set.

1:12:30 It sounds like a definition of something harsh. He wrote a paper on finite stacks. No, I don't. It's a property on the power set. The smallest part of the power set that's closed under a finite union, a binary union. Definitions of finite are always there with a double negation of two. Binary unions, and then you have the smallest part that's closed under that and contains all the same things. So if that's, if that, maybe it's something, if the whole set itself is a member of that. So, I think it was maybe something, let's see, well this work, if you take a linear subordination of the power center, then is that, just saying I had a large belt in other things, everything is the same. I'm not saying that's what the Sierpinski's definition is. I'm saying I think that's what the Tarski's definition is. I don't know. I don't work. I'm just trying to think. I'm thinking a bit groggy. I think it works. Something that's simply expressible. It turns out to be equivalent to all of them. That there exists a natural number. The interval in the natural number comes through to that as another object.

1:15:00 And that there exists a morphic math there. Yeah, but I mean that's the counting definition that we all know about isn't it? The analysis, within the counting definition, you're not saying it's an isomorphic. Right, you're just saying it's an entity. Yeah, a big difference. Because you can count sort of... You can count beyond it, somehow. I've got more than enough counters. But there's no way of counting the structure. The thing has some kind of structure, you see. To prevent you from having a one-to-one count. Yeah. If you count them all, then it's finite. But the structure may prevent you from finding the simple. That's the equivalent to the unit. It's the smallest thing contained in a single unit. Well, I think that the thing itself, which is an element of the fellowship, in that smallest thing. To say that the fellowship itself has this property would mean a lot more. Of course, that would certainly apply. But I think it's, in general, as I recall, very tough. And so there's a lot of papers on topos theory and related matters. Just take that as a definition of finance. From a logical point of view, it's sort of analogous to an extension class of things, but from a geometric point of view, it's completely ridiculous because even the simplest use of natural numbers here, the semi-continuity of the vector model is not coming under that. Because an obvious failing here is it does not follow a sub-object as you find that in the primate.

1:17:30 Because if you take a given map, pull it back, then you get a subjection, but it's a proper sub-object to the integral, a proper sub-object to the integral, or most likely not another integral, or even isomorphic to one. And so, why is not finding it easier to do that? So that seems to be something that's sort of left out of the whole theory. It's self-circumstance. But probably that's too general. There's some subtle thing that corresponds to the ordinary mathematical usage. There's a lot of mathematical usage of these things. You're talking about vector bundles and the like. A lot of the fact that in the union category of the vector bundles, you've got chain conditions. The upper and lower chain conditions are more closely related to finiteness, especially for the relevant finiteness in this definition. So it's really something that Neon and Topo's theorists should do to figure out the internal concept which is equivalent to, or involves hundreds of times a day, Complex analysis, kind of extended. Coherent sheaves. Coherent sheaves exactly. Finiteness conditions. Sayer and Schauer. These are things that arose out of the geometry. All about finiteness. They probably, you know, very rarely even think of terms as a sort of universal measure for that. I was wondering if there is some characterization of coherent total systems among all total systems, apart from, you know, there is the well-known which is that it's coherent, you can graphify it to be a theory, but that is not, you know, a geometrical characterization. I was wondering if...

1:20:00 It's more about the objects. The objects? The objects. Never heard anything of that, but I don't know. Did you say? No, there's the one that you wrote about. Maybe you called it Sukeski or Kurakoski. Kurakoski, Feynman. Kurakoski, Feynman. So... Maybe that's a very foolish... Okay, so... That's the theory of the kind of analogy, not the one that's relevant to... Yes, no, no, it's not, yes. So... There are a few contributor types as far as you would know. So all the objects are the same? No, no, no, no, no. Some of them. There might be. There might be, okay. So do you have a characterization for your total distribution? No, that's what I'm saying. Okay. There are two even topologies that are all the same. One can also talk about two even topologies in the same space. So in that case, that is the notion of quantum mechanics. I don't know. I mean, you're considering . You can have module . Thank you. So, in the constant world, yes, we have an idea of the dimension of this. The modular project is fundamental. How will we measure this dimension? How will we decide upon that dimension?

1:22:30 The dimensioning will not be a very natural number, but it varies from point to point. Yes, and you may think... On SGA-4, the definition of coherent is coherent telcos or coherent sites. These are things, again, of mobilization, the notion of coherent objects, coherent objects, and it's basically like covering conditions, where we cover these finite subsets, again, and then in Robach's key style, the Hausdorff axis. Just saying that the diagonals are compact, quasi-compact, quasi-compact, quasi-separated, quasi-compact, quasi-separated, quasi-compact, quasi-separated, quasi-compact, quasi-separated, quasi-compact, quasi-separated, quasi-compact, quasi-compact, quasi-separated, quasi-compact, quasi-compact, quasi-separated, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi-compact, quasi- A tower of concepts for more and more universities and areas in terms of a single idea theory. And that's pretty much the original category of the issue. I mean, I have to suspend it. I mean, I'm very aware of this mistake. Stay out of that sort of mess until the next year or so. But, bear in mind... That thing is not what's being measured by the natural number up there. It's the contiguous number that we're measuring, and then something else.

1:25:00 Well, except, you know, the gauge. You say that the vector space has finite dimensions. How do we say that? We usually say there exists a finite basis. So, first you have the whole thing about basis, and then a big existential problem. There's a lot involved in that. The traditional way around this type of problem is that the chain conditions actually deploys the conventional method of finding those steps. So the chains, the chains, the chains, that's sort of like, there's some word for serre about what Aaron Sheeves probably stated, chain conditions. Yes, after all, it will be properly stated, but the amazing thing to me is that correct topological objects used by geologists many times found it very difficult. What is the chain function? It says that all chains terminate. What is terminating? There exists a place, so they always put in. What is covering the terminating? They know that permitting is not just a place of covering, but topological. If there exists, if there exists a covering, what really should be really something really important.

1:27:30 That's the basic, that's the fundamental clause of what the attribution is to the topology. The point you made the other day about systems being a covering. Yeah, right. Global statements do not mean they're not being real global statements. But what I'm saying is that if you're stating this change condition, they already knew that. They already realized that film is supposed to be terminated on each element. It may not be terminated at all globally. In many cases like this, Huzel's definition of a locally convex, morphological algorithm is a kind of community-rounded morphology. And again, the question of being both in the context is that it's more multiplicative, excuse me, multiplicative in the context. And again, I mean, roughly speaking, that between certain things there exists something else. So this existence is, again, a covering, and automatically they think that's what they have to do. And I always find this, I mean, it shows that the Chokos logic somehow really... It's very striking, the dedication definition of the law. How did you arrive at this two-sided, appears to neglect me, self-rehearsing definition of the law? It's really discreet. It's quite straightforward. What he was getting at was that there are no gaps in the line. So you can't split the numbers into a left side and a right side without having one of them. Either a biggest left side number or the smallest right side number.

1:30:00 So you can think of it as just being about saying that two lines are everything. But you've got to think that the line is composed of the side of it is to get back to the short of the rap. Well, yeah. Yeah, that's what he was thinking. Yeah, that's how he was thinking. Well, that was from a bit of a game. It was. I think that was something slightly different. I think there's something very interesting about this. I mean, how they are. I'm not trying to recall what he was. But historically, how, there again, was driven. I mean, to some extent, against his own... Well, my idea, again, on some fragmentary correspondence with somebody you probably know, the idea was, as I said, that he, basic construction, he started with the line, and then he made this analysis of the path throughout the line, and then somebody told him, oh, well, you can use that as a definition, call it a construction. He hesitated slightly, and then went ahead and did that. It's not in his monograph, he says, I mean I think it is, he's thinking geometrically, he says that there's, he's put them into two sets like this, and, you know, I think he's talking about the rationals, if he's put the rationals into two sets like this, and they don't, not neither a greatest on the left or a smallest on the right, then the thing that's separating these two sets of rationals must be real. Anyway, I've got to get moving. So, I've got your number. I'll be back Sunday night, probably. Maybe I'll give you a bus. Do you go to bed early? How about tomorrow night? I don't want to ring you at 10 o'clock and find out you've been in bed for half an hour. You've only been in bed your whole life. Okay, so if I read you when I get back from Oxford and arrange to meet you on Monday, I'm going to Oxford to where my son is going, and then we're going out to the countryside outside, and then coming back on Monday.

1:32:30 I'm going to bring you and see what's happening on Monday. That's the preparation we're going to do on Monday. And that's it. Do you stay? How long are you going to be here? I'm thinking Tuesday as well, and then... Bill, you're leaving on Wednesday. No, Tuesday. Oh, okay. Then we're all three leaving. Unfortunately... You're leaving now. No, I thought you were leaving tomorrow. No, not now, but tomorrow. Okay, but I won't see you, because tomorrow I'll be on the road... So, it's been really nice meeting you. Yeah, thank you. Well, thank you for coming. It was really nice. Thank you. So, just stay on for a second. Is there a session tomorrow? The subordinate phases of one great and continuous engagement. This is a quotation. From what is one a space? What sort of space? You're holding too much shuttle. Apology of a... What are you thinking? So you haven't split it. You haven't split it. I'm thinking about you. Yeah, about you. To put all the papers together. Okay, well I'll see you on Monday then. Yeah. Thank you for your attention.

1:35:00 I'm more than happy to. I said I would go out, but I had a mishap at that point. I mean, from my selfish point of view, it would be very helpful because I may need to use the computer again. I know. I think so. It's a really good point. I've got to. I'm very enthusiastic about it. We're going at about 12 o'clock. Thank you for your attention. Can I ask you again about the suggestion that you gave last evening about filtrate? That's something I wanted to ask you as well. Yeah, let's write down your projectors. Two categories, no. No, but that's what it is. We just got one. Two categories suck.

1:37:30 I've been trying to remember the exact quotation, but I think it's from 1973. No, me too. But the fact is that, how can you study topology without an adult? I mean, they look at the curriculum. There are only toposies, yeah, but it's still... Yes, yes. Yes, please. I may get it slightly... Thank you for your attention. The left or the right? Precious aren't the finite sets. Is there a toposis? Calibrate models?

1:40:00 There's all sets. What's the reason? We're just a big world here. It may not be the opposite of what we could be. Ah, in this particular case, you mean. The left side is the points of one. The right side... Well, what about decidable sets? Decidable sets is, yes, because decidability forces the maps between models to be monopartisan. And then the models just have sets. So yes, you have a theory of decidable sets, decidable objects. So, on finite sets and in directions. So that's three shifts of one. You've changed the variance. Yes, sorry. It lies within the geometric... In the subtopos of the... Is that right? Yes, yes, yes. No, no, no, no, no, no. You're giving a little bit of a card. Actually, that draft from that window is probably not helping.

1:42:30 Let me see if I can climb up there and close it. If I have, unfortunately, no. Sometimes I take them with me, but I'm not going to do that. I may have some little sweets with a needle, a small needle, but I don't know. It might be helpful. I always use a syrup, which is very festive. I bought it in England when I was inter-desperate with Michael, and it's called bronch, something like bronchitis. Yeah, I know it is. I can't remember its name, but yes, it's very powerful. It's very powerful. It's very powerful, yeah. I mean, pharmacists should be open for us in the Sacré-Cœur. Certainly in the morning, at least. And because they change your life, actually. They change, yes. That's what I mean. So the left extension is defined as a poly. So basically the left extension is the left adjoint to the... Composition between the functions, okay, it adapts well to our case because we want to build our geometric one.

1:45:00 It looks like we have a geometric inclusion from the spin of the quotient to the theory of the equation. Then, then... Sorry, I just wish I could make a hot cup of tea for you. Well, you can't. The only way you can do that is if you can't come down. But we can't go upstairs to talk to you about something like that. Last advances. It's already a quarter to six. We're just coming up to twenty to six. Well I will have to stay here because I have to, once we're out of this place, we can't get back in, and I've got to stay here to work on Richard's computer. Okay, sometimes. So I could come on and join you later if you don't mind. It works perfectly. It's the left one which works, because we have inclusion and the left are joined to the inclusion. So it's a kind of extension, yes, and... There are a number of preserving factors that are determined by the value of the caliber sets, the caliber infinite sets, and sort of, although among the caliber infinite sets, sort of, you're in a finite intersection, even though the finite intersections are not there, they are determined by certain completions of the caliber set, so somehow that is an installation that can be preserved.

1:47:30 I think we have, we are at a very good point because we have built the sum of all of our original sources, so it should correspond to a quotient, then this quotient must... In some way be our original one, at least in the case of enough points. So I think we've got something. Yes, we'll remember whether or not points actually need to verify the equations. Yes, yes, yes, yes, yes. We should be satisfied because, yes, of course we need to check that. The worry that the base was, you know, the models were too close to the base, Just one quick question, because I, as I said, won't have to stay here, I don't know how long. What's the definition of the topos of the signable set? Ah, it's the classifying topos for the theory of decidable objects. That is the pre-shift topos on the category, you know, the category of signs. Because that is a pre-shift type. You don't lose your being of pre-shift. So, for example, if you have linear order, you have the decidable. And in that case, in order to get the geometric inclusion, so since the direct image is the fusion of...

1:50:00 And so we have to prove that actually the inverse image is like this. So that is another thing to do. But then we should be okay. If everything works, then we have a subtopus and that is very likely to work. By the sufficiency of models? If we have enough models, yes. No, because in general, if you don't have enough models, definitely, that doesn't work, because you can't have something that makes only, that depends only on what happens instead, you have, because that doesn't, doesn't return.

1:52:30 Yeah. But, well, we can't be sure, but... You have to transfer it with the general thing. I'm not sure. Central, of course. And so you're going to have dinner at the usual place, what, at 6.30 or something? Well, just say it to me. As you like. Or do you want to go straight there? Because I may not be able to join you. I really think I simply have to finish what I'm doing. Ah, okay. I might not finish for almost an hour. So if you're going to, I mean, why don't you go ahead and, especially since Bill's, you know, should get back this. Well, yes. Okay. Assume I'm not going to come. Although I'm only going to come and join us.