What is the true relation of logic and set theory?

0:00 And he's going to speak on how are set theory and logic related. I have a question. The attempt to make explicit laws of thought or specifically... The laws of thought in constructing mathematical concepts. There are two aspects to that. One is the statements that we make when we comment on or specify the integer object or the construction of the object and the inferences between these statements. The laws of these inferences is divided by the narrow sense. It's often defined to be that. The question is, do we need to know something explicit about that in our struggle to understand the laws of thought and constructing mathematical objects? There also needs to be a background of already accumulated concepts in which the new concepts can be interpreted more precisely. Usually individually in particular have two aspects, the two which have been able to call abstract general and concrete general.

2:30 The concrete general applications of the abstract general in what? In the background of the already accumulated into which new concepts can be interpreted. I dreamed for a long time, and it always seems to me we're getting closer to the goal, that requires a lot of serious work, namely that part of the explicit is just of describing directly the category of objects, and we've made many steps along that, but few people have yet taken this up as an attitude. Let me just review, in a way, the very standard way of analyzing a given subject. It forms a category. It consists of lots of concretes. It's a concrete general. It consists of all these concretes, and by the very virtue of... The fact that these are representations of the same abstract chemical, there is an induced notion of comparison between them, and so there are morphisms, and so it's a category.

5:00 It consists, if you want, in terms of operations. What are the collections on which these operations are operating? That comes out of the thing itself. It comes out in the following way, that by studying, in particular, what M is all about, we isolate a very small subcategory, A, and then we say that for each object X, we can consider the maps and the catamorphisms in the category from N to X from the objects in our small chosen reference points, from the term figures. In the extreme case of the category of abstract sets, we just take a one-point set, very small compared to the whole. On the other hand, practically any non-trivial mathematics, I mean, in contrast with the constant and discrete sets, will need more types of figures than just points to distinguish between the general pair. We have to take into account the fact that the transformation of figures into figures is compatible with change of figures.

7:30 Again, you don't see this in the extreme sets. But in general, in the small category itself, there are particular maps, and these will operate on the figures on the right, and so the transformation F, when it operates on a figure F, it will automatically satisfy the following naturality condition. Transforming the figure X by means of alpha and then applying the operation F will give the same figure in Y as would first apply in the operation. So this naturality property becomes a way of introducing incidence relations because different figures might compose different alphas in certain ways and that would express incidence relations between them. So the general strategy of analysis then is that we only understand the evidence well enough to take a large enough notion of basic figures out here, a complete analysis in some sense of the objects and maps will be obtained by considering figures of that shape and abstract transformations on them, which are... Abstractness is our restricted, or constraint, by this condition of naturality. More exactly, we consider sets, abstract sets, in most cases. So the idea is that the totality of transformations in X to Y,

10:00 while it has an infinitely rich structure, we could apply the Cantorian abstraction process to that, and just consider it as a cardinale. So each X and Y. So that's an object in this category. This category is defined to consist of those. That much we knew already. That much we accumulated already, since understanding that there is such a... And so what we then... But then, of course, we were saying that to each X we assigned not just the one cardinal, namely the cardinal maps from A to X, but the cardinal maps from A to X to each A in our class of script A. So what this mouse did was constructing the functor into the functor category, s to the a-op. So what this functor does is a script f for a figure. The script f of x is that functor from a-op to abstract cardinals, which assigns to every a. The cardinal-op assigns to every map in this tiny little category and assigns the corresponding transformation of A map between cardinals automatically will do that then in a way that satisfies this translation into naturality. The same equation, which initially looks like a special case of the associativity, and then why a special case, because we're only talking about maps that start at A,

12:30 that goes over into the naturality, A naturality, But on the other hand, we can then consider this as a condition on a transformation, whether the F really comes from M or not. So we do have the abstract idea of natural transformations defined up in all of the concrete realizations of that imaginable, which we can compare with those that actually arose from M. So far, we haven't assumed anything about A, but to somehow describe it is most directly expressed by requiring that this map be a bijection. All of these terms are used to define the language of the language of the language of the language of the language of the language

15:00 We say that A is adequate if and only if script types of A is cool and faithful. This definition was first given by Isabel in 1960, a long time ago. I'm sure it wasn't taken as seriously from that time because it was pointed out to them pretty shortly after that, but it needs to be taken much more seriously. Now, I emphasize that this sort of analysis can be... To consist of all the multi-sorted structures that obey some particular theory, in other words, if the abstract general has been specified, then if you're looking at the corresponding concrete general, the category of structures, for short, as we just said, then you can apply this method. And often in many examples it's applied, and you see this in geometry and algebra and so on. Or you can also imagine that this M is something that you're trying to achieve. Maybe it's a category of all possible elementary particles, or all possible smooth spaces, in the sense that we haven't yet said exactly what that is. Arriving at an over-determined description of what it really is, is approached by many different modes of study, some experimental, some historical.

17:30 And some by various kinds of mathematical calculations. But one of the many roads to arrive at is implicitly and should be explicitly thought about what M itself is like, in particular, to what extent M could be analyzed in this way. So, in other words, even though, in principle, one should consider such categories M as, quote, given, whether or not they're given as categories of structure. If you can find a category A subcategory which is adequate, then you have represented them as structures, because this is a pretty general notion of what a structure is, a diagram of shape A, but made of arbitrary particles and arbitrary mass between them, and a map of such structures being a natural transformation. That's a pretty, not the most general possibly, but a pretty general notion of structure. If you manage to find such a thing, then you have represented the category as I think structures basically should be thought of in that way. They arise from the analysis, not necessarily just as we started with. Both of those directions are important. Now, of course, it should be clear that here again we have this kind of ambiguity that we have with bases and vector spaces, etc., etc. At least unless we put much more conditions on A than I've said so far, you might find one script at A and another script at A which are quite different, which are both adequate. If they're both adequate, it means we have two ways of representing this one category, M, as consisting of abstract structures. So to say that the objects are abstract structures may be, in the extreme case, the

20:00 Just as inaccurate as to say that the point in three-dimensional space is a triple of real numbers. You can put more conditions on A so that it becomes less true, but that is an aspect, especially from a philosophical stance, so it has to be... Okay, so to analyzing, when we speak of abstract general and concrete general, it means that these become, A itself serves as, the category of all possible... Abstract, proximal, grounded, punctured, or automated serves as a concrete structure. So that itself arises from the concept of whatever you need. This method of analysis, of course, like many things in category theory, can be dualized. I think of this as the geometrical analysis in category analysis. After all, it's in terms of, it is a structure, but more specifically, it's a structure which basically consists of figures and instruments. And the idea of a map is that it should preserve, that a map figure should be preserved. An important example of this, as you know, is the synthetic differential geometry. The idea that the synthetic, what the synthetic basically means is that we at least partly take this stance that there is some idea to be specified, And we analyze more closely what that might mean through such, through postulating what there is and then going in to look more closely at it.

22:30 So in that case, to give a specific example, the types of figures that are needed in order to be adequate are not just points but also smooth paths and also tangent vectors. So basically points, smooth paths, and tangent vectors are adequate in some conceptions of... Also, the combinatorial categories are often described directly in this way already. A pretty common way or geometric way of analyzing any category. So the dual way is algebraic, right? So the algebraic way of analyzing M is to choose again a small category, maybe a different one, maybe the same one, call it B, U, O, V, X, Y, Z. The idea is to instead of functions rather than figures on an arbitrary object X. So functions on X are maps from X into the objects in the some chosen small category B. Well, again, again, one finds that there are maps inside B.

25:00 Beta is another function if that was a function. So this is typically, this is called an algebraic operation. In this relation we have an algebraic operation. Very often these functions will deserve to be named with quantity, because the size means what quantity is, but as variable quantities on x, when we apply an algebraic operation to a variable quantity on x, we get an x. Can I just ask, is there a significance to why you call it functions? Well, it's just a choice of words, but I think that that is at least one of the traceable traditions to... Use the word function to mean those special kind of maps which happen to have a real or complex or truth values. Oh, yeah. See, special. Right, so logic in the narrow sense is often objectified in this way, that you take just the two elements set here, and then they're called propositional. You swallow that and the truth is a quantity, so it's in that. So propositional functions, real functions, have that significance. You can certainly contemplate general domains that are very restricted. The word function you might use in some other way. You know, complex functions. Well, it's that sort of thing. The domain, the complex metaphor here is not necessarily one of these. Narrowing down at the totem. In the same way, the word figure is chosen arbitrarily in that same spirit, except that within the general space we could consider triangles.

27:30 Triangles are specific. This, of course, includes the, if script B has Cartesian products, this includes everything that's normally called an algebraic operation because the math might be going into 2 to the power of 3 instead of just into 2, et cetera, so the possibility that script B has finite products is very important both technically as well as understanding specific situations, but plays no great role in what I'm saying now. I just throw that in to reassure you that by construing an algebraic operation in what looks like a unary map, I'm not really restricting myself because the B might be the third power of the crack, more like the multi-operator, multi-argument operator, the special case of the unary, everyone wants to handle it. There are attempts. So in any given small category, one can make this attempt. You look at covariance of set value structures on B, but the opposite of that, should we call this, we aren't using F for figures, we could say, we used A over there, Q for quantity. All of these values quantities or functions on the vertical space q. The op here signifies the fact that obviously if we change x now on some general map of v, that the q of x will get mapped to the q of x prime.

30:00 On the other hand, the operation over here is contravariant. The alphas operate on the right on the figure. Whereas, uh, knowledge really operates on the left, as the moral of the assembly. I think there is a moral there somewhere. The notational balance, just as an aside, the fact that we have these two conventions for writing composition is no doubt rooted in this dialectic that I'm describing. There's even one author, Emily Hairstein, who actually says, you see, that in algebra we write composition one way and in other parts of mathematics we write it the opposite way. And it seems to me that that's justified. This convention is the opposite of yours. That's right, yeah. I mean, globally, it's a pure convention. But within a given context, you see, the fact that one should really be the opposite, even though you're talking about the same logic effects, that perhaps deserves to be incorporated into our notation. Well, sometimes it is. Sometimes, in effect, it is. For example, when you write the integral of that Vx, Vf is really, I didn't want to get into notation, but just to point out that these two variances, these two aspects of the same thing, the same space x, or the same change in one space. Alright, so again, Isbell of course didn't fail to note that there's also the code concept.

32:30 Now, there's a very concrete, well-known example of an encapsulation, which comes to every logician or philosopher, namely, where M is the category of abstract, finite sets, and where B is taken to be a two-element set, a three-element set. What we're getting here is the truth tables. The truth tables are just these betas. The point is that by looking at maps which are natural with respect to the truth tables, Also known in the jargon as movement algebra or homomorphism. The math which are thought of as operations on the propositional quantities, we do achieve this fullness, faithfulness, of course, also, especially the fullness, namely any math between which is in fact natural with respect to the truth tables, really does come from C and P. That's what this bijection matrix is really underlying. The math of the abstract structures actually comes from the real math, the real meaning in the M, or the same thing over here. So the category of finite sets is fully embedded in part of this. Now the way the truth tables are usually understood, script B is actually the category of all those finite sets which happen to be powers of 2. There's 2, 2 squared, 2 cubed, and all the maps between those in M are restricted to B.

35:00 However, it's an interesting, I find it interesting, using Marx, that it suffices to take one set, a three-element set, and all the endomaps of that, and you still get the, you just take two, then the maps which are homeworked with respect to all the self-maps of two are quite more general than those which are produced by actual maps of finite sets. Consider three true values, namely yes, no, and yes, what not, interpreted in a different way, enough to be that even that small b is adequate, or more exactly, co-adequate, that monoid of, what is it, 27 elements, we don't need the infinite, the usual, in a category, of logic, to achieve this. What you might like, you might like to achieve more than full equations, both in the geometrical style of Analyzing, as well as in the algebraic one, namely you might like to achieve not just the full and faithfulness for those objects M that you've accepted, but some kind of condition on these functors from B to A, or from C to S, from B to S, will characterize, or nearly hope to characterize, the actual objects. Full and faithfulness only refers to the fact that we've managed to completely characterize the image of... For maps, for any given pair of objects, another important issue is to characterize which objects here could arise from some object x, or the same thing here, which abstract structures of the kind figured in instance relations could actually come from object x, and so on and so forth. So I think that's probably, we could do that with the usual, but anyway, the usual answer is, going back to the b consisting of all the powers of two,

37:30 We look at the product-preserving functors. Again, it's obvious from the setup that if B has products that agree with those in M, and we twist it around into this functor, this comparison functor, we'll get values which are product-preserving. So we look at the product-preserving functors. Well, those are exactly the same. That would be also a characterization of the algebra over an algebraic theory in general, which is being product-preserving. And in this case, the finite sets and the finite ruling algorithms. We get the, that's the vision to characterize the, well, I'm thinking of finite sets here. Several comments I wanted to make about this very simple-minded, but very powerful point of view. First of all, what's topology anyway? Topology has to do with cohesive sets, continuous maps between cohesive sets. Approaching it, you see, from a vague formulation, trying to go for more determinants. So at the end of this... Continuous, somewhat continuous. People thought about, Cantor thought about, he said, well look, a very important thing about cohesive sets and continuous maps is that there are these things called open subsets, and the inverse image of an open subset is a definition of continuous.

40:00 This was a specific determination of what collision and continuity should be or might be. A very productive determination, but not necessarily the final one. Notice that it means a very strange thing. It means that topology as we know it from housework is really algebra. I like to think of it as geometry, but instead, no, it's algebra. More exactly, what you do is you take the Sierpinski space, a space that has two points, one of which is open, and its powers, then by the very construction of the Sierpinski space, A continuous map from X into the Spensky space is exactly the propositional function corresponding not to an arbitrary subset of X but to an open one or a closed one, and which of these two you take as true. Take this one and the open one as true, and then these are the characteristic functions of an open subset. So of course the inverse image of an open set is open because that's just saying If this is the characteristic function of an open set B, then the composite is again the characteristic function of some open set, and it's easy to see that fundamental logical operation of substitution is what corresponds, substitution into class, property, or subset is nothing else but the composition at the level of this characteristic or property or function. So in other words... P sub V composed of F is the same thing as the characteristic function of F inverse of V. So the inverse image of open is open by force. But see, thinking of the M as given and then isolating, there's a general vague idea of the M and then there's this very specific determination and then we get...

42:30 But the point is that if we make the analysis of the topological spaces nothing but that, then effectively what we're doing is representing it as the opposite of the algebraic category. You take all the powers of S, of course not just S, and so the operations of beta include things like S to the power of I is rematched into S by means of I for the union. So that's one of the betas. And likewise, the best process, matching to best by hand, is another one of the latest. And so, in other words, the kind of algebras that we're getting here automatically outgives the algebra of open sets. It has operators corresponding to finite intersectional union, and naturality over here just means nothing but preserving those. So, in other words, there's the analysis of what's now called frames or locales. Frames are the algebras, frames are the locales, and the locales are very closely topological spaces, not quite. In a way, this concept of locale, which was isolated in the last 20 or 30 years by people like Isabel, Raymond Booth, Coyote, etc. Belker, many other people contributed to it. It really shows in much clearer light what this particular determination of topology was always about. The fact that you also had the points running around as figures, but points as the only kind of figures, gave you that little handle on geometry, so it was still kind of justified to intuitively think that was geometrical. But that was by far the inferior part as far as the actual calculations were concerned. So the formulation in terms of frames, which are these algebras, which are homomorphisms, and the opposite of that, which are called locales, really throws into a sharper light what the Hausdorff and Polish et cetera et cetera topology was really always about.

45:00 On the other hand, Doyle is quite wrong in saying we should call locales spaces. This is appropriating. For the general philosophical notion of space, one even more particular determination, which is precisely the wrong direction of the film, according to me, according to me, we have, there are lots of things, lots of precise determinations of space, there's combinatorial space, there's smooth space, there's probably other determinations of continuous space as such, which are better, maybe, an open problem for research. So, really, a truly geometrical analysis of the... There's a vague notion of continuity and cohesion. One way of doing that is to say, okay, a figure is a continuous path. In other words, in the category of spaces, there's something called, let's say, the closed unit interval, which has certain endomaps that we study in calculus, or we don't study them in calculus. Anyway, there's a certain object, a specific one, and with any arbitrary object x, I will associate the system of all paths. The reason it becomes a system is because we can operate on the right with these alphas. So if the alphas are just endomaps of the interval, they amount to arbitrary re-parameterizations of the paths. So that would be the structure that a system of paths would have. Given a path, given an alpha, what's the result of re-parameterization? Now that includes the constant paths, so it includes the determination of points. Yeah, very important. Further remark that I should have made at the beginning. Many people seem to miss the basic point about this comparison, which is that it preserves points. This triangle commutes.

47:30 Where we take points, which are the maps from one to blank, and here we also take points in the sense of this one perspective, this topos is actually a topos. In other words, the notion of points should be taken to be the same over here as it was in this. The general idea is maps from one, maps from one. So, when we assign the set of all continuous paths, let's say, then take the points of that, the points of that are not continuous paths, they are actually the real points, you see. There's no funny business going on here that I'm telling you you should believe that paths are points. No, it's this. From the system of paths, I can extract, in particular, points, more concretely, as the constant paths. So that's a very interesting idea about how to model topology. I first thought of this because I read in Walter Knoll, who works at the Foundations of Continuum Mechanics, the basic purpose of the topology on a state space, in particular, state spaces for continuous bodies moving. The basic point is to be able to talk about continuous paths, which would be imagined as time paths, but possible processes. So that's sort of the main use of problematical stages in practice, and then many branches of practice. It's by at least a... And especially in, actually, it's much more natural when one's thinking of manifolds to think of this side and not on that side. I mean, this is completely the language that's used by anybody. The plots, there are atlases and charts and plots. They're not necessarily monomorphic. They're the arbitrary... So sometimes it's called singular figures. In fact, the term singular cohomology or singular homology, which you might not have heard many times, refers to nothing else but the fact that at a certain point in history, it was decided to drop the idea that figures should be monomorphic and allow them to be possibly non-monomorphic.

50:00 Right, oh yeah, the last one was topology. Right, so in fact, you could imagine in 1940, the headlines in the newspaper, or again in 1960, topologists reject topology, meaning, and this is literally true, meaning that people who were actually interested in working on space, as opposed to those who were developing the consequences of cost-force axioms for their own actual sake, I could realize that there was something wrong with taking that out of our determination as the final response, essentially. Because basically, this came from a geometrical reason. It came from the reason of homotopy theory, divine homotopy groups, the components of these path spaces, their function spaces. In topology, there's not only the path space, my god, there's also the space of functions on the path space. All these things are cohesive spaces in their own right. This is fundamental. Already contained in Frechet's idea that space with functions is not an abstract set. It too is cohesive. It has its own tools. It goes back to the early days of the calculus of variations. Varying a function means the fact that there is some cohesion and hence possibility of variation within a function space. So if we're going to do algebraic topology and functional analysis, which are the main ways in which Abstract topology gets into real mathematics via punctual and algebraic topology and their interactions, but we certainly shouldn't have any thoughts about a space like that. That should be as natural as we can get, and the classically determined category doesn't exist.

52:30 It doesn't have a well-defined meaning. But in the style of Fox and Friends, they invented another category, which it did have a meaning, My several spaces didn't really change, except for what was taken as fundamental in the general definition of an arbitrary object. I think this is an important, this is in some sense an important point that you might say, well, some anthropologists might say, I'm only interested in subspaces of Euclidean space, and I wouldn't say that nowadays, but you can imagine that sort of statement being made. But it does have an influence because basically what a concept like M combines is a particular and a general. The idea that the general is very important in how we deal with the whole thing. So you can keep the same particulars and modify the mode, in this case literally changing from the algebraic back to the geometrical mode. ...determination of the general which is better. In this case, there are several slogans for what better means. Cartesian pose. The existence of spaces like this in a category is called the Cartesian pose. Jim talked about this one. It's very well known nowadays. The classical category of topological theory. Basically because it's analyzed in an algebraic way, I think. Because it seems to be, we don't have a precise theorem, but it seems to be a recurring phenomenon. But not only in that case, but in others, when the basic definition of a general object has algebraic sense, you won't get Cartesian closure. On the other hand, when the basic definition of a general object is geometric sense, then Cartesian closure is very easy to achieve. It's either a comedic term or an analysis of Norman Zane.

55:00 The slogan convenient, pay too much attention to it, because the slogan convenient suggests some kind of utilitarianism, whereas it was actually a fundamental re-expression, or clearer expression, of the same sort of demand by the general definition, primarily on which was so important to make the population of algebraic topology go smoothly without these unknowable questions. Not at all that one is excluding the other, it's a matter of what's taken as primary. 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 Basic geometric objects within algebraic categories, but it's a difference. It's a difference which is primary, which is secondary. Secondary means you accept anything that's natural with respect to the first. If you have one kind of structure, then with respect to that, there's a notion of naturality and putting all the natural maps into our algebraic and our secondary categories. Both determinations work as well in the outer reaches. The outer reaches are sometimes in a condition called pathologies.

57:30 So there's a difference what kind of pathologies you're willing to like those which block simple qualitative properties like Cartesian closure. Can I ask a simple-minded question here? Is your motive for choosing, for distributing the descriptions geometrical and algebraic either historic or what? I mean, well, figure and instance relations are what geometry is about, operations and quantities are what algebra is about. That's historical, historically established. Each taken in a narrow sense, right? You could say that algebra is everything. I mean, categories are part of algebra. Yeah, that's what, that's what we're supposed to do. You take algebra in a very general sense, but I'm taking it in a narrow sense, namely operations on quantities, the system of operations on quantities, the comparison of systems by math that are called homomorphisms, and so forth. That type of applies here. Abstractly here and more concretely in the way that it's used here, because we have algebra of functions, the algebra of continuous functions, the algebra of C and W, or the algebra of propositional numbers on a particular set of space.

1:00:00 I was trying to, as I said in some recent papers, I think that category theory has many different kinds of advances that have been made in it that it's rich enough to... These are all comparable with most general philosophical questions. In other words, I'm putting that in contrast with the situation 50 or 60 years ago even, 100 years ago, when in mathematics we had real numbers, we had string in space, we had quivered algebras. We had a lot of very interesting things, but they obviously had some kind of very particular significance in physics or something. Couldn't say that they're... A fifth dissonance to attack the most general ancient philosophical questions. By contrast, I'm just speaking very loosely and intuitively, it seems that the panoply of concepts and powerful methods we have been calculated through would permit us to actually build some determined models, some determined models of ancient philosophical questions. Naturally, there's a great reluctance to do this. There's reluctance to do more than one thing at a time, and you should actually advance the philosophy that becomes so eventual that all you do is you tell the same old story that you told 30 years ago or 300 years ago, more or less, and the general conclusion in this discussion is, well, it's very difficult to know. The proposal to make actually determined models using... We can advance methods and try to advance something that the natural information for some people might be to ignore or reject or try to avoid.

1:02:30 Now, another that's progress, I'm thinking of the tradition of Christian de Molte and Suarez and these people who don't say, I don't know, at the end of the sculpture. They say, I know it already. There's a kind of ready-made theory of methods to avoid... It takes the form of what seems to be called ontology and metaphysics. It means that somehow, by pure thought, we can figure out which concepts are real. I think that's what ontology and metaphysics are. I never studied philosophy. I keep looking into it. I'm shocked by these things. That's the impression. I'm hoping you will help me here. Yes, granted, all this pragmatic stuff about how you can't know anything, then the alternate response is, oh yeah, granted, you could have objective concepts. You could try to find objective laws for building up objective concepts. So look at all the objective concepts there are. Now, don't you agree that we want to talk about the real world? And don't you agree that some of these concepts might apply to the real world and some might not? So let's sit down and speculate and try to figure out which of these objective concepts are real and which are not.

1:05:00 That's ontology and metaphysics. The philosophers don't like to put it that way. Well, this is a plea to ignore what they say and actually make some progress. It's a plea. You can see that it's a plea, right? To sit down and say, well, okay, we could imagine a boundary like that, but we couldn't imagine one like that. So that's excluded. I could imagine this, or you could imagine that. I could not imagine this so easily as I could imagine that, blah, blah, blah, blah, blah. This is pure speculation. This has nothing to do, really, with making the arsenal of mathematical objective concepts apply to reality. It's a pretense of doing so. But it's a major diversion. You could easily spend the rest of your life doing that if you want to. I mean, it's up to you. Having said that, I do think that some things are more real than others. But there's a basis for it, but something which is probably less real. Measurable cardinals. In the 1960 paper, measurable cardinals would be a better name for objective mathematics. I say it that way because There are results that Chrysler used to tell me about to the effect that measurable cardinals are good for subjective mathematics, i.e. for languages. But anyway, what is the point with this that if we can find co-adequate, can we find co-adequate subcategories of the categories of mathematics?

1:07:30 It turns out that this question... We'll have the same answer for abstract sets as it does for uniform spaces, for monological spaces, for a whole series of categories that arise in the analysis that actually have some cohesion going on. So, having noted that, that had been noted long ago, it's significant to consider it when you take the abstract themselves. What Isabel showed was that there is a small category which is co-adequate in the category of sets. There are no Houlan cardinals, no measurable cardinals. The whole construction reveals the word measurable to be quite a misnomer because it's almost the opposite of what's going on. The idea is that we should be able to determine maps between sets, arbitrary sets, arbitrary small sets, by knowing all about the functions on these sets and how they get transformed backwards. But this is a this is just a small kind of fixed and what this more precisely comes down to is that one has to calculate anything works here the real numbers will and it's better to the abstract set underlying the real numbers but if we look at the at the natural maps with respect to category b which is just the enum maps of itself

1:10:00 Every point of X determines such a map, namely the evaluation. So there's a canonical process which, to every point, assigns the process which, to every observable function, whatever you want to call it, assigns the value of that function at that point. Anything of that specific form, coming from the evaluation of X, will certainly be as natural as you like. The actual points, it turns out that this would be the way to do it, but look at those things and walk and talk like them in some ways. This will precisely fail if X is non-cardinal. Those will be true for all X, if and only if. And as I say, the same principle is applying to other categories of the whole of nature, built over the sets, structured categories built over the sets. Is it just the measurable ones themselves, or is it pertaining to one of these things enough to screw things up? It's the ones that are big enough. Yeah, the big enough ones. In other words, if those big enough ones are precisely not measurable by functions, if I think of a sort of more naive idea of what measuring means in terms of these functions...

1:12:30 If X is too big, it fails. You fail to be able to measure. Now, Ismael's theorem is actually about not just the real, but any fixed set. So, if we have faith that we're going to be able to eventually measure everything, at least ideally, then the existence of these measurable cardinals is thrown into doubt. The world of objective mathematical concepts. There's some reason for doubt in thinking that they'll ever come up in a real situation of which that property is a reasonable reflection. In other words, putting that in contrast, I would find my main bias in advanced ontology and mathematics, the experience seems to be really that in general, almost any reasonable mathematical... We'll eventually have some real application, in some way or other, maybe unforeseen. The objective of mathematicals really are as far as we know possible, and it's difficult to draw any upward line. But, certainly the mathematical kernels, for this reason and many related reasons, seem to be... Of course, screwing things up is a use of sorts, isn't it? Well, there are people who want to screw things up. The set theorists did make, of course, concerted attempts to prove them inconsistent. I don't know if it was for this reason, as a matter of fact, but nonetheless, they certainly made concerted attempts to prove them inconsistent, and they just failed, all that was. I know, I know. That's what I'm saying. So they may well be consistent, too. That slogan makes sense, right? Immeasurable cardinals are good for subjectivity. In other words, if you talk about huge languages...

1:15:00 Yeah, yeah, no, no, because it was compacted. The terminology also, of course, in large cardinals goes in the opposite direction. The larger the cardinal, the more constrictive, you know, constrained the term. Strongly compact, compact, strongly compacted, even bigger. Extendable, it's even larger, right? I mean, it's completely averted the actual order, duels the actual order. But what I'm really trying to suggest is that, in other words, that other measurable cardinals don't exist either. Small omega. Yeah, that's what I'm... Do you have to go to, for such a non-set theorist, where a sure-share subject matter is measurable cardinals to make this point? Isn't it perhaps already there in the Banach-Sipinski-Tarski paradox? I mean, is that ever going to get reflected? Well, yes, but is it going to? Well, yeah. If your point is about these things being measurable, no. No, I don't care. It's a bit different. Yeah. There is this general vague notion of set within which there are things that are more variable and cohesive than those that are more constant. Those that are sufficiently constant, and that's what Brittle showed, those that are sufficiently constant satisfy B equals L. So therefore it's true in that sense for the constant one. The fact is that they're both reluctant to accept the equal cells because they have some vague feeling about those variable ones, even though they don't recognize it in that form, that's what my feeling is. But in the same way, you see, I mean, the action of choice, which is a particular fragment of that, okay, well, that's true for constant, and that's an ideal. Extreme within the phase of maximum choices, and hence Bonhoeff-Tarski is true. But the reason we feel Bonhoeff-Tarski is false is precisely because of course it's false for the real world or anything even remotely like, you know, imagined like the real world, because there is some trace of some collision, some variability somewhere that prevents us from making those choices. So in that sense, the Bonhoeff-Tarski paradox is kind of an accurate description of what's going on for the constant steps. It should not be.

1:17:30 On the other hand, certainly, you can see that without talking about the alternate world, this is one conceptual world that contains both constants in the area of the self. Is there some analogy of this pathology for the smallest measurable part of a lingual thing? That's what I just said. I was just going to say that. Yes. I mean, I don't know if it's an analogy, but I don't like it. In other words, it seems to carry over from the constant sets to other more structured things. In other words, it seems to be the idea of such an object itself, which is a little bit out of place. It's saying we're hindered from measuring things. Well, the Balatowski paradox presumably doesn't carry over. I mean, in the sense that you're describing, it really doesn't carry over to these other situations. No, it certainly doesn't. I mean, because the actual choice simply fails. That's right. I mean, of course, there's all that wonderful technical work that you and Cohen and everybody else did, you know, back in the 60s, right? I mean, once you grasp the point that most sets are variable and constant, it's obvious that the continuum hypothesis is false for them, and so on. By the way, just a footnote to this. I noticed that there was at least one guy, there was a paper by some guy in Chronicle Physics who was actually trying to Did you see that? Yeah, some guy. I have to say, it looked very dubious to me. It was a coincidence between the numbers of the quark theory and the numbers that come up from the dominant... Quarks prove the axiom of choice. Yeah, I mean, there's some extraordinary... Empirical proof of the axiom of choice.

1:20:00 That's what I had in mind. It's so very, yeah, it's so very devious, but the smallest measurable caramel seems to be devious also. In other words, there is the Piano space-filling curve. See it? Here it is. And many other related results. For example, nowhere differentiable functions. Constructions that occurred around the turn of the century, the last century, same time. These things exist only because of natural numbers. I guess even by the authors themselves, it's rather paradoxical. For some reason, the mathematical public didn't stand up and demand a better foundation to exclude them. It seems to me that they should have. This is a much more outrageous... Distortion is the way the mathematical world ought to be, and some game with words like Russell's Paradox or something that allegedly motors behind attempts to make foundations of all the world's systems. Couldn't you even say that it was worse than that, that they demanded and got a foundation designed to include them, precisely by taking as fundamental in the definition of an arbitrary object this objective?

1:22:30 I think so, yes. That would be one of the two points. This is very much contrary to physical geometrical intuition and that people should have been more expert and sensed in their work, it seems. But then the analysis of why, what do you think, what do you think it's come from? They depend essentially on the natural numbers. In other words, my teacher Eilenberry, for example, was fond of pointing out how simple are the approximations to the Keanu curve. So you just make some little triangle and then you iterate the process at each step and you can calculate the complexity of the machine required to calculate the next step given the previous step was on the lowest possible. There's a lot of complexity. But you see, the assumption is that the process can be completed, that the limit actually exists and then it becomes a... So it's a process that's indexed by the natural numbers. As you say, I think the fundamental thing about it is it's a subjective process. It wasn't a physical process of going from one step to the next. It was a mental one. If we can imagine one step, we can imagine the next one. So that the idealization of conceptual into the real geometry here, saying the same thing could be said for this and many other similar paradoxes. This sort of thing is sometimes, when it's sort of talked, is sometimes construed as anti-infinity. It's not anti-infinity, it's anti-calibration. It's freedom infinity. It's taking place within a continuum which is not, which I'm not calling into question. Now, but now you see, the fact is that it's hard work to try to, it's hard work to be more precise on this.

1:25:00 So it's easy to say, well let's accept the natural numbers, let's accept all these bad consequences. It's possible to, careful work, to avoid them, to avoid these problems. I wanted to cite in particular the work of Adam Dries. And several other clinicians working on Karski's problem about the definability of the exponential function theory of real closed fields. The theory of real closed fields in itself constitutes a kind of category, you can view it as a category, in which you have lots of continuous maps and you have no space-filling curves because you have no natural numbers and that's why it's decidable. There's a strange correlation that I don't understand between the subjective question, is it decidable or not, in the usual logical sense on the one hand, and the existence or non-existence of these geometrical monstrosities on the other hand. So it sometimes results about this, you might not realize that that's what they are because they talk about decidability and non-decidability. You have a certain continuous map, but you don't have the natural numbers. And therefore, you either have a cyclically or a space-forming curve. But of course, it only consists of polynomial numbers. And that's obviously not good enough. I mean, it's good enough for algebraic geometry, but it's not good enough for all the different kinds of analysis that we want to do. So to push further and find categories in which your effort might seem to give you a real number of them, The type needed in analysis, and yet still avoid the natural number object, the first measurable cardinal, with the concomitant geometrical on strategy. These are requirements of long, long work. In many degrees, it achieves a certain singular breakthrough in that direction in a few years to come.

1:27:30 People are still moving out of the consciousness of that. There's an old result of Peter Pryor, basically just formalizing the setting of toposes, an old result, that a topos, which contains a non-iso, endo, mono-endo, has a natural number of it. What's a non-iso, mono-endo? Well, an endo would be an endo of some object, a mass of some object itself, which was injective. And yet, not invertible. It's a prime, prime dedication. A dedication, prime. That's what I'm saying. Well, I mean, it was something to show that it works. Oh, I know. I said it. It's dedication. Probably doesn't need my help. Because that wasn't the point. I'm signing a definite Australian paper. We find this result and many more. Yeah. So, if you have an object that's not convertible, and some object, except the topos contains this, so I call the object R, but it could be anywhere, then basically what you do, of course, is you look at the subset, the power set of R, and the portions of that which are closed under this endo, and you take the intersection of all those. You take the intersection of all the subalgebras, but you don't have a starting point if you...

1:30:00 The crucial point is that this is true, and the crucial part of the construction is that it uses omega and arbitrary intersection. The arbitrary intersection is a consequence of the definition of omega, as Jim pointed out once again this morning. It's defined for all explicitly, algebraically, using the property of omega. What's this got to do with this book? Well, any category can be embedded in a topos, right? So we can embed Van Andries' category in a topos. I've respond to all this work to exclude the natural number object, in effect, and put it back in. There's Van Andries' category, and you put it in a topos, and maybe in a... The point is that there are lots of such things, right? Take the half-open line and shift by one. So, surely, any reasonable category, unless it's very specialized, that has any sort of pretense to generality, it will contain endomorphisms which are monomorphic but not infertile. On the other hand, well, it's very good. I advocate embedding all categories in toposes because it provides a very convenient and useful in many, many ways. So there's a contradiction between that and this, dealt with natural numbers, but any topos were essentially important. So, in other words, there is this idea that, well, somehow, abstracting away from what Mandegris did, which is a beautiful thing to do because it's very serious, particular work,

1:32:30 We want to try to make a general conception of what the end result here could be. This is just an example of what I've called at various times bro-tokos, after using one of Rodenbeek's terminologies, or just general tokos, in the sense of being a general category of spaces, or in a recent paper as a category of being. In the sense that there's smooth being, continuous being, combinatorial being, algebraic being, that way, that space is particular beings that obey one of those more specific, that sell on color-specific ideas of cohesion, it should look like that. In other words, it's got this damn to make it object. Why do I say damn? Because on the one hand, it's very, very nice, the algebra of prophecy. On the other hand, it conspires the non-invertible to produce the natural number object. There are these things related. For example, this is like a spot of light, you see, within all this darkness. This is the picture that I'm trying to promote here. Anyway, you see that in here you might have an object which is a line. These are the real things. These are the subjective, these are the tools that we use. This goes through the inside of our brain, really, because we've gone through all this process, mental process, to construct this vision of reality. So, in our brains, we've got this vision of reality, but we've also got surrounding it all the machinery that we use to derive it, including things like abstract truth, not just a mental construct, but also the natural numbers.

1:35:00 The N is not a bad model, maybe, of the subjective process, but it has to be excluded from, but it could be a sub-object of the line. The usual proof that you find in topology books, if you've got the line, you've got the natural numbers, namely by infinite intersections, it produces it as a sub-object. So in the big category, you have that inclusion map, but this object is outside the band of light, whereas the line, that's what you were trying to do, might be the benefit. This raises, for example, the question, well, certainly in most cases, omega has got to be excluded from this. In some other cases, it might be included. It's not something that's decided once and for all. It depends on nature, whether omega is contractible or not. There's a comparison between two different notions of higher power sets built starting with Omega and just plain function spaces which are built up not necessarily starting with Omega. These are certainly two different things which have a lot of interaction. Well, now, we're talking about isolating subcategories of toposes. Topos is thought of as sort of the developing... This, by the way, you see, this clearly inverts the relation between thinking and being. The idea is within being, but when we reflect inside, that gets inverted. The vision of being is within the vision of being. So, when is this closed under exponentiation? Is it ever or is it some?

1:37:30 Does mere, in other words, to put it in a more technical term, does mere exponentiation in conjunction with some other semi-critical property of ours comply with the existence? I don't actually know the answer. Maybe it's a different question then. But coming back to ontology then, finally, ontology and mathematics, which I'm just seemingly promoting, my idea is that, yes, okay, within this conceptualization, which includes these clearly subjective aspects, like omega and n, and on the other hand, some clearly objective ones, depending on what the application was, then there's a borderline. What's the borderline? Should we argue about the borderline? I say rather we should we should be aware that the borderline may shift that there is this there is this I have a pre-ordered set just like just like you see we have that picture the whole compost and the quote unquote objects within it that's an important that's an important conceptual guide even though we don't make it precise and we want to make it precise We probably find it has all kinds of different answers depending on the particular situation, but we should not expect to find, should not expect to find, some once and for all answer that can be answered by speculation about whether angels can dance on the head or not, only by rather serious mathematical work can we give a definitive answer. How about a Cartesian closed, pre-ordered set? Yeah, but pre-order, I mean, you know, because otherwise you don't satisfy, and that condition is going to give you, I mean, if every one of those things, if every Bono-Endo is probably, you know, actually partially ordered or something like that.

1:40:00 What I mean is that the condition of being Cartesian closed is fairly weak for, I'd just say, in the case of a hiding algebra. Yeah, but on the other hand, if the Cartesian closed category is also opposed to that, it's unlikely to have a natural number object. I mean, no, just think about it at once. In that, you see, since you mentioned our work with Ikea, there we derived a notion of a smooth natural number. These are, so to speak, the majority of the things. But here it seems to be that you can also do the same thing, because you said that in this function, you use sinus function. Is that a permissible operation? No, no, no, no. No, no, that's okay. Sandman-Drees' very particular termination, hopefully, with all of us, is that there are no analytic functions. There's a category of real analytic functions. That's it. They, at infinity, they are asymptotically polynomial. On any bounded piece of Rn, they are more, they are arbitrary, theoretically, but, so, so, the sinus function, the idea that the solar system will really go on literally forever, that's excluded. You can have the sinus function on any bounded interval, because it's an actual math in its category, but not the sinus function on the whole thing. But you see, you don't actually need the sinus function of your conception. You use higher types, as I tried to point out to you before. Instead of coordinate-izing the circle, you could just use it as such. In other words, the circle S1 contained in R cross R. So R is any ring object, and then you look at the part defined by the equation.

1:42:30 That's an abelian group. And since you're working in a Cartesian closed category, the base of homomorphisms from S1 to S1, well classically that's the integers. And that is the same as your smooth integers in your particular class of topology. The smooth integers would be that. In other words, these are the ways of zipping around the circle in such a way that preserves the addition of angles. And intuitively, you can only double or triple it or be discreet, at least, of course. But the totality of those, classically, it's the integer. The natural average is really a very negative thing. In your case, it's not. It's discreet, and that's very interesting. There's a model. But you see, the point is, it uses, instead of the coordinateization, which you use, but it's not really needed for the particular construction, This is contained at the pumpkin type, that's one. So you're using the Cartesian closure, but not in any serious way to tell all these questions, to simply equalize them. So that's problematic. In other words, can we formulate an exact question? If you've been in a topos and done some research, when you're basically guided to take any written object in any Cartesian closed category with pullbacks, then you can make this construction. Is this a natural number operator? In other words, does it satisfy the recursion property? Notice the big difference. This definition is as an equalizer. This is a sub-object of that that's defined by the equation that says it's a homomorphism. It preserves additional angles. And so, therefore, by its construction, I know everything about maps and other tests and objects into it. It's an equalizer. By contrast, the recursion action of the...

1:45:00 The way in which the idea of a natural number of objects concentrates and idealizes the idea of repetitive subjective process is in terms of maps out of it, so the mutual statement is that if we have, generally formulated in a more general way, but as we pointed out, any Cartesian closed category this way is sufficient, if you have just a way of going from one step to the next and if you have a starting point, Then the recursion axiom says that there will therefore exist the sequence that's defined by that. So the recursion properties in terms of the maps out of it, it's not at all obvious in the simple-minded plane view anyway, how you could go from an object that's characterized in terms of maps into it, over the one that's characterized in terms of maps out of it. So, but the understanding of which criteria can close chemicals is which brain objects will tend to produce things that really do satisfy the recursion that we don't. So this is a definition of how we describe it. Ways of, well, unideal coverings of the circles, things like that. Objects collected together into one, but that doesn't in itself say that you could... Is the same analysis applied in multiplicity for GM? That's what I'm saying. GM of the complex numbers. That's what's there. Yeah, I think it would be essentially the same idea. But didn't the... ...continuous exponents until everybody agreed we're winning.

1:47:30 That only cured the speed ones. By the way, homology and chroma outside. Homology groups are often equal to z. So are they supposed to be interpreted as purely subjective records of our qualitative understanding of the shape of that space? Why do they have more objective interpretations which would be possible to use this food here? But then homology achieves the problem. But they come out of these spaces themselves. I think homology achieves the problem of homology groups. You mean the values of homology groups? Kind of a naive question. I'm really surprised that you don't like things like piano, space, scale, and curve, and you're dismissive of the Russell paradox. I guess I have the orthodox intuition that it's exactly the opposite. These things are amazing, even pleasing, and just surprising. But can you tell me what your objection to the piano space something curve is? Like, you said something about, you know, you've got an iterated process, and you have to think of it as a completed infinity, and there was something about natural numbers indexing this iteration. But isn't that sort of typical of almost any limit process? Your objection sounds like you didn't allow sequences to have a limit or something. No, I think a typical limit situation is you have some function that you know is real and you want to approximate its value. So you're approximating something sequentially with something that you already know is there. In other words, the principle of these function spaces are Cauchy complete just using the one metric, the suit metric.

1:50:00 That's the thing. That's because I can imagine the process of coming to know it, rather than, you know, there is a thing, okay, and then I come to know it. Is your problem anything? Is your picture that we've got two objects on either side of an equal sign, that the infinite side is just an approximation, but it's not really an... Equality? Like if I've got pi equals infinite series, does the infinite series really mean, and the equality really mean, not really equal, but can be approximated very closely? Well, it means that there is a definite process which does approximate, and that process is indicated by the right-hand side. Pi already exists for geometrical reasons, as opposed to the... But the equal sign doesn't really mean identity anymore. It's a disguised something else, it sounds like. I don't see your point. I think calculus is not an identity science because it's a limit. But the limit is not a real identity. No, I think it is. It is an identity. That statement is followed by an epsilon of any ordinary calculus. That statement is defined by a complicated epsilon of Delphi conditions. So it's not just saying... Here are two things that happen to be identical. It's an identity that's saying that two things come out equal. There's also two different maps coming out equal. One map is of the nature of let's look at ratios and blanks or something like that. The other is there are certain processes which make the limit. Now the issue is really how big is this domain of things for which we know the limit should exist. There are certainly some domains. For example, constant sequences obviously all have limits, right? I'm saying it's not all Cauchy. At least not all Cauchy. Cauchy is always with respect to a given metric or a given uniform structure.

1:52:30 The usual idea is we merely take the soup metric and we have Cauchy. And that's what technically would be called the domain of sequences for which there is limit math well defined. I object. The reason I object is because my aspiration has always been to be a physicist. I believe in things like the electric field. I don't see anything in there which would be the remotest idealization. Obviously, if you make idealization, semi-idealization, I don't even know. So that's the angle from which I— Well, I think a related point in the theory that Eudoxus did not really—it's sometimes claimed that Eudoxus actually invented the num set of real number in some kind of 19th century sense, but of course he didn't. For precisely the reasons I think that you described. I mean, what he did was to provide sufficient underpinning of the notion. Of a procedure for calculating ratios of length, which would include something, for example, like the square of the diagonal on the side of the, the ratio of the side of the diagonal of the square. But there wasn't the notion of an arbitrary iterative process. It was a process which would be carried out in particular cases. Well, which was justified because they were striving for knowledge. And also they were not, no, that's the other, that ratio is not numbered. But the things that were being, the ratios that were being calculated were given antecedently. Of course, of course. They were given, well they were given, they had to be, with the ratios of geometric quantities that were already given. I mean the whole issue was not the irrationality of the square root of two for the five vector rings, as you well said, I mean it was the fact that you have two definite and clear geometric entities whose lengths can't be

1:55:00 All of these terms are related to the previous, within the previous number, the number theory, arithmetic. So it wasn't a kind of arbitrary invention of the, although there was, but again, that's another interesting point as far as the processes are concerned, because there was, of course, also a process that led to An argument, an additional argument, which led to the, I'm not sure whether it was actually contemporary with whatever argument was given usually in the first one, but that one can make a kind of iterative geometric process to show the incommensurability of the solid of a square. What about the method of exhaustion? I mean, isn't that really, you set up a process in... ...areas of polynomials that you have deduced that there must be a line of a certain kind, because you're saying that circles are to one another as the squares are their diameters, and you have... Aren't you presupposing the existence of that thing first? You're just trying to get at it, you're trying to come to know it. Yeah, yeah, but you are. I think it's... But I do think that there's a sense in which I think the idea was that it was some sort of process of completion that occurred in the 19th century, an attempt at idealized completion of these sorts of processes that led to all of this idea, for example, of the Dedican reels or whatever, and the Koshy, it was this attempt to make the whole thing in some way full. So that it couldn't, it need not be extended any further what, you know, come what may. Of course, they didn't naturally, as you've identified, because of the fact that one had to use the, employ the notion of an arbitrary iteration in a completely general way in order to make that completion, of course, then you end up with these curious, you know, these monsters. Yeah, that's obviously a very important step to make. Let's take everything. You will always treasure that stage, mysteries. Then you have to come back and look more closely and say, ah, I wanted to remark that van den Dries' result is in many ways the same as Grotendieck's, that is to say, Grotendieck from a totally different angle, not caring about decidability, undecidability, that sort of thing, but wanting to have a topology, what he called tame topology, that wouldn't have pathological properties, also arrived at rhetorical functions, unpublished work.

1:57:30 But in the same decade. But it's a process. It's a process of approximation. First we know nothing. Then we learn something. We know everything. We step back and say, well, we better look at it more carefully. Again, I don't want to take a permanent ontological position in that sense. I'm saying that that big loop, that big weave into... Simple-minded fullness was a big leap that was an important stage in coming to know more exactly space. We have to step back with more refinement. Well, Aristotle was good with the... Space and bodies. Right, right. But Aristotle also had this notion, of course, that the continuum is infinite. I mean, the continuum, it has parts and they're just more continuous. Yeah. It's just, they're coethic in that very strong sense. Yeah. The real parts of a continuum are exactly the same, in some sense, as the original continuum. You can find it in book whatever it is. You read Aristotle, I know it's my reading here, John. It's in book six, or somewhere in the physics. I mean, there's quite a famous definition. And also, it's plain to go further and say that this is already divided into atoms. They kind of subject it by the way they see mythology, basically. But Aristotle then calls the result of the discrete, because he calls it, I mean, as a kind of potential. The infinite in that discrete sense is potentially there. It isn't actually there, because you don't, it never will be, but it's potentially there. Nobody has mentioned Zeno, I mean, it's huge, so that'd be a similar thing. I said, what? This morning, I said, yeah, let's see.

2:00:00 But I don't think that the word, the compactification that's involved there, though, is not such an interesting thing. It's much more idealistic. Yeah, but what kind of curve is it? That's quite a different matter. Thank you for your attention.