Morning talk / Part 1

0:00 It's often discussed upon the 10-string model as you get, just to join all the symbols that subjectively, whatever material you're into the model for, you've got out and made all the combinations, identified two such combinations. Whereas, conceptually, you want to, you want to adjoin to the ZF model, to the ZF model. You want to adjoin all those things that behave as classes should. There are a number of natural subsets in the model. The subsets should be called classes, but not all of them. That would lead to the fact that the ones that are natural, the ones that are compatible, the structures that are at the heart of the idea of a class should be called natural subsets.

2:30 They often describe the upper star as a column also, which is a different sort of column. We read into that, describing the relation between space and quantum. We realized that in this case, this was actually something that had an add-on, but also a by-line. Spaces and cogs may be accomplished using an algebra program. These are the many ones that I want. This is an actual project. We've got this and this and that.

5:00 The repertoires for that are the right edge of the hump. The cogs are very durable. They have no inverts. Each following the inverts is being comparable to x. For this reason, this is called the spectrum, and it figures the homomorphisms of the algebras into these subspecies.

7:30 X is the space, and the algebras are functions. There's just one way to use this. It's another... Well, if you could say anything. It's the first part of the suggestion. There's a graduation. You could say, why haven't you guys used this? It's obviously the simplest way to do everything. That's too extreme. I mean, you can't send something. Young agents. You're 20 minutes late most mornings, and here I come. But of course, part of the reason I'm late is that I stopped to answer an email, which was mine.

10:00 It might have been, but it took me five minutes, because I can't bloody see when I'm out in the sun. And I was too lazy to go find glasses. I had to send off a message. And it took me about five minutes. No, it's not necessary. It was on your head. Oh, okay. Well, it was okay. It was just to say that it was on my roll. I was trying to get Alex Richards. So, can you just, sorry, but can you just remind me where we are? Roughly. Exceptional and formal. Very often, mathematically expressed by something like a gesture or something like that. I'm just conjecturing to see the relation between Verlach-Nerdl models and CF models. However, anyway, the proof of probability of consistency is merely the fact that this category is not a given model at that time. But the existence of the arrows, of course, tells you that if one is not a key then, so is the other. So the idea of the formal of the scheme, in this case, is that any model with CF... All of these symbols have to be given to you, and you're just enjoying an algebra you just took them all paired in an extensible way. Of course it can be an actual model and not just a formal thing, but over the given model it's generated by symbols.

12:30 And I think that's what people usually have in mind when they assist in this direction. These three are essentially triples. They will extract out of any model of d, d will extract elements of d like so. I'm saying that this process, this will sort of be a left-back one. There should also be a right-of-right, a conceptual nature for the hound. You take all the natural subsets, given a model, some other classes there would be subsets in the model. You take all subsets, that's not various blocks. For example, that model might have been countable. In which case, the bijection between the whole model and the natural numbers is one of your admissible classes. So you have to take only the natural classes. But you see, that's a conceptual thing. And all the things that behave like classes shift, rather than all the things that have names. And I just feel that there must be such a discussion. I mean, conventionally, you get something called Morse-Kelley, which is kind of inter... It's a kind of technical interest. It's not a full second order, but what it does is it allows comprehension in respect of arbitrary form, just formulas with set quantities. The thing about BG is, what makes it unnatural is that it doesn't even satisfy induction in respect of arbitrary form. So there's something deeply bogus about it. Somehow the power set, or something like a power set, is there. You can name something with itself, or whatever. I studied Kelly's appendix, that project, so I'm pretty familiar with this idea.

15:00 Kelly introduced the idea before. Is that why it's called Morse-Kelly? Maybe this is actually, that's what I was, if you start with a model as you get. And of course that model is itself at some kind of set theory. So you sort of automatically have a model of a far stronger theory. So the question is, in fact, what that theory is. It's not got sort of a bridled comprehension, right? You've gone into, on the other hand, it should be perhaps bigger. See, there's often that. There's often, of course, a coincidence. As long as the definability is there. It's not quite possible that this kind of thing would actually be a model of force-telling and then be sort of restricted back to a kind of model of VG. Doing more is required by the diagram. My own take on it is VG was a kind of artificial construct designed to make the proof of the continuum problem, an easy so-called. Actually, it's much harder. The following. And then the National Academy patrons on the film. That's why I fit in entirely. I entirely agree. And the point is that this Morse-Kelley, or something like it, the proper theory of the power set of the model, is relatively consistent, and yet the theory is finitely axiomatic. Wait a minute. How does it become finitely axiomatic? Because you've got infinitely many instances of complications. So for every formula whatsoever, whatever quantum mechanics, you have to have an instance of comprehension for that formula.

17:30 See, those formulas are merely special cases. They're all there in this branded model. Yeah, but not only formulas, but these things. But I still don't see how, and maybe I'm misunderstanding what you're saying, I still don't see how you, I mean, just thinking of Morse-Kelley as the first one. Well, maybe not the old Morse-Kelley, but just a theory in which... Ah, there's something above BG, but still below. Yeah, somewhere in between. Well, you know, the minimum requirement that restricts the language of the person. Without saying. Well, unrestricted language means that... You have a map like this, and you have a map like this. This is so elementary. Without saying anything about what functionals are, that's a big problem. Functionals are a big problem, but there are always just the natural ones. The ones that are there naturally because of what you started with. The actual ones of actual mathematical interest. You sort of think BG is quite an unreal hybrid. It's an unreal truncation. So if this works, this ought to be a general thing. BG is a general thing. That is to say, the business of adding classes and finitely axiomatizing the resulting theory. It doesn't have to be set very good. Well, isn't that what you said? It could be number theory. It could be anything, yes.

20:00 I think John O'Rythmiak, probably, seeing this really quite a reaction about it. Why that kind of thing? Because you just have to introduce a variable to it. So, so, yeah. I was told by a secretary there's something called the EG class. I've never heard of it. I think his assessment was it has this kind of rule without any restrictions. I think you're right that Morse-Kelley is not quite a theory. Well, I mean, in between, yeah. The other pathology, of course, is the reflection method. And it's not considered a pathology by scientists. They think it's a tool. But it is a pathology, because what it's doing is kind of externally scolizing. And if you had the machinery that's going with this, you could do reflection for set form. Yeah, that there exist sets which reflect properties of the universe. Right. Given any finite number of propositions, then there'll be a set in which the truth values for those propositions, I mean, propositional functions, the truth values will be the same as in the universe as a whole. I mean, you're just restricting the quantifiers of this small set, but it turns out this set's got enough closure properties. Right. It's goal function closure properties. Right, okay. Something like that. If you look at it, that's the whole thing. So, I mean, I can remember when I first saw that, I thought, shit, this must mean the theory is inconsistent. Are ruts, ruts, are ruts the favorite ones? Well, the big ones. I mean, they try to incorporate second-order notions and stuff.

22:30 Full second-order reflections. If Morse-Kelley were finitely axed, then you'd have a set model of it in ZF. I just believe there'd be a straightforward proof that Morse-Kelley can't be finitely axed. But the curious thing is, in some sense, we're talking about ZF not being enough, but in some sense it's too much already. It's not, it's not just that. I don't know if you get these simple why didn't you say. No, but it's, it's, it's, I mean, apart from anything else, it is a curious phenomenon that, that you get, you can mimic Dedekind's arguments about induction sort of formally. And that's what gives you your humanistic hierarchy. Because you're doing inductions on these globally defined functions. Wouldn't the general theory satisfy that? You have to make use of machinery, machinery, the full quantifier machinery and the full comprehension axiom with respect to quantum quantifiers and everything in order to convert what is really a simple set thereof into this formal, in some sense, unique replacement for it. Maybe that's what people feel, doubtful about replacement, but in some sense... You're exploiting peculiarities of the formal logic and not intuitions about sense. That's the idea. I mean, I think in your terms you'd say you get what looks like a subjective, let me translate that to Kurdish,

25:00 because when you teach it, you say, well, you can go along the originals and apply something to that. You can pass the batara to God. Yes, yes, exactly. Exactly. Yes, but I mean, it's building in recursion as a fundamental principle of this theory that's supposed to... Oh, there's something dubious about that. By shrinking ZF down to what it's actually justified by practice. I mean, how many mathematicians actually did mathematics up to 12-hole coordinates? I can't think of. Well, that's exactly what I know. Now I'm saying ZF for this intermediate thing between GB and GK. Yeah. Well, what I'm suggesting is... The formulas have bound variables. See, what I'm suggesting is that maybe the natural theory, I don't know whether this will affect the category of theoretical instructions, I doubt it, but what you ought to start with is something that Amad had said this, but which is more important than you might be able to, I mean, I think, for example, the question of how much induction, or whether, indeed, you can do classifying induction for anything in general.

27:30 I mean, the way replacement comes in is weird, because replacement actually is as formulated in ZF, partly just saying that the image of a set functions, and partly saying that any formula in ZF with the functionality of any formula, it's the two things together, but you don't notice the function comprehension part of it, but it's there. It's there. It's not really the image. Your image is sort of trivial. It's the fact that the image might happen even though it's a set of sets. And after you've got that image, you can take communion out of it. That's really where the power comes from. The fact that that little set itself is all much, which might be... Yeah, the idea is you build up a set of huge sets, which contains things that are much bigger than it. Thank you very much for your time. An unbounded summation. Big category. But you're talking about a weaker theory. So, in other words, for example, my theory is the category, the unbounded theory of the category itself. Yeah. So it's, the compost is the best factor for the unbounded sum. Yeah. Depends on the action of the choice.

30:00 That's it. Oh, that's the natural number of them. No, they're power sets. And you, but you could probably introduce an unbounded sum. But are you thinking of this as an abstract set theory, or is this just... this is abstract set theory, where the sets are just bags of dots. Well, the category of sets, even if you start with a model of ZF, you can extract the category from it, because the definition of math is satisfying such and such. There's a functor extracting from any model that has a model of this category, of this model with a stronger version of it. So, no matter how much junk you put in the original set theory, as long as you've got sets and maps between sets, this will, as it were, abstract. I don't know if that's a good word or not. No, so it converts it into a category of abstract sense. Yeah, absolutely. So there's something funny about replacement. I mean, naïve replacement. Infinite sum, but naïve replacement just makes no sense. If you just state it in terms of image, you can see that it doesn't properly capture the context. The fact that you have something like the sum or the union of all the elements. Is this just because you're dealing with global, globally defined functions? So you're not dealing with arrows? I mean, of course it's trivial that the image of a set under an arrow, on anybody's, because the arrow's already carried the code away.

32:30 And there still are, you know, definable operators, even functors, lots of them build functors. Other operations that are functors, like x to the x. In some way, you see that the funny thing about topos is that this is sort of all you need for it. Without any replacement of topos. It's hard to get an image of this. Here's a topo, for example. We have these mice that can identify the x-index families of the fibers. These are already, they've started very sputantly then, these could be actual manifolds, radii, the fibres are circles, certain families are called circles, it's that kind of a thing. But most of the families that are in mathematics are of that form, there is an actual case, and the constructions, the summing, the summing is just the topological operation of the general and other math. If you're going to do math, you can just compose this is the time we're going to try to find y.

35:00 y is the sum of all the factors here, which have to match b to 1. So if y is 1, then this is just summation. So in other words, summation is just the operation of taking something which is variable, variable extensive quantity over x, and taking its total, its object e itself. So let's consider that the non-verbal script E is an non-verbal discipline, where the summation itself is nested in a monological process, throwing away the fact that it's a variable family. That's the left adjoint of the left, which we get by pulling back, which also then is the right adjoint of pi, the product of the parametrizes of that product. It's another object. Yeah, another object. Another object. Y over 1 is just another object. In the case where Y is 1, you can put the other side of X to the X, and then pull back a set of all sections, and so on. If you have an actual internal category, then you have the internal, the topos of internal actions, or the change of category has left and right answers.

37:30 Because we, the point being that there are essentially no abstractly given families, you know, if they come about in any sort of mathematical way, they're almost always going to be construed in this way already, without applying, whereas replacement is saying that you could have these families given by formulas outside this, but yeah, I mean, that's my point, I mean, it's essentially my point that you've got these... Comprehension axioms are functions built into the usual axiom replacement. So it's got more math. It's not the same. So you see, in some sense, whether the piece of x is documented internally or all the points of x are calculated externally. I say externally, I mean considering the script E as an object in the casualty category. If an object in a category is a category, then they're going to be typically not be the same as some category that's set by a telemetry act. It's a matter of being full in the universe. There's something like that that's said very well. See, the set of points is another category. It's another object in a category's category, I suppose. You could have a math there at that level of content which could be used in categories, but it was not necessarily used by a math-based product. We say that the school of the universe, in fact, is different than the every math way, because it is, or it is internally, in the, you know, that's a lot of the first order theory, but sometimes it, you know, that's at that point where, for example, these might be described by the external formulas.

40:00 But I think, you see, I don't think the school of the universe is... No, but do you need, can you do a similar stunt on policy? Where's, where's, in effect, where do you say that the power set is? The power set is y equal to two. That's all that would be. So you're saying any external subset would be outside? Yes, an external subset would be, I mean, the relationship is slightly more complicated in general. Assuming that one separates from the other, it would be different. It would only exist in that point. Yeah, I'm thinking about a more general problem. I can try to read yours. I probably spent more time on chapter one than most people would.

42:30 I mean, I think I had the advantage of most people knowing where you were coming from than the target audience of the book. So I knew the story about Cantor. Okay, so I mean, I saw what it was coming for. And I think this is a problem with Greece as well, and maybe for Cantor. It's a kind of question of which comes first, the chicken or the egg. I could find it in mine, because mine's all marked up. You're talking about, in the first chapter, you're talking about a set. It's a remark, I mean, it's a remark about, there are too many examples where you're talking about, I mean, these examples are pretty illuminating the business about. There's essentially not much difference between going from a set of 1,700 down, assigning the numbers in order of rank. You know, these are essentially the same kind of thing. They're listening. Yeah, okay. But, I mean, the idea that was bothering, the thing that was bothering me is you referred to, maybe you remember this, Richard. It refers to agglomerations or something like that. There's a set consisting of Leibniz, Newton, and Descartes, and then there's this set consisting of three dots.

45:00 So this is kind of a conceptual problem, but my way of understanding this was the same. Okay, you got sense, but then you ignore what their members are. And how do you do that? That's the problem. What do you mean by ignoring what their members are? And the trick is to do the, as I saw it, is to let the arrows carry you. So that way there's additional structure in such a set. And the structure can always be learned on a map, yes, if you don't stop with the three parts. So I guess the simplest way of putting it, is the set consisting of Newton's library set and Descartes' abstract three elements set, or is the abstract three element set an abstraction resting on those three guides? What's the, what's the fundamental way, yeah, so what's the fundamental way, what's, what's fundamental here? You know, it's a structured version of, it's somehow a structured version, you know, and certainly the, the three dots are going to map into the, the initiative. Right, so really what the concept is, is the concept of a three element set, because the fact that they're people, the three dots are worth any three, you know. And to get a particular set of three, that concept gives rise to the lack of dots definitely.

47:30 It's important. Somehow, intuitively different dots will arise from abstracting from different sets. It doesn't matter whether the sets overlap or not. I mean, if I say... The point is, once you do the abstraction... Each set is separated out just on its own. So the dots, I mean, if I'm acting from a set of sheets, funny dots or whatever they are, are associated, intuitively they're associated with those funny dots. There's another aspect that I'm not into precisely. Precisely reflects post-hack mathematics, the fact that we still don't know how much there is. We've already got some structured calculus. But the point is that you get, I mean, the puzzle is... And probably for Cantor as well. What are the dots? How do you distinguish between dots in different sets? And the point is, in this theory, there's not even any way of saying it. So you could say, I'm adding five dots to three dots, and I'm getting eight dots, and how do I know... The answer is, it doesn't even make sense to ask whether, if they were just given sets, like, the animals, the sheep that you own and the sheep in this pasture might be two different sets.

50:00 If you said that five elements and three elements overlap and two elements, then you're actually given a little diagram. Yeah. Instead of just... So they're not... That given data gives rise to... Oh sure, so... The approach of push-out is covered by the sum of the two, the thing with eight dots that has the structure of two and one, so it still comes down to the fact that in the abstract sense, the arrows are doing essential work, and it's crucial that the membership, I think it's in this book, somebody even rewrote that in the book, corrected it. The modern notation for an English model, one point set, we say that basically this category, the notion of coordinate set, that's where you say objects exist.

52:30 So anyway, within that, I think I've noticed that one word is monomorphic. That's, by the way, I think the category is a sub-object. Is that there? Sorry, it's not B. It's a math B. Yeah, no, I know that you could get that. Math is a sub-object. B is the empty value. But anyway, the thing is, it actually makes sense. Oh, yeah. No, I mean, just the fact that the notion of category itself objects the math. Well, category proves... It's called the category of groups, named after the objects here. And there's some, you know, narrow-minded logicians who say, well, actually, the actual material in the model of this is a map. So they give the names of the maps instead of where... So it's the category of homomorphic. Yeah, and see, this is, this is what we're looking for. Somehow, if you just look at one group at a time, you don't see that, but exceptionally well. I think that the usual terminology is...

55:00 It's correct to name the categories. Even though the maps are the main thing. But given that the maps are always the main thing, it's kind of superfluous to say more than that. Yeah, go ahead. The other special case is where this map is still somewhat special in that here we typically have some subcategories that refer to bigger shapes. Those are automatically run over the global element. It's not that we don't distinguish between these symbols. The point is that there's a natural transformation of all the symbols. The idea is that as an element of capital X, there's a rise to an element of the power set. So this is the element of the power set. The fact that anything in math has a unique codomain. Anyway, so that's how we deal with subordination and membership. Some treatments are described when there is some sort of inclusion only, and that may be what you meant by that.

57:30 No, no, it's not what you meant by that. The thing is that the singleton operation, the piano, is defined by the usual term, inclusion, together with... Singleton. Singleton. Not in this sense, but in the... It gives rise to... It's a member of the singleton, it's a subset. I mean, I've tried to, I've tried to organize this... I'm talking about the ordinary set of the universe. The powers that we have to adjoint this, both considered just as order-preserving maps. To me that's obvious, I never thought of it. It's obvious, but I never thought of it. I didn't think about it if you were asking about adjoints. Yeah, yeah. But by the way, this shows that topology, any right-hand way to preserve topology, in a sense, as a topology, p of it has to be one. It's sort of a very nice idea, but after all, by contrast, p is always increasing, so that can be topology.

1:00:00 You may have pointed to things about the entire universe. For example, the largest particle. The largest particle, obviously. So you don't want to talk about them without the fact that the right-hand one has to preserve the other one. Okay, so that's the preserved intersection. This has to preserve the actual... Yeah, but that still leaves me with a feeling of unease. And the unease consists in this. Things should we understand first. We have agreed to something. Ah, yeah, yeah, yeah. Historically, yeah. Not so much like in historical spirit, but in the spirit of straining out what's... The Greeks that you would start off with what you call agglomerations, which are ordinary areas, what remains after you know that's exactly what it's the same And of course, abstract equations.