Essential ambiguity of natural number concept / discussion

0:00 We're giving some cracking meditation on what led to the position that by 1930 the opinion-forming classes in its country were able to assume that Christianity could just be completely, so completely ignored, so completely ignored that they weren't even aware that it was ignored. It was not entirely a matter of state policy. No, no, no, but it's just me it was. Good God. No, no, no. No, but, um, no, it's a kind of general intellectual history of the 19th century, but written up from an extraordinary, sort of a dysentractic youth one. And there's a series of essays. The first one is in some ways the most interesting, so that's the most author of my family, and that's just called Recognition. There's a chap on one page which is very interesting. Alfred, yeah. And even more of a sort of attack on Churchill. Churchill famously said that he wasn't a fellow of the Churchill, he was more like that. Oh, great. What was he going to support him? I'll tell you. Yes, well, no, Churchill was a classic Yvonne and he certainly had ceased to have any religious beliefs of what was said about the time he did Bank of Law and sort of educated himself in an extraordinary verse at the age of 22, 23, after having done nothing about the education. He became a late 19th century. Including even given himself the word deus. I don't think they were. The judge will probably have said that he was some kind of qualified deus. He had sort of woozy thoughts about the first principle. About first being. But he was certainly a religious skeptic. He certainly didn't believe in the survival of the personality of the dead man, you know, like that, because that's really clear from the very beginning. Nor did Hayden, nor did Jefferson, and that thing is a complete KF. Nor did both of the founding fathers, but now we've got an epipodic of believers. As Macon says, what did he want to do? He probably didn't want to have a clown dynasty. He was wrong, he had the wrong target. he's talking about my god if he'd lived if he could be brought back now i predict that they certainly have a republican revised opinion the next republican you really think it might be jeb of course it's going to be jeb of course and there's another one in the wings george bush

2:30 Yes, it is, it is. I'm not justifying what they did, but they didn't see it as covering up. Look, of course they saw it covering up. Where else could it be? Well, first of all, I'm inclined to think that the allegations The idea that some kid in private parts fondled by a vicar 30 years ago has been traumatized for life. That's a bloody lot of bollocks. I actually don't agree with you on that. I think there's nothing more, it doesn't matter, there's nothing more disgraceful than to somebody... It's an abuse of a... of course the priest should be deflocked, but the idea that dragging it up in a sort of list, let's say 25 or 30 years later... Well, read Gary Wills on it, and read what Gary Wills has to say on it. Gary Wills is a Catholic ex-Jesuit, and he still believes... Well, it's not so daft as to have a celibate clergy. I mean, this is the sort of thing that's almost bound to happen. Yeah. But, I mean, it's always been around, and the fact that it's now become the focus of such... I have to say, largely... I can't really say I'm delighted to be a principal. I am pleased that I was invited to give this address. I want to talk about, begin by talking about the notion of natural number that we all have, and when I say we, I include myself. My feeling is that it is hopelessly and dangerously wrong, actually. Let me call it, I'll refer to it as the iterative conception of number.

5:00 Okay. And it's basically the idea that the natural numbers are the successors of zero. So, we've got this operation, x goes to x plus 1, and a natural number is anything that you can get by starting from 0 and iterating this thing finitely often. So, it rests on the notion of finite iteration. Now, that's taken to, most of us assume that that's well understood. We think we've got a clear idea of what the finite information is. And the fact that this is really what lies behind here explains why we don't care much what, we don't have to specify exactly what zero is are what the details of the operation x goes to x plus 1 are, because it's the actual iteration itself that grounds this stuff. And the feeling is, it doesn't matter what these things are in detail, because of the absoluteness of the idea of finite iterations. We can be sure that any two, if we take different choices for the zero and for the successor, we're going to end up with essentially the same structure that is strictly speaking, technically speaking, isomorphic structure. Okay. Now, we can infatulate this iterative conception is a number in two, in two technical, in two principles, that is, proof by mathematical induction, okay, and definition by recursion. And notice that both of these things are based on the concept, the primitive concept of iteration. I'm trying to find one of these things that's actually still voiced. These are based on the iteration as well, because in this case, iteration of inferences.

7:30 So the proof by mathematical induction is just a sort of recognition, as it were, that instance of the thing being proved. And here, in the case of definition by recursion, we're talking about iteration of calculations. So we just keep on calculating using the same we keep on calculating using the same function. Okay, and now, all of this is unexceptionable in a way, except that, except for the fact that it's assumed that this concept grounds the whole business. So let me describe what I mean by the operational balance. And I think it is a genuine fallacy, and it pervades the whole of our thinking about mathematics and its foundation. nor definition by recursion needs justification. So this, because this notion of finite iteration we take to be naively face the absolute so that it produces essentially the same result when we do it, it's widely held that proof by abduction just follows from the definition of what a natural number is, and so does definition of that recursion, so we don't actually have to establish this same thing. Now, this idea that this is a fallacy is not original with me, in case of the same. It was both Dedekind and Frege, both of them recognized it as a fallacy. Frege set up his Begrift shrift as a kind of prophylactic against falling into it. The Begrift shrift was supposed to, was intended to prevent his making surreptitious appeal

10:00 of the idea of iteration in the course of the exposition of Riffin's circle. And Dedekind famously actually gave a mathematical definition of finite iteration in his, in Dedekind's notion of chains, his theory of chains, if you've got a set S, the distinguished element A, and a function F from S to S, and then you can form you can say what you mean by the iteration of the function f by forming the chain of the element a so when you form a chain you're essentially that's Dedekind's technical definition of a chain which is a purely static set theoretical definition technical definition of what finite iteration means Well, it's possible, therefore, to, as it were, resurrect this iterative concept of number a la Dedekind, because in Dedekind's case, you can actually prove these principles. You can actually define the natural numbers, and you can establish both proof by induction and definition by recursion as theorems. In fact, Dedekind's realized that this was the central result here, because this is what allows you to show that the definition is absolute or categorically. The problem is that in order to get Dedekind's axiomatic definition of number going, you have to make powerful sets of theoretical assumptions. You have to assume, A, that there's a transfinite set. Of course Dedekind thought he could prove there was a transfinite set, but he was just wrong on that. So in modern, in modern method, we have to assume the existence of the trans-finite set with a full-blown power set. So once we've set up enough machinery to give dedicated count of natural number, we've got the machinery, we've already, we're already confronted with the continuum problem, okay, because we've got both, as it were, we've got both n and p of n.

12:30 Or, perhaps more precisely, S and P of S, where S is trans-finite. So, the continuum problem is addressed. But we've also got other difficulties. We know that approaching things this way, we run up against the intractability of second-order logic. So even though various things have determinative truth values, we haven't got any systematic way of discovery from this truth about it so long. Okay? And then we also have the incompleteness phenomena of Gödel. So Dedekind's solution to this, Dedekind's way of avoiding this fallacy, and Trages' Trages' quote, one's charitable, is a perfectly legitimate, mathematically legitimate way of doing it, but it saddles us with these really powerful infinitary assumptions. about the existence of a trans-finite set with a power set, okay? So, what I'm going to do is to present a way of approaching this problem that is both set-theoretical Well, a dedication is done in the spirit of dedication, but at the same time, strictly finite. Okay. Now, anybody who thinks that those are contradictory attributes for a theory is quite obviously sunk in the dark night of this operationally spallus. I'm going to try to light a candle in your garden over the next hour, no, three quarters of an hour, whatever time I've got. Okay, so the first step, the first step in disabusing oneself of the, in getting rid of this hole that the iterative notion of number has on us, is to notice that far from being, as most people suppose, the most ancient notion of mathematical notion we've got, notion of natural number is a relatively, is a relative newcomer. It was invented in the seventeenth century along with the notion of rational and real number. The original

15:00 notion of number, which I've set up here for you, is that of Euclid and Aristotle and Greeks. And taking these definitions right out of Euclid's book seven, we get that a number is a finite plurality composed of units, where a unit is whatever it is that each of the entities composing the number is. So, in other words, the original notion of number, the notion which our notion of natural number replace, is the notion of finite set. There are difficulties here, but you want to take them up in the discussion. The big difficulty is that Siegelton's in the empty set. It's not fair that they're aeroport, but there are ways around them. Okay. So what I'm going to do, I'm going to present, I'm going to present a version of arithmetic, which I will call you video-video. So, we're going to be talking about Euclidean arithmetic. And really, it is. It's a kind of modernized version of the arithmetic, the classical arithmetic, and it's really the theory of the quantum test. Okay, so notice that once you've resolved to study this theory, then the axioms of Zermeyl and Frankl with Murrell are self-evidently true, the most obvious anyway, except for infinity, obviously, if we're talking

17:30 about finite sets, we can't have that. Except for infinity, which we can replace by Euclid's axiom, which says the whole is greater than the part. So what does that mean in this context? Of course, it means that all sets are ready to be fine, because if F maps S, S, uh, onto, then, sorry, if it does it one to one, then it, then, at, it's on. If you take elements away from a set, it gets smaller. If you add them to it, if you add elements to it, it gets bigger. Of course, I'm not thinking of the formal first-order axioms for Zermatt and Frankl here, but I'm thinking about the intuitive principles which those axioms translate, as it were. Indeed, it's obvious, isn't it, that no formal axiomatic theory can possibly serve as a foundation for mathematics, at least not if you want the foundation of mathematics to account for arithmetic, because the syntax of first-order languages is a theory that's equivalent to a rhythm technique. So, you haven't got the option of doing a formal theory, although what I'm going to talk about can be formal, I suppose. Okay, I'm going to, um, my exposition therefore is going to be informal, but, um, I'm going to use, I'm going to have certain guiding In principle, the foremost is what I call Brouwer's Principle, and that is, when we're talking about propositions over infinite domains, and I'm thinking primarily here as a proposition that's often quantified, then we'll apply constructive or intuition to speak logical. The second principle, the other two are kind of corollaries of the first discipline. life. I'm going to distinguish local functions from global functions and local relations from global relations. What I mean by a local function is what we ordinarily mean by a function in set theory, that is a set of ordered pairs. Of course, in the circumstances where all

20:00 sets are finite, that means a finite set of ordered pairs. Now, by a global function or operation, I mean something like the power set operation, which is defined everywhere and therefore doesn't have a set as its domain. It has the whole universe as a theory as its domain. And finally, because we need to talk about constructions when we're talking in intuitionistic logic. I'm going to identify constructions or arithmetical constructions in this theory with explicit definitions and point-basics that they're in the collaboration. Now, Brauer's principle has important consequences for the definition of subsets by the principle of selection or comprehension. So let's think about that for a minute. This is Zermelo's Althondro's axiom. The idea is this. Zermelo allows us to be given a set S. We can form the subset of S comprehended by a property phi So we can form a set of X and S that satisfy the property required, provided, says Zermelo, this is definite. A defini to Heidenshaw. So I'm interpreting Brawler's Principle to mean that if we have to use quantifiers and specify this property, directly or indirectly, then it's not a definite problem. So this has knock-on effects with induction. Okay, induction along a finite, linear, of course they're all finite, linear order. So, if we've got a linear order L, which I think is this way here, starts with the first L, and ends with the last L, And we've got a property phi, such that phi holds the first L, and for all x in the field

22:30 of L, except for the last element, phi x implies phi of the next element. In those circumstances, we can compute 5x for all x in the field of L. Okay, so that's the principle of induction along a finite linear ordering. But it only applies if you look at how the proof goes, because of this restriction in comprehension, this must be definite as well. So, in particular, it can't contain unbounded global quantifiers. Okay, so that means that has profound consequences for the principle of definition by recursion. Okay, so for example, if I'm given a set S, an element A in S, and a function F, I'll Okay, then there's a unique F, mapping S, sorry, mapping the field of L to S, satisfying the recursion equation, F of the first L equals A, and F of next L of X equals G of F of X. And that's the form of X is not equal to the last one, but in G of X. So, we can easily prove, that's fine. We can, using the standard methods of proof, at least, we're getting these recursions, right? We can prove this, okay? So we'll call this local recursion.

25:00 Intuitively, of course, what I'm doing is doing, I'm taking F, I'm taking A, F of A, F of A, so enough to F last, sorry, F of, F of A, where there are, this word, last L of these things. Of course, I can't really say this, but in two of these, that's what I'm doing. I'm iterating this function f as many times as there are elements in the field. Actually, the definitions involved in setting this up give an explanation for what this means. So the explanation is that we're actually saying what this means in setting up local recursion. Now, global recursion would be the same, except you just pick an arbitrary element a, it doesn't have to belong to any particular set, first L equals A, and F of next L of X equals some global function, GAML, of F of X. So here it looks like we're just iterating the function GAML. And so on. Now, it turns out that, except in certain special cases, this is not a valid mode of definition, okay? And the reason is that when we look at the proof, we see that the induction, although the induction here is with respect to a local, that is, a formula, that is to say, a formula that doesn't involve quantifies over an infinite set. But the induction down here would have to be with respect to a sigma-1 tangent. So it's not the j. Okay, so the idea that it's... The point is that the idea of iteration, carefully analyzed mathematically, turns out to be a much subtler and more interesting notion than you had in mind at first. Okay. Okay.

27:30 Okay, so conventional, in this setup, conventional natural number arithmetic corresponds to the theory of arithmetical functions and relations. Now, an arithmetical function is a function whose, the cardinality of whose value depends only on the cardinalities of its argument. So, in the case of a binary global function, it's arithmetical. If we're all set to S, T, U, and V, if S and T are the same cardinality and U and V are the same cardinality, then its value at S and U is the same cardinality as its value at T and T. Let's look at some examples. The power set function is arithmetical. So is the successor function, the von Neumann successor function. Disjoint union also, Cartesian product, set exponentiation. All these are defined in the conventional way that they're done in the zephyr or what is reticolism all familiar with. And notice that really what we're looking at here is, in terms of natural we're looking at the functions x goes to 2 to the x, x goes to x plus 1, x goes to x, x, y goes to x plus y, x, y goes to x times y, and x, y goes to x to the y. So if we think of these as natural number variables in the conventional dispensation, these things correspond to those. If we formalize this theory, which can be done in a straightforward way, what we get is something, we get that the arithmetical theory corresponding to it is Kalmar-Elerventry arithmetic, or sometimes it's referred to as I delta zero plus X. So these theories are mutually interpretable. Harvey Friedman has conjectured that any sensible combinatorial propertistic or principle that that can be proved at all can be proved at all. Well, arguments from authority aren't really de rigueur amongst philosophers and mathematicians, but Harvey knows a lot about other stuff.

30:00 It's worth mentioning. Okay. Okay, now we can define natural number systems in Euclidean Divinity. And in doing so, the whole business becomes I mean, for a start, as you'll see, we end up with a natural number system. We don't have, we can't recapture the absoluteness of the natural number system, which we kid ourselves into believing that there's a necessary possibility, the definition of natural number, if we use this iterative conception of number. So in order to get this procedure of defining simply infinite systems or natural number systems off the ground, the first thing we have to do is to give a definition of what we mean by iteration, or generating a linear ordering. So here's we've got a function defined everywhere. an arbitrary set or an individual and a finite linear ordering L. Of course, they're all finite. They're just sets of ordered pairs satisfied and appropriate. Conditions and sensors sets are finite. We'll say sigma generates L from A if and only if either L is the empty ordering or, in fact, we get from one we get from one third to its successor by applying, to its successor term in the ordering by applying sigma. So, this gives us a set-dioretically static idea of what we mean by iterating or generating these elements. Okay, so now we can say what a natural number system is. suppose we've got a unary global function that's defined on all sets. We'll say that it generates a natural number system from A. That just simply means that if I have a particular linear order generated from A by sigma, then if I apply sigma to its last element,

32:30 I end up outside the field of the order of the question. When I arrive by sigma to the last thing, I get an ordering one thing longer. Notice that the definition of sigma generating L is delta zero, so that this definition is pi one. Okay, so let's look at some examples. Some familiar, some not. Okay, the von Neumann simply infinite system. I should point out one thing. I'm trying to capture the idea, Dedekind's idea, that we're looking at a simply infinite system like this, and so on. But what I have to do, instead of, because my global logic of this theory is intuitionistic, I can't define, in general, I won't be able to define an inverse function or a global function, even if it's x. So I need to identify the thing with its initial segments, so the empty one, the one consisting of that, the one consisting of the first of a and sigma a, on. So I'm looking at initial segments of the thing as the numbers, rather than these terms themselves. So in the case of the von Neumann system, of course, we can define a global inverse, so that extra precaution is perhaps unnecessary here. In fact, it's true in all three of these cases. But in the von Neumann system, we've just got the von Neumann successor argument. The Zermelo system, we take the Zermelo successor. We can talk about the cumulative hierarchy system in which the successor function is the function that generates the cumulative hierarchy. Now, I'm going to explain all this technically in a minute, It would be the case that both of these things are strictly longer than this, that is to say. We can measure the elements in here by elements in von Neumann or in Vermeyer, but not conversely.

35:00 Even more interesting, these two number systems are incomparable. I mean, neither is longer or shorter than the other. I'll say something about that for a minute. Now let me give you another example which suggests why the naive proof that all natural number systems are isomorphic breaks down. So the naive idea is that if we've got two of these things, we can pair off their beginning elements, their zeros, if you like, And then, having paired off two elements here, we can then pair their successors, and so on. So that gives us the idea that we can actually construct an isomorphism. What we actually get is, by this procedure, is what I call the infimum of the number system endog and endog. So this is a number system whose initial term is the ordered pair consisting of the initial terms of the two systems, and whose successor functions applied to a pair X, Y takes us to the successor in N0 of X and the successor of N1 of Y. In N1 of Y. And it's called the infimum because it's easy to see that it's measured by both of these, N0 and N1, so it's less than or equal to it. Anything that's less than both of these is measured by this. So it's a kind of, so it's a kind of, uh, infinite. Greatest thought about it. I want to tell you, your operation requires that your input is a pair, right? Yeah, that's okay. It's a perfectly good... I mean, I'm not... These global successor functions, it doesn't matter what they do outside the... Yeah, but how do you know that? How do I know something? That the inputs always will be pairs. They won't. This is defined everywhere, not just on pairs. In fact, it's not just defined on pairs of things in N0 and N1. The point is, when they are in N0 and N1, then so if that's in N0, then that's in N0, and if that's in N1, that's in N1. I didn't say in the definition anything about what sigma does outside the... Okay, so the definition of sigma outside pairs is... It doesn't matter. No, it doesn't matter.

37:30 Okay, but you do have this definition like this, so... No. I've got sigma-0 and sigma-1 by virtue of adding these two. And I just define sigma this way. It doesn't make... They're saying that it's not defined unless your x and y is in all the pairs, so it's sigma-z, where it's not in all the pairs, strictly speaking. Oh, I see which... Oh, okay, I see what you're saying. Okay, we can give it a don't care definition. Okay, but that's practicing your system. There's no problem with doing that, because you can always tell whether something's in order there. You can always tell whether something's in zero or in one. Okay, you'll have to do that definition by basis. Yes, yes, yes. You could say it's the conditional definition. Yeah. And you really do not have to care. It doesn't matter what its value outside. Okay, so I've been talking about these things, about comparing these things as the length it's on. I better make this stuff precise. First of all, I want to say what I mean by the closure of the natural nervous system under a global arithmetical function. So let just give it by means of an example suppose we've got an arithmetical function of two arguments say the power set function that's that's right that's not that or say there's a disjoint union for example okay this is for two arguments okay so what we'll say is that the global function phi represents eta and n so that n is closed under eta means that if you take any two elements of n and applied phi to them, that also lies in n. And its field, which intuitively is, you know, if I've got a up to sigma to the k up to n, this intuitively represents the number of k plus 1. So the field of this thing is the size of it. The size of the thing is that. So the field is the same size as eta applied to the field of x and the field of y. So that's what we mean by closure under eta. To show closure under eta, we just have to exhibit a function, phi, that represents it in a given system.

40:00 Would eta have to be arithmetical generally, or just when it's restricted to anything? I don't know. It never occurred to me to consider anything other than arithmetical, generally arithmetical functions, but I suppose you could do it now. Maybe it doesn't have to be. Yeah, it does have to be arithmetic. No, you don't have to. It's not necessary. I don't make that restriction. Okay, so let me give you a couple of examples under which closure fails. So let's take, um, let's let the binary global function be this one's union, and let's let n be either one of these three, any one of those three systems. So if I'm given any global function, it's impossible for all elements x, y, and z, y, First of all, these are just repeating the definition that was in the other case. That is, it's not possible that we can show of any, we're saying that of any function whatsoever, it's impossible that it should represent addition in these number systems. Now, the status of this fact is somewhat difficult, and I may have to come back and say more about it later. In fact, these facts sort of take the status of an axiom. The point is, if you go back to what I said under my guiding principle, that these things have to be, these functions have to be composed, built up from set theoretical operations, and you can show classically that there are no such functions. So at least you won't run into, classically the theory won't run into difficulty if you assume these things. And it seems, in fact, true if you take this

42:30 definition that I just gave, if you take this definition of what you mean by an arithmetical function, representing another, it's just true that there are people who are, classically. Okay. Now there's one other notion I need to explain, and that's the idea of a measure. So, I'll say that a global function is a measure for N0 with respect to N1. If given any linear ordering Ly from N0, phi of Ly is in N1, and its field is the same size as the field of L. So, in other words, they represent the same part now. So, if N0 measures, is measured by N1, we can think of N0 as being less than or equal to N1, but we'll also see that there are examples where the measure doesn't go in the opposite direction. I'm going to give you a series of examples now, which will show that. So that definition allows us to compare natural number systems as the link. Now I want to describe a method of construction, which, a rather straightforward and indeed sort of simple-minded method of construction, which will provide us with longer and longer natural number systems closed under stronger and stronger functions. Okay, so I want to talk about, I'm going to think of S here, and let's think of it as an ordered alphabet of digits. So I'm going to talk about s-ary numerals and the s-ary extension in s of a number system. So let me make these notions precise.

45:00 Okay, a numeral has got to be a finite sequence of digits, okay? What's a finite sequence? Well, ordinarily, we say a finite sequence is a function defined on an initial segment of the natural numbers, but we haven't got natural numbers, okay? So, a necessary numeral is a pair, L, F. Well, this is a linear ordering, and we can think of that as the length of the numeral. Okay, and then F is a function from the field of L. S, and we insist that if the ordering has more than one element in it, then we insist that F of first L is non-0. Okay, so it's just an ordinary S-ary numeral. Okay, notice that you can have isomorphic linear order and cognate representations of the same numeral. as it were. Since they depend on a particular choice of L, if I pick an isomorphic L prime and then, as it were, lift this function over to L prime I get, in effect, a numeral naming the same size, but it's strictly speaking a different numeral. Okay. So now let me see what I mean by the S-ary extension of a number system. You can think of S as being 10 We've got a good concrete, a good concrete feel of what's going on here. Okay, so what an S-ary extension, what this S-ary extension is, it's in effect a species of all linear orderings, with this form, whose terms are S-ary numerals, okay, so say the term N here is a pair, L to N, F. And these numerals are arranged in their natural order without any gaps, right, so it's just the numerals up to K. Now, the crucial thing is that the length, L sub n, of each of these digits must be a linear ordering line inside the number system n.

47:30 So intuitively, we're allowing, we're taking s-ary, or in the general case, decimal in this particular case, numerals whose lengths are measured by the number system n. Okay, now I'm going to state some properties of this thing, but in order to do so, I first have to give a definition of what I call a logarithmic exponential. Let me make two remarks about this. This is cognate. These definitions are definitions where the variables are ranging over finite sets. So in particular, I've got to have a notion of the log sub s, the s-ary log of y, log to the base s of y. And that'll intuitively just be the length of the S-ary digit that names y, but you can define it as the smallest power of s that's smaller than y and so on. So there's no, it's sort of technically tedious, but it's quite straightforward to give these definitions set theoretically. The point is they correspond to these natural number functions, x to the log of y, And I'll take logs to the base S here. X to the, the next one, that's, that's the first. That's next one. The next one corresponds to X to the log Y to the, to the log, log Y. The third is x to the log y to the log log y to the log log log y. So we're getting a tower of exponentials, but we get an extra logarithm stuck on every time. This is a familiar function to people who work with weak arithmeties. So now let me state the general theorem here. The general properties of these things.

50:00 I'm not going to attempt to prove it. The truth is messy, but relatively straightforward. So I've got an actual number system. and a suitable system of digits. Okay, so the SRA extension of N measures N. So it's at least as long as N. The SRA extension of N is always closed under addition. The third principle is a necessary and sufficient condition for N itself to be closed under addition is that its s-ary extension be closed under multiplication. I see some smiles in the audience. There's been a certain amount of controversy on how these functions are. The next condition is a necessary and sufficient for n to be closed under multiplication is that n of s be closed under the first of the logarithmic exponentials, lex 1. And finally, a necessary and sufficient condition for n to be closed under the nth of these things is that n of s be closed under the n plus first. You know, there are two things I've got to remark on, which I gloss over. If I go back to this definition of the logarithm exponential, we'll go back to that for a minute. The subscripts here, as it were, are metamathematical. what I'm giving here is that these aren't ranging over natural numbers we don't have natural numbers these are these are as it were indices and as many as I can write down that I can repeat the definition so what I've got is a recipe for carrying out I've got a recipe for carrying out an unending succession of definitions it's possible to as it were turn this into a variable It's still not clear to what extent it's not... I'm working on this with my students right at home. It's still not clear to what extent we can make these things work. Wait a minute. We can turn the system of hierarchies. We can parametrize that. We can be sure we can parametrize the actual... We can do the log hierarchy. Yeah. Yeah.

52:30 Okay, now... Okay, so let's notice what happens with, if we think of these extensions, if we think of, and if we can extend, if we start out with something that's not even closed under addition, we can get closure under addition by extending it once, and if we extend it twice, we get closure under multiplication, and if we extend it three times, we get closure under this function, and so on. ever going to get it, are we ever going to arrive at exponentiation this way? And the answer is, given by this result, if I take the use of the setup where the number system and the appropriate set of digits, then the following three conditions are equivalent. I get exponentiation when I expand it. That's equivalent to n itself measuring n of s. And that's equivalent in turn to n itself being closed under exponentiation. So I'm never going to get exponentiation this way. But now the whole theory is in some sense closed under exponentiation if we look at things from the point of view of arithmetical functions and not insist on representing each cardinality by a particular example. Okay. So what this leads to is the following. I'll call it the omega-n hierarchy of s-ary extensions for n for reasons that will be coming here in a minute. Again, Now, as things stand here, these are, as it were, meta-variables. It's just a succession of definitions. These subsets here are not ranging over the natural numbers. They say they're ranging over inscriptions, if you like, or whatever. the point is we're getting successive I'm giving a recipe for giving successive definitions. Now it is possible to make this into a variable but again we're not sure how far out theory that I just stated goes. That's a question we're working on. Okay, so let's start with a number system that flows under successive but not under addition on one. Okay. Then if I carry out one extension, I get closure under successor addition, but

55:00 not under multiplication. So this kind of corresponds to what we might call Crest version. This is closed under successor addition and multiplication, but not under the first of these logarithmic exponentials. This is closed under successor, addition, multiplication, and lex 1, but not under lex 2, and so on. Now, the crucial thing to notice is that if, since the first one was not even closed under addition, a fortiori, not under exponentiation, each of these things must, each of these number systems must be longer than its predecessor. It's closed under successor, but not under addition. Where is the negation? Is it in the theory, or is it a meta-negation? No, no. You can state it as not closed under addition. You can state it as a not for all xy. Oh, really? It's in the theory? Yeah. Okay, very good. Yeah, right. So it's negation pi 1. It's negation pi 1. You've got to keep negations. When you're talking about hierarchy, you've got to keep track of negations. Okay, so that's, so the interesting thing about this is that this corresponds to the, this sequence of number systems corresponds to the sequence I delta zero plus omega n of theories in weak arithmetic. So these are theories. These are, as it were, models of those theories. But rather surprisingly, you're getting more closure by getting longer, which is the opposite of what you imagine when you're doing monetary or weaker images. Now, I think, let me just finish off by remarking that this setup is intuitively closest to rather crude ideas we have about what we can actually effectively do. So, let's think about the

57:30 problem of actually writing down names for numbers, okay. And let's, so we're talking about numerical notations, okay. So let's begin with the basic system, n. Let's start with n to be the stroke movement. So n is just one n times, right? So it's clear that this system, although it's closed, if we think about these as being things we can actually write down, or that our artifacts can actually generate, we can see that this thing is closed on the success, but not under plus. Because if I keep on writing down strokes until I drop dead, I could conceivably have lived another second and put that another stroke on it, but I clearly couldn't have repeated the whole performance, right? Okay. So, now if I take let's take N of 10. Okay. So these are ordinary decimal digits. whose lengths are measured by stroke, stroke neurons, okay? So these represent the actual, the decimal digits that I get actually write down, okay? Now these are clearly, these are clearly, uh, these are closed under plus, but not times. And finally, n of 10 of 10, these are decimal numerals whose lengths I can actually name by decimal numerals. So the numbers are getting more and more attenuated. They're getting less and less graspable. Notice here, if I can write down a name for a numeral, I have to, in the course of writing that down, write down names for all smaller numeral, in building up that. Here, it's rather more subtle. If I've got a digit, say, 9, 7, blah, blah, blah, blah, 3, 2, 5, and this is a humongous thing which I've been struggling for days to write down, right?

1:00:00 Now, any other number, any number below this, I can, any particular number below it, I can write down. Let's suppose it isn't all that long, suppose it's just, say, 100 digits long, I could do that here. If I wanted, it wouldn't be very edifying, but I could do it. If there are 100 things, I can write down any numerals below it, but I can't write them all down. Okay, with this thing, I've got, I can only, if I write down a very large one, if I, the only way I can write it down is give a rule, write a little program, say, for writing down the particular numbers, the particular digits occupying each place, and it turns out that I'm not going to be able to write down below this will be mostly gaps where I can't actually write down. So these things get more and more attenuated. Well, I think I've gone an hour. I think maybe I'll have to stop at that point because there may be some questions about what's going on here. Let me just remark that questions about computational complexity become more real, sort of, set in this context, rather than in the context of assuming that there's just one number system. Okay. Well, I think I'll stop at that point and go to the plural and general questions. Thank you. So, what's the power of that act?

1:02:30 And then I introduce cooperatives. I mean, I can certainly do the first three of them, okay? I can do as many as I can do. things are variables ranging over the natural numbers. So we've got some ways to generate some more and more and more. Yeah. Well, you've got ways of generating, you can generate names for people. You start out with A, you can write down C with A, and you can write down C with A. And you can carry on doing that until your backside drops off if you want. But it's nothing to do with recursion. Recursion has to be explained. Okay, that's the idea. So, where did this idea of acuity and unionism go? Oh, because these things, each of these things has a claim to be... The idea about a natural number is that you iterate a successor function, right? So what this shows is that depending on what successor function you pick, you get different number systems. So it's essentially ambiguous. You're not happy with that. All I need is two. All I need is two. So if I go from n0 to n1, I've got two different number systems. Now, the crucial thing is you've got to understand what longer that means. It means that although I can define a global function which takes an element in this system and produces one of the same length in this one, there's no global function going the opposite direction. Where global function means, as we just settled here, term, function defined by a term. Of course the problem there is that the notion of term is just as ambiguous as the notion of number. So there's a problem there, a philosophical problem, which we've argued back. Dickon and Richard and I have argued over this stuff. But the fact is, brutally,

1:05:00 that if once arithmetic, natural arithmetic, is up for grabs, so is syntax. So is the notion of what a proof is, and so on. You're not happy? Are you happy with that? Well, I don't know whether I'm happy or not. Well, you should be unhappy I'd prefer you to be unhappy, but feel you had to accept it nevertheless. Here's how I could be unhappy. I read my Freg, and I convinced myself that nothing that gets me any less than second-order piano arithmetic would even count as a number concept. I see all these bits and weak fragments of arithmetic as ways to do interesting things in complexity theory or computability, but I could speak to my Fregian guns and say we're not talking about the Milders yet. Yes, and of course Craig is not going to work because Craig is serious, but Dedekind's analysis is perfectly okay and you get a unique number system everything's fine, you're happy except in order to get this stuff going you've got to set up assumptions which allow you to ask the question, what's the power of the continuum? And there are all kinds of propositions which are according to Dedekind, either true or false, but which we have no way of getting For example, propositions, which are equivalent to pi-1 propositions and arithmetic, which are equivalent to this sort of unfeasibly large cardinal axiom, which we don't know whether they're consistent or not. It's unimaginable that we could stare at them long enough to convince ourselves they were true. There might be certain, some of us may have psychological peculiarities that allow us to do this because we don't want to step on anybody's toes. But ordinary mathematicians are not going to be convinced by these things. But they'd have to be convinced if they accepted, say, the truth of the corresponding pi-1 proposition, they'd have to be convinced at least of the consistency of this. So what I'm saying is that if you want the standard numbers, you can have them, but the price you have to pay is you've got to be able to ask all kinds of questions which you can't possibly answer. Now that suggests, when you're confronting some things that don't seem to have answers, It seems to be parallel, say, to the problem of the luminiferous ether. If you believe in the luminiferous ether, you have to think it's there, but nature is conspiring to hide it from it. So if you believe in the standard natural number,

1:07:30 the unique natural number system, you have to believe that there's a cardinality of the power set, but I'm not sure it's nature hiding from it. Maybe it's God or supernatural or whatever. is hiding, is conspiring to hide the facts from it. That suggests that maybe we've extrapolated beyond what's legitimate. I really like your formal results, but I would like to question also the philosophical motivation a little. You started by saying that the concept of natural number is a kind of newcomer in the history of mathematics, but I would like to turn that around. It's a newcomer because it's so primitive, you know? before people thought that they should have to make that explicit, something like that, because it's so natural. That is what I want to argue. It's so natural that it took a long time for anybody to think about it. Yeah, if something is too obvious, it takes usually longer than something that is obviously a problem, you see? But nobody thought that conventional arithmetic, up until the time the natural numbers were introduced, was problematic. Of course, there were a lot of things you couldn't answer, that's a difficult problem but nobody thought there was a difficulty about what these things were yeah I think so because I would argue that there's nothing more primitive or basic than the natural numbers and there's nothing else in mathematics that we have as clear intuitions of but let me go on with my thought I would say that you are clearly sunk in the dark night yeah I thought you would say that make an effort to get out of it admit that. Okay, but then you quoted Euclid. And the quotation really seemed to indicate that he was referring to different representations. So, take that bunch of seven things together, then you get one seven. Take that bunch of seven things together, then you get another seven. But I think that's wrong. I think if he would have asked him, he would say, no, no, I think of indistinguishable units that we put together. The seven is that, what those bunch of things, that bunch of things, and that bunch of things have in common. And I'm sure, I mean, he was a mathematician, I'm sure he got the concept. And that's the concept, I think. Well, what I'm claiming in the seven is a species thing, in the category, in the category of discrete quantum. So, if you want to say, so seven, so you say things like,

1:10:00 S is a 7, or T is a 331, and so on. That's like saying Bowser is a dog, because that's the species naming the category of substance. But that's Aristotle's explanation of that stuff. But not Euclid. Yeah, it is. Euclid is more terse, but it seems to be clear that that's what he's talking about. For example, he talks about taking a number of numbers. Now, what does that mean? If it doesn't mean a set of finite sets. That's all it could mean. You won't understand book 7 of Euclid if you stick with the idea that he's talking about natural numbers. It just won't work. This is historical. Yes, you surely know more about that than I do, but when I read this quotation, I had this feeling more that he's describing some kind of structuralistic account of natural numbers. So there are these indistinguishable units, and you put them together, and that is how you get natural numbers. Those things are called mathematical numbers, and these are a special case of what Aristotle and Plato call intermediates, or mathematicals. Now, it seems likely that Plato just thought, the way Aristotle explains it is that the intermediates differ from objects of sense in that they're all exactly alike. But they differ from ideas in that there are many of each kind, whereas each idea is unique. So our notion of natural number would have to correspond to not mathematical numbers like this, but to ideal numbers. But now nobody that actually did mathematics actually used anything but these intermediates. And Aristotle gives a gloss on that. What's a mathematical number? It's an ordinary number, like the number of people in this room, but in which we ignore the individuating peculiarities of the units and just think of them as qua units. So it's this qua construction.

1:12:30 So this is called the multi-set of one unit? There can't be multi-set of one unit. the singletons and the empty set are deeply problematic from this point of view so you have to figure out some way to get around that there are very technical ways I can recommend my little quote let me finish my thought then you went on to Flege but here again of course that's about logicism I mean your account has nothing to do with logicism that's not what you're aiming at but that was obviously what Flege was aiming at count of full infinitary set theory, in which case I would set logicism on its head, I would say logic is part of arithmetic rather than arithmetic being part of logic. And a set theory is just transfinite arithmetic, as Kantor claimed all along. Okay, but what I wanted to say is that the motivation for Frege was, why did he like not to take definition by recursion or proofs by induction as basic. But why is it that he wanted to derive that? Because of his logicistic approach. I mean, that's a totally different motivation. But for the mathematician, it's induction and definition by recursion that's primitive. It's not, you know... No, you should read Dedekind's letter to Keperstoff. It's in that papers called From Frege to Gödel. It's clear that, Dedekind's explaining what he's trying to do to a school teacher named Kepferstein, and it's clear that at some point he sort of loses his temper. But he says that saying that if you start so-and-so and eventually get to such-and-such that means nothing. He says eventually get there means no more than the syllables karam sipo tatura, which I make up on the instant. In other words, he's saying that the idea of explaining things in terms of iteration is completely naff and out of the question. Now, I know a lot of people think the message of Dedicant is just that you can mimic the natural numbers inside set theory. I don't think that's right. There's no intelligible notion of natural number outside of set theory. But why is the notion of set more intelligible than the one of natural numbers? That is very simple. Because if I say

1:15:00 the set consisting of Now, you may be puritanical and not want to say there's such a thing, but you know what thing it is I'm asking you to find, right, saying it's the plurality consisting of us three. So that's intelligible, even though the claim that it's a thing is intelligible, even though perhaps full. It doesn't have to have attached to it, the notion that you're referring to acquire people well, I'm supposed to acquire atoms, for instance. Yeah. Yeah, that's simply a dichotomy of inclusion and execution. No, no, that's... That's why I don't think there's a problem with Singleton's, because once you take some dichotomy of the universe for inclusion and then say, well, I want to specify the universe and define the Singleton's... But let me finish saying what's... Okay, so, you know what that 3 is, even though you deny it exists, right? You know what it is, what I'm trying to say. But what the hell is that? name. Well, I have neither problems with number three, nor with sets, you know. I'm a classical mathematician. The point is that sets are much, much more straightforward affairs than this because they're just collections. And the reason you don't have trouble with this is just familiarity. And in my opinion, first of all, the most damaging thing about the iterative notion of number is because we don't know what those things are, we force the idea that mathematicians are given a kind of fool's license to posit any kind of abstract object they want. Philosophers don't argue with them because of the prestige of mathematical accomplishment, but of course mathematicians talk the most egregious nonsense. Well, I have to say that I... Philosophers should take them to task, but they won't. Yeah, but you quoted Aristotle. I really, I would rather stick to what mathematicians tell us than what Aristotle tells us about numbers, I have to say. But let me just finish I don't want anybody to tell me about it. I want to think it out for myself. Yeah, yeah, that's true. Let me just finish my thought. I mean, Dedekind. Dedekind is great. He told us that real numbers are cuts of Dedekind. No, he didn't. Yeah, but that's one... Okay, then he... No, he told it. I mean, if we take Dedekind augmented by Russell, what he showed was that using cuts in the rationals, we can give a model for the theory of complete ordered field. Yeah. So that's not saying that real numbers are cuts in the rationals. It's saying that there is a complete ordered field. That is how I think of that.

1:17:30 But that, of course, doesn't show that real numbers are such cuts, as you just said, right? Real numbers aren't anything. There are no such things. Individual things as real numbers. There's no such individual thing as the identity element in S5, because there are as many copies of S5 as there are five element things that look at the permutation group. So it's a confusion to think that we're talking about individual entities when we're talking about structures. That's what structuralism is. So even in orthodox mathematics, forget about all this finitary stuff, in orthodox mathematics there's no such thing as a natural number either. And if you think there are, you're just suffering under an illusion. Yeah, I'm sorry, but you still didn't convince me that I've got this illusion. I would say that there is a unique structure of natural numbers. every mathematician in the history of mathematics has really referred to the places in that structure and no mathematician ever worried I mean there is no problem arithmetic has been done a long time before it was axiomatized you know natural number arithmetic has only been done since the 17th century yeah but that's a long time isn't it and I'm sure we're going to get an argument and I would say natural numbers They are not identical to certain sets. Those are representations of it. Just as the real number line. Mathematics referred to real numbers a long time before we had a formal theory of real numbers. Real numbers are not to be identified with Dedeckin-Katz or something else. Before the set theoretical revolution, real numbers were defined as abstracted ratios of length. So they rested on a concrete notion of a line. Okay. Relatively concrete. Nobody thought they were just things of an unspecified nature, but definite things. Do you think, for example, what do you think about the identity element of the Klein IV group? Is there a unique such option? Of course, yeah. There is? Yeah. But there's many Klein IV... When you talk about the Klein IV group, you're really talking about an isomorphism class of groups. No, if you're a structuralist in mathematics, group. And there are many different representations within set theory of that thing. Okay, so you claim a license for, you claim, perhaps on the basis of being a mathematician or on the basis of mathematicians talking this way, you can claim a license to coin,

1:20:00 to construct objects which you wouldn't allow a theologian to do. Right? Just because you're a mathematician doesn't mean you can talk nonsense. And what you're doing is talking This abstract thing, yes. No, it's not, it need not be nonsense. Maybe it's meaningful, but it is not about anything. At least, I think one should... Yeah, but it seems to me you're getting really desperate here. You're getting really desperate if you start saying, well, these are things, but we can talk meanfully about these things. Frege does that, and then of course he gives up on it. And famously, he started out by saying you can't ask for the reference of a term outside the context of a sentence. And then, certainly, you can't ask what a number is outside of the context of a sentence. And then, 25 pages later, he tells you exactly what a number is. So he gives up on that, this rather naff idea, that all you have to do is assign truth values to sentences in which terms occur, and that, lo and behold, the references of the terms fall into existence. there's no problem on the Dedekind account of number theory there are all kinds of simply infinite systems they're all mathematically indistinguishable because they're isomorphic just get out of the habit of talking in this old-fashioned way about mathematical entities the whole point about the said theoretical revolution is that it allows you to dispense with all that nonsense yeah, final point tell mathematicians that they have to choose among those systems Don't you tell them? Oh, no, of course they don't. What I'm saying is that if you take a finitary standpoint and analyze their notion, it turns out to be ambiguous. So they don't have to pick one of these things and say, these are my natural numbers. What I'm saying is, don't talk about natural numbers. Do your arithmetic, yet you're pretty in arithmetic. Yeah, but you replace talking about natural numbers by talking about members of those structures. That is what you do. within your finitary set theory. That is what you do. They're also ones that are exponentially closed, so it's not quite as restrictive as the first class. No, but you know, tell a mathematician that, I'm sorry, your system with natural numbers is not closed under that function L-E-X-P-N plus T. Okay? For a certain end. Then they would say, what?

1:22:30 What are you saying? That's an arithmetic function? Of course it's closed. I quite agree. If I try to get somebody who's caught up in the error of thinking of natural numbers way, he's going to have difficulty coming in terms with it. But, you know, you just have to face reality. Okay. The whole point is, yeah. I have a question. I have a question. One was about, generally about, do you remember the Archie Symposium with Mike Hallett? The question was about how you reconstruct the foundational program, how you reconstruct analysis. There's actually a sharper question I want to ask you. Does the number Goodstein property. Goodstein property says that the sequence starting with a number eventually converges to zero. Yes, this is also a... Excuse me, that's true. I was going to mention this in my talk yesterday because it involves fast-forward functions. No, I... Are you sure you do think the number 10 has a Goodstein property? There is no number 10 in this case. So you've got to formulate the Goodstein property. Well, the Goodstein... Well, this one clearly doesn't. Yeah, that's what I was going to say. Nor do any of these. In fact, I don't think... Well, they all have a good slide at number 10, insofar as you have a being who's capable of living long enough to write down the number of strokes that represent it. But the witness for the function for it to actually... the length of the sequence for it to actually get zero is going to be... Too big. It's going to be too big for many of these. Absolutely. It's huge. But your point is... way. The thing to think about is closure properties. So I don't want to ask, is 2 to the 10 to the 729 in N1? Because that, because... I mean, are you just prepared to assert the proposition for number 10, how the good sign property? It's false in some of these systems, and it's true in a sufficiently large part. You've got to tell me what the good sign property is first. You've got to tell me what the Goodstein's... There's a number such that the results of the base pumpkin operation... Okay, but you'll have to say how this is done. Well, it is done... Remember that... In the basis... Yes, yes. ...the global recurrence has a copy to find in this system.

1:25:00 Sure, that's fair. I mean, this is a competition. You know, I'm saying a good piece of math, Goodstein's thing, uses a little bit of infinitary set theory, proves an analytic proposition. Mathematicians all say, that's good. And I say, what do you think of this? I'm not sure whether it's true. What I suspect is that the Goodstein thing is not a recursion, but it's one of these situations where you have to repeat something. And then that's vague. There's a natural limit to this, the kind of simply infinite systems you can construct, which is essentially at the level of exponentiation, i.e. the worst you can do is take power sets and write down the term with a large number of power set operators, which will give you a sort of this big size of entity in a system. And the good start property goes beyond that, so it's unlikely to be borne out of existence in terms of these hierarchies. It uses an infinite. It used to concern that. Well, looking at a particular instance of it, say, 10, you wouldn't think. So that's, I think, the more interesting question is whether you can prove it for an instantiated meaning. Yes, but those proofs are, if I was going to mention this yesterday, proofs of those that each instance is very huge. The only proof that we know is via the universal proof. The long and short answer to your question is that it's a really interesting thing. I mean, and it's not just the good side thing. It's like the whole key business about the recursive function, what you can do, you know, what are true in Piano, you can prove in Piano. Because it becomes genuinely problematic about, for example, how long the definition can be. So, that doesn't require the instantary function. So, then you do it. In Piano, yeah. So you can do it, you can prove it. You can instantiate it kind of generically in Piano. One should be careful to distinguish between... I think there's still a couple of questions. Oh, yeah, come on. Let me speed up a little bit. Speed up. Yeah, we've got here. Every one of these is Michael. This is where you moved. Yeah, but I mean, individual is kind of a meaningless question. Sorry. Sorry. We can discuss this after. What was the question again? It was something about strict finotary, right? I didn't quite know. I was asking, I missed where you moved from finotary to strict finotary. Because you started with a system of finite set theory, and then you end up with systems which you say are closed-unless set, but not unpositioned.

1:27:30 You're not really saying you can't prove that it is closed-unposition. Well, yeah, that's kind of what I do mean, because what I'm saying is, what it means is that it's impossible that for all x blah blah blah blah of phi, which means that, for example, say, if this would be a two-place function, and what I would be saying, to say it's not closed under addition, is to assert this. It says, in fact, no phi is... You must have an axiom which moved you from being finitry to being strict finitry. That's a dedicated finitry, yes. You didn't... I do have a dedicated finitry, this axiom. But this axiom... Dedican finitry, that's just finitry, that's not a strict finitry. is saying, not about things in general, but about, say, von Neumann, okay? It's saying that it's impossible that phi is a... It's an axiom, is it? Well, you have to take it as an axiom, although you can prove it classically. It's not going to follow. All that's going to follow from... All that's going to happen is that you're not going to be able to produce a phi that works, okay? Yeah. Unless piano arithmetic is inconsistent, okay? But I can show in piano arithmetic that by looking at the acronym, or whatever, or just the theory of hereditary finite set, that none of the terms which give rise to the functions in here can actually define this function. Okay? That's definitely what you're saying. Well, I'm saying that that is not the case... No, no, I'm saying, that's why I'm saying, take this as an axiom. We can get consistently at this, and I suggest that you ought to. Yeah. Okay, so that's why this is a framework, it's not a fixed theory, right? There are all kinds of possibilities for adding axioms to this kind. I think of this as a kind of axiom of infinity, if you like. Okay, well. Should we? Oh, I have one more question. One more question. Ah, okay. I have to make a choice. Oh, well, so... Maybe highlight one of the...

1:30:00 The role of constructivism in your framework. So I really have two questions, but they can be... So one of them is, so if you... One second question is, if you extend your system to classical logic, does that really make large differences? So, your explanation, so you're from Ausländerung, right? Yeah. You explicated the definiteness in one particular way, maybe by... Formally, it means only about the class. By choosing a specific class of which you do know that it's definite, right? There is another way of doing it, which I think would be more natural, to simply add the definiteness as an assumption. Yeah. And so you would say, then, for every x, phi, x, phi, x, phi, x, implies, also. Yeah, but that's conservative. I proved that ten years ago on a paper in the JSI. I did it for the classical theory, but it's a proof. It's a classical proof. Well, but in classical logic, of course, it must be. No, no, well, of course, if I ask for all x, phi of x, or not phi of x, implies comprehension for phi, then that is a conservative extension of the theory I've already done. As long as you stick to the principle. Yeah, oh yeah, obviously. If you put full comprehension in, you've got something equivalent to piano rhythm. But the fact is that this theory itself is the theory, as I put it, was bounded in the comprehension. It's bounded quantifiers in the comprehension thing. Oh, no, that's the global one. That theory, the classical version of that has got a Gerdel, there's a Gerdel-Gimson translation so that the things are equivalent. Okay. So the classical one, I mean, it just seems kind of unnatural, but disturbing to me. The phenomena continue to... To the constructive standpoint, that's much more natural, right? Oh, it did? Well, you can prove this, as it were. If you have a definite, you don't have to prejudge what is definite. You could put this in, that's true. Because if your system is growing, more things can grow definite. You won't be able to prove anything to you.

1:32:30 Well, in the mental system. Yeah. But you yourself said that this system is kind of a gross system. Yeah, I don't know what happens, I'm not sure what happens when you put in acts like this. Or a sample. But if you add classical logic, the whole thing... If I add classical logic to the theory I've described here, I don't get anything new. Okay, yeah. So it's still identity over those eggs. Yeah, yeah. So... Yeah. Okay, so, thank you. Thank you.