Alain Badiou. Infinity and Set Theory: How To Begin With The Void.

Alain Badiou. Infinity and Set Theory: How To Begin With The Void. in: EGS. 2011. (English).

I begin by summarizing in three points the lesson of yesterday. First, we define the third sequence, the third period, of this very long history of the concept of the infinite, the sequence I name modernity or modern sequence. I just say now that this sequence begins by a conflict between science and religion, a conflict between science and religion, a half struggle between science and religion about some many questions, but about the question of the infinite. For modern science at the beginning of new physics and new mathematics, at the end of the sixteenth century and the beginning of the seventeenth century, the world can be infinite itself all what exists is determined by the laws of nature. So it’s the birth of complete rationalism. And the human destiny is finally something which must be understood in side the laws of nature, the laws of the world and the scientific determination. So there is no strict connection between the One and the infinite under the common name of god. It is the set of effects of the scientific revolution of the seventeenth century, and for religion only god is really infinite. The world cannot be something like that, of if the world is infinite it is something completely different from the infinity of god, and the human destiny is not in its totality a natural destiny because the human destiny is also composed by the divine grace. So there is something of the human destiny which is not determined by the laws of nature but by the laws of grace. And we have many beautiful texts of the seventeenth century, philosophical texts concerning precisely the relationship between the laws of nature on one side and the laws of grace, divine laws, on the other side. So all these philosophical texts are like something of a struggle itself, the struggle between precisely modern science and religion. And it is why the church and the pope sentenced Galileo. The struggle really is also a political and ideological struggle, a historical fight and so on. That is the first point.

The second point is that today maybe at the end of the third sequence, it’s a possibility, we don’t know really what your present is. We live in an obscure present certainly. Certainly we live in the present which is between different things, between different possible orientations and so on. So maybe we are at the end of the third sequence. The question is not in fact the conflict between science and religion. I propose to say that he new dominant contradiction is between a reactive classicism and a reactive romanticism. Reactive classicism, the rule is to find an individual harmony between our existence and finite possibilities. So we have a reactive classicism because the goal, the sense of existence is to realize our proper possibilities, our individual possibilities, to have a good life in this sense. And so it’s the invention of modern classicism, something like that. And on the other side what I name reactive romanticism is to say that there is no mediation, there is no possibility to find really something absolutely new, but the reactive classicism is a bad thing. So the reactive romanticism is the first reaction against modern classicism, which is the idea that we cannot have a good life only by the realization of our internal possibilities, but that the good life is always under the thing of something which is much more than the pure, harmonious realization of ourselves. So the reactive romanticism is a reactive romanticism of transgression by violence, by sex, by drugs and so on, all forms of transgression which are like a false infinity, something which is apparently against the closed harmony of classicism. But in fact it is finally a form of nihilism, that is, a form of destruction of ourselves by radical transgressions.

And the question is that reactive classicism and reactive romanticism probably have the two inside something like the dictatorship of the monad, inside the world the law of which is the dictatorship of the monad, but it’s two opposite ways. But there is a relationship between the two, which is that finally there is no real access to the infinite. In the reactive classicism it’s a good thing, no access to infinite. That is the idea of the death of ideology. The death of ideologies is in fact the modern name for the end of every hope to have an access to the infinite. And it’s the idea to be inside moderate realization of ourselves without transgression, without excess, with measure, to live with measure. And on the other side there is the imaginary idea of access to something else, maybe to something infinite, but it’s the idea that in fact it’s impossible because there is in the world as it is no mediation between finally the dictatorship of the monad and the possibility of really something new. And so the common point between the two is finally the impossibility of access to the infinite as a good thing in the first orientation, classicism, because orientation to the infinite is hubris, excess. Pure ideology is the modern name of all that or false politics. And in the second orientation it’s not a good thing but a bad thing to have no access to the infinite. Transgression is the only possible way, but transgression is always, finally, a failure, and there is a closed relationship between transgression and death.

And so to recapitulate all that, at the beginning of the modern age we have a rational struggle of science against religion. And at the end of the same sequence we have an irrational struggle between a reactive classicism and a reactive romanticism. So our moment, our sequence, is something like the negative part of the third sequence, the finally negative effects of the third sequence. After the beginning, after all there was something like rational positivism, which was something like the victory of reason by physics, by new mathematics, by new development of technology, by creation of a new productive economy and so on. All that is in the third sequence, and all that is the affirmation that finally all is infinite. Science is infinite, artistic creation is creation of novelty and so on. So we have a beginning of the third sequence concerning the question of the infinite, which is a very terrible struggle, but this struggle is under the idea of a victory. A victory of what? A victory of human beings in some sense. It’s the creation of a new humanism where with science the human being can really be victorious in all directions of creativity. And today we are at the end of all that, and we are in a very different conviction, which is that we cannot change the world, that the world is something like a terrible necessity, and inside this world we must find a way without any fundamental modification of this world. And so we are really in a very strong conviction of impossibility concerning the infinite as such, impossibility to find a new way to open the situation really, to find in the situation as it is something which is maybe the opening of a new sequence. It is why I think that probably we are really at the end of a third sequence. When the situation is…when the world as it is is not very good, but it’s impossible to change. And it’s a subjective disposition. And it’s difficult to be in this world. And my conviction is that it’s difficult to go beyond today. You can say something about that more than me. It’s better to be old finally because we have some memory of something else, some vivid memory, some active memory of movements where the idea of the impossible novelty of the world was active in many different directions, that’s not the problem. Now it’s not exactly the case. We can find something new, naturally, but the general context is very oppressive in some sense, oppressive not concerning our personal freedom. In some sense we can do what we want, but precisely it’s not really a satisfaction If what we can do is inside a closure, in fact, and not inside something which is really a new possibility.

We can recapitulate all that by saying in any case with the struggle between reactive classicism and reactive romanticism the idea is that there exists no mediation between precisely our concrete life and something like an opening to the infinite. But if there is no mediation in some sense there is no future, no real future. There is a small future inside the present. There is something like the movement of the present, but the movement of the present is not something like the representation of a future. The movement of the present is the present itself. Everyday many people are saying the world changes. Every minute it changes. It’s true, but this change is a change of the same. It’s a change inside the same because our world is a world, the law of it is to change. And so it is the law of change, and if the change is the law of the situation, in some sense it’s not a true change, but the continuation. In some sense a true change would be much more to stop all that and not to continue, but precisely it’s impossible to stop because the law of the world is to continue absolutely, to continue to produce, to continue as the development of capitalism, to continue the bonds and so on. We must continue. It’s the law of the world as it is. It’s a different…that there exist old worlds where the idea was not exactly to continue because there was no idea of change. With today’s difference there is an idea of change, but the change is precisely the law of the world as it is. And all that is in relationship with the question of the distribution of the finite and infinite in our representations. When there is no mediation between our finite existence and something infinite, in some sense what exists is by necessity always the same thing because there is no mediation for something really different.

That is the second point. And the third point was the presentation of the structure of choices today, of choices to change, naturally, of choices to do something new. And I just recapitulate, so the choice to find something new in the context that it is, and I propose to say there are two fundamental different orientations, the first, to maintain that there is a closed relationship between infinite and the One, and the second to affirm that there is no fundamental relationship between the infinite and the One. Naturally, in any case, we have in the hypothesis of the relationship to the infinite, the choice is a choice outside the dominant contradiction between classicism and romanticism because in classicism no infinite is god, and in romanticism there is only the imaginary and the tragic infinite of transgression. So if we must find something new we must maintain that we can have an access to the infinite in our experience, that we have an experience which is in some sense an experience of the infinite as such. But the question is the relationship to the One because in tradition I don’t return to this point, in tradition the relationship to the infinite is generally in a close relationship with the question of the One. And so there is a division of the two options, first, the connection of the One to the infinite is transcendent, or this connection is not transcendent like the classical god was, but is immanent. And in the first possibility the question is a question of a new god. By a new god generally we must understand the old god in a new form naturally. It’s not the death of god, but he new form is a new creation. That’s why I say a new god, a new vision of what god is for our self, or, if you want, a new relationship between the idea of god and our proper experience.

And in the case of immanency we can say that the question is a question of life, potency of life, potency of desire, potency of becoming. So the first hypothesis is the hypothesis of modern religion. That is a religion which is not a continuation of the old form of religion but is really something like a rupture, something like a novelty, modern religion. And in the other case we have vitalism. That is a new philosophy of life. The second option is also divided. The first idea that there is something divine in the world, maybe the aesthetic vision of new gods but not in the sense of modern religion, but in the sense of modern paganism in nature itself we can find something divine, this is the general idea. It’s not necessarily gods in the mythological sense. There is something divine in nature. And in this tendency we must go outside the world as it is to find in the world itself something new which is the divine part, the secret divine part of this world. And we must abandon all the characteristics of the contemporary world, money and so on, and we must go in profound nature without anything, naked in nature. It’s a strong idea. It is the idea of the perfection of a new humanity, something like a return to the beginning of humanity itself without technology, money, corruption or any forms of modernity. It’s the destruction of modernity inside modernity, something like that. And we can assume poverty, something like that, no cars. All the products of the big market must be destroyed finally. And we must find a new relationship to the infinite in this radical experience of pure abandon of the real of the world of today.

And another possibility, but in the same, what I name the weak god because a weak god is without any power so the relationship to the weak god is also a poor experience, and experience without glory, without media, a personal experience of something of a new way for the good, and it’s also a radical negation of the world as it is. But it’s not a revolutionary negation, but and intimate negation, an individual negation. Maybe it’s a personal destiny to go outside the towns, outside the big towns in the secret and profound nature where there is something divine, which is maybe destroyed by the human world of today. But we can first fight against the destruction of nature, but, profoundly, we must find in nature the divine part.

All that is a first possibility where there is no question of a One. The divine part of nature is in many places. There is no unification of this destiny, and it’s a choice for everybody. It’s not a big One of conviction, ideology, party and so on. It’s a dispersive law. It’s a dissemination of the divine. In the second position, being and nature, being is neutral. There is nothing divine in being as such. Being as such is under it’s proper laws and is completely indifferent to our existence. The world is the world under its proper laws, and we are inside this world, naturally, but we are not the destiny of the world. We are a part of it, some small animals on a small planet and nothing else. And so it’s why I translate all that by saying being is neutral. The life, the world, nature has nothing to do with out destiny. We are inside all that, but we must decide our proper orientation without any indication from being as such because being as such is indifferent to our orientations. As I repeat, nothing works for us in nature. Nothing is on our side, much more on the side of all what exists. The great fire of the sun, galaxies, all that exists in nature as we are in nature, but no part of the nature is much more than another form of nature itself. And so we cannot find in nature directly our access to something new or something like a new infinite possibility. And it is our situation. You must choose. You have chosen before, naturally. The choice is done. But the clarification of the choice is of some interest.

So now I want to clarify some points of the fourth choice and some points of the fourth choice which are also of interest for the general clarification of our situation and not only arguments for this choice. So we must begin to examine the very concept of the infinite as such. We begin by the modern concept of the infinite, to show if, really, there is a possibility to open an access to something infinite in a dynamic sense. For example can we say that something like a new truth, something like a new work of art, something like a creation in painting, in cinema, something like a new vision of politics, a new movement, something like a magnificent patient for somebody, if all that constitute or not an access for us to a new form of the infinite in a world where being is absolutely neutral, where being does not work for our satisfaction, our being as such, our nature, but is indifferent to our satisfaction. So our way now will be from the idea of the infinite with some details, some precisions and so on to a question of the new possible relation to the infinite in a dynamic sense and not as a pure knowledge, but as a new possibility for our life, our existence. So we go from knowledge to existence with the infinite as the stable concept of this movement, from possible knowledge of the infinite to a possible use of the infinite as a norm for our existence.

So, what can be the beginning of this examination? Our idea is to present a form of knowledge of the infinite, of the concept of the infinite, a modern knowledge of the infinite, but how can we begin? It’s a difficult task, the beginning. It’s always a difficult task in philosophy, the beginning. I have said yesterday, I think I begin by being, but precisely it’s not the case today because if I begin by being I must say immediately the relationship between being and the infinite. And so I must return to the question of the sequence, which is precisely is being the prescription of finite possibility or is being the prescription of the infinite as such? And naturally if something like the infinite is real, something like the infinite exists must be. So by necessity, being as a concept, which is not before but in some sense after the dialectic of finite and the infinite, and it is a modern conception. You must have a dialectical thinking concerning the relationship between finite and the infinite before saying what, finally, existence and being are.

So the point is that we have the conviction that maybe we cannot begin at all and that we always decide something before this beginning. The other possibility is to begin not with being but with nothingness. We have here a very classical problem of philosophy, which is the beginning of philosophy and not only the beginning of philosophy concerning the infinite and so on, but the beginning of philosopher. Generally the answer is that philosophy begins in nothingness. It is a signification of the duped in Descartes. Descartes begins by the duped, and the duped is precisely the creation of a subjective state of nothingness. The duped of Descartes affirms that I don’t know anything. So I am from the point of view of knowledge in nothingness, nothingness of knowledge. And the idea of Descartes is to say that the nothingness of knowledge is precisely something. I think so I am. But “I am” is the first being. But you understand the Cartesian process, we begin in nothingness, we begin by saying we know nothing, but the fact that we know nothing by necessity is something. The fact that I know that I know nothing exists as a fact, and so there is something. And after that I can begin, not by nothing, but by this thing which is finally the consciousness of nothing. But it was the same thing with Socrates. It’s very striking that Socrates a long time before said the same thing. You know Socrates was saying, “I only know that I know nothing,” but it’s a very complex sentence. And so it’s a strange sentence because it’s a false sentence. If I know nothing, I know nothing, and I cannot know that I know nothing because if I know that I know nothing then I know something. But the beginning of philosophy is always something like a dialectical irony, if you want, because it’s always saying that I begin so when I begin there is nothing because the beginning is precisely when the thing which begins is not here. So when you begin you begin in nothing. But if it’s absolutely nothing there is no beginning naturally because nothing has no beginning. And so when we begin we begin in some sense always in the void, in nothingness, and we must transform the fact that we are in nothingness in something. And we can say that, “I only know that I know nothing,” you can say, “I duped, so I am,” and so on. We can also say that Hegel, that to be, nothing and being are the same thing in some sense, and so we begin by this paradox. But it’s an obligation.

And it’s the same thing to the infinite. Naturally the other way is to begin by the infinite itself, to affirm the infinite. I think that this is not the philosophical way but the religious way because we can begin by the infinite because the infinite has said something to us in the Bible, the fact of the Bible, the sacred book. The infinite says something by itself. So we can begin with the infinite but because the infinite has said something to us. So there is something external, you see, when we begin by the infinite, something which is not the pure beginning but the beginning with something else, something which is outside, something finally transcendent. And it is why in this conception we must have something like a revelation and not pure knowledge. There is a revelation. Something has been said, something has been written outside the beginning itself, the beginning of human being as such. And we return to this problem after when we have the first concept of the infinite.

But how we can begin by nothingness? We must affirm that nothingness exists, something like that. We cannot begin with something which does not exist, you know? So we must affirm in some sense that nothingness exists. But it’s a strange affirmation, that nothingness exists, because nothingness is the non-existence of something. In the case of Descartes and Socrates we say that they affirm the existence not exactly of nothingness but that we know that there is nothingness. For example, Socrates is saying “the only thing I know is that I know nothing,” so the affirmation of existence is the affirmation of the existence of the knowledge of nothing. It’s not exactly the affirmation of the existence of nothing, but the existence of that fact that it can be said that I know nothing. So it’s the affirmation of the existence of Socrates in some sense, exactly like in Descartes, in the affirmation of the existence of Descartes. Nobody can say that nothing exists because saying that nothing exists is to exist.

So the first way is to affirm the existence of a subject. The subject affirms nothingness in some sense. It’s, practically, you can read the complete history of philosophy as the history of this sort of beginning. What is the possible affirmation of nothingness? You are the affirmation, the rationalist affirmation, of nothingness like in Descartes or Socrates, that is the affirmation of the thinking of nothingness. But you can have also the existential affirmation of nothingness as an experience, the experience of anxiety in Kierkegaard or Sartre. The beginning is also the consciousness of nothingness but not the rationalist form of knowledge, but in the existentialist form of a pure subjective experience, in anxiety from Kierkegaard to Heidegger by Sartre. We experience that there is something like nothingness when all the world is destroyed, when we are anxious of our proper existence and so on. We discover the existence of what? Precisely the existence of nothingness, and from anxiety we can reconstruct after all the forms of existence of nothingness and knowledge. Naturally, if philosophy must become with nothing, the experience of nothingness is a central one, the experience of the life or experience of the knowledge.

But another possibility is…so that is the subjective possibility, to affirm, to begin in nothingness by a subject who is in nothingness. The other possibility is the possibility of a trace, of a pure inscription of nothingness. There is no vivid subjectivity but the possibility to name nothingness and to have as the first point, as the pure beginning, a name for nothing. The smallest thing possible is the name of nothing, and it will be for us the beginning. The beginning will be to assume the possible existence of the trace of nothing, of the name of nothing, of a name for nothing. That is a position which is the inscription of the void. And, you know, it’s the case where the inscription of the name cannot exactly differ from the thing itself, a point where the name of nothing is nothing as a thing, nothing in the form of something, which is a trace, which is a name, so in the point where we have a sort of dialectical ambiguity between being and nothingness. If we have really the possibility of a name of nothingness, if I say “nothingness,” I say something concerning nothingness, the name “nothingness,” which as such is something and not nothing, but the signification, the meaning of the world, is immediately in relationship to the idea of nothing.

And so our beginning will be the void, and our name will be [Badiou writes the symbol for void, ø, on the note pad] - the absolute beginning. There is no subject like in Descartes or Socrates. There is no vivid experience, living experience, like anxiety. There is a pure trace, a pure symbol. Why this symbol without any reason? Only because it has been chosen by mathematicians of the last century, but if you object to this symbol you can choose another, no problem. The consequences will be exactly the same. So it’s a very interesting point. The name of nothingness is indifferent in some sense. It is why this letter - which is a Danish letter, very often mathematicians write in Greek letters, it’s a Danish letter - precisely has been chosen because there is something strange in this Danish letter, and it is because nothingness is strange, really. And the name of nothingness must be something strange. So if you choose another name choose a strange name. But naturally it’s the name of pure indetermination. It’s not the name of something which is determinate because precisely nothingness is the complete absence of determination. So without any determination we cannot have a name which is determinate, why this name or another name if the thing which is named is without any determination. So there is absolutely no reason to choose a mark or another mark because what we want to have only is that this mark is in some sense the first mark, the first thing, which is precisely the thing of nothingness, that is, the thing of pure indetermination.

Technically, and after we shall say, we can say, that this trace is the set without any elements, the void set. But we can say that after, in some sense. At the first level we say only that is the first mark, the first trace, for something which is purely without any determination and which is, really, nothingness. And it’s a classical Greek sentence that nothingness, the nothingness, has no property, and in fact nothingness as such has no property. Nothingness, for example, is not something black. I say that because in romantic poetry nothingness is always in the form of a black night, and it’s very difficult to have representation of something without any determination. You are immediately in a metaphor and not in a name. so the name must be purely abstract. The name must be something which ahs no meaning at all, only the meaning that it’s a trace for the presence of the absence of nothingness.

But it’s exactly like the cogito because in some sense nothingness is here, in some sense. So this is a thing and not nothing. The name of nothingness is not nothing but something, but this something is a something of absolute indetermination, absolute absence of determination. And so it’s the pure presence of the pure absence, the presence of the absence. And in some sense it’s the being of nothingness, not as such but as represented in an abstract name. And so it’s clearly a beginning, you understand. It’s clearly the same dialectics for a beginning that is always to extract from the void, from the nothingness, something like a pure trace of nothing, precisely. It’s possible to say that this mark is zero too. There is complete freedom on this point because there is necessity to have this name of another name.

So that is our beginning. How we can continue after that, the possibility of rational continuation? We have this sort of mark and sometimes after the beginning nothing else. So we close the seminar by saying we have a Danish letter at the end of the sequence. What we can say concerning the void, mathematically we return to the name of the void, but it’s also nothingness or absence and so on. A point which is of interest is that we can affirm that there exists only one void, that nothingness is by necessity unique. We return to the One. Certainly we return to the One. There is one void. Why is there only one void? Because if they are different there is something in the first void different from something in the second. But there is nothing in the void, so there is no possibility of difference. Nothingness cannot differ from nothingness because the characteristic of nothingness is that there is precisely nothing, no determination at all, or, if you want, that nothingness differ from nothingness, we must find a difference between the two. But it’s impossible to find a difference between the two because the two are without any determination. So one nothingness cannot have a determination distinct from another nothingness. So the pure nothingness, long after that we can find some determination of relative nothingness and so on, but at the point of the beginning, naturally, there is nothingness. So there is no possible difference between two different voids because in the void there is nothing, and when there is nothing, there is nothing. There cannot be something different.

So there is only one void. So this name is a proper name and not a common name. It’s exactly the name of the void, and we progress. So we know something which is that the void as such is the proper name, so it is the name of nothingness as the name of the unique nothingness. This is the first point. After that we can consider that the name of the void is one thing. It is the one thing which is the name of the unique nothing, the one thing which is the proper name of nothing. but this thing, as one, can be the element of something else, not nothingness but the name of nothingness. It can be an element of some set of things for example, the name of the nothingness is the name of the rose, is the name of the European Graduate School and so on. We can put all that together and you have a strange set. But for the moment we have nothing like a rose, the European Graduate School and so on. We have only nothingness. So we can have the set, the only element of which is the name of nothingness. We can have something like that. We have no other things that are the name for nothing, and we can have the set with only one element, which is, precisely, our beginning, the pure name of nothingness. Step one.

As step two we can propose something that we can describe like that [Badiou draws the void set symbol, {ø}, on note pad], which is not the pure name of nothingness, but the set, the multiple with only one element, which is precisely the name of nothingness. It’s not the same thing because by itself the name of nothingness is not a set of anything. It’s the pure name of nothingness, but in step two we have something which we can think and we can realize, which is the idea of the set with only one element. Naturally, that element cannot be nothingness because nothingness is not an element of anything because nothingness has no determination, but the name of nothingness can be considered as an element of some set, and at the beginning w have no other possible elements, but we can have this one, that is, the set, the only element of which is the name of nothingness. And in some sense it’s the One. The One is a very small meaning, a small One, but it’s the One because we can say that the set which has only one element is a realization of the One exactly as nothingness was the name of nothing, or, if you want to be very simple, this sign, {ø}, is the sign of nothing. This sign is the sign of one thing because the name is one thing.

So when we go from the first step to the second one, we go from nothingness to the One by the mediation of the name of nothingness. And it’s a pure process. We don’t introduce, for the moment, something else than nothingness, only nothingness. We do all that with only nothingness. The only rational gesture of all that is to have a name, a proper name, for nothingness, but we are in the radical Cartesian duped in some sense. We duped ourselves of everything. We have no table, no sun, no mountains. We have only nothingness. And we have a name from nothingness, for the moment, nothing. But if we have the set composed of the unique name of nothingness and the name of nothingness, we have two things. We have created two things because this thing is the one that is the set, the unique element is the name of nothingness and this thing is the name of nothingness. So we can continue. We have two things.

It’s extraordinary progress! We have two things. We can put together these two things so we can have the set, the elements of which are, first, exactly like before, the name of nothingness, and second, the set with a unique element, the name of nothingness. And this second thing is the two because we have two things always only with nothingness. It’s a very simple machine, a name for nothingness and after that we can continue. That is the point. We can continue. We can continue because naturally we have three things, the name of nothingness, the set with only the name of nothingness for an element, and the set with two elements, the name of nothingness and the set where the only element is the name of nothingness. I continue a little too much, the new thing which is the set with three elements, (1) the name of nothingness, the zero, if you want, (2) the set with only one element which is the name of nothingness, the one, if you want, and (3) the set composed of two elements, the name of nothingness and the set of the unique element is the name of nothingness. So, zero, one, two, and at “zero, one, two” is three because we have three elements.

I give you a very difficult task, to write the four and the five, to continue by writing the four and the five. Do that for tomorrow. It’s really amusing in fact. If I said to you to write the twenty and twenty-one, it’ long completely written. But we can see immediately that there is something very simple. We have in fact zero. After it we have one, which is zero. Then we have two, which is in fact zero and one, and three, which is zero, one, two, and so on. That is a drastic simplification, naturally. My advice is to not say that the four is zero, un, deux, trois. It’s too simple! I want the complete inscription. And why is it important to return to the complete inscription? Because in the complete inscription you that there is nothing than the void. When we see “four is zero, one, two, three,” you have numbers. It is like the beginning was that we know numbers, zero, one, two, three, and so on, so that we know the multiple as such.But it was not at all our process. Our process doesn’t have as a beginning knowledge of numbers, and we can write something like that. Our process is only the pure affirmation of the existence of the name for the void, and when we write the three like that, we see clearly that we have only the void. The only proper name we have here is the name of the void, nothing else. If we write the three like that, your conviction is that we have three different proper names, “zero, one, two.” It’s not at all the same thing. And it’s a philosophical question. Naturally, finally, this one and this one are the same, and when we do mathematics it’s much more easy to do with that and with a collection of proper names, “one, two, three, four,” and so on. But when we do philosophy, and it’s a philosophical relationship to mathematics, the fact that we have numbers is not exactly interesting for us. The very interesting fact is that we can construct all the successive numbers with only one proper name, one mark, the mark of the void. And so in some sense all numbers are variation concerning zero. All numbers are made of the void, are composed of the void, and so the real of numbers is in some sense a composition with only the mark of nothingness. Numbers are made of nothingness.

And, naturally, so you write the four and five, but we can naturally see that this process can continue without an end. We can continue to write after four, five, and so on. And I take immediately after that the simplified notation. So first we have this one, zero, and we can construct one, two, three, and so on, by the repetition of the same process, that is, to have three we take zero, one, two. To have four, we take zero, one, two, three. You observe that to have a number we take, in fact, the number which is just before with itself. To pass from three to four we take three, zero, one, two and three. To pass to five we take first four, one, two, three and four itself, and so on. And this production is only a simplification of the process of construction with the only antecedent or the only name of nothingness, and it’s just now that we enter in the realm of the infinite because I have said that we can continue. Okay, but what is the meaning of we can continue? We can continue precisely indefinitely. Nothing in the process is in the form of the prescription of a stopping point. There is no stopping point, so we can continue indefinitely. So we can say something like that, the void process in the direction of the infinite, because we are only [in] composition of the void.

So all what we are making here is disposed between nothingness and something without end, so between nothingness and the infinite in fact. Numbers are disposed between nothingness and the infinite, and it’s what sort of plays between nothingness and the infinite. It’s the place of the finite. All these numbers are finite numbers, naturally, five, six, seven, thousand and so on are finite numbers. So we have constructed a place between nothingness and maybe the infinite, but for the moment it’s not precise. It is the place for the existence of the finite. And so the construction of the finite is by itself a thing for the infinite. That is the point because if we can continue it is because there is no limit. If there is a limit we must stop. So if we can continue, the process of the construction of successive numbers has no limit. So there is something which has no limit. We cannot for the moment understand clearly what it is, but there is something by necessity which has no limit because we have the process without any limit. If we take a very, very big number, many millions, it’s a composition of many millions times the name of the void. We cannot hide a number of many millions with the mark of the void. Remember a big machine cannot do something like that. But in fact it is a composition of the name of the void, a very big one, but after this very big number there is another number. This number plus one. So there is no limit.

But what is exactly this sort of being with no limit? It’s, after all, the first definition of the infinite by the Greeks, apeiron. We return to apeiron, without limit. But the strange situation is that to construct finite numbers we must dispose of something infinite because we must continue without limit. So there is no clear opposition for the moment between finite and the infinite. That is my conclusion for the moment. There is no clear distinction because to completely understand the finite we must have the idea of a continuation with no limit. After a finite number there is another finite number without limit. And if all was closed, if all was finite without exception, we cannot have that sort of concept of the finite itself. The finite itself is in relationship to the infinite because the finite must continue in-finite always because the beginning number is finite, and the continuation as such is without limit. And so in some sense the finite is without limit because we continue in a space which is without limit. So I shall stop here, but it’s a very profound idea, that from the very beginning the finite is in relation with two fundamental forms of being, first, nothingness, which is the material of numbers. The name of nothingness is something like the material of finite number, and so the material of the finite in general. So first the finite is in relationship with nothingness, and on the other side the finite is relationship with the infinite because the continuation. And in some sense we can propose the formula, the finite is the dialectical result of the relationship between nothingness and the infinite. So in some sense at the beginning we have nothingness and the infinite, and only after the finite, but we can explain all that in the second part.

Alain Badiou. Infinity and Set Theory: Repetition and Succession. 2011

So we are here, we are between nothingness and maybe the infinite. So we are between, for the moment, two forms of being, which are negative, on one side nothingness, naturally, purely negative, and on the other side we have the infinite but for the moment only under the form of no limit, that is, the process of number must continue without limits, but we are not, for the moment, in a positive vision of the infinite. So we have the finite, which is a form of positive existence. It is existence which is composed positively with the name of nothingness. We have real differences. We can say why one is not the same as two. We can write exactly “one” and “two,” and you see that there is something in two which is not in the one. So we have created in some sense the realm of the finite, which is also the realm of differences, from movement and of continuation, of repetition without limits. And the finite is between two negations. The finite is the positive existence of something which is between two negations, the negation of nothingness and also the infinite as pure negation because the infinite is not something properly, but only the absence of limits of the process. It is why in the philosophical tradition we have not here exactly something like the infinite, but, much more, the indefinite. That is something which is negative and not affirmative, which is not being as such, but which is the absence of limits of the process. So we have the infinite as the possibility of continuation without stopping point, but without stopping point is naturally a negative determination.

So it’s true to say that in some sense the construction of the finite is between two negations, in relationship to two negations, the negation of being, finally, nothingness, and the negation of the limit. To completely understand the point we must exactly see how all that continues, what is exactly the process of continuation. Why? Because the process of continuation is precisely the point where all that is open to a form of infinity, not a negative form, but the form of infinity that is the absence of limits. We can continue without stopping point. There is no end. And to understand that we must exactly observe what we do when, for example, we pass from the number ‘n’ to the number ‘n+1,’ when we do this movement. What is exactly the construction of this movement? You know, here we have ‘n’ because we have ‘n’ terms. We have really four composed terms because we have ‘n’ is one, two, three, but we have ‘n’ is zero too. So we have exactly ‘n’ terms, okay? When we want the successor of ‘n’ we take…and we put something like that. So you understand exactly this process we say something like that, to pass from a number to the number after we take the content of the first, and we take the name of the predecessor because ‘n’ is the name of all that. So finally we take the name, and this writing is composed uniquely of names, finally, of the unique name of nothingness, but with new names that we introduce to substitute to the very long writing with only the name of the nothingness. So to pass from a number to the number after the operation is very simple. It’s to take one element after, and this element is uniquely the name of the number before.

So, you know, in all this job we have only the…only material, if you want, is finally names, first name which is the proper name of nothingness and, after that, composition of names, and we give to this composition of names new names. For example, we decide that this writing, which is made with the name of nothingness, we give to this writing new names, “one,” and after that, for example, if we have something like that we give this the name “two.” So finally there exists only names and nothingness. Names and nothingness. One of you was speaking to me of that just a moment before. It is why all that is near the prose of Samuel Beckett because in Samuel Beckett the world is composed of nothingness and names, absolutely. And there are many references to arithmetic in Beckett too, like here. Why? Because arithmetics is evidently the possible metrics of a complete understanding of a world which is composed of names and nothingness. We have nothingness and names. And, look, it’s not because we give to this the name “one” that we must forget that the name “one” is applied to something which is composed of the name of the nothingness, and the name “two” is also composed the same. The name “two” is also composed with the name of nothingness and so on. And when we do the passage from a number to the other number we take the first number, which is a collection of names from zero, and we put the name, the new name, and we have one element. So it is why we pass from one number to the successor number which has one element more.

So we can finally say that all that can be summarized by two things, first, what I name a primitive name. But if a name is absolutely primitive it must be the name of nothingness because if it is not the name of nothingness it is not primitive because there is something before the name, okay, the primitiveness. And (second) an operation, which is to succeed or to come after. And with these two things we have all the process of the construction of the finite because ‘s’ or the number ‘n’ is composed exactly…the content of ‘n + n.’ So the operation of to succeed something is only to put inside the something the name of the something, okay? You have the number ‘n’ and to have the successor of ‘n,’ that is ‘n + 1,’ you put ‘n’ itself, the composition of ‘n’ and the name of ‘n’, okay?

There is a point which is very subtle in all that and that we must absolutely understand. All that is possible because the name of something is not the same thing that something. From the very beginning it is our law. The name of something is not identical to something. For example, the name of the void is not identical to the void. It’s a very important principle because if the name was identical to the void the thing would be nothing and not a thing which exists, and the name of a number is not identical to the number because the name, this name, is absolutely not identical to this set of names, which is the thing for what this name is a name. ‘n’ is a name for this set, and in the set we cannot find ‘n’ naturally. So we must have a name which is not inside the thing. So when we put the name inside the thing, we have something new. That is a very important point. So it’s why it is not at all a pure repetition. It’s a creation, the operation of something new. ‘Two’ is not reducible to ‘one.’ The name of ‘one’ is in ‘two,’ but the name of ‘one’ is not in ‘one.’ So we cannot have the name of something inside the thing because we have a complete trouble and impossibility to understand the situation. So the name something is not inside the thing, and it is why when we put the name of something inside the thing it’s exactly the operation of to succeed. To succeed is to put the name of something inside the thing. We have something new. This something new is named the successor of the thing. What comes after, what succeeds to the thing. So what succeeds to the thing is the thing and the name of the thing.

It’s very important because this operation is a very fundamental operation of thinking when we don’t have a reflexive notion of all that, not the necessity, it’s the necessity for the philosopher who is a strange man. But we are often doing something like that, to put the name of something or somebody inside the somebody and to create something new. It is exactly like when you write a novel and you decide the name of one of the persons of the novel. You create something new concerning, naturally, the person in question., you have all the characteristics of the person, the color of the hair, the manner to speak and so on, but when we have the name, the name creates something new because it’s the recollection of all the characteristics of the person and the name. And so it’s very important to understand when we have the construction of something it’s always a possibility to construct something new by putting the old name of the thing inside the thing. And so we have something new. It’s exactly the only operation of succession, to succeed, to create the successor of something. It’s the only operation that we have in all this. And we can represent all this process of the finite in the form by the name of the void then the successor the name of the void then the successor of the successor of the name of the void and so on. If successor is to take what exists just before and to put the name inside what exists just before.

And all that is another presentation of numbers. This is zero and one and two, and what is two? The two is that we have the operation to succeed two times, twice, and after that three and so on. So we can also say that a number which is a measure for the finite is the result of a repetition, not the repetition of the thing, but the repetition of the operation. A number is a number of succession. Three is three times the same operation, the name of the void, the successor of the name of the void, the successor of the successor of the name of the void and so on. And if we do this operation five times you have the number five, the result is the number five.

And so we can say something like that. The succession of numbers in the finite is what we can name a creative repetition, which is a paradox, but it’s exactly something like that. Why a creative repetition? Because a new number is really different from the number before. We can give a proof that four is not three. It’s another composition with the name of the void. So when we go from a number to another number to another number, it’s not a repetition of the thing because thing is not the same, the number is not the same. But in another sense it’s a repetition because it’s the same operation. You go from two to three or from three to four by exactly the same operation. So in some sense it’s a pure repetition. It is why to say, “one, two, three, four, five,” and so on is so honoring, because it’s really the same operation, one more, one more, one more. But in another sense it’s some creative process because all numbers are different, and we know perfectly that because if we do arithmetic we know that there are very different properties concerning numbers. We have prime numbers, and we have numbers which are not prime. We have numbers which are composed of very many primes and so on. And the field of arithmetic is a field of extraordinary differences, which is why it’s so difficult. You know that arithmetic today is still a very difficult field for mathematics, transcendent mathematics concerning the numbers that exist today. So we cannot understand that arithmetic is so difficult if arithmetic is a science of the same. But no, there are many very complex differences inside the realm of numbers.

It is why it is really the diversity, the complexity of the finite. But in another sense behind all that there is a complete repetition of the same process, the same gesture, the same succession. And so the finite, we have the first definition of the finite. The finite is between nothingness and the infinite. But our new definition of the finite is that the finite is a mixture between novelty and repetition, creative repetition. So we can define the finite as the insistence of succession, to succeed always. And this is why it is without limit, and you repeat the succession, you go from three to four, from four to five. When we are very in the direction of the infinite numbers have no names. The names disappear. At the beginning we have different proper names, “one,” “two,” “three,” and after that we have names so long that we cannot write the name, and even a very strong electronic material cannot take the name. And we can continue with the machine, and there is always a point where we cannot continue because our brain is too small. But the number continues. It’s a pure insistence of repetition.

The finite is under the repetitive law “once more again,” in French, “encore,” to succeed. Encore is the title of one of the seminars of Lacan, seminar XX, and, in fact, in this seminar there are many things, mediations, concerning repetition precisely, the very concept of repetition. And effectively in the concrete life repetition exists too, and it is why we are finite. We are always in a field of repetition. We must do very often the same things, but as a subjective repetition the repetition is also very often a creative repetition exactly like a number. So there is something common between our subjectivity and the number, and what is common? It is the finite precisely. The finite, on the side of numbers and on the side of concrete existence, is fundamentally a creative repetition. There is something which is like a law of insistence of repetition, but there is also something which is new inside the repetition itself. It’s possible a definition of a subject. A subject is always known by some repetitions. We know somebody, we can say of somebody he does always the same things, he has the same opinions, I know him perfectly, I know the repetition. But we can have some surprise. There is a possibility of something new inside the repetition itself because precisely as we know by numbers the repetition is a structure, but the novelty can be a result. The structure is not by itself the production of the same. The structure can produce finally a world with many differences, many novelties, many surprises and many complexities. But the finite is under the law of encore.

So if we say something like that we have with the first possible definition of the infinite which is the infinite is the space of repetition itself because the space of repetition must be infinite if we must continue without limits. So, okay, we repeat the operation of to succeed, of succession, but if we can do this repetition without limits there is a space of repetition which is without limits. And so, I was saying this before, we cannot understand completely the creative nature of repetition without something infinite, something which is without limits. If we know something infinite, we can have repetition, but with this form, and we return. If creative repetition was strictly with limits repetition would be like a cycle. And in fact we have strictly something finite, but we have no real creative repetition because we return to the same, and after that we do another turn, but we return to the same.

It has been in fact sometimes the vision in the first sequence, the Greek sequence, the idea that we always return to the same things. We find that, for example, in some famous sentence of Anaximander. But it’s not the case practically in number. In number we have not returned to the old. We have a linear continuation. So we have an obligation to assume that there is something without limits under the repetition itself of the operation of succession, the space of the repetition. But when we say the space of the repetition, it’s a metaphor. We don’t understand really what it is because it’s not the space in concrete signification. It’s the place where the repetition can continue. It is a fact that there is no limit point. If there is no limit point there is something infinite virtually. But the fact is that we never be in the point without limit. If we continue the succession of numbers we are always in the finite. We don’t encounter the limit or the absence of limit. The absence of limit is only the possibility to do the succession once more, but this once more is only the without limit and not the positive presentation of something infinite.

There are many thinkers in the classic but also sometimes in the beginning of the second sequence who think that’s the only form of existence of the infinite. It has been assumed, something like that, that there is something infinite, but only virtually, not really. That is those who say that the infinite exists inside the possibility for the finite to continue, so in the absence of limits. But what exists is always finite because the absence of limits is purely negative. So nothing exists which is the absence of limits. What exists is the unlimited process of the finite. This point reduces the question of the infinite to the question of the absence of limit of the finite. So the infinite is only a negative determination of the finite itself. The finite can continue without limit, but it’s not the affirmation of the existence of something infinite. And it has been the result for a while during a long sequence even in mathematics. The great mathematicians don’t admit what is the name of the actual infinite, the real infinite. They admit only something like the virtual infinite of something which has no limit, but which is never finished. There is never the end of numbers. So we cannot have a real infinite in the form of something, for example, as an infinite set. We have only this inevitable relationship of the finite to the infinite, which is the necessity of the continuation without stopping point.

If we stop here we have finally a compromise concerning the relationship between the finite and the infinite, which is that the infinite does not exist as a separate being so there is no real being of the infinite, but the infinite exists in some sense inside the finite as a negative necessity in the form of the law of succession but not at all in the form of a real being. So you know the infinite becomes in this vision a law of the finite, an internal law of the finite, and you can understand by a sort of repetition. If the finite is a creative repetition there is an internal law of the finite, which is in relationship to the infinite. But it’s the law of the finite. It’s not being as such. So there is no infinite thing. So in some sense infinite is a part of nothingness. If we return this picture, the finite in between two forms of nothingness, first form, pure nothingness, nothing at all, so there is no question of finite, infinite, and so on. In nothingness there are no distinctions, no difference. And on the other side the infinite, but the infinite which does not exist properly. In this sense the infinite is the relationship with nothingness. We have only the without limits of the finite. This without limit cannot be realized in the form of a thing. So the infinite is really no thing, in two words if you want, no thing. It is not a thing. We cannot present something infinite. We have the infinite as a virtual law of the finite, but it is no thing. So the finite is between nothingness and the infinite as no thing. So we can say that the finite is the real between two forms of nothingness, the pure nothingness and the infinite as a law, which is no thing. It’s clear, all that. It’s true, no problem.

So if we want to go beyond this very strong vision, I insist on the point that it was a part of, not only the classical vision, but also during a long time of the second sequence. If we could go beyond we must realize the infinite in the form of a thing. There is no other question. So if we must go beyond we must affirm the existence of some thing infinite and not only the infinite as the pure absence of limits of the insistence of repetition in the finite. So the infinite must be a point which is beyond repetition itself and not inside repetition. If we say without limit we stay inside repletion. It’s the repetition itself which is without limit. If we want something else, we want to go beyond repetition. So we must think of something which is not in the form of a law inside repetition, but of something which is really by itself infinite. But how we can think something like that?

So if we have something…we return to our scheme. So I repeat once more, we have in some sense pure repetition of the same operation, to put the name of the precedent term inside the new term, and also we have something new because there is a complete difference between all the successive numbers, and the final number here is completely different form the other. And here we have the possibility to consider the without limits of the process is the only possible definition of the infinite, and so the infinite, in fact, is a dimension of the finite, the negative dimension of the finite. If we want something else we must go beyond the repetition itself. All what is inside the repetition is finite by definition. So we must affirm the existence of a term which does not succeed at all, which is not inscribed in the repetition, and a term which in some sense is not only after one term like all numbers, all numbers the number is always the successor of another number. So we must affirm the existence of something which is not only after a term, but in some sense is after all terms, something which is after all numbers.

But what is all numbers? That is precisely the question. All numbers cannot be a result of the operation of succession. We cannot say, for example, the infinite succeeds to all numbers because succession is not a possibility to go beyond the operation of succession. If you want all numbers we must assume the succession inside the production of numbers, and there is no point where the succession produces something which is beyond the succession, naturally. If you do the succession you have a new number, and this number is finite. We cannot have something infinite by the succession as such. So the repetitive process as such, the repetition as such, cannot produce any term which is infinite. It is impossible. So we must affirm the existence of something completely new which is a term which does not succeed, so a term which is outside the scope of the repetition. So the imperative…this sign, ‘~’, is the sign of negation. I give you tomorrow a paper for all symbols and so on. If we have to repeat, a negation is too long. So the law for a possible, positive conception of the infinite is the matheme for the infinite. So the infinite does not succeed at all. We, for a reason which becomes clear after, we chose for the supposed infinite, we have no reason for the moment to accept the existence of something like that, we name it omega, the Greek letter omega. And so omega is non-exists ‘x’, omega is the successor of ‘x’.

Our first possible concept of a positive infinite is in fact a negative one once more, you know. So we are not absolutely at the end of our process. It’s not, this time, the negation of a limit. It’s not the without limit of the process, it’s the negation of succession itself. There is no ‘x’ for which omega is the successor ‘x’. So it’s not the without limit of the space of the succession, of the space of the repetition, but it’s the negation of the repetition itself. So we can say if something infinite exists the first infinite, the beginning of the infinity, is always the interruption of repetition, not the possibility to continue, but something which is outside the continuation. It is precisely what is inscribed here. The omega, the first infinite, is outside the repetition. It’s not the successor of something. It does not succeed in the finite.

But you know, it’s a problem of a completely new negation. When we say the infinite is the without limit of the repetition, like here, we have a negation which is a negation concerning the possibility of the process, the affirmative possibility of the process. The process must continue. If we say that our supposed infinite has nothing to do with the repetition in some sense, it is outside the process of repetition. It is the revers of the first negative definition. The virtual definition inside the finite was no limits at all. But in this new affirmation we have no succession at all, and no limits to the succession. So, you know, it’s very interesting that the same word, infinite, in the first sense, is a name for the strengths of repetition. Repetition can continue without limits. If you say without limits, we say something which is infinite in some sense, but which is the strength of the finite. The finite is strong because the finite can continue, precisely. So the infinite is the name of some strengths of the finite. In the new definition, on the contrary, the infinite is supposed don’t be inside the repetition. Don’t be a result of a succession. And so this time the infinite is the name of the weakness in the finite because it is beyond the possibilities of the finite. And it is very striking that the same word, the infinite, can first name the strengths of the finite, the without limits of the finite, and, second, the weakness of the finite. That is the possibility of something which exists without any relationship to the law of the finite, what is repetition. So we can say that we go from the very weak infinite to, first, a strong infinity, and it’s a crucial moment. If we stay before we have only the infinite as a name for the strengths of the finite. If we go beyond we have the infinite as a name of the weakness of the finite. And in the common use of finite/ infinite we have always this ambiguity. The virtual infinite is always something in relationship to the strong infinite. But if we have not an unlimited process of the finite but something which is outside the process of the finite, which is beyond the repetition, which is a sort of interruption of the repetition, we have, naturally, the weakness of the law of the finite itself.

This point is a very instructive point concerning the dialectics of thinking, that the same name, the same concept, can be two completely opposite visions, the strengths of the finite, the weakness of the finite, when we change the perspective. So the point now is to examine the sort of decision. Can we rationally accept the existence of something which is beyond the repetition, something which is beyond the successive construction of the finite numbers, and how can we do something rationally like that? This point has been the great creation of Cantor. Cantor in this sense is the father of the modern conception of the infinite because, before, the infinite as only the secret strengths of the finite was the dominant conception, and the infinite was absolutely outside all that. The infinite was god and so on. Cantor decided first to affirm the existence of something completely new, the name of which is omega and which is really beyond all finite numbers. How we can be beyond the idea of all finite numbers, the idea of Cantor, to explain completely this idea is not so simple.

The idea of Cantor admits to say that we can take all finite numbers, with the recollection of all finite numbers, we can do as if the repetition was finished. So the without limits, we can put the without limits in a new form of limitation. By saying that we can have a complete recollection of all primitive numbers, and we can decide that this set of all numbers has a name, which is precisely omega. All history is, as you see, between two names. So we can write, but it’s a writing which is complex, that omega is the name for the space of the repetition as such, for the space where all numbers are defined by successive repetition. So omega is this set. The point is here because the point is the closure of all that. We are not in without limit. We are in where there is in fact a limit, and this limit is omega itself as the complete recollection of the total process of numbers.

What are the difficulties? The difficulty is that, for the moment, we have a pure decision to produce a new name because we don’t precisely understand what is this story, the closure of something which is without limits. It’s really not very clear. We begin by saying that the strength of the finite is to be without limit, and this without limit is precisely our idea of the infinite. All that is clear. But when we say finally we go beyond the repetition, we put a term which does not succeed to any term of the series, and we have the problem of the closure. What are exactly the consequences of the closure of something which, by definition, is unclosed. But the unclosed was unclosed in relationship to the operation of succession. It’s the operation of succession which must continue, but we put omega precisely as something which does not succeed. So the closure here is not the idea of a limit for the succession. There is no limit for the succession, but it’s the idea of a pure interruption of the succession. It’s not the same idea, something which does not succeed at all but which is not, naturally, in the infinite continuation of the repetition itself.

So there is no clear contradiction between the no limit of the succession and a new term which is outside the scope of the succession. It’s not a pure contradiction, but it’s a difficulty. It is a difficulty because we must think this new reality, this new closure, in terms that we don’t have for all the precedent numbers, all the finite, because, you know, we construct the finite with the numbers before. It’s the same thing here. We construct also omega with the numbers before. But what is the signification of “before?” When you go from a number to the successor of the number you know exactly what is before. “Before” is the finite numbers which are before. But here what is exactly “before”? All the sequence, but all the sequence is by definition without limit, unfinished, there is no final point. What is the meaning of all that is before omega? It’s certainly not before in the same sense that two is before three because we know exactly the signification of “two is before three.” Between two and three we have the succession, but between all that and omega, what is the operation? What is the mental activity for going from succession of numbers to a point which is somewhere outside the succession, you know? And this difficulty cannot be clarified without a return to the notion, to what notion? In fact, to the notion of set, of what is a set, because the only rational means to understand to understand this operation is to say that omega is the set of all numbers, not the succession of all numbers, but the recollection, the set, of all numbers. And so we must continue tomorrow on the same line, as a beginning to understand the passage from a constructive conception of the finite by a clear operation to a new construction, completely different, of omega by the idea of the process of recollection in a set which is also without limit in some sense.

And with that we shall have what we can name the first infinite, the first infinite not in the sense of a law without limit, a virtual infinite, but a first infinite in the sense of a real thing, the infinite not reduced to nothing, to a form of nothing, but the infinite as a real thing. Just to speak something less abstract, you know, to speak of another field of the question of the infinite, there is always something, a use of infinity in the classical infinity of what is a woman. And I would say some words concerning this strange point, this point which is absolutely present, for example, at the end of the Faust of Goethe where the figure of femininity is practically on the side of transcendent infinity. But we find also that in Lacan where the feminine jouissance is also something infinite and also the question why the idea of the infinite has been so often associated to the representation of what is a woman. It must be an exercise for you after the three and the four, the three, the four and then the woman. In fact, it’s because, and naturally it’s a male position, the woman is represented as the point where the man, the male, doesn’t understand where are his limits. A woman is represented as the without limits of the male process, of the male existence. And there are many forms of all that, but it’s a very old idea which is inside our image. In some sense, it is a classical representation, the man, the male is on the side of the consistent process of numbers. There is a succession in fact. It is the human being of succession, and there is something like a symbolic strength of all that. And it’s true that the terrible representation of the woman in this classical conception of woman is much more like omega. It’s the moment where somebody is the representation not only of the succession and of the without limits of succession, but of the without limits of the total space of the repetition and something like an interruption of repetition, which is something which exists outside the succession of numbers. And it is why, in modern terms, we can say that a man is a finite number, and a woman is maybe an infinite number, that is not only the without limits, but much more than the without limits, the point where the question of the limit is dissolved. And it is why men are so afraid by woman because inside their representation of all that a woman, especially in love or sexual relationship and so on, but a woman represents the moment, the point where we cannot continue exactly. We must do something which is not inside the non-limits of the repetition, but we are convoked in the direction of a type of infinity which exceeds absolutely the law of repetition.

And is all that true or is all that pure mythology of the male part of human existence? It’s a question. But certainly the question of the passage where we are, the passage to the quiet infinite which exists only in the negative form of the continuation of the finite, which is also the domestic vision of femininity, if you want. The woman is only this opening of the limit to the continuation of humanity, finally, the humanity which is composed by the solid numbers which are the male figure. It’s a passage of that to the point where we have something which is not at all reducible to this pure continuation, but which is something like a transcendent law of the continuation itself, which escapes repetition, which is a true infinite. It’s by necessity a moment of trouble and a moment of difficulty. We have this difficulty in the intellectual form, but the difficulty exists in existential form in many representation of the function, the ultimate function, of the feminine in the history of human beings, or the history of pure and poor human beings. Thank you.