Discussion — 0-Minimality program, tame topology, ramifications & other topics (contd.)

0:00 I mean, Zobar has looked at it certainly in an unrelentingly classical way, and in a sense he's gone into infinite logic here, but there are remarkable things, I mean, just from a classical point of view, in what he has seen. The idea, somehow, is that you look at Riemann-Salzburg-Chanel's conjecture, and it's in a perfectly good sense, I don't care if it's zero, is of course, Chanel's conjecture is essentially an infinite condition from a standpoint. But the idea is to construct the universal domain somehow for... Shannon's conjecture. Now, it's not relative to simple embeddings of exponential rings, it's relative to embeddings which are respecting the Shannon dependence. The idea is that he introduces a pre-dimension, an abstract notion that occurred in Plushovsky before. You take n complex numbers, or n numbers in whatever you are, and you take the following to be a measure of their... ...independence. ...independence, yeah, independence of complexity. It's the difference between the transcendence degree of what we get by taking them and their exponentials and their linear dimensionals. So, Shannon's system is always greater than or equal to zero. I don't construct exponentials where it's negative. Well, it's fine, so it's defined without any assumptions. I always like that kind of definition. Yeah, and then you see, but then... Of course, if you just play around with ordinary embeddings of rings, it doesn't work out. You have to consider what are called strong embeddings. That is to say that they are rings. So, you have a tuple of elements and it's got a certain three dimension. Now, if you go to a bigger ring, you might well insert this tuple into a bigger tuple and bring the three dimension down. Of course, you can't bring it down too much because it was

2:30 an integer at the start. It can only come down finitely much. Go to extensions. Well, this pre-dimension is dropped for some, and typically this happens, this happens only at the level of minus one. If you just E on its own, you get a certain Shannon thing, but it changes if you, if you go to a more joint target. Basically, you have to consider the category of exponential rings with these, what you might call, Shannon invariants. They should not mess up the Shannon pre-dimension of traditionally strong invariants. It's again, sort of infinite precondition. And then you go after the universal domain in the following sense. You're interested in this category, but you're interested in elements in this category such that any system of exponential equations, which could be solved further out in the category along these embeddings, is already solved in the thing you've got. These exist by general analysis, and they have variable properties. For example, there's a criterion for deciding whether a system of equations has a solution. And it's just, if something is formally shining, then it's quite a deep notion, and it relates to deep things in geometry. So Zellweger constructs this thing, and he shows that there are models in all cardinalities, etc. And he shows that under some mild assumptions, there's a unique one, a continuum. And you can get extras of this. It turns out, if it is, the rules can't be defined. They stay semi-tame. They're tame over the integer. Continuous pathology, space from currents, and then it's... I mean, what's your point about the other integers? Well, let me look at it this way. I've been trying to... Same idea, but if you have this theory of the exponential ring,

5:00 where it's defined as kernel, can you prove that it's the integers? In other words, this would mean, to me, this would mean you can iterate... If you have anything like a function symbol whose values are provably invertible, then there should exist a function of two variables. Yeah, yeah. But he, of course, in this situation, one thing I admitted, he insists that the periods of the exponential remain standard. You've got them to start with, Y, I, Zs. The category doesn't have... It doesn't allow non-standard ones. Of course, if you allow non-standard ones, the thing degenerates very rapidly, because it's coding non-standard ones of arithmetic. The moment you have non-standard periods, essentially all hell breaks loose, I would say. So, he's going to play this incredibly game, omit non-standard periods, and omit, for example, Shannon, for example. But then that might not be models of arithmetic at all. That's the other part. There would be models, in fact, typically in this generic thing there would be models, there would be models of tiny fragments of virtual techniques, open induction or something like that. That's what would happen. That's what would happen. That's what would happen. Yeah. I mean, I think it probably has not been written down, but then there are all sorts of other possibilities. That's already in the results. Yeah, yeah. So, I mean, there's any of it that's going to be scientific. Well, that's the problem, but... There are lots of other more imaginative things going on. There's a very interesting student of Sergei's second name, I mean he is really, he's trying much harder in terms of getting, taking models and constructing the universal coverages, general notions of paths, and this connects the deep things of Shaparevich and complex geometry. He has some quite, I don't know what theories he has, but he's certainly got some very good insights and connections.

7:30 Zumbel is trying to look at it from the standpoint of color, and that changes things, because the undecidability tends to, you know, it depends on whether you're going to think of your exponential as defined or your logarithm as defined. You only get undecidability because it tends to vanish if you take a human surface quality. I mean, it's not really clear what is the correct way to formulate things, but just the string and the classical affine situation, whether you can define the real, that seems to be enough to find the rules. With the exponential, with the real life, with the complete exponential, one doesn't know. I mean, I admit serious efforts to try to do it. I don't know what Shannon's opinion is on this. You do get, you're actually not sure whether you can. No, one doesn't know. Zilber certainly believes it can be defined, but that's because this beautiful conjecture of his demands that. The key thing in Zilber's conjecture is to verify this criterion that comes up for when you can solve exponential equations. It's interesting work has been done on this. Unfortunately, it's a criterion for equations of many variables. It's much harder than equations of one variable. But the one variable version of it is quite interesting. It basically tells you, okay, here, you guys know this a lot. Many people are exposed to this. But there are complex numbers that satisfy me. It's also easy. I mean, you can do it by bare hands. Now, Zilber's conjecture predicts more than that. It would predict, first of all, that this equation was solvable with many zeros, and it would predict, in fact, that each n, you could find n of them which were algebraically independent, so a Shannon type conjecture has been able to prove, using Shannon's conjecture, that, in fact, the zeros

10:00 He spoke various other things of this kind involving what we call curve situations. You don't understand the equations of the form f of z, e to the z equals zero, where f is a polynomial over q. We want to figure out what the zeros are, what the independents are. And he's essentially understood all of that, using Shannon's condition, and sometimes tricky things should be announced. But this is not enough, you see. This is one case where we do not expect to be able to understand higher dimensions. Wilbur's conjecture is consistent with some distinctly non-trivial facts in classical complex function theory. It also relates a tiny bit to Neville Leonard theory. For some reason, it's quite an inspiring conjecture. Conjecture implies that you can't define it. It implies, in fact, it's what they call... Going back to the very question we discussed in this session today, Wilbur calls a structure of this kind... In this case, we'd either be finite, and I've found some such theories, complex theories, for example, there's a work he invented, a class of functions called the Liouville functions, in fact, they are just the complex analytic versions of the Liouville numbers, which were the first numbers to be put in class at Edinburgh, the numbers which just, you know, grow too fast for them to be algebraic, and you do the corresponding functions. Wilkie was able to show, using suggestions, the theory of these.

12:30 I mean, they're not interesting functions, but somehow too transcendental, you know, something like that. But nonetheless, one knows it does in MML, but Zilberstein predicts that it's a tamedness problem, isn't it? Well, yeah, I think you could be able to prove you would not be able to do it. And your sets will still be a bit measurable. But, I mean, you can't, you just, you can't get the reals of it.

15:00 It's not just for the reals, I mean, it's a general probability of probably several. Yeah, in some sense of the original origin and all. And it's certainly that, I mean, almost immediately when Cohen did it, essentially, when he's constructing a model of set theory, you can get it, your model will continue. That's one of the reasons why he's actually on it. The very fact that he didn't squeeze that onto something global.