Theory of simply infinite systems

0:00 So if we're going to deny this theorem, we've got to deny some premise on which it rests. So we'll have to take a careful look at things. So let's try to get a perspective on the problem by looking at some different views of numbers. So I'll be reading from page one of the little handout I'm using. I think the first thing to get straight about is what the problem is and I'm rather convinced that the whole Freudian the whole Freudian program of trying to figure out what things natural numbers are is perhaps based on a historical mistake because the notion of natural number is a relatively recent goes back at least not much further than the 16th If we look at the classical Greek definition of number, a number is simply a finite plurality composed of units, where a unit is anything that counts as one for the purposes of the number in question. So, for example, there obviously are numbers because there's a number of people sitting in this room, and the units are the individual people. From this point of view, the things we call numbers, like 3 and 17 and 29, are really names for species of number, just like, say, horse and man are names for species of animal. So from that point of view, they aren't really things, they're kinds of things, which you'll find. But if you look at a definition of number from the 17th century, which I've written down here, in fact, it's Newton's definition in his lectures on arithmetic, he says, by a number we understand multitudes of unities, that is to say, not so much what everybody understood it up to that point to be, not what you could do it to be. We understand not so much a multitude of unities as the abstracted ratio of any quantity to another quantity of the same kind, which we take for unity. So Newton is trying to get the notion of number out of geometry And I think this idea probably was first really clearly articulated by Descartes, but I'm not really sufficiently okay with the historical interface to tell that for sure.

2:30 But let's look at a recent characterization of the notion of number. And we had a Michael Dunn in his book on intuition. He said, what is essential is to regard the natural numbers of mental constructions generated in a determinate manner by the repeated operation of the successor operation to zero. Considered as an infinite structure, he says, the totality end of natural numbers is uniquely determined. There cannot be non-isomorphic structures with an equally good plane to represent N. That's putting it rather boldly. Maybe, I'm not sure whether this is dumb in its own view or whether it's just he saying this is how the intuitionists think of it. But if you take the business of mental construction out of it, it's still the idea that the natural numbers are what you get from zero by continually adding one. and so on. And according to Douglass, and I think probably according to most people, that description itself actually characterizes what a natural number is. We might, I think of it as the iterative or maybe operationalist concept of number. And for the rest of my talk, I want to consider the possibility of rejecting this. And maybe the motto for the rest of the talk ought to be back to Euclid and Aristotle. Or maybe even back to Pythagoras. That is, we should take the old idea of number and see what might as you can get out of that. In fact, I'm rather convinced that this notion of number is ultimately incoherent because it's based on a notion which, it's based on the assumption that simply by describing the process of generation we can grasp the notion of finalness. I don't believe that to be the case. But let's leave that kind of thing aside and just take it. That we won't start from that notion of number, But rather, in going back to Euclid, we'll actually be founding the theory of natural numbers in the theory of sex.

5:00 So I've now reached page two. And what I want to do is consider classical arithmetic in a modern guide, mainly, as the theory of finite sex. which is essentially the theory I'm talking about is just conventional set theory but without assuming the existence of delicate infinite sets replacing that so-called axis of infinity by an axis which says every set is dedicated to finite so I need to make some distinctions in order to make clear what I'm trying to say so I've listed them here we want to make of course the basic notion is the notion of set and by a set I just mean a finite a plurality finite in size composed of definite well defined objects and of course the idea is that a set is itself an object using Craigian terminology so it can also be a member of further sets individuals are just things that aren't sets, but which can occur as numbers of sets. So these two things, he said, are to comprise the Catholic objects. I'm also talking about species. I'm using this Brauerian terminology terminology of class for reasons that will become clear later. So a species is determined by it just comprises all those objects which satisfy some property. So they may not be finite in number. But if they are, we say that that species forms a set. And the final piece of jargon I have to introduce to you to make this discussion clear function in contrast local function.

7:30 A global function, intuitively, is a function defined on a whole universe of sets. So it's on a whole universe of options right there. And you define it simply by specifying if sigma is a global function, you simply define by specifying for each x what object sigma x is. The contrast with the notion of local function is that a local function are basically just a set of ordered pairs of the appropriate sort in the usual way. So this distinction between global and local is absolutely crucial here. We'll see later on in the discussion. Okay, so the basic laws of the theory are the Zermelo-Frankel axioms. Without the axiom, the counter's axiom in Tennessee. The crucial point to notice here is that if we look at the central axioms of ZF, they're quite clearly they take the form in this context, they take the form of finiteness principles for example, the axiom of power set just says that if you've got a set that is a finite plurality, if you look at the species of all of its subsets they form a finite plurality, so these are as it were closure principles for finiteness, and if you look at sort of character of self-evidence that we would expect axioms to have. The replacement axiom, for example, says that if you take a set S and you replace every element X in it by X's image under a certain global function, sigma X, then the totality of these replaced things forms a finite totality as well. So it's perfectly straightforward. as a finite principle in these circumstances. But the one that we want to look at that will play a crucial role in the rest of the discussion is comprehension. And so the axiom

10:00 of comprehension the idea here is that we have a set S as a definite, that's a set And a definite property, phi, a definite property, phi, we can form the set of X in S such that X has a property of phi. What that means is that if we single out those elements of S, which satisfy that property, since they're obviously going to be contained in S itself, so if S is finite, this is finite. obvious as a principle of finance principle what's crucial is the meaning of definite okay i'm going to be a little bit vague about it the way uh zermelo was because i don't want to give a i don't want to present this stuff formally but the intuitive idea is definite means if you pick an element at S, either 5S holds or 5S, either 5S is true or 5S is false. So it's decidable, if you like. Now what constitutes a definite property is going to play an important role in what follows. But the two additional principles that I'll take outside the Zermelo-Frankel system itself are Euclid's axioms of finiteness, which is just a reformulation in modern terminology that is common notion five, namely that the whole is greater than the part. if you look at that set in set theoretical terms it just says you can't put a set in one-to-one correspondence with a proper subset so we take that as an ax and finally I'm going to assume what I call Brouwer's principle that is to say that conventional or classical logic of how you would call it only applies when the domain of quantification is finite So in particular, if we have a property that requires an unbounded quantifier to express it, a property like that, or a property like that, we cannot assume that it has a definite, if I think of this thing as a property of Y, say, I cannot assume that these things have definite truth values.

12:30 The idea is that the presence of the unbounded quantifier means that these propositions are indefinite. And in particular, we can't apply comprehension to them. And so one thing that that means is that there might be, you might have a set, and a substitute, well in general you will have, for any non-angy set except there will be subspecies of that set which don't coincide with subsets. So we'll see that that has an important consequence. Now the idea is that approaching arithmetic in this way through the theory finite sets we're going to reject or iterative view of operations view of the dominant express so here in that passage I read out though we can simply take the natural numbers to be the successors of zero in some sense we're going to have to give a set theoretical analysis of that and in particular the key notions here are induction and recursion. Okay, now there are lots of different set theoretical variants of this thing that we can look at, but I thought the simplest thing would be to look at these with respect to finite linear order. So let's take the linear order, R, and then defining these things, one uses the usual semi-theoretical definition of things. So let's suppose we've got a linear ordering part of the first element. And then there's a successor operation, as it were, on R, next R, first R.

15:00 And so on, down to the last element. now you can it's not difficult to show from the axioms that it's not difficult to prove that any such leaving or ordering is doubly well ordered that is to say if you take a non-empty subset of it it's got both the first and the last element so how do we then state induction If we're given phi, a definite property, and r, a linear ordering, and we assume that phi of inverse r, and we're all x in the field of r, minus the last element. get phi of x then phi of So the conclusion is that for all x in the field we are phi of x. Okay, so that's just induction along a finite linear order. Now, you might think that this is a sort of operationalist way of proving this, namely, well, five holds of that, therefore by this assumption it holds of that, therefore this, but it holds of the next one, and so on. And so the argument goes, since this is finite, you've been exhausted by enumeration. So it must hold of the last. But the whole point is that we don't accept this idea of we can't fall back on this idea of these supposed intuitions of iteration. And we must give a proper set theoretical analysis of what's going on here. I mean, what I'm doing is not all that unusual. It's what Dedekin did in his monitor act. This was Oh dear, I think I started pulling his own. I can't get right now.

17:30 Did you see how you were going to say it? All right. Well done. Okay, so how do you prove this? Well, essentially, you take the set of the field of r such that 5x and you just notice that because it's got first R in it. So it must have a largest element. And if that largest element is not the last one, then we get a contradiction to this statement here. So what it really depends upon is being able to comprehend those elements of the field which satisfy its property. So if this property isn't a definite one, then the argument doesn't go through. So, for example, if I've got an unbounded quantifier here, I can't form the corresponding instance of comprehension. And consequently, I won't be able to conclude that this thing belongs to it. Sorry, what's the field of art? The field of art, by that I mean, it's just the set of things that belong to art. See, these are ordered, the notation is intended to suggest that this is a linear order, and so on. So the field will just be these things that appear in the order as a set. So what I have to do is pick out the subset of this field that satisfies this and shows that it's got to be the whole field. But the crucial operation is being able to pick out this subset. If I can't do that, I can't get the proof. So this argument won't work unless the property phi is definite. Now what does that mean for recursion? So this is the crucial thing because as Dedican knows this, as Dedican points out in his monograph, it's recursion, it's a crucial thing here, not induction. we can get local recursion going. So we've got, let's suppose we've got a set S

20:00 an element A in S and a function mapping S to a function that's called G mapping S to itself. So we've got the word priority for our recursion here's the set S and here's the element A and we've got this function F mapping S to itself. Now local recursion then says we've got this function G right local recursion says there it is exactly local function function such that first bar equals a, this distinguished element, and f in R beyond X is G of F of X if X is not equal to the last one. Okay. Now, how do you prove this? Again, you can't appeal the so-called intuitions of iteration, but essentially what you do look at the set of X in the field such that the theorem holds or the recursion works, let's say let me put it this way such that let's say this on the on the initial segment first bar up to X there's a unique there is a unique F satisfying well actually we don't have to say unique there is an F satisfying the recursion conditions.

22:30 Okay. And then we show by induction that this set is the whole root field. Now, the crucial point that makes this work is that all my values of path are going to end up inside this set S. So I've localized. target for my recursion. In particular, what it means is that this quantifier here, which I've written out in formally, quantifier over a set of multiple functions. Namely, all of these functions from initial segments, from the fields of initial segments to S, are subsets of the field of Cartesian product of the field of R and S. So I've got a, so my, this, that's where this existential quantifier is bounded here. So it's got a, it's got a, what happens if I try to do this on these things for global recursion? So this time, I've got this linear ordering, R. And now I've got a global function, sigma, global. and the theorem in question would be the following there is a unique local function sorry and I've got an element A saying it's an arbitrary object any object so the theorem would say there's a unique local function satisfying the first element of r equals a

25:00 and f of the next element after x is sigma of f of x as long as x than the last problem. Now, the curious and interesting thing is that even though I can prove this local recursive theorem, I can't prove the global one. That's critical. The reason I can't is because when I try to carry out this part of the argument, I have to replace this quantifier here is bounded in the local case because I fixed the target for all by recursion with an unbounded property. So I have to look at something like this, the set of x in the field of R such that there exists f satisfies the equations on the other first r up to x. So it satisfies the recursion equation on that initial segment. The problem is, of course, you can see that there's an unbounded quantifier here. So this whole property is what we might call a signal 1 property. It's got an unbounded bonfire in it. Consequently, I can't form this cell. Now, notice what's going on here, and this, I think, is important. What's going on here in both cases, what you're doing is iterating the function in question. In the local case, you're looking at A, G of A, G of G of A, and so on. In the global case, you're looking at A, sigma of A, sigma of sigma of A, and so on. And of course, it's implicit in the setup that I'm allowed to write down as many of these names for successive elements as I like.

27:30 But the point is that there's a profound logical difference between these two things. And if you don't make the underlying operationalist assumption that you can, as it were, write out in principle, as people say, write out in principle all of these values up to the last ones, in either case, then you have to fall back on this purely set theoretical argument. I don't know how many people are familiar with Dedekind's letter to Kephristine, but exactly the situation here has come up that Dedekind's addressing in that letter. He's attacking the idea that these three dots actually have a precise mathematical meaning. So what we're engaged in here is converting that so-called intuitive argument into a precise mathematical one. That's how Dedekind proved his result. The difficulty here is that unlike Dedekind, we can't actually bound the values here in advance, and therefore we can't be sure that this thing works. Could you just restate what it is to be a global function? A global function is a function that is defined everywhere on the universe of sets, like the power set function. So, what I'm saying is, I can prove in the case of a local function, that is where I've got a function moving around inside the set, that I can integrate it. but the proof here I believe is the explanation of what we need by iteration. It's not that the iteration convinces the truth of the theorem, that's what I'm saying with the global function like Kauer said, you don't know how far out it's going to go in advance so you can't because we're bound Why is that function is that quantifier unbounded down there? This one Yeah, this is, I'm sorry, maybe it's misleading. When I write this quantifier like this, I could have said there exists x, x satisfies it, right? This is just a quantifier over objects, over sets,

30:00 because all I'm worried about in the case here is that I'm looking for, the condition I'm looking for is that there's a local function defined on this initial segment. Yeah. Okay. but I can't bound that you can't bound it to the power set of the field of R or the power set of the power set there's no way if I had the power set of course I could bound it to the power set of the power set of the power set and so on as many things as there are in R if that made sense but I maintain that doesn't make sense because all I don't know I don't know all I know is that r is a linear order what I'm trying to say is to get your head around this stuff you've got to disabuse yourself the idea that finite means you can count it out on finite time it's still a sub-function it's a function from r to r right no this function will be a function defined on subset of the field of art, but no, because for example, if sigma was the power set, it wouldn't be. Okay, so now the question is, what does arithmetic look like under these approaches from this point of view? Well, that I think is rather interesting and surprising. Okay, so now we're on the next page, whatever that is. I've written down these definitions. I think I'm going to have to try to explain these things intuitively. If you're unhappy about it, you've gotten to look at the actual pocket definitions I've written down here. So there are two approaches to natural number of arithmetic.

32:30 I suppose one could pull in the classical and the modern approach the classical approach says what are natural numbers they're names of kinds of finite pluralities, namely they're names of equivalence classes of finite sets under the equivalence size. So they're not things on that view. Unless you're a raging Platonist, for example, you don't think the horse is a thing over and above particular horses. So in that same way, we've got lots and lots of tens, but there's no ten in itself, as it were, detached from particular channels. So what we need here is the idea of an arithmetical function. These are global functions, by the way. And the idea is this. Let me give you a definition. A one-place global function is arithmetical, yet whenever x is the same size as y. sigma x is the same size as sigma y. So it's a function, the cardinality of whose value depends only on the cardinalities of the argument. Now it's easy to see, maybe not that easy, but it's essentially straightforward to see addition, multiplication, exponentiation, with somewhat more difficulty bounded sums and products. For example, is just Cartesian product because the size of the Cartesian products maybe I should use S and T so I should see that these are sets

35:00 the Cartesian product of two sets S and T the size of it depends only on the sizes of S and T not on anything else exponentiation you just consider the set of all maps from T to S maps depends only on the size of g and the size of s. We prove that these things exist. Yeah, but that follows the meat because of the Zermelo-Francoil axis, the usual order. Exactly. Okay. As I say, bounded sums and products are harder. But if you're familiar with the literature of weak arithmetic, what you're looking at here is Calamar, elementary. arithmetic. And sometimes it's called I sigma 0 plus X. So that's the conventional national number of arithmetic that corresponds to this. So there's no surprises. Now what becomes surprising is what happens when you look at the alternative. of doing this that is the second way of doing these things is the theory of simply infinite systems ok so naively the idea is this you start up with if you think about what the notion of simply infinite systems you start off with an L of A and the successor function sigma, so you apply sigma to A, then you apply sigma to that then you apply sigma to that well, as you can see by the give away dots of ellipses here, I'm not giving you a proper definition of it because I'm telling you to do an operationless term so how do you define this notion? And it turns out that the easier to define not this sequence itself, but this one, the empty segment A, A, sigma A, A, sigma A, sigma, sigma A.

37:30 So you define this species of linear orderings, which are, in fact, initial segments of this. The reason you need to look at the initial segments rather than the original thing is because you don't have, even global injections don't have inverses in general. That is, you can't say, even though something's one-to-one, you can't look at something and look at a sigma of something and find out what sigma of it is worth. So you have to look at it this way. and you can define this without appealing to these dots of ellipses simply by saying that sigma generates r from a I've written it down in the notes I think it's, I don't know, where is it this definition is on this definition is on page 7 the number the definition 1 the idea is that the first element of R is A and the successor element of R as far as it goes coincides with sigma and now to say that sigma generates a simply The system from A just means that sigma, it just means that if sigma generates R from A, so if it generates some initial, if it generates some sigma like that, then sigma of last R doesn't line the field of R. so you get outside with the charge you've been looking at so let's look at some examples for example take A equal to the empty set sigma x equal to x union sigma x and that generates what we might call the von Neumann

40:00 simply in a system if I take A equal to the empty set sigma x Equal to singleton X, I get the Zerbella symphony of this system. A couple of other examples. if I take A to the empty set sigma x equal to x union power set of x I get what I call the cumulative hierarchy in the infinite system of the stages in the ordinary cumulative hierarchy of your sense. Up to omega. Ah, that's right. Well, that's a moot point, actually. How far it goes is a moot point. We've got to get onto the idea of what we mean by a measure or a scale. let's suppose we've got a species and a simply infinite system and the global function function mu of one argument. We'll say mu is a measure S in N means that is in S, then mu of X Y is in mu of X is in mu of X is the same size as X. So a measure sorry, I'm a little missed out of that the field we're going to be careful here of mu of X it's the same size as x. See, the things that belong to these systems are linear orders,

42:30 but intuitively the nth one in the system has n elements in the order. what I'm saying is, given any set line in x, if I apply mu to it, I get an element of n, has exactly the same number of elements in its field as X has. So in these circumstances we say that we say N is a scale for S. And I have to be a little bit careful about saying when one simply intimate system measures another, I have to say that, I'll say, N measures M, say, simply intimate systems. That is to mean that M measures the species of all fields of elements of M. Okay, under this definition. But it means to say that M measures M is to say we can exhibit such a global function. Now the fact is system measure human hierarchy converse cannot established that is to say you can't you can exhibit a measure for symptom system with respect to this one but not conversely so I guess back to your question Stuart does it go all the way to omega well there's no such thing as omega it certainly never stops

45:00 well it doesn't go beyond omega I don't from this point of view there is no such thing as omega, that's part of what I'm trying to tell you I mean basically once we look at this theory of symphony of an assistance what you find out is the idea here that you iterate something and go on forever doing it is that that's ambiguous There are lots and lots of different ways of going on forever, not all of them. If you interpret going on forever as going on inside one of these systems, then depending on which one you pick, you go on forever further, and some then you do in others. And sometimes you can't compare them. It's very strange. Okay, so let's suppose Phi is an arithmetical function of, say, two arguments. Any number of arguments we do here. Let's do it with two. We'll say, and let's suppose that N is a symbiote that says, we'll say phi is definable means we can define a global function function such that if x lies in n, psi of x, sorry, if x and y really a binary, if they lie in n, then psi of x lies in n. the field of psi and xy that is intuitively the set of things that you want to make up a linear order

47:30 is the same size my original parapetical function of the field and the field of y so for example multiplication will be definable in n if whenever I pick two elements of n if I can define this function psi which will correspond to multiplication so if I pick two elements of n psi of those two elements will land me up in n again and that will be a linear order whose field is the product of the numbers which give us the field of x and the field of y. So closure, the point is closure of these things under arithmetical function is not something that we just take for granted. We have to prove it. And in fact, I can tell you that you can't prove under even addition in this system. You can't. No. But don't despair. There are ways of getting around this. And that's the idea That's the idea of an S-ary expansion of the Synthian system. I better, rather than write all this down, I better draw your attention to it in the notes. I actually wrote these things down to avoid having to write everything out, but I'm afraid that professional habit means I can't say anything without writing the bloody thing out on the board, which is down. Perhaps everyone's relieved, actually. I don't know. So is good. Okay, so let's look on page eight. So the idea is, we want to

50:00 generalize the idea of a decimal representation. So the idea is you take a set S, which consists 1SK, let's say. I'm doing this intuitively. Don't be fooled into thinking that these things are natural numbers, okay? I'm just trying to use notation. Okay, so it consists of a no-element set, a one-element set, and so on up to a K-element set for every cardinality less than S itself. Okay, so a typical, if the set has that property So, e.g., a von Neumann ordinal, but von Neumann natural number has that part. It's a set which contains numbers representative of all sizes of sets smaller than a cell. so what do I mean by an S-area numeral I mean a sequence defined on the field of a linear ordering order in R with values in N. So formally, it would be that sequence would be so formally we actually have to say what the order is otherwise we don't know what order the sequence is. So we'd identify it with such a sequence such a function defined on the field of R together with the linear order in itself, which tells you the order is the things in the secret that's supposed to be told. So it's an X. And we have to insist that F of last is not the empty set unless last R equals first R. So the only

52:30 so intuitively it's just a finite sequence of elements of S but these things represent numbers less than S so it's clear the relation to ordinary decimal representation the only difference is that for technical reasons it's much easier to write numbers starting from left to right rather than from right to left Of course, if we had invented decimal notation instead of the arrows, that's the way we would have written them down in that order. We always write it backwards. The way we read, if you think about it. Units. The first thing you see should be the units, then the tens, and so on. But we do it backwards because we stole it from the arrows. Okay. So what I mean by the S-ary expansion, N of S, of a synchrony infinite system, N. The idea is the initial term is the zero neural, and sigma of x is the next numeral after x. But the crucial point is that all the linear orderings underlying the numerals must lie in this simply infinite system. So if you like, it's the decimal numerals whose length can be measured by the simply infinite system in question, or the S-area numerals in general. Now, let's look. I'm not going to write all these down because I've already done it. It seems odious to write them down again. Let's look at it in the notes.

55:00 It's on page 9. So these properties of symphony infinite systems and their necessary expansions. okay the first one says that no matter what simply infinite system I start with its S or expansion is closed under addition in fact a necessary and sufficient a necessary and sufficient condition for the original system N to be closed under addition so plus is definable here is that n of s is closed under multiplication. Now things... So if you start out considering n of s, n of s of s, and so on, this may be... As far as we know, this is closed under nothing but plus. But plus one. definition of a simply infinite system, that there's always a next thing. But as soon as you go to the SRE numerals over that, you get closure under addition. As soon as you go to the SRE numerals over that, you get closure under multiplication. So where does it go after that? One might guess the next thing up is exponentiation. Wrong. Okay. The crucial point is, look at number I should say that In each of these cases, the next one along measures the previous one, but not necessarily the other way around. So intuitively, these things are getting longer. So if you look at condition 6 down here, the property 6, the following conditions are equivalent. We've got N, say, and N of S. The following conditions are equivalent. this is closed under exponentiation this system measures this one so I haven't got longer in going from there to there and the third condition is that this is closed under exponentiation so if I start out with

57:30 something that isn't closed under exponentiation to start with, these things get longer and longer and longer and are closed under more functions. In fact what happens is what happens is that the functions that are closed under these, x to the log y I'm writing this natural number of functions because it's easier to see. x to the log y that's what you get when you go from here to the next one Then you get x to the log y to the log log y. Then you get x to the log y to the log log y to the log log log y. and so on, keep creeping up with more and more exponentials, but every time you have to add a log on them. These are really weird-looking functions, but in fact, it turns out that these things have been studied extensively. I was rather surprised to discover, it turns out that these functions are the functions that you add to I sigma zero arithmetic to successively take you up to a hierarchy of functions lying below I sigma zero plus exponentiation. Now, that doesn't mean anything to you. That's okay. But what essentially it means is that we're crawling up towards exponentiation, but we'll never get there by this method of S-ary expansion. And you might ask, is there a way of diagonalizing out? There is. It's this acronym system that I mentioned earlier on in the notes, which I didn't explain. But it turns out that when you try to diagonalize out here to grow faster, you just collapse back. It doesn't get any longer. It gets shorter, in fact. So these things have rather interesting, mathematically interesting properties. of course, each of these things has got a claim to being a number system of some type

1:00:00 up there, but they're all of different lengths. These things are getting longer and longer and longer. Which is interesting, because if you think in terms of non-standard models, the way you get the standard model is by cutting down. So, you can't think of these as non-standard, because they're going in the wrong direction. Well, maybe I should just mention a couple of other things. One way of thinking about these simply infinite systems is that you can think of a simply infinite system as a systematic attempt to pick out a representative member of each, In fact, if you had a function chi, such that chi of x, such that whenever x is the same size as y, chi of x is the same size as chi of y, and moreover, x is the same size as chi of x. And this is imagining a situation where you define a global function which selects, which given a set X, gives you the sort of canonical set of that size, right? Well, it's easy to see that from this you can define a simply infinite system that's a universal scale. If you've got a simply infinite system that measures all finite sets, you've got such a global function easily. so the facts of the matter are that if you formalize the system suitably then you can construct ordinary ordinary non-standard model in which for example if you look on the last page you can get models in which the von Neumanns and the Zumalos the von Neumanns are non-standard but the Zumalos are standard vice versa. None of them have, none of these initially, none of these familiar systems I've described are closed under a teaching.

1:02:30 You get models of that sort. You get models in which no simply infinite system forms a universal scale. So I think that's actually, those models actually indicate what's actually going on here. So I'll leave you with this picture. Set theorists always draw this nice sort of V-shaped picture of the universe of set, right? But I'm going to draw a different one. I'm going to draw a picture of the cardinality classes in the universe of sets. So there's, I guess, there's only one element in zero, but there's one, two, and so on. You can think of this as a block of wood with all these cardinality classes as layers. These simply infinite systems are kind of like wormholes going through here. But the crucial point is that they all peter out before they get to the end. you get this, you get all these different natural number systems in terms of these, in terms of the infinite systems when you're talking about generalizing iteration. But underlying it, the block of wood itself as it were, by definition, goes all the way to the top. But what you can't do is systematically select, describe how you can select a representative at each time. So any attempt to do so will give you one of these wormholes, which indeed doesn't have a last element in it. It keeps on going forever, but it keeps on going forever in a different way. Even though it keeps on going forever, there are parts of this thing that it never gets to. so I see quite a few puzzled looks on people's faces so maybe that's the point at which I ought to just stop and ask if anybody's got any questions I was wondering if you could say a bit more about the philosophical motivations behind this? They're quite simple and brutally stated, right? There's no such thing as a natural number. The natural numbers are a myth. They arise

1:05:00 because we confuse names for types of finite sets, the size of the finite sets, with things. arithmetic is really the study of the sizes of finite sets. So then you have to convince yourself that the notion of finite sets is actually what Zermelo and Frankl tell you it is. But that's quite straightforward. I try to indicate why that is. Most of the principal axioms, well, all of the principal axioms of Zermelo and Frankl are statements to the effect that certain species are, in fact, finite and size. So they're all finite assumptions. assumptions. Moreover, they're all I would have said self-evident ones. So, add to that the further assumption that the logic of dealing with genuinely infinite species has got to be you simply can't assume that classical logic applies there, so I'm following Brauer and assuming that the logic of these quantifiers of the whole universe of sets have got to be constructively understood rather than classically understood. So that's about it, those few slogans as it were. Yeah? Just a really naive question to understand what's going on here. I mean, if you take the hereditary finite sets and you think of them as each having a notation in binary, the binary notation corresponding to an actual number, which gives you the, which can be read off as the code of that. The acronym code, yeah. Right. Now, that looks like a scale that goes all the way through the natural numbers. Well, what it does. Why is that? I mean, obviously that's not... No, no, no, that's a very good question. I mean, I actually mentioned this. I skipped over it because I despair. If you look on page... If you look on page 7, you'll see I've got something called the acronym simply into the system, right? And I despair of...

1:07:30 I mean, technically, it's a mess to say whether it's what the successor function is. But think of it as starting from the empty set and generating sets in the order of their acronym codes. So what you get there is something that, indeed, as you say, as it were, sweeps out the whole of the cumulative hierarchy. The point is it peaches out. That's what I don't understand. There are other sequences that are... It's easy to see, for example, that the humanity hierarchy is smaller. That's just the one whose terms are the empty set. That's the one that's generated by x goes to x union power set of x. I can easily define a measure for this inside this. But not the other way. But not the other way around. So, I mean, somehow I mean, why isn't it why isn't the Ackerman sequence co-found more than this here? Because, intuitively although you can tell if you've got a term in the if you've got a term in this cumulative hierarchy system. You can't necessarily find and determine the acronym system which measures it only the other way around. It's easy. Because however far up you are in the acronym sequence, you can always tell where you're going to finish off the next power set operation. if you think about it, the map, let's take an easier case. it sounds like it's backwards from what you would expect. So take the von Neumann in comparison with this cumulative if I've got one of these, then it's got a von Neumann in it. Right. It's not going the wrong way. So I want to be able to say that I can measure, I want to get a function that, from here to

1:10:00 here, that gives me a von Neumann ordinal, the same length as this hierarchy. But the levels of this hierarchy are just indexed by the von Neumann ordinal. And this von Neumann, the von Neumann ordinal, which measures it, is right in there. So my measure just sends me from this point over to here. But going the other way is much trickier. You see, I've got to start with the von Neumann ordinal and find out what the thing in this cumulative system is. There must be one. well the reason you say is iterate the von Neumann iterate the cumulative hierarchy of the instruction along that ordinal you can't do it so you can't exhibit the function of the method now something similar happens between the cumulative hierarchy one and the acronym in fact the acronym one was the diagonalization S-R-E if you take S to V2 then it just and if you start If you start with a base set of size N, then it will be N area expansions that you get out of the acronym thing. It's funny because it looks like the acronym thing is laboriously chugging its way up this hierarchy. And the feeling is that what's happening, this can't be the right way of thinking about it, But it strikes me as, one way of thinking about it is it's peaking of so many things, it's going too slow. It just can't make it all the way up to the top. But that's obviously just a matter of view. Fred? I want to go back to your, for some more, Mark. You said that thinking about objects, this thing, was a mistake. I believe there are names for kinds of things, these species have been described. Yeah, thinking about things like 172 as a thing is a mistake. Right. I agree that you can certainly model our medical talk in set theory and you can give precise descriptions of it. Why should we be able to think of it another way? I mean, why is it a mistake to think of it as any other one you've done? Well, I mean, in some sense, it isn't a mistake if I can do this simply for the systems, right?

1:12:30 because they might pick out a particular thing for each number. That's okay. And if we make the usual assumptions in set theory, if you get that against the arm, there's one natural number system, up to isomorphism, etc., etc. So there's nothing wrong with thinking of it. It's just sort of as if we're arbitrarily identifying the natural numbers with the members of some synthetism. Mind you, the embarrassment there is that Whatsoever is the number three in absolutely infinite, an absolute infinity of simply infinite system. So that doesn't single, that doesn't, there's nothing, there's no freeness that these things take out of. But that's okay there. But the trouble is when you don't have it, when you don't have a simply infinite system that's a scale, identifying the things is equivalent to doing, identifying the terms in a simply infinite system depending on which system you think you get different answers for what things they are or how many of them are, which is even more serious. The main thrust of my point there was not that there isn't some way of thinking about it, but that the whole approach to arithmetic The basic problem is to say what the natural numbers are, it's just been steady. The basic problem is to say what a finite sentence is. There is something basic I don't understand. The thing is this, so we have that for second order arithmetic, each model of an arithmetic must be isomorphic to any other model. Yes. Now, you have here, I mean, these systems that you, if they are, if they are modulates of arithmetic, they have got to be isomorphic to one another. So you must be able to match up structurally. But if you think about the proof that all simply infinite systems are isomorphic goes by the theorem that says you can always define functions by recursion on a simply infinite system. But you can't prove that here.

1:15:00 I took quite a bit of time trying to show you that you can't prove recursion with respect to global functions. and they have to be global functions in this case because there isn't a huge universe of sets sitting on top of this on this view this Euclidean view that's everything there is so that argument just doesn't get off the ground it's the problem that it relies on excluded middle that is applying excluded middle to a global thing that's essentially what those I mean the idea is As Dedican noticed, you can prove things like induction and recursion just from ordinary set theoretical principles. But you've got to be able to form the sets in question. And when you're in a situation where you don't have transfinite sets, I mean, with ordinary set theory, you know from Dedican, if there's a transfinite set, there's a simply infinite system. It's as simple as that. That's all you need. But there aren't any transfinite sets here. Semi-consistence have to be proper classes, or species, in my terminology. But because of the global logic of these theories, species don't have the property of the proper classes. They aren't clearly individuated because there's no sort of extensional identity for them. They don't have the properties that would be required in order to get that dedicated argument. So, I mean, I said at the very beginning that the part of the talk when everybody is just sort of shuffling their feet and looking down at their papers, I said there's a theorem that says that all simply infinite systems are isomorphic. So if you're going to, and I did also say that I was just going to try to show you how it made sense to think they might not be. So, then I said, well, we've got to drop some assumptions. So, what's the assumption we've dropped? It's the axiom of infinity. Well, you drop, yeah, in a certain sense you are, but the way I would pitch it is that what you're dropping is that excluded middle holds for... For the universe of... Well, for infinite collection. That it gives you full-tronnal equivalence. Yeah, that's right. So, I mean, think of this as the set theoretical analog of Heiting Erythritic. Yeah. Except, I think, actually, Heiting Erythritic is incoherent.

1:17:30 Incoherent. Yeah, incoherent. There's no foundational point of view from which it makes sense. It can be interpreted in piano erythritic, so we can do all kinds of models and stuff, but starting from the bottom, it's based, essentially, as this quotation from Dunnick said, it's based on the idea that numbers are just given to you, they are what you get by starting from zero, minimal constructions or whatever, and I just think that just doesn't make any sense. I just can't make any sense. This is kind of a vague technical question. A lot of this work is kind of reminiscent for me of these classes I took in like intuition is the except theory, when you see sort of all kinds of strange things happening like two being uncountable and things like that I forgot what the theorem was but there's a lot of our sort of intuitions and the way we think which is just so fundamentally wrapped up with excluded middle and when you give that up you get to see that there's all kinds of distinctions that you have to make in there the point is excluded middle does work for finite for finite right would a natural place for formalizing this be in something like ZFI? No, because allows comprehension in respect to arbitrary problems. It does, but it doesn't allow excluded middle. Oh, okay. Even in the finite. Okay. Yeah. So, the whole point is that it's the unbounded quantifiers that must arouse our suspicion. Incidentally, my own view is that ordinary set theory exactly the same thing applies the axioms are equally self-evident except for Cantor's axiom which really the only difference between this system and the usual one no I mean the axiom of energy so I mean what I really think is the only difference between this theory and Cantor's theory is a swabble over what finite means Thus, in this theory, it means that it can finite. And Cantor takes the view that a set can be definite in size and still be the same size as servitude's proper subsets.

1:20:00 So the whole argument turns on that question of whether sets are finite in Euclid's original sense, or there's a looser sense of finite that Cantor gives us. but finite in either case so the global logic in both cases would have to be intuitionism or constructive but I'm not sure how the reason I didn't go into that in great detail is because I'm not sure how to make sense of constructive in any certain sense I mean once you start having doubts about the natural numbers then a lot of things start to go what do you mean by construction I mean, what does Church's thesis mean in these circumstances? I don't know. Just to invite you to gloss on it, I'm going to say about the Church's thesis. Couldn't be qualified like this, but here in the Church's position, he's wondering to understand really well about the proof of the categorization. So, I think you're objective to come to the thought that a determinist structure is given some other way by this structure of pure iteration. If that's dropped, and when it is fixed, you get an intuitive philosophy which just admits that. There is no, in a certain sense, better term of the structure being investigated. We did partial characterisation of what these structures have in common, but there's no single definite structure. I think that's what we're looking at. Some things that some of them say. Some intuitions say. Well, the classic... Or more extreme types, like Ascendant Bolton. But the trouble is, Ascendant Bolton takes... Instead of abandoning the metaphor of construction, he takes it extremely seriously. You know, you can't go beyond 10 to the 12 federations or whatever, stuff like that. I can't make sense of that either. So the idea here is if you look at it set theoretically, then you've got your instincts as a mathematician will keep you going, because it's all stuff that you know. And you don't have to start, you know, if you look at S and Volk and stuff, it's all, there's all these weird different kinds of tense logics and so on that are trying to capture what you can actually do and so on.

1:22:30 I think that's all irrelevant. Yeah, the interesting thing is that in the natural number case, is you start out with i sigma zero, which is arithmetic based on plus in times, plus times in successor, and bounded quantifier induction, okay. And then, to that, you add these axioms to the effect that x to the log y exists, and so on. Okay, so you get a higher, So the way these were write-ups, you get a high, by studying the proof here of these things, you get a hierarchy of these theories with these more and more things provably total. But of course, the underlying idea here is that all these things are dealing with the natural numbers, and we're just worried about what things are provably total. Whereas in my way of looking at it, you get bigger and bigger number systems, and they're closed under more and more of these functions. I mean, to get this stuff going, it requires a rather careful proof-theoretical analysis. And the way I've done it, in terms of these binary expansions and so on, or necessary expansions, it's just crude, but it gets to say, crude is simple, but it's essentially the same problem. So I found that, but I mean, the point is, they start with here, where you've already got plus and times. I start two steps below I start up with just successor and then add plus and if you start with you've got nothing but successor then you've got plus then you've got times so there are two levels as if we're back of here I mean it's the relationship of majorizing that you have between these little letters that goes through the ability now. Yeah. Something like the situation

1:25:00 you get with the stronger sequences going further than these further infinite ones. I mean, even though you've got a... The successor function is applying to all natural numbers, but it will never get to... for successive functions. For example, there's the von Neumann one, as it's available, and so on. They give you different systems as natural numbers. the idea is that by doing these S-area expansions, they take binary expansions, because that's the way they were done here. These things, it seems natural to say that these are getting longer. Because anything in here, you can of equally in here, but not inversely. Okay? So, but here, they're all the same length, but they're closed, they're different things that are provably total. You've got these sort of careful estimates on rates of growth and function through theoretical estimates on rates of growth. I was just going to say it's really a very intentional view of functions, isn't it? Well, this idea of different rates of growth in, depending on as it were, where they live along this ordinal hierarchy except that it's defined because, well you made the point yourself, you don't have extensional identity for these simple, infinite systems I mean it's talking very loosely you're talking about rates that grow one's a bit nervous about saying this I think I'm going to have a very sort of close if I say this one could run and run so thank you very much for your time if anyone else was there if anyone would need