Indecomposability of the continuum / discussion (partial)

0:00 The following is an academic lecture on Ontario, for the first seminar that we have on CREA. You already know that John Bell is a very prominent specialist in the world of the theory of models, the theory of categories and the theory of topics, and the theory of CREA is at the end of the month. Alors son premier grand livre sur lequel on a tous appris la théorie des modèles, c'était son grand livre avec Slomsom, le modèles, celui ultra product, c'est paru en 69, donc déjà assez ancien, un grand classique. There's his course in mathematics of 1977, and then there's his great book of 1988, Top of Us and Local State Theories. And recently, in 2005, he published The Continuous and the Infinitesimal in Mathematics and Philosophy, the publisher polymetrica you have to order it over the over the web but it has a great merit of being exceedingly unique space yeah 39 years and giving it away and so today john va nous parler des problèmes de compositionnalité non-composition et de continue et demain je He will talk about the signage. He will talk about the séminaire of Andrej Rodin at 10h30, L.E.N.S. Salle Info 1. The new building. It's the new building for the old building. It's the new building for the old building. On category and physical. Thank you, John. Oh, yes. Well, thank you, I should say, to begin with, thank you very much for inviting me here to Chayal. It's a pleasure to be here, and I hope what I have to say here will be of some interest. Yeah, I have to put this. This is one of the most intimate gatherings I think you're going to attend to in some time. The door opened outward, it would be something like that scene in the Marx Brothers, you know, where they're all jammed into the state room and eventually they open it.

2:30 However, okay, well, I'm going to talk on the Indie Composability of the Continuum. This is actually a paper that I'm contributing to a special issue of the journal, I thought it by Michel Degla, and so this is a sort of draft, if you like, of the article that I intend to submit there. And it really falls into two parts. There's a sort of philosophical or account of the development of the idea of sort of tracing the concept of indecomposability, which will be a very quick tour of that idea philosophically. at the, more specifically, at the tapas models, I'll give you a sketch anyway, of the tapas models in which the continuum is indeed composable, and try to show you how that, or at least describe the properties of the continuum in some of these models. The continuum often is indeed decomposable models in autopsis. It happens surprisingly frequently. Okay, so let's begin. I'll have to sit down. Well, I'll have to... Well, I'll go here. Okay, well, anyway, let me then... It'd be easier for this part, let me just read it, because it doesn't have any symbols, and then you can follow it up there. Okay. So, first of all, then, so what is in decomposability? So it's characteristic of a single continuum that it'd be all of one piece, in the sense of being inseparable into two or more disjoint, non-empty parts. This is one character, sort of cohesive if you like. By taking part to mean open or closed subset of the space, one obtains the usual topological concept of connectedness, thus a space S is defined to be connected if it cannot be partitioned to two disjoint, non-empty, open or equivalently closed subsets, or equivalently given any partition of S into two open or closed subsets,

5:00 one of the members of the partition must be empty. This holds, for example, for the space R of real numbers. I'm sorry about this. It should be an R there. I'm not going to change. Something went wrong with the symbols of the printer that I was using. For the space R of real numbers and for all of its open or closed intervals. now a truly radical condition would result from taking the idea of being all of one piece literally that is if it is taken to be an inseparability into any disjoint non-empty parts or subsets if you like whatsoever so a space S satisfying this condition is called indecomposable thus S is indecomposable if for any subsets U and V the union of U and V is S and U and V are disjoint then one of U and V must be a form of indecomposability, actually slightly weaker than the version just stated, assuming that you're using intuitionistic logic, anyway, is that for any subsets U and V, each of which contains at least one point, if the union of U and V is S, then U and V must be non-emptive. Now, indecomposability can also be phrased in the following way. If we call a subset U of S detachable, if a complementary subset V of S exists satisfying union of U of V as S, and U of V, the intersection is empty, then S is indecomposable precisely when it's only detachable subsets of the empty set and the whole space, respectively. Now, indecomposability can also be provided with a logical formulation. Here's a few symbols. It's clear but small. You could say s is decomposable if and only if you take any property p, then if the property of p or not p on it exhausts the space, so if all x and s are the px or not px, remember that you're, of course, that is actually classically, of course, that's a trivial condition. But you have to think the underlying logic here. All of this is completely trivial if you work at classical logic because decomposable spaces only have one point there. But anyway, so if you have px or not px, then either for all x and s, px, or for all x and s, not px. Of course, classically, the antecedent, they're trivial, so that's classically a condition

7:30 that you can see in classical case. If you have classical logic, then that just has the most one point in it. Now, we observe, as I said, that the law of excluded middle of classical logic reduces in decomposable spaces to the trivial empty space of the one-point spaces. That being the case, how is it possible for non-trivial in decomposable spaces to exist? To get a clue, let's reformulate our definitions in terms of maps rather than parts. If we denote by 2 the two-element discrete space, then connectedness of a space is equivalent to the condition that any continuous map on S to 2 is constant. And the indecomposability of S reduces to the condition that any map from S to 2 whatsoever is constant. Supposing S to be connected and to possess more than one point, then from the law of excluded middle it would follow that there exists non-constant and has discontinuous maps from S to 2. would be decidedly otherwise if all maps defined on S were continuous, for then clearly the connectedness of S would immediately yield its indecomposability. So if S could be conceived as inhabiting a universe, let's call it U, in which all maps defined on S are continuous, then within U, S would be both non-trivial and indecomposable. This necessarily entails the failure within U of the law of excluded middle. In other words, the underlying logic of U must be non-classical. Such universe as U can, in fact, be found. underlying logic is intuitionistic, and within them the law of excluded middle fails in just the way necessary to admit the presence of non-trivial, indecomposable spaces. In such universes, R and its intervals are not merely connected, but indecomposable. I'll return to this. Below the models are what are called smooth, or the various topices on this that can be. Now, if we try to trace the concept of indecomposability, historically, we can see that while the in decomposability connectedness as we've defined the terms of course they're modern mathematical concepts, contemporary concepts, related ideas in regard to continuous entities can be traced back to antiquity Anaxagoras for example asserts around whenever it is XPC, I haven't put the data in there, 500 or so, that the things of the world order are not separated one from the other nor cut off with a hatchet neither the hot from the cold nor the cold from the hot of course this

10:00 phrase not cut off with a hatchet was a favorite of Herman Biles and he uses a lot you'll find you'll find who was also a kind of champion of in decomposability the continuum and he often used that phrase now here the one world order is the homogeneous continuum supposed by Anax Sangeris to constitute the world Although, so it's believed. Okay. Oh, sorry. Wrong operation. Okay, so Aristotle, who first undertook the systematic analysis of continuity and discreteness, and in fact, you may remember in the categories he actually defines the terms there, maintained that physical reality is a continuous plenum something like this and that the structure of a continuum common to space, time and motion is not reducible to anything else he held that continuous magnitudes are potentially divisible to infinity in the sense that they may be divided anywhere though they cannot be divided everywhere at the same time Aristotle identifies continuity and discreteness quantity. As examples of continuous quantities or continua, he offers lines, planes, solids, that is, solid bodies, extensions, movement, time, and space. Among discrete quantities he includes number and speech. It's quite interesting, really, that he chooses speech rather than written language. He says, he points out that spoken words are analyzable into syllables or phonemes, which, of course, are linguistic atoms, which themselves aren't reducible to anything simpler. It's interesting that he uses actual speech. At the time, there were horrors of the civilization. Sorry? At the time, the civilization was more horrors than somebody. Exactly. Exactly. Anyway, he lays down the following definition of continuity. He says, I mean by one thing being continuous with another, that those extremities of the two things in they are in contact with each other, become one and the same thing, and as the very name indicates, are held together, which can only be if the two limits do not remain two, but become one and the same. So in effect, Aristotle here, this is from the physics, anyway, in

12:30 effect here, Aristotle here defines continuity as a relation between entities, rather than as an attribute appertaining to a single entity. That is to say, he doesn't provide an explicit definition of the concept of continuum, at least in the sense that we think of it. He indicates that a single continuous whole can be brought into existence by gluing together two things which have been brought into contact, which suggests that the continuity of a whole should derive from the way its parts join up. That this is indeed the case as revealed by turning to the account of the difference between which he offers in the categories quote in there he says discrete are number and language continuous aligned services body and also besides these time and space for the parts of a number have no common boundary at which they join together for example ten consists of two fives however these do not these do not join together at any common boundary but are separate nor do the constituents parts with parts three and seven joined together at any common boundary. Nor could you ever, in the case of number, find a common boundary of its parts, but they are always separate. Hence, number is one of the discrete quantities. A line, on the other hand, is a continuous quantity, for it's possible to find a common boundary at which its parts join together, a point. And for a surface, a line for the parts of a plane joined together at some common boundary. Similarly, in the case of a body, one could find a common boundary, a line or a surface, the parts of the body join together. The time ulcer and space are of this kind. Okay, so I will skip the rest in quotation. Accordingly, for Aristotle, quantities such as lines and planes, space and time are continuous by virtue of the fact that their constituent parts join together at some common boundary. By contrast, no constituent parts of a discrete quantity can possess a common boundary. So let's just as an exercise, to turn Aristotle's notions into mathematical definitions, and what would we do is one way we might do it. So we suppose that we're given, let's call them quantities, A, B, C, et cetera, and we suppose that we have some kind of inclusion relation between quantities. So we say that U is included in A, this usual inclusion zone there, is understood then to

15:00 mean that U is included in A, or that U is a sub-quantity, or if you like a part of A. We assume that for any quantity A, there's a void subquantity with the property that it's included in all subquantities. Given subquantities U and V of a quantity, we suppose also that there are subquantities U of the union, which we'll call, sorry, the join of the meet. We use the union intersection signs, respectively, of the two parts, subquantities U and V, with the property, with the usual properties, that the join is the least subquantity, which includes the two ones, subquantities U and V, and that U intersect V is the largest, which is included in both of them. And central. The intersection, or the meet, I should say, the greatest subquantity, which is stuff. This is on the top. Oh, yeah. Let's put it in the right position. We can think of the intersection really as the boundary of U and V. Let's call it for the purposes of, anyway, for our purposes of the boundary. So we call a quantity A discrete, corresponding to any part U of A. There's a V, right, for which the union is A, the union of U and V is A, or the joint, I should say, of U and V is A, and the boundary between them is empty, an empty boundary. U and V are then, so to speak, constituent parts of A without a common boundary. That's one way of constraint. So boundary is just immediate. contrast we can call a quantity continuous so or an aristotelian continuum provided that any pair of constituent parts and they have a non-void common boundary is what he's saying essentially that is whatever you would be in a are such that uh that you that you and a are actually proper parts uh and they're both not empty then it and and they and they together could they together to constitute A, then they must have a common boundary. This is a very, this is a weaker version of indecomposability, that I mentioned. Well, that's one way of thinking that I've been inclined to think that Aristotle, if you thought of that it does seem to be implicit, that Aristotle does have some idea of indecomposability in mind for continuing, at least. Now, I'm going to jump forward a couple of millennia um and into the 19th century and just consider briefly the views of the geometer veronese

17:30 who of course was a champion of infinitesimals and was execrated by cantor for this cantor as you may know regarded infinitesimals as the cholera bacilli of mathematics and he spent quite a lot of time fulminating against against mathematicians such as veronese who still felt that infinitesimals were, who believed that infinitesimals were a coherent notion. However, I'm not talking about infinitesimals here, so I won't mediate on that. It's interesting that his views of the continuum, and the idea of, you can somehow tie his ideas of the continuum to the idea of decomposability rather than indecomposability, although at it that he changed my view in it, such as it is, it would appear that when you look at his discussion, or one of his discussions anyway, of the nature of points, of course the question of whether points actually constitute parts of continuous, an old one, Aristotle didn't think they did, it has been held of course that points are actually, are really anyway. Varanese doesn't think that points actually are parts of continuum. He says essentially that they're nothing more than signs indicating positions of the uniting of two parts and he's talking here about, if you like about rectilinear continuum lines So to elucidate the nature of points, Varanese offers two thought experiments The first of these, he supposes the following that the part A picture next to it, but I'll just, it's easy enough to understand. You have a rectilinear object that is aligned to the part, one part A of it is painted red and the remaining part alpha white. So I suppose further, there's no other color between the white and the red, so you just imagine you have two parts of the, contiguous parts of the line, if you like, colored red and white. Anyway, he goes on to say, that which separates the white from the red can be colored neither white nor red, and therefore cannot be a part of the object, since by assumption all its parts are white or red. And this sign of separation of uniting can be considered as belonging either to the white

20:00 or to the red, if one considers them independently of one another. If we now abstract from the colors, we can assume that the sign of separation between the parts A and alpha belongs to the object itself. Accordingly, a point can belong to a continuum through what he calls assignment, but cannot it. In the second thought experiment, Veronese invites his readers to, well, and I quote from him here, to cut a very fine thread at the place indicated by X with the blade, those are supposed to be, those boxes, those are primes, the red things that came out of boxes when I put them in. Anyway, he invites his readers to cut a very fine thread at the place X with the blade of an extremely sharp knife, so that the two parts A and A' separate, and we assume that one can put the thread back together without seeing where the cut was. This is in figure two, so you cut it when you put it back together. Without a particle, he says, of the thread being lost. One produces this, apparently, if one looks at the thread from a certain distance. If one now considers the part A from right to left as the arrow above A indicates, it's What one sees of the cut is surely not part of the thread, just as what one sees from a body is not part of the body itself. It happens analogously if one looks at the part A prime from left to right. So if the sign of separation X of the parts A and A prime, which by assumption belongs to the thread itself, were part of the thread, then looking at A from right to left, one would not see all of this part, since that which separates the part A from A prime is only that which one sees in the way indicated above supposes the thread put back together. the premise of this, it seems to me, the premise of this argument is the assumption that the thread or linear continuum can be separated into two parts, which upon being rejoined constitute the continuum in its entirety. In a word, that the continuum is decomposable. From this assumption, Veronese infers that the point or sign of separation cannot be part of the continuum the argument becomes clearer in its contrapositive

22:30 form, namely, if points of separation are parts of linear continua then such continua are indecomposable in the sense of being inseparable into two parts, which upon being rejoined, reconstitute the continuum in its entirety and then you can formulate that argument I give a more formal version of that which I will skip in the next parable. So Veronese, since Veronese seems to take it for granted that the continuum is always decomposable, and he seems to assume this, it seems to follow, he's using this to show that points can't be parts their own. Now, interestingly, if you look at the work of the philosopher Brentano, of course, who was the teacher, among others, for Sorolla, Um, and, uh, I've got other philosopher's name that escapes me, and the one with Golder Mountains. Who's that? No, no, no, no, no, no, the other one, I don't know. Minor. Minor. Minor. Yeah. Um, it's interesting to see what he has to say in the continuum. And this is, uh, of course, this is work that he wrote, uh, quite late on, uh, apparently after he, I think he was blind, and he dictated this to Emanuensis, I think sometime during, maybe just before the First World War was around there. But anyway, like Aristotle, Brentano considered itself evident that a continuum can't consist of a point. In fact, Brentano's view on the continuum, as in some other aspects of his philosophy, was quite Aristotelian. He says, points are just boundaries, both for Aristotle and certainly for Brentano, and not to be regarded as actual parts of the continuum from which they spring. If two continue or have a common boundary, that common border unites them into a single continuum. Such boundaries exist only potentially since they come into being when they are, so to speak, marked out as connecting parts of a continuum. And the parts in their turn are similarly dependent as parts upon the existence of the continuum. For Brentano, the essential feature of the continuum is its inherent capacity to engender boundaries. He says this again and again in his writings on the continuum. And the fact that such boundaries can be grasped as coincident. Now, Brentano took a somewhat dim view of the efforts of mathematicians to construct

25:00 the continuum from numbers, in other words, to reduce it to the discrete. his attitude varied from rejecting such attempts as inadequate to recording them the status of fictions. In fact, in a letter to Husserl drafted in 1905 Brentano asserts that I regard it as absurd to interpret a continuum as a set of points of course it's going very much view of the day that Brentano's account of the continuum entails its decomposability sorry, yes, emerges from these criticisms, sorry. Yes, would seem to, I think, emerges from these criticisms. For example, we read, it says that one has postulated something completely absurd. This is in connection with the Dedekind cut. It's seen immediately if one splits the supposedly continuous series of fractions between zero and one into two parts at some arbitrary position. One of the two parts will then end with some fraction f. The second, however, could now start only if there was some fraction of a series which was the immediate neighbor of f, which is, however, not the case. We should apparently have something that began, but without having any beginning. Well, here Brentano appears to be saying that when one divides the closed interval, let's say 0, 1, well, 0, 1 at an intermediate point c, when necessarily obtains the closed intervals 0c and c1 with a common point c. In that case, any continuous line is indecomposable. It's a disjoint interval, at least. Now, of course, none of the above thinkers really could be claimed to have asserted explicitly the continuum is indecomposable. I'm certainly not claiming that. Indecomposability was a cornerstone, actually, of course, He says so explicitly, the intuitionistic continuum. And I guess that's the first place in which the notion of indecomposability really comes into, becomes explicit. Where I, as you know, spent quite a lot of time trying to establish the indecomposability of the continuum. He spent many years trying to formulate proofs of this

27:30 course of his own principles. Now while in his early thinking, Brower held that the continuum is presented to intuition as a whole, and that it is impossible to construct all its points as individuals, in his mature thought he radically transformed the concept of point. There was some sort of evolution in his thinking there, and in this transformation of the concept to point. He endowed points with sufficient what you might call fluidity to enable them to serve as generators of a true, let's call it a true continuum. This fluidity was achieved essentially by admitting as points not only fully defined discrete numbers, such as the square root of 2 pi e and the like, which have, so to speak, already achieved being, but also numbers which are in a perpetual state of becoming in that the entries in their decibel and dyadic expansions are the result of free acts of choice by a subject operating throughout an indefinitely extended time this is of course the free choice sequences also these other rather more controversial notions as the creative subject which emerges in his later theme Now, the resulting choice sequences can't be conceived as finished, completed objects. At any moment, only an initial segment is known. In this way, Brouwer obtained the mathematical continuum in a way compatible with his belief in the primordial intuition of time, that is, as an unfinished, indeed, unfinishable entity in a perpetual state of growth, a medium of free development, as he's often called he told. In this conception, the mathematical continuum is indeed constructed, not, however, by initially shattering his decanter or dedicating an intuitive continuum into isolated points, but rather by assembling it from a complex of continually changing overlapping parts. The mathematical continuum as conceived by Browery displays a number of features which renders it bizarre to the classical eye, although I guess it's much more familiar now. For example, in the Breuerian continuum, the usual law of comparability, namely that for any real numbers, A and B, either A is less than B or A equals B or A is greater than B, fails. Perhaps more fundamental is the failure of the law of excluded middle in the form that for any real numbers,

30:00 A, B, either A is equal to B or A is equal to B. Now, while the intuitionistic continuum may possess a number of negative features from the standpoint of the classical mathematician, it has the merit of corresponding more closely to the continuum intuition than does its classical counterpart. Hermann Weill, who in the early 1920s was closely associated with Brouwer, pointed out a number of respects in which this is so. So Weill says, for example, in accordance with intuition, Brouwer sees the essential character of the continuum not in the relation between elements and set, but in that between part and whole. I mean, it's a sort of mariological conception. The continuum falls under the notion of the extensive whole, which Husserl characterizes as that which permits a dismemberment of such a kind that the pieces are by their very nature the same lowest species as is determined by the undivided whole. A kind of homogeneity, if you like. Far from being bizarre, the failure of the law of excluded middle for points by Weill as fitting in well with the character of the intuitive continuum, and he says, for there the separateness of two places, upon moving them toward each other, slowly and in vague gradations passes over into indiscernible. In a continuum, according to Brower, there can be only continuous functions. The continuum is not composed of parts. For Brower had indeed function defined on a closed interval of the continuum, as he conceived of it, is continuous, in fact, uniformly continuous. As a consequence, the intuitionistic continuum is indecomposable, a fact which Weill found thoroughly agreeable. A genuine continuum, Weill says, cannot be divided into separate fragments, and as I said, he often expressed this by quoting Anaxagoras' hatchet metaphor mentioned above. Weill also welcomed Brouwer's construction of the continuum by means of sequences generated by pre-acts of choice, thus identifying it as a medium of free becoming, which, he says, does not dissolve into a set of real numbers as finished entities. Weill felt that Brouwer had come closer than anyone else

32:30 to bridging the unbridgeable chasm as he liked to call it between the intuitive and mathematical continuum yes, well let me go on then because I think I have some time now the uniform continuity of functions defined on a closed interval of the intuitionistic continuum closed the door, was shown by Brower to follow. From certain intuitionistically plausible principles, he held choice sequences should satisfy. As you may know, a number of Brower's principles were, which, of course, were formulated in ways which made them kind of incomparable with ordinary mathematical practice in a way at the time he formulated them, were provided with, if you like, more precise, given more precise formulations or versions of them by mathematicians later on. And so it's possible to develop formal systems. Brouwer, of course, loathed formal systems, but his successors did, of course, formulate intuitionistic mathematics, devised formal systems as very familiar for intuitionistic mathematics, although he didn't approve of this at all. versions of these later principles of braher that continuity principles i'm going to mention the creative subject and so on were provided uh formulations of these to be given in the formal systems that were developed by by logicians and mathematicians later on now one such principle is the continuity principle which which is which is boils down to saying that if you're given a relation between alpha and numbers n. If for each alpha, number n may be determined for which alpha, for which there's a relation, for which r, the relation r between alpha and n holds, then n could already be determined on the basis of knowledge of a finite number of terms of alpha. The idea being that if you have the knowledge that if a natural number can be assigned,

35:00 you've given a choice secret, which is something that's developing in time, if you like, and isn't fully determined, then if you have enough information to, so to speak, to assign a natural number to that, you already have to have had, it can't be on the basis of the whole of alpha. It can only be given on the basis of a part of alpha, in other words, a finite part of it, which is already given. Now, from this principle, it can be shown that every function from R to R, I hope I put that in there. It isn't in there, but it should be dropped out from reals to reals, is continuous. actually how the continuity principle is established or the principle of continuity. Another such principle is bar induction, a certain form of induction for well-founded sets of finite sequences, which I won't go into here. It's somewhat more complicated to describe. Brouwer used bar induction on the continuity principle to prove that every real valued function defined on a closed interval was uniformly continuous. You need a bit more than the continuity principle to prove uniform continuity. Brouwer gave the intuitionistic conception of mathematics an explicitly subjective twist, if you like, by introducing what is now called the creative subject. The creative subject was conceived as a kind of idealized mathematician for whom time is divided into discrete sequential stages, during each of which he tests or may test various propositions, attempt to construct proofs, and so on. In particular, it can always be determined whether or not, at stage N, say, the creative subject has a proof of a particular mathematical proposition, P. Now, while the theory of the creative subject remains controversial, there's no, and there are a variety of different ways of spelling it out, and there's still some argument going Its purely mathematical consequences can be obtained by a simple postulate, which is entirely or temporal elements. This is what Krippi did, actually. Creative subject allows us to define for a given proposition P the binary sequence, a binary sequence A, N, by saying that A, N is 1 if the creative subject has a proof of P at stage N, and it's 0 otherwise. Of course, it's assumed that those are mutually exclusive

37:30 and exhaustive possibilities. I mean, this is, you know whether, the creative subject know whether he has a proof or he doesn't. So that's, in that level, sort of metatheoretical level, the law of excluded middle holes there. Okay. Now, if the construction of these sequences is the only use made of the creative subject, So if you just hobble the creative subject to put him in this cage where he only does these things, then references to the latter may be avoided by postulating the principle known as Kripke schemes. It's a sort of special case of the creative subject which admits a very nice formulation that Kripke produced some time ago, and said that for each proposition P an such as p-holes, if and only if an equals 1 for some n. Now, if you put all these principles together, you can show, and it's quite well-known, there's a nice account of it in various accounts of it, for example, by Van Dahlen. You can show this has remarkable consequences. These principles give you very strong consequences for the indecomposability of subsets of the continuum, as von Dahlen showed back in the 90s, not only is the intuitionistic continuum in-decomposable, I mean, as we've already observed, but assuming the continuity principle in Kripke's scheme, it remains in-decomposable even if you prick it with a pin and a point. You remove a point from the continuum. The result's still in-decomposable. This is not the case, actually, for smooth infinitesimal analysis, for example, which I'll be discussing later. So, as Van Dahlen says, the intuitionistic continuum has, as it were, a syrupy nature. One cannot simply take away one point. It kind of gets filled up in some sense. If, in addition, bar induction is assumed, as other principle, then even more surprisingly, in decomposability is maintained even when all the rational points are removed from the continuum. So if you actually take all the rational, just take the, you remove all the rational points and call the result the irrational points. Of course, it's not decidable whether it becomes rational or irrational, but nevertheless, what's left is still decomposable.

40:00 It's very, sort of like molasses. This is a, so if you take Brower's, these later Browerian principles and formulate them in this way, then it has much stronger, you get much stronger forms of decomposability for the continuum, and I don't think all of these consequences have yet been worked out. Okay, well now I turn to a sort of little tour. of the, um, of in decomposability of various objects, particular continua in, in, um, uh, in the tapas theoretic setting. Um, well, the, uh, it's very interesting that, um, there are many tapases. Tapases, of course, tapas can be thought of in a variety of ways. Peter Johnston in his enormous compendium which is still in progress on tapas theory I think two volumes have now been published the total already of something nearly a thousand pages calls it sketches of an elephant and the idea is that there's so many ways you can view tapas or the idea of a tapas that it's a bit like the blind man the story about the blind man and the elephant one gets the tail and thinks the elephant's long and thin, and he gets to the trunk and thinks the elephant's long and thick, and so forth. And so there are various ways of thinking of tapas, and there's a bewildering variety of tapas, which could be thought of as sort of models, if you like, of mathematical universes, in which certain types of Bill O'Bear long ago introduced a nice way of thinking about this, namely that in a tapas you're really thinking of a universe, or at least in many cases, of a universe of variable sets, that is, of sets or objects, if you like, necessarily sets, you don't have to think of sets, but objects that are undergoing some form of variation. And if you construe variation, the term variation sufficiently broadly, with sufficient liberality, then you can think of any tapas in that way.

42:30 And interestingly, when one introduces one of the most important cases, and of course these are the cases that provided the basis for the definition of a tapas, it goes back Deacon and his school, was the idea of sheaves over a topological space, where one is, and this was, of course, the sort of locus classicus, really, for the idea, the idea being that you take a topological space, doesn't matter what it is, real numbers, for instance, will be considering, well, briefly, the case where the space is in real numbers, and then you space, and this forms a tapas. You can think of the objects there as sets or objects which are undergoing variation, or continuous variation, if you like, over the open sets, or if you like, over the space. And interestingly, when you do that, when you introduce continuous variation, you often find that objects that are connected turn out to become actually indecomposable. This is a striking fact. I don't think there's no general account, as far as I know, of indecomposability of objects and tapas. I mean, in other words, I don't know of a kind of general condition, if you like, that you can impose on a tapas, some sort of axiom that ensures that objects, that you have indecomposable objects. I mean, there of objects like the real numbers or various spaces, I don't know of any sort of general conditions that you can impose that ensure that the objects are the real numbers. You look like real, varied sheaves. Yeah, yeah. These are just sheaves over the, yeah. But anyway, it turns out that the Brower's, if we think of in decomposability as something arising from Brower's principles. In decomposability arises, it's quite natural, it arises automatically, right, when old maps are continuous. And you can force that to be the case in certain tapasas. So you get in decomposability for free there. But interestingly, even when you don't force old maps, well,

45:00 it's not clear that you forced old maps to be continuous. You have to prove that, as in certain topazes. It's not immediately obvious that, by any means, that... Just to understand, what does it mean? It seems to be separable, meaning that you force to, how to say, to respect your separability condition, to respect the variation, right, over the base. Yes, and then internally, you can prove that all maps are continuous. But it's not obvious. In some cases, you force that to be true, and then it automatically comes out to be the case, because that's the, you know, as in the smooth terms, that's the intention. But there are other ones where it's actually not so easy to show that that's the case, and it's not so obvious. But surprisingly, it happens quite frequently. I won't say the norm, but the case where the continuum, for example, is decomposable actually is the exceptional case. I mean, conversely, the case of decomposability seems to be quite closely allied to the behavior of choice and extensionality. Of course, of course. If you have the actual choice, I'll come back to that. Okay, so let's just look at some of the examples. One of the examples where the continuum is in decomposalism, it's discussed, it's a very nice discussion of it in the book by Maclean and Mordyke. And they actually, that's a very honest book, you know. I mean, they go into real detail. These things are not so easy to verify. on. The problem, of course, with verifying internal properties of topices is that you don't have an automatic transfer principle as you do it on standard analysis. And sometimes the calculations can be quite intricate, as they are indeed in models, in Cohen models, in generic models of set theory. Anyway, one nice example, and this is an example that's McLean and Moordyke in their book is the what's really the so-called go topos big top us top us to find actually on a large where the where the underlying

47:30 site is is big it's it's the site of topological spaces that's great and each corner yes that's great and these pointage and it's it's it's it she's on it on a large site that is not just on open sets if you like you know right in a particular space it's actually defined on on all topological spaces well you take a suitably suitable chunk if you like of the category of topological spaces anyway um it's shown there that the uh that every map if you take the dedican reels so you you take you you construct the dedican reels within this tapas as of course you you can do in any tapas that has a natural number object, you just mimic the definition of the Dedekind Reels, and you can show there, although it's not, it's a fair calculation, I mean, it takes four pages or something, you know, in the Clayton-Mordyke, it's really not trivial, that every map for the Dedekind Reels to itself is continuous with respect to the usual open interval topology. And therefore, it follows, I'm not going to go in the proof but it then follows that are in, well there's an argument that shows already by, the argument goes back actually to Alan Stout who did this work I forget when, back in the 70's that are, that the real the dedican reels are always in any tapas, right, with a natural number object the dedican reels are always connected in a certain way that they satisfy a certain form of connectivity which is given on the next page here And he shows this by looking at the Dedekind reels. If you put these two things together, you can show the fact that every map from reels to reels is continuous. And R is actually connected in this sense. It's a sort of slightly weaker sense than the usual one. But if you put these two together, you conclude that R is actually indecomposable in the weaker sense that I mentioned earlier. So, actually, McLean and Mordyke don't mention it. They don't mention decomposability in their book. They just prove that every map is continuous. And from that, plus this weakish connectedness condition, you can put those two together to get the decomposability in a weaker sense.

50:00 Now, a very interesting example that was first where in decomposability was first detected, a bar was detected actually by Dana Scott before tapas theory was, elementary tapas were invented. This is in a paper of his called Extending the Topological Interpretation to Intuitionistic Analysis, the second paper there. He effectively investigates the tapas of sheaves over the real numbers. So in other words, at continuous variation over the reals. And he shows what he actually is doing in this paper. It's before the Tapas approach, really. But what he does, what he's essentially doing is taking a universe in which the truth values are open sets. The hiding algebra of truth values is the hiding algebra of open sets in the open sets of reals. And he computes, I mean, he sees that the, if you use that as truth values, in other words, he's doing something like a Boolean-valued model, but with hiding algebra, using the hiding algebra instead of a Boolean algebra. And there he shows that every map is uniform, that every map from reels to reels is uniformly continuous. And so it follows that, I mean, later on, and approached by Hyland also, his student, Scott's student, Mike Foreman, they actually took Scott's original work and extended it to, or if you like, interpreted it within the tapas of sheaves over R, and that's essentially what, that's a very nice example of a very natural tapas where the reels are actually decomposable because you have this very strong principle held internally, namely that the Browardian principle that every map from reels to reels is uniformly continuous. Now I have to say that it's by no means obvious that that's the case. I mean, you can see whether when you look at the tapas of sheaves over the reels, certain things emerge kind of automatically from the construction. But the fact that every map is uniformly continuous

52:30 is by no means obvious. Also, it's very interesting, I think, just to contrast the... I just mentioned this point that if you... So in other words, in tapas of sheaves, the truth values are really... It's like doing mathematics you know, with truth values, which are open sets of real numbers. Those are the truth values. So, of course, the logic is intuitionistic, actually, because you're hiding out. Now, on the other hand, if you look at the corresponding Boolean value model, which is very familiar, it's the one that was considered by Scott and Solovey, you know, after co-enforcing was developed. and this is the Boolean-valued model in which the open sets sorry, in which the truth values lie in the Boolean algebra of regular open sets there we have a Boolean-valued model and it just satisfies it's got some interesting properties the continuum, you can show the various properties of the set-deoretic continuum it has more points than it does classically, sorry, than it does in a standard model But on the other hand, of course, classical mathematics prevails there. You can't possibly have every function on reels being uniformly continuous and so on, because that's incompatible with the law of excluded middles. So when you go from the top of sheaves over the reels, right, the sheaves, and you do what's called, you take what are called double negation sheaves, that gives you essentially the Boolean value model over the regular open sets, which is a very familiar thing for set theorists. All of these features just disappear. I mean, all these glowerian features, all of the intuitionistic features simply disappear. And yet both of these objects, the sheaves over the real numbers, well-known object in algebraic geometry, and the Boolean-valued model over the real numbers, over the regular open sets of real numbers. model for set theorists, you go from, and they're very closely related, you can get one from the other, and yet they have very different properties. Now another example of a tapas, which is again, not at all obvious that it has the properties,

55:00 is the free tapas. This is the tapas in which the, it's interesting, the free tapas of course is a kind of minimal one, it's obtained by As I said in my talk last week, actually, since we considered this example, it's the tapas, which is in some way embeddable with a natural number object that is embeddable in all, naturally embeddable in all tapas, and it's a kind of minimal, it makes the minimal assumptions as to the existence of objects. It only has the objects that have to be guaranteed by the tapas axioms, that is a truth-value object, a terminal object, and the cartesian products, and if you like, function spaces or, more equivalently, powers. And that's all it has. It doesn't have any other objects. It just has those and the natural number object. And it's the sort of minimal topas, if you like, in which you can do constructive mathematics. Now, interestingly there, many objects in the free topas exhibit properties called strong in decomposability, which is something I actually mentioned in my last talk. So we say that it's strongly in decomposable if whenever you have a covering indexed by set of natural numbers set a then it's actually equal to the extremely strong form within decomposability that any covering even even when even when it's not disjoint covering then it's what one of the entry one of the elements of the covering has to be identical the whole space and then you can show from that that in particular, I'll just skip over this I don't want to go on too long you can show that this is a very strong if A strongly are decomposable any map to the natural numbers is actually constant I mentioned this last time so I'm not going to not going to now, okay, so I'm going to skip this part For our purposes, so very many objects, it turns out, in the free topics are strongly and decomposable.

57:30 In fact, all power objects, I mean, anything you can obtain as a power set actually is strongly and decomposable, which is a rather unexpected feature of what appears to be a form of constructive mathematics. Now, this, perhaps more interestingly for our purposes It has been shown, as Joyal showed some time ago, although he never published the proof, and as I think I mentioned perhaps before, he's actually proved that in this top, that's all maps from reels to reels, dedicate reels in this case, are continuous. So again, R is actually indecomposable in this weaker sense there. So indecomposability of R does appear to be sort of inherent, if you like, in this setting, in which you're doing constructive mathematics in a kind of minimal way. You're making the at least in the tapas theoretic sense. So in decomposability does seem to be quite a phenomenon that arises rather frequently. Finally, I mention here well, actually not, it's an ultimate thing. The effective tapas, in which all maps are again You can show in the effective tapas, that is, where all the maps are computable, if you like. That's one way of thinking about it, maps between objects are computable maps. You can prove that the power is strongly indecomposable. More interestingly, again, you can prove that Frederic and Reels that all maps are continuous again. So again, R is indecomposable there. Well, of course, this really follows essentially from the, that's not quite so surprising there, because, of course, the continuity principle can be derived in some way by considering the fact, can be derived, if you like, from the fact that maps are computable. And you really only have finite bits of information in order to determine maps, and you can derive the continuity principle from that. And then maps from reels to reels, arbitrary maps of continuous. Now, finally, I want to talk a little bit about the decomposability and smooth infinitesimal analysis, because that's the place where we would really, so to speak, expect continua to be in decomposable.

1:00:00 These other cases, it's actually hard to prove, and it comes almost as a kind of bonus, by comparison with the topices for smooth infinitesimal analysis or synthetic differential geometry, where the whole idea is to force all maps to be smooth. So it's not very surprising that the continuum will be decomposable there, not surprising at all. It's an immediate consequence. Although it's not so easy, perhaps, to construct models for smooth infinitesimal analysis, it's more difficult, I guess, than any of these other cases. The free topis you get automatically is a free object in the category of topises. with logical maps show that it has an initial object and the effective tapas and the tapas, the gros topo and the sheaves or the real numbers are all really rather familiar and not very difficult to construct models for smooth infinitesimal analysis are quite hard to produce a big account of the well, sort of systematic account of the construction of them is given in Mordite and Reyes' published about 20 years ago now, I guess. Okay, so I'll just sort of...