What is the true relation of logic and set theory?

0:00 There's a question about decidability and non-decidability, but the point is that, as you've heard these theories, there's a category in which you have a certain continuous mass that you don't add to the natural number, and you either have decidability or a space-filling curve, but of course it only consists of polynomial numbers. And that's obviously not good enough. I mean, it's good enough for alphabetic 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 for which the ever-nicely-behaved real-number object with lots of functions, type needed in analysis, and yet still avoid the natural-number object, These require long, long work, and van der Gries did achieve a certain state of breakthrough in that direction a few years ago, and people are still looking out for the consequences of that. So, let me finally say about topos. There's an old result of Peter Pryde, basically just formalizing the setting of topos. So what's a non-iso-monoendo, well an endo would be an endo of some object which was injected and yet non-invertible.

2:30 It's been tried, it's been tried, it's been tried. I mean, it was something to show that it works. Oh, I know. I said it. I said it. It was so important. Yeah. Try doesn't mean . But that wasn't the point. I'm citing a definite Australian paper. I know. You can find this result in many more aspects of topology. Yeah. So if you have a monoendo that's not convertible, on some object, since the topos contains this, I call the object R, but it could be R. So if you have a monoendo that's not convertible, on some object, since the topos contains this, I call the object R, but it could be R. So if you have a monoendo that's not convertible, on some object, since the topos contains this, I call the object R, but it could be R. 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 of them, you take the intersection of all the sub-algebras, well, you don't have a starting point, you jack it up one so you get the identity as a starting point, I mean, there are various techniques, but the crucial point is that this is true, and that The crucial part of the construction is it uses omega and arbitrary intersection. The arbitrary intersection is a consequence of the definition of omega, as Jim pointed out this morning. It's defined through all, explicitly, using the property of omega. On the other hand, what's this got to do with this border? Well, any category can be embedded in the topos. We can embed a banded basis category in the topos. Metrics go on to all this work to exclude the natural number object. If you put it back down, here's Van Der Riese's category, and you put it in topos, maybe in a, oh, the point is that there are lots of such things, right? Take the half-open line and shift by one, that's a lot of work. Any kind of line, I think. It would be pretty extreme to exclude these from mathematics, right? I mean, to exclude these from analysis. So surely any reasonable category, unless it's very specialized, if it has any sort of pretense to generality, it will contain endomorphisms which are monomorphic but not infertile.

5:00 And on the other hand, well, it's very good. I advocate inventing all categories and topos, because it provides a very convenient and useful in many, many ways focus on the objects that you've got there to sort of normalize categories which are more profitable than making long topos. So there's a contradiction between that board and this. Now the natural numbers of any topos are essentially the same. So, in other words, there is this idea that while somehow abstracting away from what Van de Dries did, which is an evil thing to do because it's very serious particular work, but we want to try to make a general conception of what the end result is to be. This should be an example of what I... But I've called it various times drug topos, after using one of the drug-based terminologies, or just general topos, in the sense of being a general category of spaces as opposed to a particular space, or in a recent paper, as a category of being, in the sense that there's smooth being, continuous being, combinatorial being, algebraic being, net way, spaces, particular beings that evade one of those more specific, but still not totally specific. Ideas of collision. It should look like that. In other words, it's got this damn Lumeca object. Why do I say damn? Because, on the one hand, it's very, very nice, the algebra, to be able to objectify the algebra of propositions and parts, and one should certainly do that. On the other hand, Lumeca conspires with many non-invertible models, To produce a natural number out there. But all of these things relate. For example, if you... This is like a spot of light, you see, within all these directions.

7:30 This is the picture that I'm trying to promote here. Anyway, you see that in there 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 is like a picture of 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 arrive at it. Including things like abstract truth, that's just a mental construct, but also the natural numbers. We wanted to idealize some process of thought. Well, we can do that. Again, this 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 lines. I mean, the usual proof that you find in topology books, well, 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. So, on the one hand, and this is very much the point of the other thing, and without saying so, but 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 the nature, whether omega is contractible or not. But, so omega should be excluded, but probably. What about exponentiation? What about, in other words, function spaces? Traditionally there's a comparison between two different notions of higher types, right? Power sets starting with omega and just plain function spaces which are built up not necessarily starting with omega.

10:00 These are certainly two different things which have a lot of different reactions. Well now if we're talking about isolating subcategories of topos, the topos is thought of as sort of the enveloping This, by the way, clearly inverts the relation between thinking and being. Thinking is within being, but when we reflect inside, that gets inverted. The vision of being is within the vision of thinking. So, when is this closed under exponentiation? In other words, to put it in more technical terms, does mere exponentiation, in conjunction with some other reasonable semi-trivial property, like endometriosis or non-endometriosis, Implied existences. I don't actually know the answer. Maybe it's certainly a different question than what we made. But coming back to ontology then, finally, ontology and metaphysics, which I've just seemingly promoted, my idea is that, yes, okay, within this conceptualization, which includes these clearly subjective aspects, like omega and m, 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 be aware that the borderline may shift, that there is this, there is this, just like, just like, you see, we have a big oval and a blue oval, and that picture, the whole compost and the real uncoated objects within it, that's an important, that's an important conceptual guide. Even though we don't make it precise. If we want to make it precise, we probably find it has all kinds of different answers depending on the situation, but we 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 of a pen or not. Only by rather serious mathematical work can we give any definitive answer. How about a Cartesian closed, pre-ordered set?

12:30 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 everyone, if every ISO, you know, if every, one of those things, if every, if every Bono-Indo's ISO is probably, you know, actually, properly ordered or something like that. So I mean that's a weak, what I mean is that the condition of being Cartesian closed is fairly weak for a, just say, in the case of a hiding object. Yeah, but on the other hand, if the Cartesian closed category is also opposed to that, it's unlikely to have a natural number of objects. When the successor is closed, it's really different. Well, I mean, let me think about it a bit more. These parts of the space, right? What you are just saying. Here it seems to me that you can also do the same, because you said that you use function. Is that a permissible operation? No, no, no, no, no, no, no. Van den Duys' very particular termination arrives abruptly, but always. Neonatalytic functions is a category of neonatalytic functions. Except... They, at infinity, they are asymptotically polynomial. On any bounded piece of R and I, they're more, they're arbitrary, really. 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, and there's an actual math in this category, but not the sinus function of the whole thing. As I tried to point out in the report, instead of coordinate-izing the circle, you could just use it as such. In other words, the circle S1 contains an R cross R.

15:00 So R is any ring object, and then you look at the part defined by the equation. That's an abelian group object, and since you're working in a Cartesian closed category, you can form the space of it. So the 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 tuples. The smooth integers would be that. In other words, these are the ways of zipping around a circle in such a way that preserves the addition of angles. But the totality of those, classically, is the integers, the natural numbers together, which are negative, which is a discrete thing. In your case, it's not discrete, and that's very interesting. There's a model, but you see the point is it uses, instead of the coordinateization, which you knew that it's not really needed for the particular construction, but what you're using is higher order, you see. But not in any serious way in the topology question. This is simply equalizing. So that's problematic. In other words, can we formulate an exact question? You could always invent an etopos and then restrict. Basically, the idea is to take any ring object in any Cartesian closed category with pullbacks. Then you can make this construction. Is this a natural number object? In other words, does it satisfy the EMI? The recursion property. Notice the big difference. This definition is as an equalizer. This is a sub-object of math that's defined by an equation that says it's a homomorphism, reserve division of angles.

17:30 And so, therefore, by its construction, I know everything about maps from other test objects into it. By contrast, the recursion axiom, the way in which the idea of mathematical number of objects concentrates and idealizes the idea of repetitive subjective process is in terms of maps out of it. All right, so the initial statement is that if you have, Jim formulated it in a more general way, but wasn't pointed out in the Cartesian clinical 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 a sequence that's defined by that. Obviously, how you could go from an object that's characterised in terms of maps into it over into one other object. So, the understanding of which partitions close categories, which ring objects tend to produce things that really do satisfy the person and those that don't. This is the definition of how he described it. Five idyllic recoverings of the circles, things like that, objects collected together as you want, that doesn't, in itself, say that you could, I mean this might, yeah. The same analysis applies to the multiplicity, the same analysis applies to the multiplicity of the VM? That's one. VM of the complex numbers. Yeah. That's one that's a bit artificial. Yeah. Yeah, I think it would be essentially the same idea. You don't have continuous exponents, really continuous exponents, only pure discrete ones, by the way homology and protonology are not on that side, homology groups are often in the same group.

20:00 So are they supposed to be interpreted as purely subjective records of our qualitative understanding of the changes? I'm really surprised that you don't like things like the piano, the space-filling curve, and your 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-culling curve is? Like, you said something about, you know, you've got an iterative process, and you have to think of it as a completed infinity, and there's something about natural numbers indexing this iteration. But isn't that sort of typical of almost any limit process? Your objection sounds like you wouldn'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 approximate its value. So you're approximating something sequentially to something you already think is there. The principle that these function spaces are Cauchy-complete just using one metric, the supermetric. That's the thing that's interesting, because it's saying that something exists. Just because I can imagine the process of coming to know it, rather than, you know, here's the thing, okay, and then I come to know it.

22:30 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 a geometrical reason, as opposed to, you know, a curve. But the equal sign doesn't really mean identity anymore. It's a disguised something else, it sounds like. How many times can I see your point? I think it's a good comparison. I think calculus is not an identity science because it's a limit. But it is. The limit. The equals limit f is not a real identity. No, I think it is. It is an identity. That statement is defined by an epsilon, I mean in ordinary calculus, that statement is defined by a complicated epsilon delta condition. 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. The results of two different maps come out equal. One map is of the nature of, let's look at ratios of lengths or something. The other is, there are certain processes in which we can take the limit. Now the issue is really, how big is this domain of things for which we know the number should exist? There are certainly some domains. For example, constant sequences obviously all have limits. I'm saying it's not all quotients. At least not all quotients. Quotient is always a respect to a given metric or a given unit constant. The usual idea is you merely take the sequence and they have quotient. And that's what technically what we call the domain of sequences. The domain of sequences for which there is limit math well defined is smaller than the domain of sequences.

25:00 My aspiration has always been to be a physicist. I believe in things like the electromagnetic field and the continuous matter, the ocean, and all those things, and I don't see anything to do with the remotest idealization, but we need to, obviously, make idealizations, but some idealizations are way beyond that. So that's the angle from which I... Well, I think a related point, at least here, is that Eudoxus did not really, and it's sometimes claimed that Eudoxus actually invented the concept 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 a square. But there wasn't the notion of an arbitrary iterative protein. It was a process which was carried out in particular cases. Well, it was justified because they were driving for knowledge. That's the other thing. They were ratios, not numbers, of course. 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 the rate that 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 2 for the 5-factor rings, as you well said, I mean, of course, the factor rings have two definite and clear geometric entities whose lengths can't be compared by what had been the previous, within the previous number theory, 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 iterative process is concerned, because... There was, of course, also an iterative 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 incrementability of the size of the square. What about the method of exhaustion? I mean, isn't that really...

27:30 You set up a process in areas of polynomials and you deduce that there must be a line of a certain kind, because you're saying that circles are to one another as the squares are to their diameters, and you act... 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... See, that... 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, an attempt at idealized completion of these sorts of processes that led to all of this idea of the, for example, of the dedicated reels or whatever, and the kosher. It was this attempt to make the whole thing in some way full. So they need not be extended any further come what may. Of course, then naturally, as you've identified, because of the fact that one had to 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 long spaces. Yeah, that's obviously a very important step to make. Let's take everything. You can't just treasure that stage in history, you see. Then you have to come back and look more closely, like, how did this happen? Well, one of the remarks that Van Andries has resolved is, in many ways, the same as Grodendieck's. That is to say, Grodendieck, from a totally different angle, not caring about decidability, undecidability, that sort of thing, but wanting to have a topology, what he called a tamed topology, that wouldn't have pathological properties, also arrived at The Automatic Function, unpublished work, but in the same decade. You know, successive approximations. First you know nothing, then we learn something new. You know everything. You step back a little bit and look at it more carefully, deeper and deeper. But it's not, again, I don't want to take a permanent anthropological position in that sense. I'm saying that that big leap into this simple-minded fullness was a big leap to an important stage in coming to know more exactly.

30:00 Space. We have to step back. Well, Aristotle. Space and bodies. Right, right. Aristotle also had this notion, of course, that the continuum is an infinite, I mean, a definite continuum. It has parts and they're just more continua. It's cohesion in that very strong sense. The real parts of a continuum are exactly the same, in some sense, as the original continuum. You can find it in the book, whatever it is, if you read Aristotle, I know it's like reading him. It's in book six, somewhere in the physics. I mean, there's quite a famous definition. And also, it's clear... To go further and say that his order divides into atoms... Right, no, but that's the... ...to be a kind of subjective idea... Well, no, that's the word... ...mythology, basically. Someone was talking about constructors, right? Who is that? But Aristotle then calls the result of the description, 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... Right, it never will be. It never will be, but it's potentially there. Most important. And this is... Nobody has mentioned Zeno. As I said this morning, I say again, what the mathematicians do is such and such. But I don't think that the word compactification that's involved there, though, is not such an uncontrolled one. It's much more ideal. Yeah, but what kind of curve is it? An analytic curve. That's quite a bit of a matter. That's totally controlled in comparison with the total control of something much more tight, regrettable, in contrast to what we saw in the official handbook.

32:30 The end is somehow determined, I guess. Yeah? I mean, why should... Why should we be looking for things that are compatible with the kinds of things we experience in the physical world, and why should that be taken as some kind of guideline for what we might want to do in mathematics? Well, I'll be trying to do that. In mathematics, we know historically that mathematics is a cycle. I mean it. It seems to me we need not be confined by it. We don't have to deny it in order to also accept perhaps unbelievable possibilities. We contemplate all kinds of possibilities. Beautiful. The question was, let this concept of ontology and metaphysics make sense, which I interpret to mean, does there somehow exist a predicate that through speculation we can define, which could be applied to the whole realm of algebraic concepts, which is fantastic if you can imagine it more so, which could single out the quote-we-all-unquote line. And I was saying, in general, no, I don't believe that there is such a clear thought. On the other hand, within that richness of concepts, we can see these two aspects. We can see those general theories which are trying to idealize merely our subjective process and getting to know things on the one hand, and we can also see those theories which are about the construction of more and more precise images. And we can note that there's a correlation. Things that seem so fantastic seem unreal, like space-building pyramids and measurable cardinals, and the origin and the subject of the objects.

35:00 So there is a kind of longitude that's partly determined by that distinction, things that are mainly subject to the object. On the other hand, it's a precise... Logicism. I mean, this work of Peter Johnstone, building on Peter Fry, about how all logical distinctions can only be made into what he called QD topologies, namely those in which the object is a quotient of a decidable object, precisely excludes this movement. There are other kinds of non-QDL. In other words, the pro-general D-inclines. And quant-malogic? Logical language cannot distinguish between those in one that you get the caution of the other. Isn't there a deep reflection of this in the fact that you can think, as Colin has clarified in his work, of sets, constant sets, as the decidables of the topos? And this notion of object, this notion of object as it goes with the semantics of classical quantification theory, the thing which Has the Platonist ontology built into it, Doane actually comes out of these assumptions, well, in fact, can be seen as a reflection of the imposition of this QD condition, in that he gives you a notion of, well, in fact, extensionality isn't extensionality just a reflection of that, in the sense that it does build up this uniform separability condition, and hence imposes properties on the maps of... Speakers include mathematical physics, geometry, algebra, analysis, quantum mechanics, algebra, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory,

37:30 But in the proto-classical set theory, one tends to think of the absence of obstructions to maps as the general case, that that's the case of false generality, but it's the complete absence of obstruction to maps, to inverses of maps, which goes with that absolutely arbitrary notion of object, that of extensionality, that is seen, I'd suggest, as a kind of false, as introducing this What you call the ontological metaphysical false generality, since the title of your talk was the true relation between logic and set theory, and it's an absolutely wonderful talk, but I don't know how many logicians or set theorists would actually recognize... Either the logic or the set theory. Yes, I actually recognize either the logic or the set theory are in it. Oh, yes, the connections, of course, are profound, but I'm just saying they might be rather difficult for some logicians and set theorists to grasp. I just think that's perhaps a point that ought to be brought... They've had 30 years to work on it now. It's high time they started working on it. I couldn't agree more, but perhaps that last point is a connection that should be brought out more strongly. I mean, about the QD toposys, but also the origins of this... The notion of object in false generality. The idea that the category of objects, the category of being the ontological width of which is such that a more restricted notion of physical object or concrete object just sits down, so hence the Platonist picture which I was discussing with Owen last night, that the entire physical universe is a little blob of structure sitting at the bottom tip of the ordinal cone. It all gets unified, all the structure and unity are actually derived from The realm of objects and functions, in the logicist approach, the language-based approach, you have to introduce another ontological category as well, but that of incomplete entities, functions in the ZF-based approach as interpreted in a strongly ontological, Platonist way. You just have the one category, but either way, the way the structure and unity is built in in that approach comes out of this way of thinking of extensionality that you have in logic and set theory, or rather the way that extensionality is built in in logic and set theory without seeing its roots in the conditions that are expressed in this way, in the topos setting, in their topological aspect. I suppose you would say as a reflection of objective variation.

40:00 Would you agree with that? I think so. I'm sorry. I'm sorry it was very badly expressed. Well, I think we must thank Professor Lovell. Thank you for your attention. So we'll see you there. Yeah, oh yeah. Michael? Come with me, I'll give you a ride over to John's. Oh right, okay, thanks. Bill's going to go home and have a nap. He's going to get out of it, yeah. I'm sorry, I think unfortunately I got it right there, yeah. It was very, very badly expressed. Yeah. Well, I think you're being very polite. But I do think there is a question about extensionality at this moment, and also I think that it would be difficult for an orthodox task if there are individuals out there to actually see what the philosophical claims that are being made is, I think, unless it were directed a bit more directly, so...

42:30 Well, of course, most mathematics isn't, and that's what's at the core of the science of physics. Most mathematics isn't, and that's what's at the core of the science of physics. And, you know, some of the big stuff, because it's all spread, for sale, and it's all out there, and it's all out there, and it's all out there, and it's all out there, and it's all out there, Well, I think there has to be some contact, there has to be some contact with the origin, with him, not what he would want, you know, real requirements of objective parameterization. Yeah, good idea. No doubt a philosopher, unfortunately. That's what makes me a little bit frightened of having to open my... Well, I'm not as wrapped with people unless we've got a nice spot. Thank you for your attention.