The wholeness axiom in set theory and attempts to find a mathematical representation for a notion of generative implicate order / breaktime conversations

0:00 My basic thinking is, and I missed this out, I meant to say it right at the beginning to try and motivate us more, is that I feel that certainly quantum mechanics, and I believe also classical general relativity, have not been sufficiently explored in terms of their differences. I think we're trying to find commonality too soon we're trying to bring them into a common a common framework prematurely we need to initially focus on their differences and try to amplify their differences really and see what each of them individually is telling us before we then start to look at how they might fit together so this is my small attempt to do that and I just want to finish up with a diagram that came to my mind during this week particularly listening to Nick I think is where it really emerged I have this sort of picture of the relationship between everything and everything which sort of encapsulates my view of it write matter configuration. This is heresy for most people, to separate things like this. Over here I'm going to put space-time geometry. I mean what theory is to relate these two together and I'm suggesting that maybe the way that that should ultimately be done is in two bits there will be a thing that relates space-time geometry to matter and that's going to be some kind of extended quantum regal. And I explain to you why the arrow goes that way. And down here we have an extended general attribute. We'll forget the classic, I'll bring in the classical quantum elements if you can have it. Right, what I'm really saying here is that

2:30 Somebody, I think it was Ernst, pointed out the way that initial conditions or boundary conditions often get left out of the specification, was it? Was it Ernst who said that? His teacher told him off for doing this. No, it was... When he was talking about strange attractors... About... 10, wasn't it? Was it 10? all right so it's true isn't it the boundary conditions are not part of the theory you you you put them in when you when you come to solve the problem so what you know what kind of theory could have the boundary conditions built in it seems to me we have to be a a two-part theory of this type, where in a certain sense the space-time, if you take a cosmological view the space-time geometry is forming a boundary condition, or a background, or a basis to stand on, for your quantum mechanics it's controlling your matter configuration and the matter configuration in terms of singularities is like boundary conditions for the space-time, the space-time has to converge to these to these points, and they are therefore determining the boundary conditions for this part of the theory. So think of it as matter being the background for geometry, and geometry being the background for matter, or think of it in terms of differential equations with their boundary conditions, whichever way, I get a sense of a sort of pair of relationships like that and the reason that I think this came to me because it's very like what happens in biology where you have the organisms and the environment interacting, so the laws that are controlling the environment are very, very different to the laws that are controlling the organisms and they coexist but they don't exactly harmonise get process out of that dynamics of the sort of tension between these two things. So maybe in order to get process we do need a theoretical framework like this where we can have this sort of tension between two things.

5:00 Now you get an idea, how is this diagram changing what we do now? Well effectively when general relativity tries to do the whole job it short circuits this bit. General relativity short-circuits that and replaces it with this energy momentum tensor which is an equation to rig it so that you don't have to ask what the physics of that is. It's rigged so you don't know what the physics are. By definition, this is zero dilution. So you can take any metric you like and calculate what the matter is that produces that. it may come out as an unphysical answer general relativity has that paramedic if you like the geometry tells you about the matter you don't need any other theory that can't be right so that's the way that general relativity tries to accommodate that picture and it's obviously silly quantum mechanics tries to accommodate that picture by replacing this Minkowski metric, so it becomes transparent, it doesn't do anything, you just close the loop immediately, you short-search it. Now I'm saying you need both of those bits in them. Okay, so somewhere over here it's a quantum description, and somewhere down here it's a classical description where's the where's the join so to speak, what sort of join is it well what we've been focusing on this week is mainly inside this bubble so you might say you've got to split that bubble somehow so there's the quantum description of matter somehow you go through one of your non-commuting and you come out the other side with a classical version perhaps but if you want to do it over here you want to do it over here as well so I'm building up a much more complicated system of theory we try to just

7:30 do that that's what the theorists tend to try to do they just want a beautiful equation that will just merge these two bottles into one thing and I'm suggesting that is not the way to go you won't get anything interesting, you won't get processed, and maybe, in fact, nature is telling us, you know, I'm more complicated than that. You can't do it as easy. So, that's where I'm at. Thank you very much. Fred, where have the boundary conditions been in your world? The non-commutative part, I think. So I guess this picture is sort of consistent with what I was saying. But for me that's a non-commutative theory. I have no energy in my theory. But mathematically it seems to be fitting that you want to go to that part, and that part is going to be... The matter configuration should be expressed in non-commutative reality, I suppose. That's the real thing, and that's again the abstract thing, the space-time geometry. reflecting the matter and the reality. So in that sense, I think it is there that an objectivity is there. But that clearly applies to quantum mechanics just considered as a, just a classical quantum split is accommodated there without bringing, I don't know, do you think that you have to somehow... So if you, I don't know exactly that what in my, but when I made that other commutative manner configuration thing I don't really understand what it is physically because as I said it could be again space-time but it could be a feedback in a new way to a commutative world and I don't I said okay this can be brain space but it could also be space-time again that's why I asked you about the curbs in space-time I want it back to be it depends on how you describe that model and so it could be that you get information back there again to the same space but we had a non-commutative world and then you have another projection to the sphere. Yes, yes. And in the meantime you do some non-commutative things because that describes much better the reality of this matter, the world, the world, let's say. But I don't know that. I mean, but that's a possibility, I guess. Malcolm, going back to the early part of your talk about the cosmological expansion. Yes. Any plausible mechanism

10:00 in your theory which could suggest the observed acceleration in its manner? The fact of acceleration. No, the observed, I mean that the actual value of it. No, no, I mean the fact that since the 1998 observations, there seems to be increasing mounting certainty that it is speeding up. Yes, that's what I was saying. I thought I was explaining that. Yes, there are two ways of doing it, unfortunately. isn't it? Two ways of doing it. One is just the idea that our little bits of the universe, matter, of this fractal view, is embedded at a larger scale in antimatter. And in order that on average you've got the same amount of each, there must there be an excess of antimatter, which is expanding. That's what antimatter does. It expands away from itself. So that's dominating the global space-time, the macroscopic space-time, and therefore pulling that particular scale, in fact. Well, that's the funny thing about general relativity, isn't it? When you treat it as a homogenous space-time, the whole space-time has to stretch, has to expand. Right, but I still don't see whether that provides you the speed. anti-gravity gravity, you get attraction accelerating attraction the metaphor will fall, you said the metaphor will fall no, that isn't even a metaphor, it's just a picture but the exploration was seen experimentally on type 1a's bonobo and if you expect them to consist of anti-matter sorry, I didn't hear Yeah. The experiments showing the acceleration were looking at certain supernovae. Yeah. They're embedded in antimatter. Why would they not annihilate ordinary matter as antimatter does? There would be an empty region. I mean, they all see empty regions, actually. We have a matrix of the universe, large scales, and all things.

12:30 It's full of voids. It isn't very homogeneous. Okay, they are voids. And anyway, I don't know what scale I'm talking about here. As I said, the very large scale behavior impacts on the small scale, in general, doesn't it? you can't stretch space out there and have this little bit stay unstretched so you don't need the antimatter here as long as the antimatter is out there and it's pulling that space apart that impacts on what's happening here as well we have to follow our space has to follow that does that no, you want acceleration I want to I want to I want to explain to the observer in the first 1998 the acceleration The acceleration, just the fact of acceleration. No, the fact that it's... Sorry, you know the results I'm talking about, but I mean, I think it's now been repeated several times in the number of recent data, which shows that the expansion is accelerating. Yeah, that's what you want an answer to. Why is it accelerating? Yeah. Yeah, because it's driven by antimatter. Well, why is it just a linear expansion? if the universe is empty but expanding it expands at a constant rate if it's got some matter in it that slows down the expansion because it's pulling it in if it's got antimatter in it it speeds up the expansion it's adding to it pushing it out that's one explanation and the other one which perhaps you can invoke on a different scale is this velocity business that once you form matter and then dust and stars the unit mass is increasing so for the same temperature the speed is less and therefore if there's a speed component in the effective gravitational density the condensed parts are relatively less gravitating than the non-condensed antimatter parts and so they don't balance anymore once you get condensation their net gravitation effects don't balance and the diffuse antimatter therefore slightly exceeds

15:00 the attractive effect of the matter but that's only once you start to get condensation it doesn't explain the early epoch I don't have a no but then you don't know but then if that's precisely what you'd expect to get with the data you wouldn't expect it to, obviously, the rate of expansion would be lesser than the Epoch, because it hasn't gone out of, you know, the back and the out, and the condensation effect hasn't kicked in. That's it, that's it, yes, yeah, yeah. So, you know, so that's what I see. And the other awkward thing is, well, why are these, why are there these irregularities in the first place? Why do you have regions of excess matter over antimatter? And again, that goes back to origins, and I have no idea what to say about that. I do have a mechanism that if you've got small irregularities, they can be amplified. I do have a mechanism for that. It's an extremely weak mechanism, but in principle it is there. Once you've got some irregularities, I mean, this is how gravity is. It does tend to, over time, accentuate any non-conformity. And I can find that in my model as well, that the anti-matter tends to join regions that are anti-matter. Matter tends to join regions that are matter, so increase these recognizes. But going back to, you know, the beginning of time, I don't know how to do that. You said that plomions repels the solar and photons... No. No. No. And the matter repels everything. Repels everything, yeah. Matter attracts everything. The photons... Do nothing gravitation. But they do follow you basically. So, does it mean that they are acted upon, they do not act? That's right, yes. Yes. The latest sake of cow. Yes, I'm sacrificing. I can't remember, who invented that one? I don't know. Somebody said that you... Things, I mean, it was sort of a... this was a discussion with the African American people. Oh, it goes back as far as that.

17:30 Anyway, you see that quoted again these days, that principle, and I keep seeing it coming up. If someone could produce anti-hydrogen atoms, this would prove you wrong, wouldn't it? No, not if you did it in a laboratory, no. You mean if you found one in nature? if you say antimatter all this repels gravitation it's a theory of gravity only electromagnetic forces are not effective so they're not unified they are not unified they're not they can't be unified Were you at our table last night? Yeah, that's okay. I thought you were speeding. Built a spell. To click OK. Can't close this down. The bubble from the paper clip. Ah, the table. Wanting him to say okay. I have to say okay. I don't know. Copy break. This is a copy break time, I guess. Natural evolution. So I guess you sort of took your sort of idea of quantization, de-quantization. But maybe that in another cell. So in my opinion you can get any other big computer geometry. Because it might be important. I didn't just start the name off on the first slide, but I don't know, it's not mine, it's

20:00 not the word, it's not the word, it's not the word. As I get older, I do more and more of that word. The end of the word appears at the beginning, so get down to the word. Oh, okay. So I'm just going to give you a very short presentation. And I thought that I would try to convey just some of the ideas behind what I'm trying to do. And so I call this talk the Generative Implicated Order because this is essentially what I'm trying to look for. I'm trying to find a way to describe the implicate order that we already heard about, I think mostly on the first day. And so I thought, as a background, I could probably use some of the things that we've already seen so that you can relate back to it. And so, the implicate order, where does it come from? Well, in Basil's talk, I think, he had a question. What is basic? What should we use as the fundamental? What should be fundamental? And his answer was, I believe, the hollow movement. And the idea is that we should use process as fundamental. And this process moves from something that is implicate, where you cannot distinguish between things. All of the orders that you have in the explicate order, which is the order where we can measure things, exist in the implicate order, but you cannot see them explicitly. And so the idea of the movement is a movement of unfoldments and enfoldments. and it was depicted this way. Imagine that this is, this E represents an explicate order, then there is an unfolding and unfolding that you could describe by ends, for instance, in this, in that example. But, yes, and then if you continue, you actually come to Schrodinger or Heisenberg evolution in the quantum theory, just from this.

22:30 What I would like to do is, I would like to have the implicate order as something that is truly fundamental. And so I would, instead of starting from the explicate and going to the implicate and back to the explicate, I would like my description to start from the implicate. So that I start from and then I move out, have an unfoldment, so we can maybe draw it like this, and then because it's difficult to draw the implicate order, of course, the explicate, you have just a dot, the implicate is not anything that you could explicitly see in any way. So this is the way that I would like to try to model it and the idea, let's see, the idea here to my approach is I would like to impose a context that can work for me. so that I don't have to worry about it. Very often when we work in physics, it's more in a reductionistic way. So we tend to start from parts and then build upwards. Of course, it's not purely a reductionistic process, because we do also try to consider, as we were talking, I think, today at lunch, we do try to consider the larger picture. So we do have a certain purpose. But I feel that although we have the larger picture, we don't have the larger picture working for us. We don't have it working for us automatically. This is actually something that we do all the time psychologically. If you think about intention, for instance, if you have an intention

25:00 to do something, then that in a way orders what you are going to be doing. You may still If you are thirsty and you have an intention to drink, then that orders the whole situation, your whole dynamics. So whatever you do, you will be looking for, maybe, oh maybe here I can drink something, or maybe somewhere else. It's still an open-ended intention, but we use these intentions all the time, and they work for us, and they are implicit in a way, because we don't have to constantly think about it, we enfold them somehow into ourselves and have them working for us. And within that, so whatever we do, we do within that order then automatically. You see that also in the quantum theory, you can see it as a type of formative causality. So you have a sequential I think I was talking to someone about this. You have a sequential causality where one thing causes the next and then the next and so on. So you have a sequence of events. Or you can have a causality that depends on the context. And so then the whole of whatever is happening, the whole context influences the movement of the part. And in this situation, it is very much working. It is very much working for you. So it's not something that you have to think about. It is already working for you. So this is what I would like to do here. Can I impose this implicate order somehow from the top? So I'm not working from the bottom, I'm trying to first impose an order that will be working for me all the time. And so, what would I like? Oh, and also, generative. I would like

27:30 this implicate order to be generative. What does that mean? I mean, I would like it to do something. I would like it to change something, introduce a real change. And so, how do I do it? Well, what should my order be? Because I would like to use as few assumptions as possible. And also, my idea here is to start from the infinite, not from the finite. I somehow, the existence of this type of the dynamics, a wholeness, where things are not really distinguishable, you have the implicate, in a way, but you have a dynamics in it that generates something new. So I would just like to assume that. And the way I do it is that I say, okay, so let's assume that there is a wholeness preserving movement, you could say, that is non-trivial, in which wholeness moves from itself to itself, but it moves in such a way that all of the orders, everything is preserved. Everything that you could possibly define, everything that could be defined explicitly, is preserved in a way. So mathematically the way I do it is through the Homeless axiom which says there embedding that I can call J, which goes from the whole mathematical universe, which is to itself, in the end. But it moves in a non-trivial way. It is non-trivial.

30:00 and this is, yes, and this is the way that I try to model, or this is an analogy, a mathematical analogy for the generative intricate order. What happens is that so this part you can see as this in a way where you have first you have the unfoldment process. You can also split J into two parts if you want to and then see this first part as it is an elementary embedding. Oh sorry I didn't mention what an elementary embedding is. An elementary embedding is basically something that preserves everything that can be defined mathematically. All relations, everything, is preserved. That's all it does. And what I'm saying is, I'm assuming that it exists, and I'm assuming that it's non-trivial. So that it actually does something. So that so that at least some set here is moved to another set. Do there exist non-elementary embeddings? So does an elementary embedding preserve non-elementary embeddings? Yes, if you can define it mathematically, yes, non-elementary embedding. Yes, non-elementary embedding. I was asking, that was the first question, what is a non-elementary embedding? Oh, a non-elementary embedding. Well, it would be something... I'm not sure. You mean the opposite of it somehow? I don't know. Just a function? Any function? Yes, it depends. I'm just checking the definition because the definition is a bit vague for me. So if there are non-elementary embeddings, I suppose so, then an elementary embedding also preserves the non-elementary embeddings.

32:30 Yes, it does. What does that mean? For instance, what could be a non-elementary embedding? This is what I'm not sure about, but say it's just a function. Does it preserve it? Yes, so if we're any formula, so if you have some formula... So it preserves every function that is non-elementary embedding? Yes. Any definable formula? Yes, mathematically definable formula. If you have any mathematically definable formula that holds true in the domain... Then also, this will hold true. It will also hold true. So yes, this is just a way of modeling this. Okay, so I can do it perhaps this way, and many things can be said about it, but is it interesting really? Can it be related to physics in some way? What can we do with it? I would like to... well, is it interesting? Yes, I think it is, because the idea here is that we start with this dynamics, and what we're going to do is that we're not going to add anything new to it, but we're going to let it act on itself. So can J act on itself? Can we apply J to J? It turns out that it's possible. So we can define application. And when we have J acting on itself, this will generate we will have many different we can we can look at all possible combinations of it so we have J J acting on J J acting on JJ and so on and so what we what we turn out to have is a universal algebra which is just a set with an operation and the operation is an

35:00 operation of application. And what are we really interested in if we're interested in physics? Well we're actually interested in going from the explicate to the explicate in a way. This is the description that you would like to have. Although this might somehow be fundamental, we're still interested in describing the evolution of processes, where we first measure something and then let it develop, and perhaps measure it at a later point. So you can see that I think it was Nick who had this, and stressed that this is what we want to try to model in a way. And this is where our implicate orders would We're going from the explicates, through the implicates, to the explicates. And so, we can, in order to see how the explicate orders are related to each other, we can look at AJ, this Universal Algebra. So, what is it like? What can we know about it? Well, it turns out that this algebra is left distributive. For left distributive, so if A, B, and C are elements in A, J, this is what we have. Left distributivity. It's not right distributive and this is this is the only the only thing that we have there are no other it's not yeah so okay so this is a left distributive universal algebra and this left distributivity can now help us to see how the different processes, if you like, are related to each other. And this is where we can learn how we go from one explicate order to another. It also turns out that one can define linear

37:30 linear orderings on these elements. And there are different types, actually. But yes, so that is something else that is perhaps interesting in it. And also, application is different And it turns out that application, if we look at everything that is reversible, just at the reversible part of application, in a way, So, we say i and k are elements in this algebra, then we have this relation. And this, I don't know if you remember, I took it away if you remember, but this is very reminiscent of something that we've already seen. Namely, from this, if we see this as explicit orders and unfoldments and unfoldments, from this we can derive the Schrodinger evolution, if you like. But the only elements in your algebra are applications of J to itself? So the I and the K are applications of J to itself? Yes, they are. So why is it inversible, this i-1? Well, why is i-1 inversible? Well, the application of j2 itself, it was only an embedding. So there you use an inverse. You have j, which is a one-to-one function, you could say. It's not really a function. It's defined on all of the mathematical universe. You didn't say that it was subjective. No, but you can have an inverse on the image.

40:00 Yes, on this part where you have J, you can have an inverse here, which is not an inverse because so there are no inverses for J really that's my color right yes there are no inverses on J but you can have a partial I don't know what you would call it but but an inverse on this part on this part you can have an inverse but not on all of J but then it's not in that algebra, the inverse, is it? You don't say that or...? No, no, it's not. No, no, no. This is, you're absolutely right. So this is just to say that this is not reversible. So we have a direction. But if we look at what is reversible, if we look only at the reversible part, and why do I want to look at only the reversible part? Because normal dynamics is reversible. I mean, the normal description of dynamics. So that's why I just wanted to say, okay, so if we look at just the reversible part and so forth, if here we have x, it has to be in the image of i. But it's in the image of K, or what, I mean, or is IK on X, right? So it's in both images then? Well, it's... There are two things acting on it, well, three, I mean many things. IK and I-1, so I-1 I understand now because you restrict to the image. But there's also IK. So why is it in the image of is that trivial? Maybe it is, yeah. Yes, it turns out to be that way. Because that's another element of your algebra, right? There's a J, J, J, J in this K, for example. Yes, yes, yes. I and J are elements in my algebra. Everything belongs to any G, doesn't it? Yes, everything. Whatever you do here, we're going to be... But not everything is in the image of many powers of J, let's call it powers of J. Because you go slow, down, down, down when you take composition, but you can take applications.

42:30 Applications, yes. So why is it, is everything in the image then? Do you look at each in, this k is arbitrary, right? In that statement, k is arbitrary. k is arbitrary i is arbitrary any i any k but you only use the inverse image on on of i on the image of i right not on k yes yes maybe i'm misreading this first part because what is it in fact is it i k on x or i kx yes yes it's it's k applied to j to i Then Ix must be in the image of K. Let's see, I, first we have... Go ahead. No, no, but let's see, it's just application, so it's J, let's see, so it's I, and on it we apply K. And on it we apply k, which, and then we look at a set in... So you do it in the other order? I k, you do first i then k, right? So, yeah, so this was just to give you... Yes, to show you that there might be an interesting structure there. to maybe finish with, because I don't have so much time, is to tell you that there are many different ways of proceeding and trying to go to physics, for instance. It turns out that this universal algebra is isomorphic to special braids. And special braids are... that, well, I don't know if I should start with the braid group, but they're just

45:00 certain braids, and it turns out that they are not, they're not reversible, and these so they're not, the braids in here are not inversible, they are, yes, and so if you then have your special braids you can continue, and what you can do is you can try to embed those braids for instance into, because braids have crossings, for instance. So you can try to embed those crossings into a plane. And when you do that, you end up with a special kind of temporally leave algebra. And temporally leave algebra can be seen as towers of von Neumann algebras, which are the algebras of observables in the quantum theory. So you have that way of trying to connect, connecting it up with physics as well. So I think that I'm going to finish around here, So, the idea is, I would like to try to describe the implicate order, and I would like to describe it from the infinite to the finite. I would like the infinite to somehow act on itself, and thereby give us the structures that we see in physics. This is my idea. And this is the way that I've been looking at. I've used an elementary embedding and then I've gone and looked at this algebra which can then be connected to brain. And brains can be connected to physics in a different way. This is just the outline of the picture.

47:30 So mathematical equations are fundamental in your picture. Sorry? Mathematics, anything mathematical is some more fundamental in your implicate order. The implicate order is mathematics. Well, in this case it's modelled that way, yes. Is that inevitable? Are you saying that it has to be that? You mean, what would you like it to be? I've never thought about the implicate order. I would like to have it mathematically because if I can impose an order from the beginning that is mathematical then things will follow mathematically from it and it's easier to relate to the mathematical structures that we have so that's just in physics that's the only reason why to be able to connect it to things that we already have. Yeah. So, that was just an outline of... But elementary embeddings, how can you find them? You have to respect all definable properties. Well, I do it by saying, by just saying that they exist. Or postulate, no... Axiom? Yes, it's an axiom. But do you have examples? Of an elementary embedding? One case where, well, I suppose if you have very few, if you have very small v, perhaps it should be, but I'm not sure, because everything that you can mathematically define, that looks like an awful lot, even if you have only two elements to use, I think, I can invent them. And it also turns out that, I don't have an example, and the reason is that... Because you have an infinite set, I cannot imagine an example. It was infinite, right? Sorry? It was infinite. Oh, it was infinite? Yes, yes, yes, yes. V, I assume, is Gödel's set theory of the universe. I think you said it's infinite. And then to invent a map that respects all properties. Yeah. I mean, V is a standard notation in set theory for the whole set theory of the universe, and I think that's what it's intended to be. Yeah, yeah. That makes sense of another one. So that's an action that exists.

50:00 Yeah. That's the axiom. And, from the axiom, it also then follows, it turns out, that you cannot have an example. Is that a good axiom? Well, you cannot have an explicit example of what... Yeah, that's what I wondered about. But that's okay, because I'm not... I don't want to have... I'm looking for the structures that I get from it. I'm not looking, not actually looking for what exactly it is. But if I would say something does not exist, then I also would know that I cannot have an explicit example of God. So, can you exclude that it does not exist? Can I exclude it? When you put an action, you have to be free. And are you free? I have to be free in what sense? In making the action. Why is that? I don't understand. Well, it has to be independent, I mean, yeah. Is it proven to be independent? This is the question you said. Is the Horner's axiom proven to be independent of the other? The other axioms are set there. Yeah, well, it seems to be, yes. It's not proven to be independent, but it's... I think we would have heard from the Scepteris if it had been. I don't think it is. I mean, see, it's a reflection. So it's a Riemann hypothesis. Suppose that the Riemann hypothesis is true. Yeah. And then you prove something. Yeah. But you actually don't know whether it's independent, right? Yeah. They're very weird, these axioms, like the wholeness axiom and Scepteris, and what I've understood of them, these, you know, the so-called reflection principle type axioms. and they're very I mean there is something I mean this is very interesting but there is something I think I said to you last year that I find terribly counterintuitive about it conceptually which is that you're taking as your starting point this wholeness axiom which is connected with this kind of Gordillion and Platonist vision of the whole universe of mathematical objects as somehow existing already as a completed all-encompassing totality there of a rather static and Platonistic sort, you know, really the ultimate vision ontologically of static being, making the whole of being, you know, it's completely static, and then

52:30 out of that trying to get a mathematical, you know, some mathematical clue as to how one ought to think about the implicate order, which is really an extreme opposite in terms of the way that it leads us to conceive the whole being as something intrinsically dynamic and standing conceptually almost in complete opposition to a kind of Platonistic vision of this kind. I'm not saying that that destroys your project, but it just seems to introduce a strong conceptual tension in the guiding ideas right from the beginning. Well, maybe it's because I don't really think it that way, because to me it is very, very dynamic. I mean, the axiom itself, the axiom itself, the wholeness axiom is a movement, in a way. You can see that as a movement. So that is dynamic. And then why do I do it in set theory, for instance? Well, because I wanted to make it as fundamental as possible. categories, maybe, yes. But this is where I started, and I feel that I don't really have a problem with it. No, I mean, just maybe different ways, different styles of thinking. I mean, to me, the homeless axiom is something with which I can't imagine how you'd give a kind of catalytic version, which seems to be completely against the whole spirit of Well, because it's assuming that you start from this completed totality, V, and try to introduce, by looking at this kind of reflection principles, ways of thinking how it could reflect all the parts of itself within itself. But that's to think of, you know, there being one all-encompassing universe in which all mathematical structures live, and guiding Biodic-Cathlete theories. But you're talking about... You don't think that everything is living inside one single universe. You think it is related, right? We are talking about the completed, as it were something that had so well that it could totally define. But isn't that part of it? It is not totally definable, is it? Well, it's infinite, and it's not a set, for instance. it's obviously a proper class, but V is not a set, I mean, yeah, that's quite clear, that's the first thing. So there are those ideas, but it's the absolute universe, it's the absolute, you know, it's the absolute totality of all sets, in other ways.

55:00 But there's nothing to talk in the bag, you see, that's why. No, I'm sorry, I'm not saying... And the thing that, in category theory, the thing that I had a slight problem with was the objects. Because it seemed to me at the time that you always had to have an object. And so, right, right, right, right. So, right, so my object here is then the, you could say, the mathematical universe. Well, the absolute totality of all possible objects in the sense, as if you think like a plomb of a Platonist like Erdlethorst. No, I'm just trying to articulate what seems to me, but this just may be my subjective impression, a slight conceptual tension in the starting point. So the implicit order on the one hand, which is all about ultimate dynamicity, if you want to use ultimate flow, ultimate becoming is ultimate, being is derivative, and the Godelian vision of the set-theoretic universe in which static being is ultimate and becoming is entirely derived and illusory. But it's not. It's not to me because I'm changing it in a way because I'm changing the axioms. I'm adding an axiom and I'm adding a dynamic axiom. Yes, this is where I guess I'm just not seeing where that dynamic aspect of the wholeness axiom comes in. Well, it's a movement. Embedding itself, it's a function, you could say, but it's... Well, okay, I'm sitting up there. Is there any way to, I think, it's a set of words for the large cardinals, and it really has to do with this inaccessible cardinals and definability. When I say everything we define, we can define, is preserved, and obviously this embedding should do something where we can not define something. So, what's your idea? Is there some shifting around of this very, very large set, which we have no hope whatever to grasp. Is this what this mapping J does? And is J unique by the way? Is there only one J? Is there anything known? Or what do the set theorists say? About Jane. No, it doesn't have to be, no, it doesn't have to be unique, no. There can

57:30 be other, could be other jades. And yes, this is... But they have all the same structure, that's what I'm saying. All the jades have the same structure, whatever it is. The same structure, jade. That's an operation, that's an entity. but it says the left self-dissipity holds forever as I understand it it's a little bit like and I think in fact the way that people like Gödel thought about it was actually explicitly guided by Leibniz's ideas about the monads all the monads, the monads each as it were see the whole of reality but each from its own individual point of view they all reflect the universe there's a supreme monad which is god which reflects the entire universe because it is its own ultimate expression kind of self-reflection and it's all very pictorial language but to my surprise i mean i had an extremely interesting talk in paris about this about a month ago um these ideas of leimnitz and the monadology were a very very strong guide on girdle when he thought about set theoretic um axioms and particularly reflection principles of which the Homer's axiom is is an example so you know there is a kind of underlying metaphysics in the background there but it's one which intuitively to me seems to be completely imposed to the spirit of Bohm and the Implicate Order but that's just my own take on it but is Jay considered to be mathematically defined? No, it will. No, no, no, it's, no. What will mathematics be, no? No, it's a better not, Peggy. It's better not, otherwise. Because otherwise you could always go one better. It's the idea that you could never get, it's wrong. I mean, Jerry, as I understand it, it's really more an expression of the fact that you could never get outside the universe of sets, in some sense, although, of course, the universe is not itself a set. But you can apply that embedding when proving or disproving things in terms of that set theory, I guess. Well, yes, with the help of it you can show certain things, like within all of those axioms, with the wholeness axiom. You can show that the existence, for instance, of all large cardinals.

1:00:00 Yeah, that's precisely what it was introduced. It was to provide a kind of cut-off point so that you could, you know, instead of the various reflection principles which were used in proving large cardinal theorems, this is supposed to be some kind of ultimate maximal reflection principle which will say that all large cardinals, in some sense of all, that somehow this is supposed to pin down. It's more precise than that. Well, yes, it is more precise, and I don't know enough set theory to understand exactly in what sense it does operate as to make the notion of all large cardinals more precise, but that's as I understand the motive which was involved in its introduction by the set theorists. But it very definitely came out of the way that set theorists try to think about the whole universe in terms of reflection principles in order to guide them when they're thinking about large cardinals. And that way of thinking is so tied to the Platonist vision of the whole totality existing as, in some sense, a totality of determinate objects. And as I say, that to me seems to be completely opposed ontologically to the underlying idea of the implicate order, which is why it just seems to me to involve a strange tension that you should be trying to get a mathematical model, or guiding idea for thinking about the implicate order out of something which, you know, comes from something that is in the first place, it seems to me, so opposed to its spirit. But I'm not saying it's wrong, just that there does seem to be a conceptual intention involved from the outset. But isn't, in a way, that theory the most fundamental thing we have? Well, you have categories. Many people have thought so, and some think different. In some meetings they would say that numbers Yeah, you pay your money in a taste, it takes your choice according to taste. See, that's why I chose it. Yeah, sure, sure, sure, because you believe that it is, as many people have done, that it's simply the most fundamental notion we have in mathematics. I personally don't believe it is. But I mean, the whole point, surely, of David Bohm's approach was that he thought that the notion of order was fundamental. Exactly. So he would surely have wanted I know he didn't make specific pronouncements much about foundational questions of mathematics,

1:02:30 but surely his intuition would have been to try to find some way of re-expressing the notion of set in terms of some more fundamental underlying notion of order rather than the other way around, trying to capture the notion of order within a framework that takes the notion of set as already basic. It just seems a very unbombian thing to be doing. Sorry. Well, I just see it in a different way. There's so many things in it that remind me of the intricate order. Like sub-similarities, for instance. This is what you get pretty automatically from, say, into the infinite. it. And I don't know, there are just so many things that, because I'm, again, I'm not, the step theory is not important, I mean, it's not what I'm really interested in here. No, no, it's fine. It's the order that I get from this chain, from this elementary embedding. It's the order than it gives me. But then you can say that every other theory, even though you might, you'd be using whatever you may be using, you might say but really, all of those things exist in set of theory. Yes, that's where we part company, but that's separate. That's an issue in foundations of mathematics. Yes, it is. Directly relevant to what you're doing. Right. Something that an idea of the... That movement of Maybe it's not possible to be made explicitly, certainly. That feels right for the whole movement, doesn't it? To me too. It must have been an identity in everything you can. Thank you. so who's next time for he's talking about time isn't he he's talking about time about time

1:05:00 Oh, is this? What? Oh, is that a verdict? Really? About time. Yeah, and we left it to the last minute. Pun intended. Do you want to use this as well? Do we have a stick? A memory stick? I think it's him looking for a book. Yeah, I'll buy that. Someone has some space on the memory stick that we can transfer. Yes, I think I have. But can you check if it has got it? Because this is the thing I can do with that. Can you just check if it's working at all? No, no, no. Just check if there's nothing on it. The funny thing here is that... When I came here I thought that was... So you want this implicit order to be mathematics, yes. But in fact, when you work with J, this is not mathematically definable. Right, because otherwise it would be a combination. So that's why I was asking that. So in a sense, if you want to implicate order to be mathematics, then you want to use that J, then you have to do the mathematics of the mathematical undefinable. Right. Which is what Gödel thought he was doing, I think, you know, the whole point being that he said, or at any rate, well, in the way that he chose to think about the structure of the V. It's a little bit like when you introduce Grotendieck, you know, the Grotendieck universe into the category of theory, or the second Grotendieck universe, you're just helping yourself, as it were, to some kind of closure principle that is not really, I mean, arguably is inconsistent or teeters on the brink of inconsistency no no well but it teeters almost on the brink of inconsistency but it would be interesting to know if it's independent because that probably can never happen I mean you can never prove it but it's typical of those but it has to be given that it's a word what is it where Marx the outer edge but I'm still a little worried

1:07:30 because normally when you introduce an action and you use these actions and you have a system then you call that mathematical definable in that system now you have an action, suppose the whole thing together with the action then in fact what you do should be mathematically definable using the action and then you get a contradiction if you then apply the same reasoning again so you should stop at that moment and do not go back to the new system I agree, absolutely Is that also the issue about Greuten-Dix's use of universes, of the second new Greuten-Dix universe? But it's more like a Kantian regulative ideal, or a pure reason, than it is like an axiom in the way you think of it. But it's not the same. No, it's not exactly the same, but there's this much in common that it teeters on the outer edge. Of course, normally, mathematical definable, if you're right, you said functional, yes, depends on actions. The whole mathematics is built of actions and corollaries and theorems and lemmas. So that in my, if I understand it, that means mathematical definable. I can try it out on my computer as well. So there are actions. It doesn't work. There are actions already included. I can try it out. Now you're at independent on another action. You couldn't put anything on it. But then in that noose thing. No, it won't read on the file. Oh, well that could explain why I can't get it to read on the thing in the hotel either. Now we are in trouble if we then reapply let's say yeah yeah because then this j becomes to be mathematical definable and then you're in trouble but there's i mean there's a very simple issue you don't work in the new system you add an action to the system which you then don't really use as a new system yeah there's some kind of strange there is and i think that you see the thing is i think you're absolutely right but when you say that the whole of mathematics rests on you know axiom systems this is of course in one sense yes obviously true but what does the general semantics of the axiomatic method itself such rest on I mean why I mean in 20th century mathematics our notion of axiom has really been influenced mainly by Hilden's success as effectively as pretty well any anything for which you've got a consistent model but you know if you go back to Euclid it really did mean something which was kind of ontologically guaranteed as the starting point

1:10:00 it was something to do with the categories of being a site and that aspect of it except for people like Gödel, it's largely dropped out now, so it's just become in a sense you are saying it is mathematically not definable as long as I don't use the whole section Yeah. That's what you said. Because if I use the whole section, it's in that mathematical system. Yes. So you're not using the enriched axiomatic sets, the axiomatic system, let's say, because then it becomes definable. Yes. It's the same as, like, before we take it out. No, no. But if it's mathematical definable, then you get into trust. So if you begin to work in a new and last-time geometric system, then that j is mathematical definable. And then you accomplish that you don't write. Yes, exactly. Which is why I say it kind of teeters on the very edge of... So it's only a matter whether you call it mathematical definable if you just add it as an axiom. But all the other things that you do in mathematics where you add axioms and theorems and results, and you call that mathematical definable, should also be, that new thing should also be mathematical definable, with the new action included. Then you should say, okay, I can use the action. But then it comes to mathematical definable and it is not. So in a new system you get a contradiction then. Yes, I agree. Absolutely. And I think that's telling us something very deep about the way that we now think in current mathematics about the status of the axiomatic method itself by comparison of the way that, say, the Greeks thought about it. I think subtle here, but it's more a philosophical problem. It's not a problem internal. You wrote the phrase, everything that is mathematical definable, before you actually get the J. But if you look at it now again, after your talk, and I see that phrase, then I will use the J. Then you're in trouble. But at the moment you used it, I didn't think of using the J. You said mathematically definable, I would think of the actions in the theorems and the results and the constructions that I know, and that would not include the Holman's action. And so I don't really, there is something funny about this. It's very funny, I agree, but I think it's acting more like Kant's notion of a regulative ideal than it is like the notion of an axiom. A Holman's axiom is not an axiom in the same sense as axioms in other parts of mathematics,

1:12:30 so it seems to me that's so you're talking about self-reference really yeah after you introduce it and you look at the beginning it changed it's an uncertainty principle yes but that you should say that you should if you say that this cannot count as a as a statement of mathematical primable then you could agree with it yeah normally you would assume that when you were allowed to use the actions to define mathematical definability. You're obviously not intended. But then you should specify that, right? Because that's what got me confused. But then you can't specify it within the... Yeah, because before you defined it. Sorry, sorry, right. Hey, I'm trying to write on some time with you. You're losing time here. I'm basically a metamethical physicist. I started by doing analysis of pure analysis and then I moved to the metamethical physics, which for me is basically applying analysis of physics and mostly functional analysis. And so, which means that I belong to a group of people who try to take a mathematical view on what physicists do. And generally speaking, we are not dealing with...