Discussions Following Geometric Structures in Logic, Physics & Computer Science

0:00 There is something like that. There is something like that. Excellent. Yes, it's happening. There is. There is an idea. So, we have to stop. The structure of physics never explained to me why there is a stupid theorem of the Gaussian values. But we are able to find it. You might put that I have to show up. I have become a layer of classes that I have to tell you about. When you remark to distinguish computer science from mathematics, it strikes me your notion of a primate and a computer science. The real difference here is my primate. It's a very primitive primate. It keeps learning nothing. They have to talk to this primate, the computer scientist, and apparently it's not making any progress. And I think it's quite a notion of proof and of logic.

27:30 You still need a notion of, you know, in a very loose generic sense not to mean the natural numbers as a set or as a... How are they enjoying, how are they enjoying your retirement? That's why I can be here for a bit. Well, you also have the tonic of mathematics, you never will. That's not the main purpose of retiring. That's not true. It seems to be paying off very well. Quick question before I go. Do you know a man, an academician, Karl Skrona in Sweden called, I think, Maurice Goossens? I don't know him. Oh, you don't know him? No. Karl Skrona has no university, so I... What, he lives in... Ah, well, no, from the story I heard, he lives in Karl Skrona, so he probably teaches and stuff there. Oh, okay. Karl Skrona is... I have a doubt that he's pretty far from... Okay, well in that case there's no point in pursuing the line of questions. When the question came up concerning what Alan Cohn was saying about his collaboration with Revelli,

30:00 Who is, I think, a very serious, and he seems to be very normal by the way. Well, I've read quite a lot of his papers recently. He gave a talk in Oxford about, I think, earlier this year, in February, where I thought he had some very essential things to say about the problem of defining the end product. What the conceptual problem, what the really basic conceptual difference is. It put me in a position to work quite a lot with Chris. I do understand that having to be dense and fruitful helps correspondence with Chris. It's got some sort of neurological disorder, which they have not been able to identify, which they thought at one time was multiple sclerosis, which apparently is not, and they haven't been able to identify it. But it's not come out very bad in the last three or four months. But I was hoping at some stage that we could finally get you fixed up with properties. He was somebody that you would very much enjoy meeting because he's a very, very clever man. He certainly has thought very deeply about the problems of mathematics. There is physics where the foundation is clearly giving a great deal of conceptual application. Well, I have to say, since we are here, I would say that the discussion is out by two persons, and I mean, it could have been twice as much, but it's beginning to build into it. It's a long one, in particular, because I've had a good discussion. We could have had a good discussion. Yes, yes, we agree. Well, I thought that the general introduction to K-marking, which the channel described, that it was exactly right.

32:30 Yeah, right, yeah, that's what I had a lot of, but I thought they may not have been particularly deep, but they were certainly very... I find myself in very strong agreement with what he said about there is this fundamental problem. We do not have a framework in which to put together our ideas about number and space and the things started to come apart around 1854 and there have been these two very powerful lines of development of mathematics ever since and here we now increasingly see indications of how they might be brought back again. Now let's talk about what has been said in this conference around the line of that. And the discussion began well on those lines, and then as I said, unfortunately rail railing knocked off the rails, mainly because of the loss of gas and all of this stuff. I think that to be honest, at least the objective idealism was coming for an absolute if those rail lines were efficient. To clearly ask people to come up with mathematical ideas, we're going to have to understand better, and we're going to have to understand why they've misled us into the objective of the course. This is a question that I wanted to answer you. I'm observing him. You know, there's questions. If you ask what does it continue to match, well, the answer at Connes is nonsense, but it changes the homomorphism all the time. I mean, most of these algorithms wouldn't be safe if they didn't even have any homomorphism. Well, I put that to him. Well, it was because this was actually Marika in Puebla. Wow! This is what we thought it was. He was very evasive. Exactly. I mean, just trying to... Actually, I confused him. Can we construct a less evasive answer? But they told him privately. Yeah, that was extremely evasive. Thank you for your attention.

35:00 There are all those homomorphisms. There are bimodules which are, for every epsilon, almost a homomorphism. Now, this is starting to make some sense, but that was the first thing you said. I mean, I still don't understand. I still don't know. There is finally, once again, I have to agree upon your work and I read about it on my book. Okay, fine, now I know what to look for in the book. Thanks. The Bible, after all, is my own work. I don't believe, anyway, that this is the right notion for... Because the main example used for a non-communicable problem is simply by an equivalent equation for the age of an operator. When the two points are based on the number, you take the equivalent of the age of an operator. Identify the two points that are based on the number where you make the equation. The equation is that one point is based on... We know that, but if that is so, we have to take a real model and put maps on top of it, at least it's good if the picture we gave of our, say, career. I always assumed it was something related to that. I didn't really talk about it, by the way. You said something about, well, topos theory is an example of that, not conversely. Yeah, yes, yeah. I think he wants to say something. Yes, I didn't understand it exactly. It's really different. They never really look at it together.

37:30 I think that's probably not for me. You say so. As I've said many times, I think that this non-communitivity is very artificial in all those types of times. That's like you said, there are two approaches. You're identifying the points where you don't know anything. It leaves out really two theories. Well, yes, yes, it leaves out all the machinery. So I think if you, this is really an example we should work out and publish. If you look at the, there's a forest, right out by the Irradiable Theta, you get a Nézandu, I think, so it's a toko, which is commutative in that sense, but there should be a sheet of commutative range in there, which actually contains all the analysis. In some sense, the original ring of conclusion is the real number, or even the rule. Thank you for your attention. The generalization of the quotient space is contained in that string trope. Well, as he thinks the generalization of the quotient space, the quote geometric meaning of the construction, he sort of sees it as coming out of this algebra of functions, which is somehow something that's, you know, in itself prior to his... Yeah, yeah, I think that's why, of course, he was only able to follow a fraction of what he said. I had that impression that for all that he... It advertises this work as non-competitive geometry, giving more powerful insights into generalised geometry. It's really an exercise in abstract algebra which hasn't shown that it does have great dynamics. Theoretically, it's not true at all.

40:00 Yeah, yeah. He wants to just present a general face. So, for example, if she's running an ordinary topological space, and this is best thought of in a non-communal way, then we can use the steps in the basically of Mitchell, and looking at the open steps as indices, so that you can look at matrices, so that you can look at matrices. Well, of course it's non-community, but that has sort of grown out, so you get a faithful representation of the huge non-community of the brain, into, as large or small, this huge non-community of the brain, because it is very uneconomical or ungeometrically constructed. Yeah, ungeometrically, it's the huge non-community of the brain, which he thinks is basic. His algebraic function is separated from any of the guiding geometrically constructed. I feel it's been stated that you're presented with you by an italic keyboard, and this coupling by open sets is precisely presenting the manifold by an italic keyboard in the equivalence relation, where you develop the rest of the side. I think that's a special case. And in fact, there's two. Two to one. An example is representative enough that we can understand them, or it's just a question of matrices. Or maybe it's slightly more subtle as the representation of a very reflective draft, which is broken to the people of that logic today. It's useful. In that sense, it's the algebraic set value of something over a perfect community, in a way.

42:30 Well, thanks to Anderson for that, and once again to having helped me to understand what he can't do, what he can't do, what he can't do, what he can't do, what he can't do, what he can't do. I can also talk about philosophy, but I didn't think I'd be trying to be involved in the interaction that we collect and send to individual people. How can you see yourself as this individual? Well, actually, Tetsune's account was better than Kahn's, because Tetsune never said that it was all in one brain. Because he was constantly talking about convincing your colleagues. Yes, yes. Convincing your colleague. Who did that? Tessier. Ah, okay. He was convincing another scientist. So the key ingredient was the collectivism. Yeah, that's right. Which, I think the language lasts very much on the line of the individual and the physical world. The physical world by emotions, making it. Sorry, was Longo the person leading the discussion, the guy who chaired the panel session? That was Longo. I didn't actually know him. I'd read his books. I know now who he is. He has a book. He's written quite a number of books, so semi-complex with mathematics. I have to be, to be honest, I find myself finding... Yes, and I... Well, I would also say that what you said here, I could agree with. I mean, that was immediate, like at the early standoff in Mexico, they were looking at things to form the images of the objects in your mind. That was too bad.

45:00 Well, that doesn't directly go against what Mungo was saying, except that you're quite right, he did, I think, completely miss out the social dimension, he didn't miss out that these tools for better understanding of direction and structure are actually made by us in society, but I still think he put something quite useful into it, I guess because I'm very much influenced by my conversation with Albert there, the feeling that they are onto something, right, significantly, yes, I'll go to the woods there. With these ideas, there have been topological Urgish counts underlying the mathematical construction. Yes, it's influenced by Husserl. Husserl doesn't necessarily damage Husserl. No, no. Provided it's spitted into the... No, it's understanding the right way. There's a recent book you may have come across. It's just been published by Lakoff and Nunes. Lakoff? Lakoff. Oh, yes. Oh, God. It was those guys who came to Buffalo. Did they? Well, trying to apply, you know, cognitive semantics to understanding the epistemological process, I thought most of what they said was quite sensible. I don't think it was very deep or particularly enlightening, but I don't think I should disagree with any of it. I thought it was pretty traditional. You can ask the discussion of the hypothetical Martian or whatever. Well, I think the discussion of Wierstrass and the arithmetization analysis was on the whole, my insight. What I had to say about natural continuity, which is very much in contact with... Well, the mic is too short. You are too busy producing deep and serious mathematics because people... I thought you might find some parts of this book quite congenial, but I associate them with this notion of the folk theory.

47:30 Actually, Fatima, who was a graduate student of linguistics for a time, used to read books on semantics. So, Lakoff was one of the... Yes, he wrote about mathematics. Well, but no, we're all connected. So, I read one of his books, and there's this notion of hope-fear. What it means is this, that the example was simply that the thermostat, heating, controls for the heating. How does it work? There are two alternatives presented. Is it that the thermostat turns the furnace on and off when certain prescribed temperatures are met, turning the thermostat off? There are each snobs who are technicians who have some kind of, you know. On the other hand, there's the folks. It increases the heat, you see. And, well, now, actually, in practice, it's more or less true. We know whether to put on... In practice, it almost works. Now, isn't the folk thing much preferable to that of the late snob? This is really the way it's put. It's an extreme form of feminism. I'm very sorry I got you off on the lay call.

50:00 No, it's okay. I had a long, long discussion with him. I will just simply read his book. I just thought that his historical remarks about the Earth's class and about what was going on at the time, because the terms of all the arithmetization and analysis were quite... I would just like to look up what kind of formalist I am. I don't really claim myself as a formalist, but that I present the answer in the spirit of a modern form of formalism. ...which is not reductionist, but rather sees the foundational question as a question of seeking invariant or universal mathematical forms. Yeah. So, I felt that I should call that point. Did you get that, Michael? I saw the abstract, yes. I must admit, I was slightly taken aback by that. No, no. No, but somehow I misused the word. I'd say it's more of a form of... There should be a name for it, that's the real point. What is this perfectly dialectical material? Let's describe it as dialectical naturalism. I mean, functionalism sounds a little bit pragmatic in one sense, but you know, in the other sense, if you put emphasis on functions, this is of course fine, but that is not what the main kettle is bringing in. So I don't think you can use that word, formal functionalism. Yes, but of course, you know, what isn't stated here is that the invariant or universal mathematical forms are not... The structures that live in the appropriate category, the relevance of which in organising mathematical logic in order to indicate how it's going to be more deeply...

52:30 Anyway, it has a good feature of being slightly provocative in these days. Yes, we're provocative. Yes, yes, yes. It has to be either pragmatic and useful or else completely monistic. Yeah. I'd rather put it so, seize the foundations. The question of seeking what is universal in mathematics, but seeking it from the inside out. We're going from mathematics to what is the best tool for guiding the further development of people in mathematics. I see, it is very specifically centered here on the question of in what sense you can say that nil-potent elements exist, possibly, in this case, because of the potential properties of nil-potent elements. I don't want to be too high, too abstract. I also want to write light widths. He was one who did non-standard analysis from the constructive way. Before Robinson. Before Robinson. And with, not depending on all computers, but just the computer, was completely constructive. So this is... The infrared, for example. Yes. And which is taken up by Lohbeck and Palengrain. E.K. is working together with a young Swede named Halvgren. They have been working on the construction of the non-standard power system at the Laubitz school. So Laubitz was invited for the lecture. Tell me more about it. Oh, Sweden. Well, anyway, Sweden and Laubitz. Sweden was a collaborator. You know that we're the Blauzigs. No, no, I don't at all. The idea was to, well, I mean, from our modern point of view, why insist on two values?

55:00 Take the category of sets. All the first order, all the higher orders lie... For example, the construction of the reels. So the reels themselves will turn out to be sequences of reels modulated by the Frechet filter. That's what Logwitz did explicitly. So we're not in the ultraculture, we have a very concrete... Yes, sir, you don't have to use all that ultrapart space. No, no, the only thing is it's not two values. In fact, the truth values are two to the i mod the pressure of the filter once again. You identify two subsets of the index set and they differ only finitely. And the guiding idea is that, you know, you find the right geometrical structure to quote about them. Well, it's not really geometrical. It's not very geometrical. It's the notion of from a certain point on. Well, yeah. That's what I meant. That's right. Except for a finite set of exceptions. So, basically, he dealt with the real, the net setting. So the only thing that, you know, and standard means things that come from the diagonal, so you have some of the basic needs for non-standard analysis in your predicate called standards. That's what I meant by it's geometrical, because it's coming from the way you think of the diagonal. Oh yeah, that's right. Because you're approaching that by the right, to give you the properties you need in the diagonal. I'm sorry, I was not using the time. No, no, no, you're good. The fact that it's not too valid is not really such a huge drawback, because the things that you actually want to prove will normally be either true or false anyway. Because you only want to prove, sort of general, what the misplacement is. Sort of like the zero-one law on probabilities. See, even though you contemplate arbitrary probabilities...

57:30 When you look at these infinite product spaces and paths and so forth, most of the statements turn out to be either probability zero or probability one. You could, in fact, compute it also probabilistically. Any ordinary theorem of analysis will turn out to be actually true. You don't have to worry about the true values, but you do have the usual advantage that you can eliminate some alternations with quantifiers. Thank you for your attention. What I needed was a little bit of exact function. Another problem would be if you're taking filtered programs. I think your paper is on the rest of it. That could be, yes. It remains. I was able to make it. You know, a lot of quantifier is different to the comparison between the powers of being monic. I have that in my head. Indeed, I thought that the potential is monic inside the doomsday. You know, the quantifier is preserved. The other thing is that... In fact, I think that you're considering much more generalization, but there's a simple example of that, namely, that a stationary power of three sets is probably going to be equal to that power of four sets.

1:00:00 Technological work is a matter of course. There are also many other fields of study, such as mathematics, geometry, algebra, mathematics, physics, physics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, There are also a number of other fields of study, such as mathematics, geometry, physics, and quantum mechanics, as well as quantum mechanics and quantum mechanics and quantum mechanics and quantum mechanics and quantum mechanics and quantum mechanics and quantum mechanics and quantum mechanics Well, you certainly, I mean, you've pointed out so many times how this has happened. I mean, the classic illustration, of course, being Zermelo's misunderstanding of Kant, completely missing the whole dimension of the way that Kant had thought about men and art. But also, I think, again, in the case of this, the complete loss of the way that this work of draftsman was numbered. And nobody looked at the piano from the point of view of the structure on the end of it, supplied by the perspective of contrast advancements. And that needs to be incredibly deep because, as you've said yourself, the natural numbers are the source of so many of the far-fetched misconceptions of the way that's got to stick together in mathematics. Well, that was really just the guts of what Angus was talking about yesterday. Just how much mischief there is. And all of these phenomena. I said we're having the wrong focus.

1:02:30 If you're not familiar with the way that these, what you term in the separable one round of five, are designed, it's actually built inside. And, so much as you're not familiar with them, I know those are all very general guiding ideas, but they're not hard results yet. It already seems so suggestible. There are a lot of different types of responses, and I'm no doubt generous enough to attribute to them exactly this, but I think the number three is the fact that given each category, given each object, we can do FSS. And this is totally natural. That is, we vary the object, we vary the category. It's all going to fit together. So, it's sort of the level of classes. It's not a particular natural number. It's nothing to do with the natural operation on endomaths or any other endomath. Ah, so that has a name nowadays, what is it? The church numerals. I was going to say, it's the church numbers. Here's an example of that. Fry did some work on it, didn't he? The church construction of topos. Especially interesting is the paper by Carré and Romain. Ah, okay. There are a number of different types of numbers, which are called dinatural numbers, which I wish they'd changed the title. They started off talking about dinatural translation, which is sort of a strange notion, but then they translated it through structures operating on categories, through objects, and through mathematics. And the most terms is just what I said. And obviously it gives a deep connection with the behavior of the land of abstraction. Anyway, so they calculate for various particular categories. So for finite institutions, you can apply this idea. You of course get something much richer than the usual natural numbers. And this is kind of interesting.

1:05:00 But as I say, I think the true idea in some sense... Something that the Church wouldn't have been able to express would be this idea of iterating endomapses of any category, any totally natural operation across the Church. So, Rasmus' idea was that this kind of definition would imply all properties that you want about mathematics. You haven't collected them together in one thing yet. But you don't need any abstracts. He says explicitly arithmetic is different from geometry. You don't need any abstracts. And this is partly what he means. You have to book on a record of this. If I'm not mistaken, this idea of piano is really Zedekian's idea in a way. The idea of Zedekian infinite is intimately tied up. That is, if you find a Zedekian infinite object, it means there exists an endo-map theory. As I say, monic but not epic. Using the higher power set, the intersection of all the subsets of that set was closed. You can prove that the problem of the mathematical numbers, probably the biggest of all, is austerity. So... Well, I always took it this way. When you look at the structure quite systematically, in group theory, in fact, George Rousseau wrote a very nice paper about that, you see, when he pointed out that somehow there's these two different traditions, and group theory ultimately says that the subgroup generated is an illustration of all subgroups. Where as a foundation you have to do something else, to pretend that this is all just a struggle, but in my mind, this constructivism is the key. There's nothing, there's no content to it other than this idea of all the subsets. So that in turn rests on the nature of, you know, assuming that truth values exist.

1:07:30 Yes, yes. It rests on the nature of, well, objectifying the subjective and ultimately having the truth values existing. And that's very simple. Well, it rests on there being the right relationship between them. What we were talking about in your talk, the intensive expenses in terms of the big supports in the space for the existence of the art of physics. And again, and of course Dedekind, I think Dedekind more than Hilbert was the person who really took us down the route of this performance programme. This completes the course of the content. It's a construction of an epitome. He thought he started with the line which contained the rationales. I know you can map the line by Lunato embedding the power set of these rationales. We say that map is not monic. Thank you for your attention. I will ask Jesper Lutzen, who is a co-creator of the story of Goodman and Joseph, to talk about the history of distribution and stuff. You will be his leading lecturer. Ah, good, good. Dedicated reactions to ideas and dedications. That's all the real. Just all the real. We have an absolutely pure structural characterization of the real, where nothing is captured the pure. The terminology of philosophers used nowadays is anti-REM structuralism. Anti-REM structuralism. Anti-REM. Anti-REM, the four things, part of things. Structuralism are part of things. All this delusion that materialists have that there is actually some nature of the structure stuck in the world.

1:10:00 This is the great insight that is dedicated to all of this. And to some extent, Kant isn't part of this, but I think that's interesting. Again, bullied into meaning this. This is one of the things. The pressure here could be... I don't want to be invited to North Korea. I don't think he needs to worry about you. This is a very odd situation. I spoke to Longo. He says he's deeply involved in writing a long paper jointly with Girard. With Girard? Oh, he seems to have a pretty good grasp of... Well, I told him, I said, well, look, Girard's talk was not convincing. Even his defenders responded to the talk of the Korean, right? The new talk was to follow location and resources.

1:12:30 Then there is a huge leap. Girard's theory, which contains those names, is in fact about that. Until you run out of the talk of the world, how important these three things are, and so on. But he's just assuming it. He probably doesn't even know. He's just assuming that Girard is honestly reflecting those concepts. I could never see why they call this resources. I mean, there's nothing to do with resources. Now, this location is just a location in God's mind, but he's disturbing from that. Any way I could play into logarithms, I'd say, look, the problem I'm convincing, I agree these three words might be important, but how do we know that they don't work? Well, you know, it's crazy. Exactly crazy. But it's okay. I'm working with them and I have them in mind. Bye. There are so many, obviously so many more important things to understand. Yeah, yeah, exactly. Well, I would say that the things to understand are these things. There are also many things to understand. Oh, I think it was Constance. Right. I think we should buy back this. I mean, I have not managed to talk about this. Well, I was reminded of Cowley's famous remark about the work that one of his colleagues is so bad it's not even wrong. Well, I'm surprised we've learned that for that long because I thought that he was somebody who spent a lot of time collaborating. And I thought you set out the framework for a fruitful debate very nicely at the beginning of our discussion.

1:15:00 Question before, because I know you've got to go off on something. The single thing. Acute O in being O. And this is what gives rise to a lot of the... So-called paradoxes and such, etc., which are homologies, particularly, well, for instance, Borrelli-Forty. I remember you talking about this a couple of years ago, and I thought it sounded if you could read that. Well, perhaps don't major at the moment on Borrelli-Forty, but just on the peculiarity of it being a pre-op thing. But the fact that Burdike and Goyal chose it is free. And I just wanted to know the sources of the various studies. I took an answer from the LHCF, which I think I just might be able to answer. Right, okay. Well, that's what I'd like to understand. We want it. You know the book of Joyal and... What have I got? It's very simple. It's about three structures of various math. It gives the notion of small. They're axiomatized, they're categorized. So you have two things, one of you, but then it can be crossed by various things. It can be the three-fold set with you, the three-fold set with you, the three-fold set with you. So the three-fold one unary operating system. What gives you the illusion of global and transitive, just global? I'm sorry, when you say that this, this circumstance with being a free object is what, is what the character of context is?

1:17:30 ...on sets. You have this, you know, a way of saying set, you'll associate a new set, namely the set, the new, the Singleton set, whose only element is the old set. That's one of the things that you do in the 1960s. ...the teacher and the parents and said, what is this mysterious thing? Singleton access. Conception starts out with this idea of global inclusion, which is your idea of everything is in its place, and therefore... So, for example, global meaning for union. But then it was Piano who introduced together these two ideas. They were the singleton and the membership. So, say X is a member of Y, it's the same thing as saying the singleton of X is a member of Y. So this gave the thing some context at this mere post-set. But it has really the characteristics. I think it's important to note, suppose you want to say that this conclusion is somehow determined by elements. Now the normal mathematical idea would be, okay, what are elements? Those are like atoms. Those are things that are next to the bottom in the order. And it makes sense to require, if you wish, that... A is included in B, if and only if for every atom it's included in A, it's also included in B. So this kind of extensionality, in that sense, is perfectly easy without any membership or signature. I mean, I don't believe in it, but yeah. No membership or signature. So the role of synthesis really, I think, is pure naming, which is saying, give it any set, well, any small set, I'm going to find an atom which is going to serve as its name, because these atoms which at first were completely undistinguished have now got this odd role, but arbitrarily possible, as being the numbers in the phone book.

1:20:00 Yes, yes. They're going to be the thing which distinguishes the elements of the collection instead. Well, I mean, you know, this thing which John Labour is so hung up on about, you know, extension is just determined by what its members are. It's got to be assumed that there is some absolute global extension of extensionality. Rather than seeing what the topological content of the extensionality condition is and setting the topos. It's all connected up with all sorts of special points. So you can assume that every atom is in fact the name of some test. So then you could say, well, okay, it's... This is really just the result of these two steps, sort of the structural content of atoms generated, plus this arbitrary naming of atoms. Well, as you say, and it's important to understand that, because extensionality as such is a more meaningful condition than that, and as you say, there is more... But the point is there's clearly an issue there. It's not a complete place, that is an empty place. So the pre-destruction of the Melo-Frenkel model shows that in color names are arbitrary. There's no concept of which atom is the name of which cell. We can still find out that there is a union of the discrete registers of the human face at the same time as the people at the end of the brain. Well, pure distinguishability. Except they're not really going into that. Indistinguishable sets. Indistinguishable sets are equal. Indistinguishable things are equal.

1:22:30 Say that. No, no, no. Not something I believe in, so I said, but basically, we're getting a signal from the membership of this again. Part of the structure that they normally attribute to this post-class, unions of all the members of the given set, and on the other hand, power sets. Well, these are adjoining. There are two acts that are contained in Y, if and only if X is contained in P of Y, so those two operations have the same dignity to one another, and that requirement probably determines certain things about this naming, namely the names of the subsets of acts can deduce some sort of constraint in the name. The adjoinance itself doesn't begin. It doesn't in any way depend on this naming. It doesn't even depend on being atomically generated. And another observation is that P being right adjoint would preserve a terminal object if there were one. But the terminal object would be V. Cantor proved that P in anything is bigger. Either you reject Cantor's basis theorem or you... Well, there should be no universal set, otherwise you couldn't have this simple algorithm. And again, that has nothing to do with the Tesla paradigm of membership. And that's? That's some particular way of indexing subsets by elements. No, it doesn't. There should be no top elements.

1:25:00 If you want to develop an element that is highly invested in the field, by the mere act of it, do you want to preserve that? Is this a useful enterprise? One could further develop this. No, no, no, not seen in this light, but that's the point, you know, after getting this fixed on it. It's going to be a supply value. One-hour lecture, that was strong. They were interested in the water. It's very elegant. Then I, just to make it very concrete, I wanted to write down what is the tree representing the new order of pairs, considering 0 and 1. That really requires a system. I don't know what level it would be. I think that's one of the two things over here, at least they take more of them up there. Yeah, I wanted to look up something in that book the other day and I couldn't find it. In the book? This is the one thing they should have, but they don't have it. What book? Maybe the proof of... No, it's the proof of... Which is a problematic thing, you know, Rosebrook and I writing this book. And the amazing thing is that there is no understandable proof of some sort. But I mean, what is it? Well, why is there no proof of it? Well, because it has precisely to do with this idea of dedication. And here is the outline of the truth. George Lemon has got two aspects. One is the whole negation that you're after. I mean, a negation is more than a negation. It's a positive result, which is the...

1:27:30 So it says that if you have an endomath which is not order-preserving, it increases the depth of axis bigger than a sphere axis. Six points. Now, how do you prove it? You prove it, again, by exacting everything. And so on. You reduce it to the case of something that's outside the source of piano property, and there's no, it's minimal, there's no stuff that's closed under this map, and, you know, you have to start from a point. So there's a point in the map, and also closed under filtered soup. So now, the idea is to show that this minimal thing is itself actually filtered. Then you can take its own soup and it will be a fixed form. So suppose that it has the property of being filtered because of its minerality. That's all there is. That's Burbanki fixed form. Well, that's the building in fixed form. Your problem is like this. Suppose it does not have a maximum. You can find a bigger one. The theorem, so then I would get a contradiction of the top and then I would get the bottom. So, it's very interesting. Well, but you didn't want to build up any kind of choice at all. It's in the first step. You assume, if you assume that there is no maximum element, then by action of choice, you choose a bigger element. So you do bring in, you do use the absolute choice, but not as it were in the first step. Not in the fixed point law itself. Not in the fixed point law itself. So that's the part that I want to prove. And as I said, it reduces to the fact, to show that there's minimal fact.

1:30:00 All of this goes under an inflationary bracket, which is more minimal in respect to all those properties. It's already itself filtered, so you can take a soup and it will be a fixed point. How is this proved? So far this is a rather clear outline, although most books we don't find it that clearly outlined, nonetheless, some do. When it comes to actually proving that it's filtered, it's filtered. This is now a sort of double induction. But by induction, I mean, again, there's Jadakin-Thompson, and this is mathematics. But we have to show, you see, the pairs that are... What you want to show is two things. Actually, most of the cases, they actually show that it's totally ordered, and I'm not even quite sure what to expect in the topos, whether you can prove it's totally ordered because it's so simple, so you don't have anything about total orders in general. What you should just expect to prove is, again, which would be sufficient. It's very natural to assume that this has built its roots in all of these applications and from Toko's point of view. So I don't even know how to predict these, but it's done by doubling. That is, so if you think of it in terms of fixing one of them, I suppose... I'm sure when one of these two answers work, the other one must be such and such, but then you have to quantify with the other one also. It gets very complicated. Usually the book should be generated to a totally dense paragraph. And so I thought, well, Shirley knows all this elegant treatment. It's exactly the sort of thing they do, because these three things, they consider three things of a lot more different sorts. I think it is even, I think it is even considered a Philippine suicide, but you actually proved it.