 I think I will skip almost just because of lack of time. Here is what I'm supposed to say. And as you see, it's very long. And I won't have time, probably, to do everything. So I prefer, sorry? You can take the time. Oh, well, you will see. So the first paragraph is the behavior of vary. If you say, for example, big, very big, very, very big, and you continue, there will be what I call a saturation phenomenon. Adding one more vary won't change the value of what you say. And the language is more clever than all of us together because it can even indicate this saturation point. If you want to say very, very up to the, you say he's really very big, and that means he's very, very and no need to go. You cannot say he's very, really, very big. Really, very big indicates the saturation of vary. And this I can explain to a six years old. He will understand that. But adding one more vary serves to nothing. So I want to see to model this linguistic phenomenon. It's not linguistic because I think it belongs to all languages in French or I want to model this. Of course, the immediate idea is to say, OK, let's call x the set of adjectives to which vary applies. So x is equipped with an endomorphism, which I call T because I'm French and T is the initial of Très. And this saturation phenomenon will be given by, for all x, there exists an n, number n, such that T to the n plus 1 of x equals T of x. Tn, sorry, Tn of x. OK, that's very pleasant but has nothing to do with the language. Why? Because if you write this thing, if there is an n and being well-ordered, there will be a smallest n. Nobody will agree on the smallest n. And I didn't put this in the data. I just said. I mean, the set of fixed points is really good. The fixed points are really good. OK, so certainly this won't work. Insets. So we might try to see, because insets, the natural numbers are well-ordered. OK, so we might try to say instead of sets, let's suppose that x is big x, is an element of a growth nictopause. In that case, n is not well-ordered. And this might work. Well, even in that case, it doesn't work. Why? Because as I said, I can explain this to a five years old boy. He knows nothing about or girl. Sorry, sorry, sorry. I'm speaking English. I had the rather give this talk in French. So you will excuse me if my English is sometimes. So as I said, I can explain this to a five or six years person. And he will understand it or she. But he has no idea at all about natural numbers. He doesn't know what natural numbers are. He doesn't know infinity. This can be said without any use of infinity. Therefore, it cannot be done in growth nictopause. Because in growth nictopause, there is a natural numbers object. OK, so all this is elementary language. So what can we hope for after that? Maybe nothing, maybe this will fail anyway. But we can try to work in an elementary nictopause without natural numbers object. OK? Sorry? The five years old is supposed to know about the nictopause. Well, that was my problem to prepare this talk. Because I need, I shall show how I use them, I need elementary toposes. But nothing has been said here about elementary toposes. And maybe some people don't know what they are or how to use them. So I almost decided to cancel my talk. Because I don't like to speak just for my pleasure and with very few persons understanding. Note that I say persons. OK. So I took this as a challenge. And when I finish, you will tell me if the challenge has been met. So you don't know about elementary toposes. And I don't. I think I defined it in my talk. Sorry? You defined it in my talk and you were not there. Yes, but I mean, you didn't prove anything about what I did. No, it's true. I quoted your Mitchell-Bennabow language and saying that one could use that to. OK, but you didn't. No, OK. But people here, just on the basis of your talk, which I liked very much, could not be able to follow what I say if they don't know how to use the language. So don't panic. The language will be the language of set theory. Sorry? The Mitchell. Yes, OK. I don't like giving names to things. If in the discussion we have the possibility, I'll tell you why Mitchell has nothing to do with that. I have proof of what I say. So it's the naive language of set theory. The difference with the languages that Olivia presented to us is that it is not a first order language. It's a higher order language. You can quantify not only on elements, but on subsets. And for example, the most striking example is one of Peano's axioms, which I will certainly use, namely what characterizes N, many axioms. But the strongest one is that for every subset, so I quantify on subsets of N, if 0 is in S. And if Tx or Tn, when N in S, if S is stable by T, N in S implies Tn in S, T is usually called successor. Then I have all of N. That's the most powerful axiom of Peano's axiom. But you see, it requires the possibility to quantify over subsets, OK? There are, and I shall talk about them, other versions of natural numbers, fried or lovier. And I shall see how they fit with what I do. OK. Now suppose x is equipped with an endomorphism. x now is an object of the elementary topos, which you don't know about. OK. If x is in x, let me first. If S is contained in x, I shall see that S is stable if T of S is contained in S. And now if x is in x, I can take the intersection of all stables of objects, which contain x. And this I shall call the orbit of x. OK. What is this orbit? It's x, Tx, TTx, et cetera. OK. So we have the notion of et cetera. Take the orbit. And now the axiom, which I didn't want to write with sets or with growth in the topos, et cetera, or whatnot, is for all x, the orbit of x has a fixed point. No need to say anything about natural numbers. OK. So the axiom for vary is an endomorphism such that each orbit has a fixed point. If you have objections, you just tell me. I'm confused. Sorry? I am confused. Is this an axiom that you're putting on x? No. I say if I want to express the phenomenon of vary as it is in the language, I can express it by saying for all x, the orbit of x. What do you say for all x? You could have one x, which has a fixed point. Well, I say very has the property that very, very big is very, very, very rich stops. That's the phenomenon. Maybe a source of confusion, Jean. Sorry? That you wrote capital x, the counter power looks like. For all little x? Element of x, right? Small x. Any element of x? Small x. Small x. OK, OK. If I have to write everything, I will never finish. Yes, please. So you're looking at an object having this with equipped with another model, having this property that all objects have a fixed point. That's vary has this. I axiomatize vary by saying it is an object, very, yes, is an endomorphism of an object which has the property that every orbit has a fixed. This can be said in the internal language of a topos. Of course. That's why it needed. This can be said in the internal language of a topos. Since I have a specialist here, he might tell me what if x doesn't have any element. But this is, sorry? I say nothing if x has no elements. And I say the orbit of every element stops. I say nothing. Because x could be empty. And this would be true for the empty. No, even if it's not empty, it can have no global elements. Oh, I'm sorry, x cannot be empty because? No, no, it can be empty. For every, yeah, it can be empty. It can be empty. I don't exclude the empty. Well, the internal language permits, even if x has no global element, to write this formula. But if you don't like these things quantifying on all x where x is something, an element of capital X, which has no elements, let me say it in another matter. Because it will tell you the power of elementary toposes. I shall write for the time being, I shall write, I shall define a relation, which I write like this, x smaller than y, if y is in the orbit of x. And again, what does that mean when there are no elements? But I can define now something which perhaps might satisfy this is a binary relation, as a binary relation. It is characterized by, it is the smallest or the intersection of all pre-orders on x, such that this can be proved. OK, that now this would take intersections, et cetera, et cetera. I could use, and what does it have to do with this? This order relation, because I want to make a remark on the power of toposes. This is a binary relation. A binary relation on x can be viewed as a map from x to the power set of x. OK? And what is this map? It's the map x gives o of x, and now I don't care. This is a binary relation well-defined, so now whether x has elements or doesn't have, I don't care. OK? So I will write as an abbreviation if you want that y in order x, o of x, is an abbreviation of x smaller than y for this relation. OK? Now I think one more remark. Just a side remark. To show the power of elementary toposes, it has to do with families. There are two notions of families, families of sets if you want, or families of anything. A family of sets indexed by i is usually like this, and this notion of family you can do for any category, say with pullbacks because you want substitution. There is another notion of family, which is the one which is used by this set. So I shall call this an implicit family. There is another notion of family. Namely, a family of sets indexed by i is a map from i into sets. Or better, it into some sets of sets. That's the comprehension scheme which says, well, if I have a map from i into sets, I can, well, the power of toposes is that implicit families can always be made explicit. Why? If you have this, intuitively, for each i, you note xi inverse p minus, let's call this p, xi equals p minus 1 of i. And this is a subset of big x. So this tells me this implicit family. I can write it as an explicit family. And this is one of the most fundamental tools of toposes. Implicit families can be always made into explicit ones. OK. So orbits and et cetera, I'll have to be quick on each thing. Orbits and et cetera, orbits are exactly the notion of et cetera, of very simple et cetera. There are more sophisticated et cetera's than this one. But I won't have the time to talk about that. All right. So the idea is to study in a topos an endomorphism by looking at its orbits. I want to understand properties of an endomorphism. Well, first, look at orbit-wise properties, things which are. And then also, how shall I say? See how these orbits fit together. This could be done in sets. It hasn't be done as far as I can see. And many of the things which I will say not be done systematically. Many of the things which I will say I think are new even in the category of sets. OK. So I shall call an orbit an object x together with an element x0 and an endomorphism t, such that t, such that the whole of x is the orbit of x0. That's what I call an orbit. So you want to have that O of x0 is x. That's what you want. That the orbit of x0 is x. The orbit of x0 is x. That's now you call that an orbit. I call this an orbit. That's an abstract orbit. An abstract orbit. Note that x0 need not be unique or anything. No, it's a generator of the orbit. Well, it's a generator. But maybe there are many in how, OK. So. You no longer requiring that every orbit is fine. What properties have such orbits? Are you supposing that the orbit has a fixed point as before? No, no, now I'm talking about orbits in general. I said I want to study an endomorphism by looking an arbitrary endomorphism, by looking at its orbits and then seeing how the orbits fit together. So the first step is what property can an orbit have? First of all, an orbit can be seen as a weak natural numbers object. I shall explain in what sense. Weak. As I said before, there are many notions equivalent of NNO. There are the lovier definition. There are the fried definition, which some people may not know, but I'll recall it. And there is the payable definition. Let's see. Let's compare with lovier's definition. If this is an orbit, and if I have any object y equipped with y0 and translations, I will always call t, OK? If this is an orbit, there is at most 1f making this diagram commutative. So in lovier, there is 1 and 1f. I said weak NNO, replace there exists, 1 and 1s, is there exists at most 1. So this shows how it is a weak NNO in the sense of lovier. In the sense of piano, say. In the sense of piano, it is said that natural numbers satisfy the most important maximum of piano, the fifth of recursion. And you add 0 is not a successor. Let's call s the image of t, s for successor. 0 is not a successor. And the map t is monic. And the big recursion axon. Well, an orbit and also that everything is either 0 or a successor. In an orbit, what do we have from piano's axon? We have the big axon, recursion axon. But we also have that x0 union s by s, I said, is all of x. Another way of saying is that in an orbit, all the elements are either x0 or successors. There are no other elements. So it's, again, a weak. You mean the image of the operator t? Sorry? You mean the image of the operator t? Yes, s I called the image of t, s meaning the successor elements. OK? So again, we see that we have weakened a little bit. Well, a big bit, but nevertheless, we have weakened the definition of natural numbers of piano. Yes? Because this reminds me of the first construction. Can you speak louder? Yes, this reminds me of the construction of Dedekind beginning with an infinite set, and you get the natural. No, no, I mentioned Fried, I mentioned piano, and I mentioned Lovier. I'm not talking about Moore. Otherwise, I will be here tomorrow. What I mean is that if you have 0 is not in s, you have a monolick, so just 0 is not in s. In an orbit, 0 can be in s. In an orbit, 0 can be in s. OK? Well, since you are mentioning this, I anticipate on something which I was going to say anyway, namely this. Suppose I don't even have to talk about orbits. Suppose I have an endomorphism. I shall say that an element x is cyclic, t of x is in the orbit of x. What does that mean? Start with x, write tx, tx, tx, tx, tx, tx. Sometimes you come back to x. I want to say x is in the orbit. x is cyclic, if, sorry, x is in the orbit of tx. That means, I want to understand that, that means start with x, write tx, tx, tx, tx, and you find x. That's correct for cyclic. OK? And let's add one more thing. I shall say that t is acyclic if, not if it doesn't have any cycle. No, its negation doesn't work at all. A is acyclic, if and only if, that's my definition, every cycle, cyclic element is fixed, is a fixed point. I start with x, and then it's just at the very first step I finish. This is acyclic. What about what? That if it has a fixed point. Well, then x is not, if it has a fixed point, every fixed point is cyclic. But acyclic means that the only cyclic elements are the fixed points. That's my definition. OK. Let's call c of x the object of cyclic elements let's call fix x the object of fixed elements. This one is very easy to describe. No need of higher order, it's just fix x is the equalizer of the identity and t. But cyclic elements, you cannot define in a category unless it is a toposso or toposlike. We have always this inclusion. And to say that t is acyclic is this equality. OK. Let me just, I know everything, but I might forget a few elementary things. Now, first theorem is that an orbit is acyclic if and only if the pre-order relation on the orbit is an order relation. OK. Have to be proved, it's not difficult to come to difficult question a little bit later. For example, the natural numbers, if they exist, are acyclic because the order relation induced by successor is an order relation. Well, again, we shall see, I said, orbits are weak natural numbers. Let me give another example. If is an orbit, then it is a monoid with one generator. And such monoids are commutative. So when you prove that the natural numbers is a commutative monoid for addition, et cetera, and has a generator, that's true for any orbit. So many things, when proves for natural numbers, are true for orbits or other, for example, for acyclic orbits, then you have a total order. OK. I rush. The fourth thing is killing the cycles. It's passing to the associated order. How do you do that? If you have a pre-order relation, you can take the associated order. Let's write x equivalent to y if smaller than y and y smaller than x. And let's write x doubly equivalent to y if x and y belong to the same cycle. These things are almost the same. When is x equivalent to y if and only if x equal y or x and y are cocyclic? This relation is symmetric, transitive, not necessarily reflexive because x and x need not be cocyclic and yet they are equal. So we can take the quotient of x by this equivalence relation. This quotient is now t respects this equivalence relation. So factors as an endomorphism of this. However, this is an order relation. The pre-order becomes an order relation on x. And the orbits now, each orbit of x gives an orbit here. But now the orbits are ordered. Therefore, this is a cyclic. So what we have done is just by identifying two points which are in the same cycle, we have replaced the whole cycle by a fixed point. And this says what? Let's call by, to have notation, t of e, the category of objects equipped with an endomorphism. Let's call a of e, the subcategory of this, formed only with a cyclic thing. What I have just proved is contract an adjoint by just killing the cycle. Killing means reducing them to a single element. And in particular, if every orbit of t of x has a cycle, then for the associated cyclic object, every orbit will have a fixed point. And we will be in the situation of vary. There are many more things to say, but I have to take only a tiny bit of each paragraph if I want to respect the time. So what comes next? There is a difficult theorem which says the following. This one is really the other ones are little exercises each one takes maybe 10 lines at most, provided you give them in the correct order. But here is a difficult theorem. An orbit is finite if and only if it has a cyclic element. That's a fact. Why? When you say finite, I'm sorry, I don't want to annoy you, but just for the purpose of the talk, can you tell how you define finite? Finite is Kuratovsky finite. There is a notion of finite. But again, if you don't know elementary toposes, think of finite as finite. I want to speak things that people can understand. Sorry? If you don't have the action of choice, I'm not saying anything about the action of choice. In an elementary topos, one defines Kuratovsky finite elements, which in the case of sets are just the finite elements. In the case of Grotendijk toposes, they are a bit more complicated. But think of them as finite. That's all I can do. It would be good for the purpose of your talk. That if someone write down the notions of Kuratovsky finite on the blackboard, maybe you don't want to do it, but Olivia is ready to do it. I don't care. Provided she doesn't take on my time of talk. Well, it is, in a sense, first it is difficult, but it has some philosophical connotations. That means that this abstract definition of finiteness, or even in set, the definition of finiteness, has something to do with orbits, with, et cetera. Finite has something to do with, et cetera. Orbits are axiomatization of, et cetera. So I think I don't take your time of talk. You are. So the idea is this, is that one can describe using how you call the logic, the collections of finite subset of a power set. Because what you have are the singletons. The singletons are the singleton mass. This is given. And finite subset are generated by finite union of singletons. So you take the smallest joint sub lattice. I mean, sub object of p is closed under union, and which contain the singleton. And if you do that, you get all the Kuratowski finite subset of its. Kuratowski finite subset of its is something which is contained in the smallest, and x would be Kuratowski finite. If x itself is contained in the smallest. So the joint is also there. OK. Well, let me say I don't like your definition. I shall give an equivalent definition. No, no, why? It's not your definition. By the way, it's not your definition. It's the definition we use. OK, so I shall give another definition which will perhaps make it more intuitive why there is something between orbits and finiteness. Instead of saying that this is a joint sub lattice, that means we can suffice to say that it's stable by the following operation. If s is in power set of x and x in x, then it's stable by not taking the union of two things, taking the union of one thing and a singleton. You obtain all of x by iterating the process of taking the union of x and the singleton. What does it have to do with, et cetera? Well, I didn't want to talk about this, but the et cetera I mentioned is a very simple one. There are more complicated, et cetera. For example, suppose we have x equipped with a family of endomorphisms, not just one. The et cetera will consist in taking something, the orbits are obtained by taking something, and ti, any i, of that something, not just this is a kind of multiple, et cetera. Well, the power set of x has this structure. Namely, if x equals the power set of i, it has this structure by taking ti of a subset with i in i equal the subset union i. We don't want to interrupt a lot, I'm sorry. No, no, you don't have to push it. You don't have to push it. I added some time. So what's a finite object? Or more generally, we look at the orbits in the sense which I will not define of this ti. The orbits are all the finite sub-objects of i, saying that i is finite is that, OK? I mean, we have one definition so that you can explain your theorem now. No, the theorem is this. An orbit is finite if and only if it has a cyclic element. And that's a difficult theorem. How can you see this? Because I want you to see to have a vague idea of what's going on, but it's not trivial. Suppose you have a computer and a repetitive process. Start with one thing, take ti of that thing, and then ti, ti, et cetera, et cetera. And suppose you give the computer science, by the way. So don't. And suppose you give him the only instruction, start with some x, repeat the computation, and stop when you find something you have already, a result you have already met. And it stops, which means ti, et cetera, et cetera, has a cycle. It will stop, certainly, in a finite number, in a finite time. Conversely, if it stopped in a finite time, that means that it has met the cyclic element. That is very vague. But that's how it is. Except that in that case, it's finite in the usual sense of set theory, whereas k finite has been defined by you, now, or by other persons. But the point is that this finiteness, for example, doesn't have some properties one might expect of finite. For example, a sub-object of a finite object need not be finite. We all believe that the subset of the finite has set this finite. Of course, because we live in set theory in a Boolean topos. I'm not supposing it's Boolean. Here is another very strange and, to me, it was surprising result. Here it is. Suppose x is smaller than y. We can, that is y in the orbit of it, we can define the closed interval x, y as being all the z, which are between x and y. Now, the following are equivalent for all these intervals our finite is equivalent to all the xx, our finite. This can be proved. But this equivalence is not true in general. It is true if and only if the topos is Boolean. Now, here we get if and only if the whole structure of the topos is determined by this. And of course, if t is acyclic, these two equivalent conditions are satisfied. But if they are not acyclic, the topos has to be Boolean. That says also something. Acyclic means maybe there is no fixed point. If there is a fixed point, OK, it's acyclic. But maybe there isn't. For example, for the natural numbers, there is no fixed point. However, every interval of the natural numbers, closed interval, is finite. We know that for the natural numbers. But we know it now for every acyclic orbit. Not only the thing which goes all the way, the orbit which goes all the way to the fixed point. If there is a fixed point, it's finite. But all the intervals are finite. But the surprising thing is that Booleanness is needed unless you assume the orbits. The thing is acyclic. Booleanness is needed in general for this equivalence. So again, some properties of et cetera imply properties of the whole topos. Now, seven is connected components. In this case, I shall give an idea of the proof because I like it and because it doesn't take a long time. We have this order relation. So it's a category. So we can talk about the connected components. When are two objects, two elements, x and y, I shall put another equivalence thing in the same connected components? The answer is very simple. Namely, they are in the same connected component if the orbit of x and the orbit of y has a common element. I shall give a proof. It's not so obvious unless I miss something. Well, this relation is obviously reflexive and symmetric. It remains to show that it is transitive. Usually to get the connected component, you have to make zigzags like this. But here that says just one zigzag is enough. Why? But suppose x, y, z. There is an element u in the orbit of x and there. And there is an element v here in the orbit of z intersect the orbit. But these, u and v, are both in the orbit of y. And we have seen that in an orbit, the pre-order is a total pre-order. So we have either u in the orbit of v or v in the orbit of u. But then that's trivial to check that it is transitive. OK. So pi 0 is here is one more thing, one more application of the orbits. Suppose we have x equipped with an endomorphism. I want to talk about t, t composed with t, t composed with et cetera. Well, you look at x to be x, and it is equipped with an endomorphism. Let's call t blank for each f take t composed with f. So I can take the orbit of the identity. And the orbit of the identity is exactly what we want. So we have not only a level of elements, but of functions. For example, suppose you want to, suppose you say, OK, with nature numbers, I can write for all x, there exists an n such that tn plus 1 of x equals tn of x. But suppose I want to say something more. Namely, not for all x, there exists an n. But there exists an n for all x, which is good enough for all x. So you see I'm writing formulas as if there were nature numbers, but there isn't. How do you say this? That means the orbit of t has a fixed point. And this will give you a uniformity of t in x to the power x. For all x, such that for all x, this holds. I want to be able to say that this fixed point, there is a uniformity, a uniform bound. So you see that this notion of et cetera covers many, many, many things, even when there is no nature numbers object. And you want to do as if there was one. So this orbit, the orbit of the identity along t, I shall call it the leading orbit. And what property does it have? For every x, the leading orbit maps subjectively on the orbit of x. Take the evaluation. If you have, say, f, well, if you have in the orbit of t in the leading orbit, let me call it all x, the leading orbit, then for each x, evaluate f at x. And this is a mapping of this into of little x. And we know that such mappings are unique. And we know they are subjective. So every orbit of any element is a quotient of this leading orbit. For example, if this leading orbit has a fixed point, then all orbits have a fixed point. Now I will finish with just one more remark, important one. I have said that very has the property that each orbit has a fixed point. OK? I claim that this is a very important notion, mathematical notion. What does it mean? That t is such for all x has a fixed point. I call such an x and t rooted forest. Why? Each connected component is a rooted tree, which means all the orbits are finite, have a fixed point, et cetera, et cetera. That's the correct definition of a tree. So it's important. This says that very is a rooted forest. I think I have. Any questions? So during the lecture, it was mentioned about. If you will, please speak louder. I'm hard of hearing. OK. So you mentioned the definitions of natural number objects in a topos. You mentioned the names of Raid, Lovir, and Piano. Is it possible to give the three definitions precisely? So I want to know the three definitions of. An elementary topos? Well, I suppose it is an elementary topos. I'm not sure. The actions of Piano in a topos. The actions of Piano in a topos are the actions of Piano that everyone knows. It's rather the actions of the natural numbers. There's Lovir, it's a mini d'anandomorphism S, and an element 0, which is universal for this property. And for the orbits, what I showed you is that there is no existence, but there is unity. If it exists, there is unity. Instead of having existence, there is unity. The actions of Piano are 0 is not a successor. The successor is a mono, and everything is 0 or a successor. Which is very nice. In one moment, when I look at the successor and the element 0, it's a sum. It tells me that 0 is not a successor because it's joint. And it tells me that everything is a successor. And it also tells me that a successor is a mono, because in a sum, it's a mono. It says all this. And what's left of it? It's the fact that, although it's not said like that, that Grantin is a connex. What is said by saying that... We don't say it, it's what we write. The co-equalizer, it means exactly that Grantin is a connex. All orbit is a connex. Very good. Because each element can be connected with the starting point of orbit. What's left of it? That's Freud. I said to hear it. Piano, I've already mentioned it many times. It's not worth losing time. There's a paragraph that I haven't mentioned, but which will be very good in the discussion. A plea for the language. Pleadoyer for the language. And it will go very well. That's already a plea for the language.