Indecomposability of the continuum / discussion (partial) (contd.)

0:00 The real line, in fact, you can show that every connected space, that is, you take a connected space and you look at its counterpart, so to speak, in the smooths office, and then the connectivity goes over into its decomposability. and then you can axiomatize smooth infinitesimal analysis we just look at these axioms you have axioms for the real line which are stated in first order terms as an intuitionistic field axioms for an order relation which are the next principles I'll just skip over those they're rather familiar um they're rather familiar sorts of principle um well actually if you look at number six not all of these axioms are satisfied in every model i mean i've chosen i've chosen some convenient ones the the tri i just mentioned that the trichotomy principle goes over because that can't be that doesn't generally hold but it goes over into this comparability principle that if you take an AA, it's either bigger than 0 or less than 1. This is the usual sort of substitute for trichotomy and intuitionistic analysis. And then axiom 6 is one that is particular, peculiar to this theory because it's not satisfied by in constructive analysis, as it's usually presented, that if A is unequal to B, then If A and B are distinguishable, then one is definitely to the right or left of the other. That principle actually doesn't hold it. And sometimes you don't have that one. It varies. And then 7 is just the existence of square roots for strictly positive quantities. The key object in smooth infinitesimal analysis is the set of nil-square infinitesimals. this set delta of members of squares of zero micro quantities, and then there's the characteristic principle. I call it the micro-refineness axiom, but it's also known as the Cochlovier axiom, since they first formulated it, and that says essentially that every map

2:30 on delta is affine. In other words, it can be in this third-linear column. Yeah, the last line, because that's, linear would be D epsilon. You wouldn't have the, you wouldn't have the A in it. So, um, well, that's supposed to be equal to the last step. But anyway, you can then prove that, from this axiom that you proved very quickly, that delta can't be, has to be non-degenerate. I mean, it can't just reduce to zero. So it's inconsistent to assume that it's just zero. Of course, compatible with intuitionistic logic, and then you can prove that the, well, elements of R, which you call, one way of, one, there would be several ways of defining infinitesimals, one way of doing it that is independent of the, of the algebra, if you like, of the real numbers in this setting would be to say, and in fact is available in any model of intuitionistic is to say that something is infinitesimal if it's indistinguishable from zero. That is, if it's not the case, if you can show that x is unequal to zero leads to a contradiction. That is a very nice definition of infinitesimal. It's an intrinsic definition of infinitesimal. And unfortunately, it's not available in intuitionistic analysis because you can show in ordinary constructor intuitionistic analysis In fact, not x is not equal to or less than y implies x equal to y. That's not true in smooth infinitesimal analysis. There are some differences. So in particular, every member of delta, that is the nilpotent infinitesimals, or indeed the infinite elements whose squares are 0, would be infinitesimal in this other sense. And indeed, any nilpotent element, any element for which x to the n equals 0. one way of stating continuity at the bottom here, you can say that there are a number of ways, actually, of formulating the idea of continuity. a very natural way, one that suffices for our purposes, is that any function is continuous in the sense that if x is indistinguishable from y,

5:00 then f of x is indistinguishable from f y. Well, of course, classically, that's completely trivial. Every function is continuous in that sense, since indistinguishability classically reduces to identity. The point here is that this is a non-trivial, you know, that relation, there is a non-trivial relation. There's some very nice work that was done by Jacques Penon years ago on describing continuity and infinitesimals sort of intrinsically in these terms, and he did some very nice work some years ago on this. Anyway, a very nice way of formulating another axiom that one adds which leads to decomposability is the constancy principle I've stated in this form. It's not the form that you'll find it in more than I can raise. Essentially, the smooth infinitesimal analysis is really based on the idea differentiability. I mean, that's the sort of key underlying notion. And in order to do integration, you introduce other axes which then have to be shown to be realizable in models. And if you do that and you boil down, I haven't talked about integration here, but a natural consequence of the integration principle, essentially, which is that if a function whose derivative of zero is constant function, and that's the only consequence you need, I state as the constancy principle that if you have a closed integral, an r or d to r itself, and you have a function which is locally constant, in the sense that f of a plus epsilon equals f of a for all a, and epsilon in delta that is all micro quantities, then f is constant. In other words, universal local constancy implies global constancy. Now, it's very easy to prove it. It's actually a very pretty proof. I'm sorry, I'm running over my time. I don't know. But I think it's an extremely pretty little proof. I'm very fond of it. From that you get in decomposability. It's a very nice little proof, so let us go You can easily show, then, from the constancy principle that r and all of its closed intervals are indecomposable. So, you take a to b r, or any closed interval, so a is, right, and

7:30 you suppose that a has been decomposed right into u and v. You then define the function from a to 2, right, by setting it to be 1 if x is in the first part of the partition, in the first element of the partition, and 0 if it's in the second one. And then you claim that f is constant. Of course, if you can prove that, it automatically shows that f is, that gives you a decomposability. Well, then you show it's constant. So you take, for any, you have f of x, for any x in a, b, oh, sorry, in a, I should say, an epsilon in this, in delta, you have either f of x is 0 or fx is 1, and f of x plus epsilon is 0 or f of x plus epsilon is 1 because as I didn't mention any interval or r is closed under addition translation if you like by an infinitesimal that was a principle I skipped over there well then that gives you four possibilities using the distributive law f of x equals 0 and f of x 0 or the first is 0 and the second is 1 both 1. Now the middle two possibilities you can rule out because f is continuous. You can't, I mean, you know, x and x plus epsilon are infinitesimally close, so you can't have one being 0 and the other being, value being 1. So those go out. So that leaves you 1 and 4. In both cases, f of x plus epsilon are identical. So f is locally constant, hens constant and so forth. Very pretty little proof, I think. Now, I said it's moved infinitesimal analysis in decomposable subsets of R to correspond, Grosomoto, to connected subsets of R in classical analysis. That is, to intervals. And you can, this is borne out by the fact that any puncturing of R is actually decomposable, and of course this isn't the case Baruerian analysis. This axiom, the axiom, that in fact R, if you puncture, if you take A away from R, then in fact R is the union of these two disjoint sets, the things that are strictly bigger than A, strictly less than A. The product from O6. Yeah. Yeah, from O6. If you don't have O6. Well, it's one that's realized. It's, it's, no, axiom, axiom is one that's, that's, um, that's insured in these models.

10:00 it's just that it isn't you don't have to have it but the point is that axiom 06 is in some way to the idea that you're dealing with ordinary intervals but if you move one then what's left is decomposed into two parts it's quite a natural condition even though intuitionistically the intuitionistic continuum is rather peculiar in that way it actually seems to me diverges from intuition there. Whereas this one is closer, I think. All right. I've got too much more to say on that. Anyway, I'll just sketch over this. There are a number of indecomposable subjects. You can prove, for example, that various infinitesimal neighborhoods of zero are indecomposable. There's a figure here. yes and all of them are you have lots of them they're the ones that are things which are both equal to or greater than and equal to or less than zero that's indecomposable delta k is the is the set of kth order in nilpotence, that's indecomposable and then the union of all those that is just the nilpotent elements themselves is indecomposable and the set of elements that are indistinguishable from zero is also indecomposable. These are all indecomposable makers. I'm going to skip that bit. Now, I did want to mention finally this is something I mentioned in the last Okay, I'm going to skip that, and we'll go on finally to this. I'll skip the stuff on. There's some stuff on the decomposability of non-standard models, but I won't do that. Now, let's just look finally. It's rather interesting to compare the properties of the various versions of the real numbers in the smooth top. Of course, in every tapas you have, with a natural number object,

12:30 you have the usual constructions, versions of the real numbers available. Vedicin reels, the Cauchy reels, or we call them the Cantor reels, if you like, which uses Cauchy sequences, and various completions of these, because these aren't always complete in a tapas. now in addition to and I mean these these real versions of the real numbers these structures are not of course all isomorphic in the tapas either now in the smooth tapas you also have in addition to all of these that you have, you get for free so to speak in every tapas with a natural number object you have this smooth real line the one that's given axiomatically which is, you know is a decomposable It's guaranteed that's indecomposable. Now, does that imply that the Dedekind reels or the Cauchy reels and all these other versions of the reels are indecomposable? Well, it does under certain conditions, as I've found. It's not terribly hard to prove, but it's interesting to, you know, just to conclude with this. We can define Dedekind reels. I've given a definition here. We don't need to go through that. But the Dedican Riels are defined, of course, in terms of the cuts and the rationals, essentially. There are various ways of doing this, but here's one way. And you get a thing called R.D., which is an ordered ring of Dedican Riels. And you can show it's always... There's a sense in which the ring is always constructively complete. There's an interesting... That is, if you look at Bishop, Arad Bishop, of course, early original work on constructive analysis, formulated a version of completeness for the reels, which you'll find in his book and also in the later book, Bridges. And in fact, you can prove that the dedicate reels in a tapas is always constructively complete in Bishop's sense. On the other hand, it's not conditionally complete in the classical sense, right? of real numbers has least upper bound and the greatest lower bound because you can prove, it was proved by Peter Johnston years ago that in fact being conditionally complete in the usual sense is equivalent to the logical law, to the permeability of this law which is equivalent to the non-constructive

15:00 De Morgan law not A or B implies not A or not B on the other hand the real numbers does share some features of the constructive reels that's not possessed by R. For example, you can prove, you see, that it does differ from this, for example, not x equals y implies x equals y. That's not the case in the smooth reels at all. That if x is equal to or less than y, and y is equal to or less than x, then x equals y. These are all features that hold for the usual constructive, for the reels in constructive analysis, which are not possessed by the smooth reels. oh, and of course there are no nilpotents I mean, most importantly the Denikin reels or indeed the Cauchy reels as I haven't considered here you can't have nilpotent elements non-trivial nilpotent elements now there's a natural way I won't go to the details there's a natural homomorphism that you can define from the smooth reels this is inside the smooth toppers the Dedekind reels is defined in a natural way just to take the appropriate cut and you can prove that this is that this will embed the rational numbers as considered in the real, right, in a smooth real line into the Dedekind reels. And the kernel of this homomorphism coincides with the ideal of infinitesimals where the the appropriate infinitesimals so if you take the ring, you get an embedding of this quotient ring which is essentially the R with this new potent infinitesimals removed into Rd. And you can prove that that is something much closer to the classical. If you factor out by the ideal new potent infinitesimals which is an ideal you get something that's very close to being the Dedicant Reels. because you can then prove that it satisfies it's an intuitionistic field it's an integral domain now, when is there's an isomorphism well, this phi it's actually precisely when R is constructively complete in the sense above

17:30 that is, if R satisfies the condition itself of being constructively complete, then in fact you can get the dedicate reels by factoring the smooth reels by by this idea. And then, again, you can define the open interval topology, and in fact, you can prove that you get... RD is then always connected. We already know that the reels is always connected with respect... Dedican reels, this is the result of stouts, is always connected with respect to the topology, but in smooth infinitesimal analysis, you can get a stronger form of connectedness a form of indecomposability, which inherits from R. Namely, if you take a detachable subset, let's say A, of the Dedekind reels, then A is either disjoint from the image of R or contains the whole of it. And it follows from this that if phi is surjective, then Rd is itself indecomposable. So in other words, if R is constructively complete, then the Dedekind reels, something much closer, if you like, to the classical reels, is itself indecomposable. and I think probably I think probably that's all I've got to say thank you many thanks John some questions Do you see any of the models of the reels, the Dedekind reels in this, I mean the ones which have this property of relatively indeconversibility, as having any possible applications in mathematical physics distinct from the classical I mean there are some people I believe who played around with this idea in the context of trying to understand things like the two-slit experiment oh well I haven't thought of that

20:00 that's a good question I haven't thought of it I mean I confess I haven't really thought of the idea of applications of decomposability or these ideas to physics at all. I mean, it's natural enough it's a good question. I mean, it's natural enough well, yeah, because that's a good question some time ago, you know, work was done by Richard Yasha long time ago, a student of Roger Penrose, and he tried to show how to do quantum, get basic results in quantum theory by considering tapas, working with tapas of a particular kind, where it was the properties of the real numbers within these tapas that was supposed to generate the basic results of quantum theory. his his sort of program I didn't he didn't carry it through but anyway it does suggest that just my point of information that there is a guy in Australia who has been carrying on this work to specifically science yours workers yeah well it's it's there of course the well there as far as I remember from what he was Richard Joshua was telling me years ago he's in quantum computing yeah that's right years ago was that we talked about it I mean, that really it's the property, he wanted to show that it was the properties of the real numbers or the measuring structure, right, the quantities that you use that are the results, or at least arise in the process of measurement, or if you like, underline, I should say, the process of measurement, real numbers, and their variability that leads to these phenomena. As I understand, that's also precisely Corbett's program. Right. Well, it's a very interesting idea. It's also partly Isham's program. Yes, it's partly Isham's program, too, although he uses pre-sheaves. I mean, of course, I know he does pre-sheaves over, you know, certainly. I'm going to talk about that tomorrow. Well, indirectly. But, yes, so in other words, to the extent that it's properties of the real numbers that were supposed to exhibit

22:30 or from which these quantum phenomena were supposed to follow, at least in these sort of programs. Yes, I mean, nobody considered it in decomposability. It's true as a possible property of the real numbers that would perhaps underlie or lead to these physical phenomena. So I don't know whether anybody's considered that. I doubt it. I mean, it certainly didn't occur to me, but that doesn't. And you mean absolutely nothing. Did anyone make the computations? Oh, yes. They show that you can do certain... Working within the topos. Oh, yes. That was the original work. Yes, oh, yes. There have been a number of attempts at doing this in topos, theoretically. There was the early one by Yasha. Now there's this program of Chris Eichem and his gang at Imperial. And Jeremy Butterfield, who was involved in that, too. And as I say, there's also this group in Macquarie, in Australia. But I don't know. It's an interesting... I mean, in decomposability, it's... You see, there's a kind of physical phenomenon. I mean, it's something that... I don't know. It would be interesting to find if it showed up, for example, in some of this work that's been going on in attempting to reconcile, you know, to quantize gravity, for instance. I do have some, if I have time tomorrow, I'll mention some wild speculations of my own in that regard. I don't actually want to be written out of Paris on the proverbial rail, so I may give it to myself. But I do think that the indie composability of it, I do think, is a rather good model. I like the idea of it being a model for plot lengths. I've floated that idea around, although I don't know how to... It's just a sort of metaphor. But when you get very, very small and you can't... When the scale gets so small that you can't really distinguish points, element you can't tell whether events come before one another this is closely connected with a decomposable so yes who knows I've heard Lou Crane speculate on

25:00 very similar lines you know one. You have mentioned two or three times in this position infinitesimals, non-standard analysis. There's an old article, the Pat Nicholson, 20 years ago, something like this, about non-standard the totalization of the poppers. Pardon, la suite de ces papers comment se coure, comment se marche. How do you find? Yes, and the connection between low-standard analysis, Edward Nelson, and several analysis. Well, I don't, yes, I did, right, I don't, I don't think, two things, I did mention last Last week, my talk last week, they're too difficult. The nonstandard analysis, as it's been, you know, as it's usually presented, is very non-constructive. And there are several reasons for this. the fact that the models are constructed using sorry well the usual models you get the compactness theorem is for example the correct logic will yield you know certain forms of the law of weak law of excluded middle it's not constructive the construction of ultra powers models for example you have to you have to use the fact that ultrafilters are prime but you can get ultrafilters I mean actually the existence of ultrafilters isn't incompatible with constructive reasoning as it showed long ago Zorn's Lem is perfectly compatible with constructive reasoning but you don't really care about the fact that you're using an ultrafilter

27:30 it's the primeness of it that's important because you have to preserve disjunctions Right. And that condition is, if every ultrafilter is prime, that's equivalent to one another non-constructive principle. It's equivalent to not A or not not A. So that's one problem. So some work has been done, and I'm not sort of, I confess, not really sort of au courant with it, of trying to produce constructive versions of non-standard analysis, which is getting around that problem, in particular, in intuitionistic set theory, say. I mean, if you just take that as your background theory, how would you introduce a version of, you know, produce non-standard models of that constructively. But you can't use, you know, you can't use the only ultra filters because they have to assume those are prime. So there have been various attempts at, by by people such as Eric Palmgren and also Ike Moordyke to constructivize the right, the production of non-standard models. Now I don't know and I think the program try to produce something like a constructive version of internal set theory, that you don't need the action of choice of whatever non-constructive principles or classical logic in order to do that. If you look at Edward Nelson, if you look at the internal set theory, it's sort of riddled with constructive... Of course, Edward Nelson doesn't care about intuitionistic logic, so of course it's sort of riddled with... And it's very natural that it should be riddled with non-constructive elements there because it was designed as a classical theory. Now, the other thing is the connection between the smooth infinitesimal analysis and non-standard analysis. Well, I did mention that there isn't really a kind of systematic, as far as I know, account of the connection. But there's at least one important special case that was dealt with by, that was introduced by Mordyke and Reyes in their book, where they show, I didn't mention it here, but where

30:00 you can mimic the existence, it's really in non-standard natural numbers, it isn't an producing a whole non-standard hierarchy you know within the model but you can models and i don't know how much further this has been investigated but but in particular in in connection with the construction of the direct delta function this is what they were interested in robinson had done of course the construction of delta function well sometime before um you know in non-standard analysis the problem there is that or the fact there is that if you you um the the infinitesimal delta function trying to think of the function right and it's got it you know it's it's got that it's zero everywhere except except here then the area has to has to go to one so the basis the base for the um for the curve is a you know is an infinitesimal in robinson it's got an infinitesimal width in robinson's sense well that's fine but uh it couldn't have an infinite i mean it's the the new potent infinitesimals are simply too small right you at all. So the problem that Mordech and Reyes, very clever what they did, I don't know whether it's been systematically extended, but they, in order to get the delta function within a smooth setup, the point is that in non-standard analysis everything is classical really. I mean it's all just discrete and it's a very beautiful way of essentially replacing limits if you like by computations with infinite or infinitesimal elements. But apart from that it's just It's just the classical world. The challenge in smoothing for the decimal analysis was to do the same thing, but within the smooth world, where everything is actually the delta function, in other words, could be considered a smooth map. And they did this. I mean, you could do it by essentially mimicking the construction, well, introducing non-standard integers within the model and then carrying out essentially what the Robinson construction. As far as I know, I mean, I think that's a rather, that may be a unique case. I don't know whether any other work has been done there. I mean, certainly there isn't a kind of systematic, it's kind of surprising at first in a way. I mean, I thought so too. Well, you know, why these are two theories of infinitesimals and the, you know, the connection between them is not very intimate, let's put it that way.

32:30 so but but such attempts are being made uh but in the case of smooth infinitesimal analysis i i don't know whether there's a nobody seems to work down a kind of general way of using infinite you know of having into robinsonian infinite infinitesimals within a within a model of smooth infinitesimal now except in that one case as far as i know question sorry to set that on but you mentioned about the relevance in sheet theory of variable sets yeah why this could be isn't it and there are two things that are very relevant there and also in by having their rivals of our options themselves then you easily deal with second-order complication, I mean, that's also not a big issue for the convenience. Yeah. As for the compositivity, it shouldn't be related to the fact that you can tell it to the middle, in the logic of the problem. So it comes from free from the logic of the process of the middle. And on the other hand, I don't know, there's this connection between the use of variable sets in understanding as indexed product, you know, that's a very powerful tool, which shows that in some of those purposes, you have to interpret secondary identification as a indexed product, and the category is not complete. It contains a product indexed of a collection of articles of the category set, which is very useful, very valuable set. And this really solves any problem of creativity. Now, there's a whole, you know, that task I have no idea if this can be used to put things straight, and the other part is use of it to admit it, as far as known in the compatibility law. Yes, I don't, well, as far as the, well, was it your first, your second point, actually. I mean, I can't, yeah, I think between, you know, none of these, of course, in Tampas theory, generally, none of the, you know, the power object, the truth value object already

35:00 introduces impredicativity right to the start. And, of course, you know, the effort to try to find predictive versions of these theories. going on. There's a thing in the effort to try to find predicatives to opposite. And I don't know what the final form of this is going to look like. Mimicking the, but what you're saying seems to be, well, somehow you can get round, somehow get round in predictivity even in a, even in a topless. But it's not as a, you know... Yeah, well, I guess, although, I mean, the fact is that if you, there are certain features of impredicativity or stronger, I mean, if you like done it, impredicativity or extensionality too is another one, which were sort of taken more or less for granted in reconstructive mathematics, which are now, of course, not acceptable. I mean, for example, in type theories, theory, for example, where it's emerged that way. Of course it's not, I mean, it's predictive, but also various extensionality principles fail. So it's not clear to me exactly how these would be realized or what consequences they'd actually have when Topper's theory is modified in the appropriate way to, if it can be, to realize these futures. As far as the, yes, the excluded middle, well, yes, of course, the indie composability is a, doesn't just fall, you know, it's a much stronger form, so to speak, of the failure of excluded middle, you're quite right. Somehow, the fact, you're already one step on, you've got one foot on the ladder already, of course, when you have the failure of the Law of Excluded Middle. You're absolutely right. But what's very striking about these, in decomposability and these stronger forms of it, you know, these remarkable forms where, you know, where you have this sort of constancy principles and so forth, really is that it's not guaranteed, of course, by the failure of the law of excluded middle, let's say, you know, even, well, certainly

37:30 for propositions, that one has stronger and stronger forms, if you like, of the failure of the law of excluded middle, that is not, you know, each one of which is actually stronger than its predecessor, but it's quite true that, you know, once you're on, you know, you've got one foot on the ladder, then, you know, it's necessary to have had that first from. But I think the interesting thing is that, I mean, just from a sort of technical standpoint, I think the early days of, you know, the early days of elementary topist theory, when all this was being worked out, of course, it was automatic, but the law of excluded middle fails. It's very easy to show, of course, that the law of excluded middle fails, just in the Sierpinski talk or something, just where you have two-stage variations. It was then, came, I think, as rather a surprise, that you could actually strengthen this to get in decomposability. Of course, you had already to have had the Fendi and the Law of Excluded Middle if you have these decomposable objects, but it turned out to be rather difficult to show, you know, by rather delicate analysis, that these other forms, stronger forms of decomposability or however you'd, or universal continuity, for instance, we hold. But I agree on that, yes, Yes, the failure of the law of, well, a continuous or some kind of amorphous object really is one in which you can't distinguish. In other words, well, the law of excluded middle does fail, I mean, for identity, and already you've got something like a continuous notion there, but it doesn't follow from continuity that you have a decomposability. After all, a lot of these thinkers thought, well, all right, we know that there are problems about cutting the continuum, but perhaps not many of them would have gone further to say, as strong as Brouwer claimed, right, that just because something's continuous, then it's indecomposable. And that's why I brought up these examples. And as I said, the only one who came out with it, you know, was Brouwer.

40:00 The other is, I mean, well, you have to sort of pudge it a bit, I would say nothing of Aristotle. But the question was whether, yes, so if the law of exclusion, failure of the law of exclusion middle is characteristic, right, her identities, locations on a continuum. This point was actually, I don't know, first made by Peirce. Right, it was Peirce who made this before Broward, he says in 1903, I think it is, Right, right. That's right. Well, that's what he calls it. I think he uses the synecology, or synecism, yeah. I think he really meant sort of connectedness of all things. But, yeah, that's right. But I think there, you see, it doesn't follow, I don't think his continuum, he thought of his continuum as a decomposable, I have to say. I think it's much like, it's something like Conway, you know, like Conway's, you know, the gold numbers, you know, surreal numbers, right, but maybe intuitionistically done. So I don't You know, the indecomposibility is something that is a further, you know, more radical feature of the continuum, even when the law of excluded middle is already rejected for points of it. I have a question, but I don't know how to formulate exactly. For birth, the continuum was very huge. Yes, yes. In fact, birth was first to introduce the idea that the cardinal of R was an inaccessible cardinal. In your framework, in Christian history, logic, topos, et cetera, we don't have the classical theory of cardinals. But nevertheless, can we say something concerning the size of R and the relation between indecomposibility And the fact that art must eventually be very loud, you know, in the sense of the continual hypothesis. I've been asked that question. We don't have the other notes. I've been asked something like that question a number of times, and I've never been able to hear a satisfactory answer.

42:30 I confess. Yes. Well, in smooth infinitesimal analysis, I don't know what the answer would be. I've been asked that, and I don't know. So, of course, you can do, well, well, I mean, I'm not sure, but of course you can show that, of course you can prove that automatically that the real, you know, the real are bigger, the isomorphic to the natural numbers. But then that just follows from the fact that one is, you know, one is discrete and the other isn't. But trying to put some kind of scale on the size, I just, it's not a question that naturally arises there. I don't really know how to do that. I do think in the other, in places where, in tapasas, where, again, one doesn't have a very natural, you know, Of course, in every, you still have, of course, the power, you always have the power set of the natural numbers, you know, in Russell's paradox holds, I mean, it's constructively valid, I mean, so it's always bigger in the usual sense. Now, but that, of course, doesn't tell you, that doesn't really tell you anything directly about the smooth realign, right, in a smooth tapas. It tells you something about the power set of the natural numbers. And in all those cases, well, in the other tapases too, you do have something like a scale. Well, for example, the fact that power objects are strongly decomposable already shows they're huge, right, with respect to the natural numbers. in what sense? Well, they're huge in the sense that you can't decompose. I mean, that they're incomparably bigger, in the sense that if you try to cover, but if you try to cover the thing by an index by the set, then one of the objects already has to be the whole thing. I mean, that's a size... Yes, it's very much like that. It's an enormous, it's

45:00 this huge leap that that you you can't I know that's but that that that is it is it is it is like measurable cardinals yes and that that that's that is a notion which which which I was stronger than measure it's stronger but it's because it's it's not cardinality is equal that's right it's not as I cardinal it's very similar yeah it's very similar it's in the sense that it's a but it's it's not on the other hand I don't think that's true but that property probably doesn't hold it I don't I doubt whether it holds the smooth times smooth topics is supposed to behave very much like the real world he does everything smooth that's why you can do physics in it I mean the effect of tapas and all these other ones, the free tapas, are rather bizarre universes in certain respects, you know, where objects get blown up into, like the power set of one, the truth value object, you know, in the effective tapas or in the, it's already so big that it's incomparably large with respect to the natural numbers. You can't partition it. The only way you could you can divide it up index index of the natural numbers subsets of it is one of the elements of the partition of it cover I should say actually has to be the whole thing at one point in the discussion of the paper paper on smooth internal analysis you well you said that you thought the intuition is that I continue on diverges from intuition this kind of a side side comment in a from the treatment. Could you spell out exactly in what way? Well, perhaps I was venturing out a little in there. Yes, this was in connection with the idea that if you puncture it, it was that aspect of it, which is that if you take these principles, later power principles, it just, if you, look, if you puncture, I mean, you know, somehow the idea that if you, if you, you know, you pull a point out, then what's

47:30 left is separated. You know, I, right, I mean, our interview of the continuum is, well, we don't know that, that, of course, you can't, may not be able to tell whether two points are either the same or not, but once you've punctured it, so you cut it, then it falls into two pieces. Now, according to the intuition, if the intuition is to continue with these other extra-barrow principles, it doesn't. That seems counterintuitive to me. Well, there is a question of dimension here, because if you imagine the continuum not to be a linear one-dimensional thing. Oh, okay. You can puncture a thing and just think of it. Sure, but I mean just for the linear continuum. Yeah, yeah. I mean it seems to me that for the It seems the two things seem to be very associated. Yes, yes, yes. But I do think in the case of the linear continuum, whatever the feature were, that somehow you cut it, it's the whole idea of topology of separation. You cut it and it folds the two pieces. If the intuition is to continue, at least these extra principles, that is, with the free creative subject and the rest, right, with the Kripke scheme and so on. When you kind of continue, linear continue, it doesn't fall into two pieces, distinct pieces. This strikes me as counterintuitive. The syrupy, uh... Yes, because it's sort of, right, it's syrupy. It's sort of, you know, it's like to try to cut a strip, you know, a sort of thread of molasses. Cut it some of that old sort of, you know? Well, no, that's what, that's what Van Dahlen, when he gave, I remember didn't say to him, well, you know, doesn't that, isn't that a bit, I mean, it's not counterintuitive at all for a sort of molasses. Of course, you're right, if the world really is like molasses, actually, then sure. But the idea of a continuum, which is some kind of rigid thing, you know, I mean, where, a thread which doesn't fall, you know, where you can separate it, it seems counterintuitive. Well, in other words, the intuitionistic continuum is is characteristic, as it's enlarged, you know, in the Ferdinand Brauer axioms, principles, it's leading into a certain conception of the continuum, of a molasses-like continuum, perhaps, rather than one that's suggested by smooth infinitesimal analysis, where you could separate parts. And you just have maybe many different conceptions of what the continuum is. So, all right, let's say it's counterintuitive for the thread, conceived that way,

50:00 but not for a river of molasses. Are you happy with that idea generally that there are just different conceptions of the continuum? There isn't the continuum. I'm a fully paid out member of the Society of Pluralists, and I have the cards in the room. Was sind und was sollen die Continua. That's right. Exactly. Can I just say... Oh, sorry. Yeah, can you abound a little bit more on the idea of three church sequence? You know, Yeah, well, there, um, the, the Brauer's conception of it was that, um, the, it's, it informs that, that, that the only, that no initial segment, right, of the, of the, you're making choices, right, in such a way that no, really, either free choice sequence, it has to be spelled out, it's been spelled out in various ways, really, is that no knowledge of an initial segment of it will determine the whole. I mean, in other words, of course, then you have to say, and the idea is in some way that in the case, of course, it has to be given a precise formulation, but the idea is that the continuum for Brower couldn't consists just of uh points or numbers uh real numbers for which one had a at least in principle a full description like square root of two or e or so on that's right oh yeah complexity yes and there's there are connections which have been worked out between you know between the two ideas yes it has been worked out yes yes there they're they're they've been i mean i'm not this isn't have, there are the connections between these two notions. And the idea that if you take a free, that in order to get the, well, I call it the fluidity, the Bauer continuum, it can't be, but look, there was a tension, which Bauer and Ed Weill and others, of course, recognized, between the idea of the, of course, but they had the intuitive continuum, and the idea of the, let's call it the arithmetical continuum. Well, I mean, you know, in mathematics, this is what Weill kept saying in Das Continuum. He says, well, you know, we have this continuum of intuition, and then we have the continuum

52:30 of mathematics, which is something that arose, if you like, from a measurement that introduces a discreteness into it, right? And then how do you bridge the gap between them? Well, Brouwer, Weyl made an attempt in Das Continuum. Cantor says, well, it's not a problem because he doesn't believe in a continuum. It's just hysteria. And then Brouwer felt the feeling of tension between the two of them. He wants an arithmetical continuum. I mean, he wants to develop a mathematical continuum. right there was the program after all I mean to to to explicate the notion of a real number which was the program was the 19th century public mathematicians in the 19th century thought about this was also Broward's you know he was also concerned with this in such a way that the somehow the real numbers formulated in the correct way constitute the arithmetical continuum now as far as he The real numbers as you think of them as sequences, then, of course, one has sequences that are fully determined, and they constitute, I don't know, the part of the continuum that is, you know, a kind of familiar part that's sort of reducible, if you like, to the discrete, in some sense, because you have full descriptions of these numbers, square root of turmeric. that doesn't constitute the full continuum for him because there's distinguishability you actually have gaps between the numbers that are describable or if you like deterministic in some way you can distinguish them the sort of ensemble of all those cannot constitute the continuum which is connected but in order to stay with the idea an arithmetical continuum in some sense. It meant enlarging the notion of a sequence. And this is how he came up with the idea of free-choice sequences. Those are the things that somehow complete the continuum. They're the parts that fill the holes, if you like, between the members of the continuum, the numbers that are fully determined. Now, there are various ways of spelling out what a free-choice sequence means. For example, what about a random sequence? It's the same thing. Yes, a random sequence is a way of thinking one way of thinking of free choice. For example, sequences that constitute Borel normal. That's right, or just flipping coins, I mean, or whatever.

55:00 And these had to be part, in some way, of the continuum, right, for Brouwer, because they were necessary to be introduced them, because they introduced this, they're the source, really, of the Fede of the Law of Excluded Middle. I mean, you know, in other words, you can't tell whether two of these things are distinguishable or not. For Brower, a continuum consists ultimately of elements that in many cases are indistinguishable. And that's the reason why free choice sequence is introduced, because if you don't have them, well, there are models, you can have intuitionistic logic, of course, without free choice sequences. I mean, you know, there are recursive models of these things. But for Browery, it was important that you have this notion, because he thought of a continuum as being something, and Browery, too, it consists of parts, and you narrow it down. And how do you narrow it? Well, there's a narrowing it down. When you narrow it down to a determinant point, like the square root of two, it's predetermined. Narrowing down to one indeterminate really means a free choice sequence, and that had to be there. This seems to me to raise a very interesting connection with your reply to Richard's question, which is just how this Brauerian theme and how this whole relation of this tension between the intuitive content of geometric and kinetic intuition of continua and the arithmetic determination of continua is connected with this relativization of the notion of elementhood and separability, which one sees in the top of setting and which is relative to the deductive resources of particular theories. Is it not possible to look at a kind of general approach to this theme by classifying toposes through their separability conditions and seeing the separability conditions as related to forms of the axiom of choice of the failure of the axiom of choice takes in a topos setting, and particularly how the uniformity of the separability conditions is related to the kind of role of variation stopping that one sees in the case of choice principles. And it's always seemed to me that that's a potentially very strong controlling framework

57:30 for seeing the way in which the geometric aspect of the structure of objects in a topos is related to these logic arithmetic properties. Yeah, as far as the, there are, I mean, yes, there are sort of weak versions of the actual choice which hold, you know, the dependent choices and countable choice which hold in, you know, very useful principles which don't have any untoward logical consequences and which can hold. but there are clearly versions such as I mean in the topos theoretic context there there's a wide variety of extensionality conditions like yeah the well pointedness relatively stronger but those don't then one is a generator and these are subtly different consequences oh yes but actually those don't seem to have Those which you can formulate those, I mean, a lot of those, they are versions of various versions of extensionality. And those... And they all seem to be closely related to separability conditions. Yeah, but they don't actually play very much of a role in... They're, as far as I know, they don't play such an important role in determining the... I mean, except in the case where, you know, you have some very strong bird like the Axial everything decidable they I mean for example they're they're they don't really play very much of a role as far as I know in the in the the type of kind of topics would be considering here I mean they they they typically the extensionality conditions will tell you that you know will be sufficient to to ensure that the tapas is localic or, you know, or, I mean, or, or, or, or, or, or, or, or, or, or, right. And the stronger forms of choice, like decidability, of course, precisely, but you've got decidability of identity points, so you've got a discreet, you know, clear discreet. Well, what they usually tell you are conditions, what they tell you really is the tapas is of a, you know, it really is, it tells you that the, the tapas that you're dealing with is a, is of a certain form, base tapas. I mean, often the extensionality conditions, you know, do ensure that. In other

1:00:00 words, they're really, their importance, as far as I know, is rather more, let's say, external than internal. I mean, in the sense that you tell you, well, you know, that if you have such and such a, you know, the condition for sort of weak form of extensionality, for example, that ensures that the tapas is localic over, you know, over the given one, or another form of extensionality that ensures that it's, even that it's defined, if you like, over given one. So I don't really know whether that, I think the extensionality, there's another form in which the terms used now, well, you know, that extensionality has turned out to be more important, I think, in contexts where you're dealing with theories that are more constructive than Tapas theory. I mean, for example, functions are automatically extensional in Tapas theory, right, because it's like internal set theory. It's like set theory. It's internal logics like set theory. But if you're working with time, then extensionality could do, I mean, and extensionality is something that ensures, for example, that the axiom of choice implies a law of excluded middle. I mean, it already, you know, kind of, it sort of collapses the whole thing. Toppices without extensionality, or the whole theories without extensionality in the stronger sense, for example, that, well, if you just don't have the action mix, you know, that A equals B is not equivalent.