What is the true relation of logic and set theory? (contd.)

0:00 And the yes, no, and yes, but, not, is something I've heard of in the media today, turns out to be fair enough to be that even that small media category is no more exactly true, I think, at a moment like this, when I say that I'm not going to achieve this. What you might like, you might like to achieve more than both, basically. Both in the geometrical style of analyzing, as well as in the algebraic one. Namely, you might like to achieve not just the full unfaithfulness for those objects M that you've accepted, but some kind of condition on these structures in B to A, or in B to S, or B to S, or B to S, which will characterize and nearly only characterize the actual objects that you've created. Full unfaithfulness only refers to the fact that we've managed to quickly characterize the image of... The maps for any given pair of objects. Another important issue is to characterize which objects here could arise from some object X, which abstract structures of the kind figured in this situation could actually come from an object X. So I think that's probably the answer. But anyway, the usual answer is to come back to the B consistent with all the powers of Q. If we look at the product-preserving functions of EDS, 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 comparison function, we get values which are product-preserved.

2:30 If we look at the product-preserving functions of EDS, well, those are exactly the same thing. And that turns out to be also a characterization of the algebra over an algebraic theory in general between product-preserving and, in this case... The finite sets and the finite fields to get the, that's the vision of the characterizing of the finite sets right now. Oh yes, so several comments I want to make about this very simple minded and very powerful point of view. First of all, what's topology anyway? Topology has to do with cohesive steps, continuous math between cohesive steps, Trying to build through more determinants. So the M is continuous categories. Continuous, somehow continuous. People thought about, Cantor thought about that. I think that was one of his main engagements, abstracting. But anyway, I guess it was Hausdorff. He said, well look, a very important thing about a piece of continuous math is that there are these things called open subsets. Of what collegian and contiguity should be, like a very productive determination, but not necessarily a final one. In fact, definitely not final. Notice that it means a very strange thing. It means that topology as we know it from housework is really algebraic.

5:00 We like to think of it as geometry, but the analysis that's given is 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 suspension space, the continuous map of x into the suspension space is exactly the propositional function corresponding not to an arbitrary subset of x, but to an open one, or a closed one, which we assume you think is true. If this was the open one, it's 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 that this is the characteristic function of an open set V. And the composite is again the characteristic function of some open set. And it's easy to see that the fundamental logical operation of substitution is what corresponds substitution into Class, property, subset is nothing else but this complex, like the composition at the level of the structure of the property of the function. So, in other words, P sub P composed of F is the same thing as the characteristic function of F inverse of P, in any case. So, the inverse of P goes to the subject, right? Of course. Well, see, thinking of the M as given and then isolating... There's the general vague idea of the M, and then there's this very specific determination of the M. 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 an algebraic data. We 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 matching to S by means of... I quote, union. So that's one of the betas. And likewise, S-process maps into S by hand. That's another one of the betas.

7:30 And so, in other words, these kind of algebras that we're getting here automatically outgives the algebra of open sets. It has operators, of course, to find that intersection and union and naturality over here just means nothing but preserving them. So, in other words, there's the... There's the analysis of what's now called frames or locales. Frames are the algebras, but they're often frames of the locales. And the locales are very closely topological spaces, not quite. In a way, this concept of locale, which has been isolated for the last 20 or 30 years by people like Isabel, and I don't know if you've heard of it. On the one hand, it really shows in a 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. It was by far the inferior part as far as the actual calculations were concerned, and within the traditional determination of topology. So the formulation is that the strings, which are these algebras, and they're homomorphic, and the opposite of that, which is called a tau, really shows, throws into a sharper light what the housework and polish, et cetera, et cetera, in topology was really always about. On the other hand, 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 in the wrong direction as well. But according to me, there are lots of things, lots of precise determinations, combinatorial space. There's probably other determinations of continuous space as such, which are maybe an open problem for research.

10:00 We could try to find a really, a truly geometrical analysis of the 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 closing 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 within the arbitrary object x, I will associate the system of all paths. And the system of all paths easily becomes a system because we can operate on the right with these alphas. 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, a way of re-parameterizing it, what's the result of re-parameterizing it? Now that includes the constant paths. So it includes the determination of points, a sort of remark 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 can be used. We're going to take points, which are the maps from one to the other. Maybe we also take points in the sense of this number category, this topos. The notion of points should be taken to be the same over here because there's a general idea of maps for one, maps for 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 aren't actually the real points. There's no, there's no funny business going on here that I'm telling you you should believe that paths are points.

12:30 No, it's the, from the system of paths I can extract the particular points more concretely as a constant path. That's a very interesting idea about how to model topology. I first thought of this because I read in Walter Knoll, who works in the foundation of continuum mechanics, that the basic purpose of the topology on a state space, in particular, state spaces for systems with continuous bodies meeting, the basic point is to be able to talk about the continuous paths. And we can imagine that as time has, the possible processes... So that's sort of the main use of topological data in practice, but many branches of practice is via these paths, and especially in, actually, I mean, 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. They're very arbitrary formats. Sometimes it's called singular figures. In fact, the term singular cohomology, or singular homology, which we have no doubt heard at any time, first did 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. Sorry, I'm not excited. Where was I? Oh yeah, the last one was topology. In fact, you could imagine in 1940 a headline in the newspaper, or again in 1960, Topologists reject topology. This is literally true, meaning that people who are actually interested in working on space

15:00 As opposed to those who were developing the consequences of Haussler's axioms to realize that there was something wrong with taking that Haussler determination as his final, it was Fox's principle. Basically, because basically this came from a geometrical reason. This came from the reason of homotopy theory. Because here Avis defines homotopy theory as the components of path spaces. What are these path spaces? They're something. What are function spaces? Ah, there are also function spaces in topology. There's not only the path space. My God, there's also a space of functions on the path space. All these things are cohesive spaces in their own right. This is fundamental. It's already contained in the Crescenes. The space of functions is not an abstract set. It too is cohesive. It has its own cohesiveness. It goes back to the early days of the calculus of variation, varying function needs back into its own. Cohesion and hence possibility of variation within the function space. So, we're going to do algebraic topology and functional analysis, which are the main ways in which abstract topology gets into real mathematics via functional analysis and algebraic topology and their interactions. But we certainly shouldn't have any class about a space like that. That should be as natural as it can be. And the classifying of the early categories doesn't exist. It doesn't have a well-defined meaning. But in Scott, Fox, and France, they invented another category. They did have a meaning. But literally, the nice, simple phases didn't really change, except for what was taken as fundamental in the general definition of an arbitrary object changed. I think this is an important... This is, in some sense, an important point that you might say, well, some topologists might say, I'm only interested in the subspaces of Euclidean state. I wouldn't say that nowadays, but if you imagine that sort of statement being made, therefore, what do I care what's the foundational definition of it? But it does have an influence because basically what a concept like M combines is a particular and a general.

17:30 The idea that in the general is very important in how we deal with the whole thing. But you can keep the same particulars, modifying modes, in this case, literally changing from the algebraic back to the mode, and get a different determination in the general, which is better. In this case, they have a general slogan for what better means, Cartesian flow. The existence of spaces like this, in this category, called Cartesian flow, that Jim talked about before. These are all known nowadays, but classical categories are basically because it's analyzed in an algebraic way, but it seems to be, to have a precise theorem, it seems to be a recurring phenomenon, but not only in that case, but in others, when the basic definition of the general object in algebraic sense, you won't get. On the other hand, when the basic definition of the general The method says that Cartesian soldier is very easy to be true for Zinc. Not only in 1940, but again around 1960, Steenart, who rather than being a slogan for genius, a philosophical phrase that philosophers didn't pay too much attention to, because he was a slogan for genius, for yes, militarism, was actually a fundamental re-expression or clearer expression of the same thing Making the general evidence in space depended primarily on a particular figure to obtain the Cartesian closure, which was so important to make the calculation of the long bridge probably go smoothly.

20:00 Not at all, but one is including the other, no matter what state in the primary. It might well be that this category M has an object like the S over there, and so it will transport over here, so that the general objects that we talk about will have as a derivative of the Eulerian notion, open set, because they're independent, and so it is not the one and only binary one. And in the same way, we can find the basic geometric objects, but then on the greater categories. It's a difference which is primary or secondary. Secondary means you accept anything that's natural with respect to the earth. You have one kind of structure that, with respect to that, there's a notion of naturality, and putting all the natural mass into an algorithmic regression project is our secondary practice. Outer reaches are sometimes conditions called pathologies. There's a difference in what kind of pathologies you want to accept. You don't like those if you block simple qualitative properties like Cartesian code and take down the poor stuff.

22:30 Can I ask a simple-minded question? If you're voting for choosing the, for distributing the description of Geo-Lectical and Algebra as the historic or one, I mean, the theory is in relation to what's known as the operations and findings of the historic or the established, each taken in a narrow sense, right? I mean, you could say that algebra is everything. Categories are part of algebra, and so that's what we're supposed to take algebra in a very general sense, but I'm taking it in a narrow sense, named operating on quantities, system of operations on quantities, and comparison of systems by mass. They're called homomorphisms, so that type of thing. That's here, and abstractly here, and more concretely, the way that it's used here. Algebra, algebra, algebra, algebra, algebra. The algebras are propositional, and I'm trying to, as I said in some recent papers, I think that that very theory of all the many different kinds of advances that are made in it, it's rich enough to be comparable to its most general philosophical expression, other than putting it in contrast to the situation 50 or 60 years ago. There are a lot of very interesting things, but they obviously had some kind of very particular significance in physics, so I couldn't say that they're dissonant to the path of the most general things and most optical questions.

25:00 By contrast, I'm just speaking very loosely and intuitively. It seems that the contemplative concepts and powerful methods we have in category theory 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, and I didn't think it would actually advance philosophy, but it becomes so habitual that all you do is you... You can tell the same old story that you told 30 years ago or 300 years ago, more or less, and the general conclusion of the discussion is, well, it's very difficult to measure. The proposal to make actually determined models using methods in order to advance the thought is something that the natural inclination of some people might be to ignore or reject or try to... Avoid. Now, but there's another tradition, another tradition, not more inimical to its progress. I'm thinking of the tradition of Christian Volk and Suarez and these people who don't say, I don't know. They say, I know it already. And so this is equally as important. In other words, there's a kind of ready-made...

27:30 A plethora of methods to avoid seriously taking up philosophy 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 is. In my view I'm just, I never studied philosophy. I keep looking into it and I'm shocked by these things. That's the impression I'm hoping will happen. But it seems that ontology and metaphysics basically means, yes, granted, granted... After all this pragmatic stuff about how you can't know anything, then their 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, though, 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. That's ontology in minutes. The philosophers don't like to put it that way, but... Well, this is a plea to ignore what they say and actually make some progress instead. It's a plea. You can see that it's a plea, right? You can sit down and say, well, okay, we could have, we could imagine a boundary like that, but we couldn't imagine one like that. Now that's a clue. This is an excuse. 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 can easily spend the rest of your life doing that if you want to. I mean, it's up to you. Having said that, having said that, I do think that some things are more in deal with others, but there's a basis for it.

30:00 I'll give an example of something which is probably the best deal. Measurable carbon. I say it that way because there are results that Kreisel used to tell me about to the effect that measurable cardinals are good for subjective mathematics, i.e. for languages. But anyway, what is Bill's point with this that we can find co-addictive, can we find co-addictive subcategories of the category of abstract sense itself? 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 an analysis. We'll actually have some for reading. Having noted that, that had been very long ago, it's significantly considered in the case of mathematics itself. What I've shown was that there is a small category theme that is co-adequate in a category of sets, if and only if there are no long cardinals. In fact, the whole construction reveals the word measurable to be quite a misnomer because it's almost the opposite of what's going on.

32:30 The idea is that we should be able to determine maps between sets, arbitrary sets. So I know all about the functions on these sets and how they get transformed back. But this is just a small category, a fixed small category. All the x and y are arbitrary. And what this more precisely comes down to is that one has to calculate the, if anything works here, the real numbers will, and it's better to, the abstract set underlines the real numbers, that if we look at the natural math with respect to Category D, which is just the random math, every point of x determines such a math in the evaluation. The canonical process, which is every point, assigns the process, which is every observable function, whatever you want to call it, assigns the value of that function at that point. Anything of that specific form, when you're coming from an academic context, will certainly be as natural as you like. But conversely, if we're going to recover from functions, the actual points, it turns out that... And this would be the way to do it. Look at those things, walk and talk like those kind of things. And this will precisely fail if X is new on the target, to be true for all X, if and only if. And as I say, the same principle is applied to other categories of the whole level of nature, uniformity, precision, chronology, physics, built over the sets, structured categories built over the sets, and the same property.

35:00 The whole of them, just in case it does work, that doesn't want to be a question for the class. Is it just the measurable ones themselves, or the pertaining ones, I would say, that the students don't know? Well, yes. It's the ones that are big enough. Yeah, the big enough ones. That's a sort of more naïve idea about the measuring game, because you have some terms in these functions, and if x is too big, it fails to be able to measure. Now, even now, a theory is actually about not just the wheels, but any fixed set. Right, so, you see, if we have faith that we're going to be able to eventually measure everything, at least ideally, then the existence of these measurable patterns... Although they might be consistent, although they might belong to the world of objective mathematical concepts, there's some reason for doubt in thinking that they'll ever come up in any real situation of which that property is a real thing. In other words, putting that in context, I would find my name, I'm going to blast against topology and mathematics, the experience seems to be, really, that in general... Almost any reasonable mathematical concept will eventually have some real application in some way or other, maybe completely unforeseen. The objective of mathematics is really not, as far as we know, possible, and it's difficult to draw any upward line. But, certainly numerical cardinals, for this reason and many related reasons, are supposed to screw things up with the use of tools.

37:30 And there are people who want to screw things up. You said theorists did make, of course, concerted attempts to prove them inconsistent, and they failed. 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 at once. I know, I know. So they may well be consistent, too. However, you can recall that we want to be precise, we want to be clear. That slogan makes sense, right? Immeasurable cardinals are good for subjectivity. If you talk about huge languages... Yeah, yeah, no, no, because it was compact. 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 compact is even bigger. Extendable is even larger, right? I mean, it doesn't make, it's completely inverted, the actual, dual to the actual. Do you have to go to such a non-set theorist or a chauchet subject matter as measurable cardinals to make this point? Isn't it perhaps already there in the Banach-Sipinski-Karski paradox? Are we going to get reflected? Well, yes, but is it going to? If your point is about these things being measurable, you know. I don't care. I think it's a bit different. It's an important point to bring up. You see, I mean, the Bonacharski paradox seems to be just as reasonable as E equals L. Namely, they're both true, E equals L. Namely, they're both true about constant sets.

40:00 General vague notions set within which there are things that are more variable and cohesive and those that are more constant. And those that are sufficiently constant, and that's what Gödel showed, those that are sufficiently constant satisfy the equal cell. So therefore it's true in that sense, for the constant ones. The fact that the set is both reluctant to accept the equal cells because they have some vague feeling about the variable ones even though they don't recognize it in that form. 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 Constance, and that's an ideal extreme for them, okay, and hence Bonacharski is true, but the reason that we feel Bonacharski 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 phrase or some cohesion, some variability somewhere that prevents us from making those choices. So... In that sense, the monochromatic mechanics could have an accurate description of what's going on for the constant set. It should not be, on the other hand, strict control. You can see that, without talking about alternative rules, this is one conceptual rule that contains both constant and variable sets. Is there some analogy of this demology for the smallest measurable part of a linear rule? That's what I just said. I'm just going to say that. Yes? I mean, I don't know if it's an analogy, but I don't like either of them, so I think that's why I emphasize the fact that this particular problem needs to carry over from the constant sense to other, more structured things. In other words, it seems to me the idea of such an object itself is a little bit out of place. I mean, it's saying that we're here to measure things, whereas the Banach-Tarski paradox... Presumably doesn't carry over. In the sense that you're describing, it really doesn't carry over to these other situations. No, it doesn't, because the action of choice simply fails. That's right. I mean, once you... I mean, of course there's all that wonderful technical work that you and Cohen and everybody else did back in the 60s, right? I mean, put it in color, we were very much... Once you grasp the point that those sets are variable and constant, it's obvious to see... The continuum hypothesis was false for them and so on.

42:30 By the way, just as a footnote to this, I noticed that there was one guy, there was a paper by some guy in particle physics who was actually trying to use the Tarski paradox. Did you see that? Yeah, some guy. And it looked, I have to say, it looked very dubious to me. It was a coincidence between the numbers, you know, the numbers that get a quark theory and the numbers that come up... Quarks prove the axiom of choice, don't they? Yeah, I mean, it's something extraordinary. Empirical proof of the axiom of choice. They didn't choose our Einstein, that was... That's what I had in mind. ...so very, yeah, very, very... ...the smallest mathematical problem I've seen... ...in other words... Namely, there's a piano space-filling curve and many other related results. For example, constructions that occurred around the turn of the century, at the same time that the modern version, and these things exist only because of the natural number. For some reason, topology has been handed out to demand a better foundation to exclude them. It seems to me that we should have. This is a much more outrageous distortion of the way the mathematical world ought to be than some game of words like Russell's Paradox and the motives behind it.

45:00 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 the subjective logical aspect of it. Yes, I think so, yes. That would be, I'm going to make a few points. This is very much contrary to the physical, geometrical intuition. And people should have been more expert in the sense that they were, it seems. But then the analysis of why, what do you think, where do these things come from? Well, they depend essentially on the natural numbers. In other words, Van Heter-Eilenberg, for example, was fond of pointing out how simple are the approximations to these determinates on the whole triangle, and then you iterate the process, 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 level. The level of complexity and what we see is that the assumption is that the process can be completed. The limit actually exists and then... So it's a process that's indexed by the natural number. 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. You can imagine one step, you can imagine the next one. So that the idealization, including an iterative conceptual is intervening into the field of geometry here. The same thing could be said for this and many other similar paradoxes.

47:30 This sort of thing is sometimes what is sort of talked, is sometimes considered an anti-infinity. It's not an anti-infinity, it's an anti-calibration, a speech impediment. It's taking place within a continuum, which is what I'm not calling it to. Now, but now you see, the fact is, it's hard work to try to, it's hard work to try to be precise on this. It's easy to say, well, let's accept the natural number, let's accept all the grand consequences. But it's possible to careful work to avoid them, to avoid these problems. I wanted to cite, particularly, the work of Anton Rees. And several other positions in the past 10 years or so, working on Carsey's problem about the definability of the exponential functions of the field-closed field. Carsey's theory of the field-closed field in itself has a kind of category, in which you have lots of continuous maps. And you get 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 this subjective question, is it decidable or not, and the existence or non-existence of these geometrical monstrosities on the other hand. But, so it sometimes results about this... And I realize that that's not what they are because they're kind of on the side of the ocean, not on the side of the Earth. But the point is that I've compared these theories with a category in which you have a continuous map, but you don't have a map on the other side of the Earth. But of course it only consists of polynomial functions, and that's obviously not good enough. It's good enough for algebraic geometry, but it's not good enough for all the different kinds of analysis. So to push further and find categories where you have a nicely-behaved real-numbered object, lots of functions to try to use in analysis, and yet still avoid the natural number object, the first numerical algorithm, with the concomitant geometrical method.