🦋🤖 Robo-Spun by IBF 🦋🤖

(All parts in English)

It’s a strange schedule, but after all why not: during the weekend, I sometimes find myself writing to you. It’s a manner of speaking, I write because I know that we’ll see each other during the week. Well, last weekend, I wrote to you. Naturally, in the meantime, I had more than enough time to forget about this writing and I’ve just reread it during the hasty dinner I’m having in order to be here on time. I’ll start from there.

Naturally, it’s a bit difficult, but maybe you’ll take some notes. Then after that, I’ll say the things I’ve thought since then, thinking of you more directly.

I had written this… which of course I’ll never deliver to pubellication [wordplay: a pun between ‘poubelle’ (trash) and ‘publication’], I don’t see why I would add to the contents of libraries …there are two horizons of the signifier.

With that written, I make a brace… as it’s written, you must pay attention, I mean, don’t think you understand …so within the brace: – there is the maternal, which is also the material, – and then there’s written the mathematical.

I will be forced, I know, but still, I can’t immediately start talking, otherwise I’ll never read you what I’ve written. Perhaps later, I’ll have to return to this distinction, which I emphasize is a distinction of horizon.

To articulate them, I mean as such… this is a parenthesis, I didn’t write it …I mean to articulate them within each of these two horizons, it’s thus… that, I wrote …it’s thus to proceed according to these horizons themselves, since the mention of their “beyond”—beyond the horizon— only holds up by their position… when you get bored, tell me and I’ll tell you the things I have to tell you tonight …by their position—I write—in a factual discourse.

For analytic discourse, this “factual” involves me enough in its effects that one can say it’s of my making, that it can be designated by my name.

The wall-love [a-mur—wordplay on ‘amour’ (love) and ‘mur’ (wall)—explained only at first use]—what I have here designated as such—reverberates in various ways with the means of what is precisely called “the edge”, of this “edge-man”. The “edge-man” inspired me—I wrote this: “brrom ‘brrom -ouap – ouap”. It was a discovery by a person who, in the old days, gave me children.

It’s an indication regarding: – the voice—the (a)-voice—which as everyone knows barks, – and the (a)-gaze as well, which doesn’t “look so closely”, – and the (a)-trick that makes the trick, – and then the (a)-shit too, which from time to time graffities rather insulting intentions in newspaper pages, under my name. In short, it’s the (a)-life, as a person currently having fun says, it’s cheerful! It’s true, in sum.

These effects have nothing to do with the dimension that is measured as my doing, namely that it is through a discourse which is not my own that I create the necessary dimension. It’s from analytic discourse which, not yet—and for good reason!—properly established, finds itself needing some clearings, which I work on. On what basis? Only in fact that my position is determined by it.

Alright. So now, let’s talk about this discourse and the fact that within it, the position as such of the signifier is essential. I would still like, given the audience you make up, to point something out: this is that the position of the signifier takes shape from an experience that is within reach of each of you, to have, so you can see what’s at stake and how essential it is.

When you have an imperfect knowledge of a language and you read a text, well, you understand, you always understand. That should make you a little more alert. You understand in the sense that—in advance—you know what is being said there. Of course, as a result, the text can contradict itself.

When you read, for example, a text about Set Theory, you are told what constitutes the infinite set of whole numbers. In the next line you are told something you understand, because you keep reading: “Don’t think that it’s because it goes on forever that it’s infinite.”

Since you have just been told that that’s why it is infinite, you are startled. But when you look more closely, you find the term that indicates it is about “deem” [to judge, to estimate], that is, it’s not on that basis you should judge, because they know this series of whole numbers doesn’t end, that it’s infinite, it’s not because it’s indefinite.

So you realize that it’s because, – either you skipped “deem”, – or you are not familiar enough with English, that you understood too quickly, that is to say, you skipped this essential element, which is that of a signifier that makes possible this change of level, thanks to which for a moment you had the feeling of a contradiction.

You must never skip a signifier. It is to the extent that the signifier does not stop you that you understand. Now, to understand is always to be oneself understood in the effects of discourse, which discourse as such arranges the effects of knowledge already precipitated by the mere formalism of the signifier.

What psychoanalysis teaches us is that: all naive knowledge… that is written, and that’s why I’m reading it …is associated with a veiling of the jouissance that is realized there and raises the question of what betrays itself there from the limits of power, that is to say—what?—from the trace imposed on jouissance.

As soon as we speak—it’s a fact!—we suppose something to what is being spoken, this something we imagine as pre-posed, even though it’s certain that we only ever suppose it after the fact.

It is only to the fact of speaking that is related, in the current state of our knowledge, that one might perceive that what is speaking—whatever it may be—is what enjoys itself as body.

What enjoys a body that it lives as… what I have already stated …the “kill-able,” that is to say as addressable in the familiar, of a body it addresses familiarly and of a body to whom it says “kill yourself” [tue-toie—a play on words between the familiar “tutoyer” and “tue-toi”] in the same line.

Psychoanalysis, what is it? It is the locating of what is understood as obscured, of what becomes obscured in understanding, as a result of a signifier that has marked a point on the body.

Psychoanalysis is what reproduces… you will find the ordinary tracks again …is what reproduces a production of neurosis.

Everyone agrees on that. There is not a single psychoanalyst who has not noticed it. This neurosis that is attributed—not without reason—to the action of the parents, is only accessible to the extent that the action of the parents is articulated precisely… that is the term with which I began the third line …from the position of the psychoanalyst.

It is to the extent that it converges toward a signifier that emerges from it, that neurosis will be ordered according to the discourse whose effects produced the subject: every traumatic parent is ultimately in the same position as the psychoanalyst. The difference is that: – the psychoanalyst, from his position, reproduces neurosis – and that the traumatic parent, he, produces it innocently.

What is at stake is—to reproduce this signifier from what was first its efflorescence. To make a “model” of neurosis is, all in all, the operation of analytic discourse. Why?

To the extent that it removes the “quota” of jouissance! Jouissance, in fact, demands privilege: for each person, there is no second way of attaining it. Any reduplication kills it: it only survives as long as repetition is in vain, that is to say always the same.

It is the introduction of the “model” that brings this vain repetition to completion. A completed repetition dissolves it, because it is a simplified repetition. Of course, I am still speaking of the signifier when I speak of the “yadl’un” [wordplay: “il y a de l’Un”—there is One].

To extend this “dl’un” to the measure of its empire… since it is assuredly the master-signifier …one must approach it where it has been left to its talents, to bring it, itself, up against the wall.

That is what makes useful, as an incidence, the point I have reached this year, having no other choice than “…Or worse”, this mathematical reference, so called because it is the order where the matheme reigns, that is to say, what produces a knowledge which, in being only produced, is tied to the norms of surplus-jouissance, that is, of the measurable.

A matheme is what is properly—and solely—taught. Only the One is taught. Still, one must know what it is about. And that is why this year, I am questioning it.

I will not continue my reading any further, which I have read—I think—slowly enough—and which is difficult enough that, for each of its terms that I have spelled out well, a few questions for you may attach themselves. And that is why now, I will speak to you more freely.

There was someone, the other day, who after the last event at the Panthéon… perhaps he is still here …came up to me to ask “if I believed in freedom.”

I told him he was funny, and then, since I am always rather tired, I broke off with him, but that does not mean I would not be ready, on that subject, to make him some personal confidences.

It is a fact that I rarely speak of it. So that this question was of his own initiative. I will not regret knowing why he asked it of me.

What I would now like to say more freely is that, making reference in this writing to that in which, to that by which I find myself in position, this analytic discourse, to open it up, it is obviously as I consider it as constituting, at least in potential, this kind of structure that I designate by the term discourse, that is, that by which, by the pure and simple effect of language, a social bond is precipitated.

People noticed this without needing psychoanalysis for it. It is even what is commonly called “ideology”.

The way a discourse is arranged in such a way that it precipitates a social bond involves, conversely, that everything articulated within it is arranged by its effects.

That is precisely how I understand what I am articulating for you about psychoanalytic discourse: that if there were no psychoanalytic practice, nothing I could articulate of it would have any effects I could expect.

I did not say “would have no meaning”. The very nature of meaning is always to be confusional, that is to say, to form a bridge, to believe it forms a bridge, between – a discourse in which a social bond is precipitated, – with what, from another order, comes from another discourse.

The annoying thing is that when you proceed, as I have just said in this writing that “it is a matter of proceeding”, that is, to aim in a discourse at what functions there as the One, what do I do on this occasion?

If you allow me this neologism, I practice unology. With what I articulate, anyone can do an ontology, according to what is supposed beyond precisely these two horizons, which I have indicated are defined as horizons of the signifier.

One can, within university discourse, take up from my construction the model, supposing there at some arbitrary point I know not what essence that would become—no one knows why—the supreme value. This is especially suitable for what is offered to university discourse in which, according to the diagram I have drawn, the point is to put S2—where?—in the place of semblance.

Before a signifier is truly put in its place, that is to say precisely recognized from the ideology for which it is produced, it always has effects of circulation. Meaning precedes in its effects the recognition of its place, its instituting place. If university discourse is defined by knowledge being put in the position of semblance, this is what is checked, this is what is confirmed by the very nature of teaching, where, what do you see?

It is a false ordering of what could have “fanned out,” if I may say so, over the centuries, of various ontologies. Its peak, its culmen, is what is gloriously called The History of Philosophy, as if philosophy had not… and this is amply demonstrated …its impetus in the adventures and misadventures of the discourse of the Master, which from time to time must certainly be renewed.

The cause of philosophy’s shimmering is, as is sufficiently stated from the very points where the notion of ideology emerged, as if then the cause in question did not lie elsewhere. But it is difficult that every process of articulation of a discourse—especially if it has not yet recognized itself—should not give rise to a certain number of premature puffs of new “beings”.

I know well that all this is not easy and that it is still necessary… this in the good tradition of what I am doing here …that I tell you more amusing things.

So let’s talk about “The Analyst and Love”. Love in analysis… and of course it is due to the position of the analyst …love is talked about. All things considered, it is not talked about more than elsewhere, since after all, that is what love is for.

It is not the most delightful thing, but after all, in this century, it is talked about a lot. It is even prodigious—after all this time!—that people still keep talking about it, because after all this time, one might have noticed that it does not succeed any better for all that.

It is therefore clear that it is by talking that one makes love. So, the analyst, what is his role in all this? Can an analysis truly make a love succeed? I must tell you, as for myself… [Laughter], that I do not know of any example. And yet I have tried! [Laughter]

For me it was—a wager, of course, because I am not completely born of the last rains. I hope the person in question is not here, I am almost certain of it [Laughter]! I picked someone, thank God, whom I already knew needed psychoanalysis, but on the basis of that request… you can see what filth I am capable of to verify my claims …on the basis of this: that at all costs he had to have conjugo with the lady of his heart.

Naturally, of course it failed—thank God!—in the shortest time! Well, let’s shorten this, because all of these are anecdotes.

It’s another story, but one day when I am in the mood and dare to do La Bruyère, I will address the question of the relations between love and semblance. But we are not here tonight to linger on such trifles!

What is at stake is to know this, to which I return because it seemed to me I had opened the way, it is the relation of all this that I am in the process of restating, which I remind you with a brief touch of truths of experience, it is to know the function, in psychoanalysis, of sex.

I believe that on this point I have struck the ears, even the deafest ones, with the statement of this which deserves to be commented on: that there is no sexual relationship. Of course, this deserves to be articulated.

Why does the psychoanalyst imagine that what underlies what he refers to is sex? That sex is real, there is not the slightest doubt. And its very structure is the duel, the number “two”.

Whatever one may think, there are two: men, women, as they say, and people persist in adding the Auvergnats! [Laughter] That is a mistake! At the level of the real, there are no Auvergnats. What is at stake when it comes to sex is the other, the other sex, even when one prefers the same.

It is not because I said earlier that, as for the success of a love, the help of psychoanalysis is precarious, that one should think the psychoanalyst does not care, if I may put it that way. That the partner in question is of the other sex, and that what is at stake is something that relates to his or her jouissance, speaks of the other, of the third, about whom this “parlance” around love is stated, the psychoanalyst cannot be indifferent to it, because the one who is not there, for him, is indeed the real.

That jouissance, the one that is not in analysis, if you allow me to express it that way, functions for him as the real. What he does have in analysis—that is, the subject—he takes for what it is, that is, as an effect of discourse.

I ask you to notice, by the way, that he does not subjectivize it. That does not mean that all this is just his little ideas, but that as a subject he is determined by a discourse from which he has long come, and that is what is analyzable.

The analyst, I specify, is by no means a nominalist. He does not think of the representations of his subject, but he must intervene in his discourse, by providing him with a supplement of signifier. That is what is called interpretation.

For what is not at his disposal, that is to say what is at issue, namely the jouissance of the one who is not there, in analysis, he regards it for what it is, that is, certainly of the order of the real, since he can do nothing about it.

There is something striking: sex as real… I mean duel, I mean that there are two …never has anyone, not even Bishop Berkeley, dared to state that it was a little idea that everyone had in their head, that it was a representation. And it is quite instructive that throughout the history of philosophy, never has anyone thought to extend idealism that far.

What I have just defined for you on this point is this: that especially for some time now, we have seen what sex is under the microscope… I am not speaking of the sexual organs, I am speaking of gametes …realize that we lacked this up to Leeuwenhoek and Swammerdam.

As for sex, people were reduced to thinking that sex was everywhere: nature, nous, the whole caboodle, all of that was sex… and female vultures made love with the wind.

The fact that we know with certainty that sex is found here: in two little cells that do not resemble each other, from this, and under the pretext of sex… of course, long before we knew there were two types of gametes …in the name of this the psychoanalyst believes that there is a sexual relationship.

We have seen psychoanalysts… in literature, in a domain that cannot be said to be very filtered …find in the intrusion of the male gamete… of the “spermato” as one says, and “zoid” too …into the envelope of the ovum, find there the model of I do not know what kind of fearsome intrusion.

As if there were the slightest relationship… between this reference which has no relationship at all, except for the crudest metaphor, with what is at stake in copulation …as if there could be anything here that refers to what comes into play in what are called “relationships of love,” namely—as I have said, and first of all—a great deal of words. That is indeed the whole question.

And it is here that the evolution of the forms of discourse is for you much more indicative of what is at stake—it is about effects of discourse—much more indicative than any reference to what, totally, even if it is certain that there are two sexes, to what totally remains in suspense, that is, whether what this discourse is capable of articulating includes, yes or no, the sexual relationship. That is what is worthy of being questioned.

The little things I have already written for you on the board, namely: – the opposition of a : and a /, of an “there exists” and a “no, there exists” at the same level, – that of “it is not true that Φx,” and, on the other hand, a “every x is in conformity with the function Φx” and “not all”—which is a new formula—“not all,” and nothing more, “is susceptible”—in the right-hand column—“to satisfy the so-called phallic function,” it is this around which… as I will try to explain in the seminars to follow, that is, elsewhere …it is this, that is, in a series of gaps that are found at every point of presuming that according to these terms—that is, here, here, here, here—various gaps, not always the same,

…it is this that deserves to be pointed out in order to give its status to what is at stake around the subject, the sexual relationship.

This shows us well enough to what extent language traces, in its very grammar, the effects called subject, this quite sufficiently covers what was first discovered from logic, so that we can from now on attach ourselves, as I have done in some of these appeals I make, to the hearing of a signifier, so that I can try to give it meaning, for this is the only case—and for good reason—where the term “meaning” is justified, in stating: “there is One” [y a d’l’Un—wordplay].

Because there is one thing that must nevertheless appear to you, it is that if there is no relationship, it is because—of the two—each remains one. The unheard-of thing is that psychoanalysts, whose mythology is denounced more or less rightly, it is strange that the very one that fails to be denounced is the one closest to hand.

When the gametes join, what results is not the fusion of the two. Before that happens there must be a hell of an evacuation: meiosis, that’s what it’s called! And what is One, new, is made with what we can quite rightly call… why not, I don’t want to go too far …I will not say the debris of each of them, but after all a “each of them” that has shed a certain number of debris.

To find—and my God, under Freud’s pen—the idea that Eros is based… in the subjunctive [thus: melt—fondre]: see the equivocation, but I don’t see why I shouldn’t use the French language, between foundation and fusion …that Eros is based on making One out of two, it is obviously a strange idea, from which, of course, proceeds this absolutely exorbitant idea that is embodied in the preaching to which, however, dear Freud recoils with all his being… he lets it slip to us in the clearest way in “The Future of an Illusion,” in many other things as well, in many other places, in “Civilization and Its Discontents” …his repugnance for this idea of universal love.

And yet the founding force of life, of the “life instinct,” as he expresses it, would be entirely in this Eros which would be the principle of union!

It is not only for didactic reasons that I would like to present before you, on the subject of the One, what can be said to counter this crude mythology, besides the fact that it might allow us, not only to exorcise Eros—I mean the Eros of Freudian doctrine—but also dear Thanatos, with which we have been bothered for quite a long time.

And it is not pointless here to make use of something that has come to light recently, not by chance. I have already introduced last time a consideration of what is identified as set theory. Of course, don’t rush ahead like that!

Why not also… because we can also have a bit of fun: men and women, they are “together” too. That does not prevent them from each being on their own side. The question is whether, on this “there is One” being discussed, we might not, from “the set”… from a “set” of course, which was never made for this …draw some light.

So since here I am launching some trial balloons, I simply propose to try to see with you what, in there, can serve—I won’t say as illustration, it’s about something quite different: it’s about what the signifier has to do with the One. Because of course the One did not just arise yesterday.

But it did arise all the same concerning two quite different things: – concerning a certain use of measuring instruments, – and at the same time something that had absolutely no connection, namely the function of the individual.

The individual, that is Aristotle. Aristotle, these beings that reproduce, always the same, that struck him. It had already struck another, a certain Plato, of whom I actually think it is because he had nothing better to offer to give us the idea of form, that he ended up stating that the form is real. He had to illustrate, as best he could, his idea of “the Idea”.

The other [Aristotle] of course, points out that, after all, “form” is very nice but what distinguishes it is this: it is simply what we recognize in “a certain number of individuals who resemble each other.” Here we are setting out on various metaphysical slopes. This does not interest us in the least, the way the One is illustrated: – whether it is as the individual – or whether it is a certain practical use of geometry.

Whatever refinements you may add to so-called geometry… by considering proportions, what is manifested as a difference between the height of a stake and that of its shadow …for a long time now we have realized that the One poses other problems, and this simply because mathematics has progressed even a little.

I will not return to what I stated last time, namely on differential calculus, trigonometric series and, generally speaking, the conception of number as defined by a sequence.

What appears very clearly is that the question is then posed quite differently as to what is at stake with the One, because a sequence is characterized by this: it is set up like the sequence of whole numbers. The task is to account for what the whole number is.

Of course, I am not going to give you an exposition of set theory. I simply want to point out this: – firstly, that it had to wait quite late, the end of the last century—it has not been for more than a hundred years that an attempt has been made to account for the function of the One, – that it is remarkable that the “set” is defined in such a way that the first aspect in which it appears is that of the “empty set”, and that on the other hand this constitutes a “set”, namely the one for which the so-called “empty set” [Ø] is the only element: that makes a “set with one element”.

This is where we start from, and last time… I say this for those who were not at the Panthéon, where I began to tackle this slippery subject—that the foundation of the One, for that reason, proves to be properly constituted from the place of a lack. I illustrated it roughly by the pedagogical usage in what is at stake in conveying the so-called set theory, to show that the so-called theory has no other direct object than to show how the proper notion of cardinal number can arise through bijective correspondence. I illustrated it last time: it is at the moment when—in the two compared series—a partner is missing, that the notion of the One emerges: there is one missing.

Everything that has been said about the cardinal number comes down to this: if the sequence of numbers always necessarily includes one, and only one, successor, if in so far as what is realized in the cardinal—from the order of number—what is at stake: it is properly the cardinal sequence inasmuch as, beginning at zero, it goes up to the number that immediately precedes the successor.

In formulating this for you—improvised as it is—I made a small error in my statement: that, for example, of speaking of a sequence as if it were already ordered. Remove this which I did not affirm: it is simply that each number—cardinally—corresponds to the cardinal that precedes it by adding the empty set.

What is important and what I would like you to sense tonight, is that if the One emerges as an effect of lack, the consideration of sets lends itself to something that I believe is worthy of mention and that I would like to highlight, from the reference to this, that set theory has made it possible to distinguish, in the order of what a set is, two types: – the finite set, – and to admit the infinite set.

In this statement, what characterizes the infinite set is precisely that it can be posed as equivalent to any one of its subsets. As Galileo had already observed… who did not for that matter wait for Cantor …the sequence of all the squares is in bijective correspondence with each of the whole numbers. In fact, there is no reason ever to consider that one of these squares would be too large to be in the sequence of integers. This is what constitutes the infinite set, by means of which we say that it can be reflexive.

On the other hand, for the finite set it is said, as its major property, that it is suited to what is exercised in properly mathematical reasoning… that is, in reasoning that makes use of it …to what is called “induction”.

“Induction” is admissible when a set is finite. What I would like to point out to you is that in set theory, there is a point that I, for my part, consider problematic. It is that which concerns what is called “the non-denumerability of parts”, meaning subsets as they can be defined from a set.

It is very easy if you start from this: to take the cardinal number: you have a set composed, for example, of five elements. – If you call “subset” the grouping in one set of each of these five elements, – then the groups formed by 2 of these elements out of five, it is easy for you to calculate how many subsets this will make: there are exactly ten. – Then you take them by 3: there will again be ten. – Then you take them by 4. There will be five. – And you will finally arrive at the set inasmuch as there is only one, there, present, to contain 5 elements. To this must be added the empty set which, in any case, without being an element of the set, can be shown as one of its parts. For parts are not the element.

What is ordered from this… if someone wanted to write on the board in my place, that would give me a break …this is written like this: 1, 5, 10, 10, 5, 1.

What do we find that we have defined as part of the set? – The empty set is there. – The 5 elements α, β, γ, δ, ε, for example, are there. – Next are αβ, αγ, αδ, αε. You can do the same starting from β, you can do it starting from γ, etc. You will see that there are 10. – And next here you have (αβγδ) with the lack of ε. And you can, by leaving out each of these letters, obtain the necessary number of 5 for grouping as parts of the elements.

By which you find, what is certain… it would suffice for me to complete this statement of a set with cardinality 5 by the sequence that we will put beside it, which is the one referring to a set with 4 elements. In other words, picture it as a tetrahedron. You will see that you have a tetrad: you have 6 edges, you have 4 vertices, you have 4 faces, and you also have the empty set.

The remark I am making has this consequence: I only alluded to the other case to show that in both cases “the sum of the parts” is equal to 2^N, N being precisely “the cardinal number of the elements of the set”. There is nothing here that in any way undermines set theory.

What is stated about countability has all its applications, for example in the observation that nothing changes in “the category of infinity of a set” if any “countable sequence” is removed from it. Nevertheless, the contribution made by non-countability, in that, certainly and in any case, one cannot apply to a set, a finite set, the sum of its parts as just defined—does this, I ask, represent the best way to introduce “the non-countability of an infinite set”?

It is a didactic introduction. I contest it from the moment when the property of reflexivity ascribed to the infinite set and which entails that it lacks the inductivity characteristic of finite sets, nevertheless allows it to be written, as I have seen in some places, that “the non-countability of the parts of the finite set” would result—I emphasize—by induction, from the fact that these parts would be written as the infinite set of whole numbers is written: 2^א0. [that is, 2 to the power of the cardinal of ]

I contest it! And how do I contest it? I contest it from this, that there is a certain artifice when it comes to the parts of the set, to take them according to their scale, whose addition indeed gives 2 to the power N. But it is clear that if you have on one side: a, b, c, d, e—to Frenchify the Greek letters I wrote on the board, I had a reason for that—and if you bring to them what corresponds: – a, b, c, d, correspond to e, – a, b, d, e, correspond to c.

You see that the number of parts, if you substitute a partition, leads to a formula which is very different, but whose reason for interesting me you will see: it is that the number is 2^N-1. I cannot here, given the time and the fact that after all not everyone here is absolutely interested in this, but on this point I would like to, I appeal… I appeal, I must say, as I usually do, in a desperate manner …I appeal to grammarians from time to time to give me a little tip… they send me some: they are always the wrong ones …I have appealed to mathematicians—many already—to answer me on this, and truthfully they turn a deaf ear.

You should know that this “countability of the parts of the set”, they cling to it like a tick to a dog’s skin. Nevertheless, I propose this, which has its own small interest, I am going straight here to a goal that will leave aside a point on which I would like to finish afterward, but I am heading straight for a goal which has its interest.

Its interest is this: that, substituting for the notion of “parts” that of “partition”, it is necessary… in the same way that we have admitted that the parts of the infinite set would be 2^א0, that is to say, the smallest of the transfinite, the one constituted by the set, the cardinal of the set of integers[ ] …instead of having 2^א0, we have: 2^א0-1. I suspect that this—anyone—can make one sense what is abusive in supposing the bipartition of an infinite set.

If, as the formula itself indicates, what is called “the set of parts” leads to a formula which contains the number 2 raised to the power [of the cardinal] of the elements of the set, is it entirely acceptable… and especially from the moment when we call into question induction when it comes to the infinite set …how is it acceptable that we adopt a formula which so clearly shows that what is at stake is not the parts of the set, but its partition.

I will add something that is of real interest: that א0, of course, is only an index… an index that is not chosen at random, and an index forged to designate… for there is the whole series of other indices in principle admitted, the whole series of whole numbers can serve as indices for what is at stake in the set inasmuch as it founds the transfinite …nevertheless, from the moment when it is a question of the function of exponentiation, and it seems that we have abused induction in allowing ourselves to find there a test of the non-countability of the parts of the infinite set, is it not, on closer inspection, possible to find here, in this zero, another function, the one it has in exponentiation, that is to say that whatever the number, the zero exponent as regards the power, equals 1, whatever the number.

I emphasize: any number to the power of 1 is itself [n^1 = n], but any number to the power of zero is always 1, for the very simple reason that a number to the power of -1 is its inverse. [1/n = n^-1, n^1 . n^-1 = n^0 = 1] It is therefore 1 that serves here as the pivotal element.

From this point on, the partition of the transfinite set leads to this, namely that if we equate aleph zero in this instance to 1, we have, as concerns the partition of the set, what indeed seems quite acceptable, namely that the sequence of whole numbers is supported by nothing other than the reiteration of the 1, the 1 that emerges from the empty set.

It is from its reproduction that it constitutes what I gave last time as being at the principle manifested in “Pascal’s triangle”, as it is at the level of the cardinal of the monads, and that behind the supports, what I have called… I say this for the deaf who wondered what I had said …the “nade”, that is, the 1 in so far as it emerges from the empty set, that it is the reiteration of lack.

I emphasize very precisely this, that the 1 in question is quite exactly what set theory substitutes, as reiteration, for the empty set, in which it manifests—it, set theory—the true nature of the “nade”.

What is indeed asserted as the principle of the set, this under Cantor’s pen… certainly, as they say: “naive” at the moment when it cleared this truly sensational path …what Cantor’s pen asserts is that, as for the elements of the set… this means it concerns something as diverse as one may wish, on the sole condition that we posit each of these things, which he goes so far as to call objects of intuition or of thought, that is how he expresses himself. And indeed, why refuse him that, it means nothing other than something as eternal as one may wish …it is quite clear that from the moment intuition is mixed with thought, what is at stake is signifiers, as is clearly shown by the fact that it is written as a, b, c, d.

But what is said, it is very certainly precisely this: that what is excluded… thus in the belonging to a set as an element …is that any element whatsoever be repeated as such. It is thus as distinct that any element whatsoever of a set subsists.

And as concerns the empty set, it is asserted as a principle of Set Theory that it can only be 1. This 1, “the nade”, as it is at the principle of the emergence of the numerical One, the One from which the whole number is made, is therefore something that is posited as originating from the empty set itself.

This notion is important because if we question this structure, it is to the extent that for us in analytic discourse, the 1 is suggested as being at the principle of repetition, and thus here it is precisely about the kind of 1 that is marked as being, in the theory of numbers, – only of a lack, – only of an empty set.

But there is, from the moment I introduced this function of partition, a point of “Pascal’s triangle” that you will allow me to question. With the two columns I have just made, I have enough to show you where my question mark is aimed. Here is what I state.

If it is true that we have as the number of partitions only the number that previously was assigned to the set (n-1), to the set whose cardinal number is lower by one unit than the cardinal of a set, look at how, to generate from this number corresponding to the “presumed” parts of the set we will call more briefly lower, lower by 1, as an element, to find, as Pascal’s triangle has already taught us, the parts that will compose… they will be found in a bipartition …that will compose as parts, according to the first statement, the higher set, we must each time add what corresponds in the left column to the 2 numbers that are: – [1)] immediately to the left, – and [2)] above the first, to obtain on this occasion: here the number 10, here the number 4.

What does this mean, if not that to obtain the first number, that of the monads of the set, of the elements, of the cardinal number of the set, it is only by having, I would say: by an abuse of procedure, placed the empty set among the monadic elements. That is to say, it is by adding the empty set to each of the four monads of the previous column that we obtain the cardinal number of the monads, the elements, of the higher set.

Let us now simply try, to make the thing more visual for you, to see what this gives on a diagram. And let us take, to make it simpler, the previous column again, let us take here 3 monads and no longer 4.

The set, we represent as this circle. But the empty set, I do not insist at all that it must be at the center, but simply to represent it we have it here:

We have said that this empty set, when it comes to forming the tetradic set, this empty set will take its place among the monads of the previous one, that is, to represent it like this, by a tetrahedron… of course, it is not about a tetrahedron, it is about numbers …if it is designated by the Greek letters α, β, γ, we will have here, as the 4th element with “one element” in the order of these subsets, we will have the empty set. But nevertheless, the empty set, at the level of this new set, still exists, and it is at the level of this new set that what has just been extracted from the empty set, we will call it otherwise, and since we already have α, β, γ, we will call it δ.

What does this lead us to see? It is that at the level of the element of the penultimate subsets [n-1], that is, to designate this one, namely that… let us say, to stay with the intuition of the five quadrangles …that one can bring out in, let us also say, a polyhedron with 5 vertices.

There too we have to take—what?—the 4 triangles of the tetrad. In what way? In that in these 4 triangles, we will be able to make three different subtractions, with this added to them, which constitutes it as a set, or more exactly as a subset.

How can we reach our count… except at this same level, where we would have three subsets …by adding the single elements of the set, that is, α, β, γ, δ, as not taken as a set, that is, as defined as elements they are not sets, but being isolated from what includes them in the set they must be counted, so that we have our total of four, to provide the part of the number 5 at the level of the set with 5 elements, we must introduce the elements as 4 simply juxtaposed, but not taken as a set, “subset” on this occasion, that is, what?

We see from this, that in set theory every element is equivalent. And that is indeed how unity can be generated. It is precisely in that it is said that the concept of “distinct” and “defined” on this occasion represents this, it is that “distinct” means only “radical difference” since nothing can resemble anything else, there are no species. Everything that is distinguished in the same way is the same element. That is what it means.

But what do we see? We see this: that if we take the element as pure difference, we can also see it as the sameness of this difference, I mean to illustrate that an element in set theory… as was already shown in the second line …is completely equivalent to an empty set, since the empty set can also play the role of element. Everything that is defined as an element is equivalent to the empty set.

But in taking this equivalence, this “sameness of absolute difference,” taking it as isolable… and this, not taken within that set-theoretic inclusion, so to speak, which would make it a subset …this means that sameness as such is, at a point, counted!

This seems to me of extreme importance, and very precisely, for example, at the level of the Platonic game which makes similarity an idea of substance, in the realist perspective, a universal inasmuch as this universal is reality.

What we see is that it is not on the same level, and this is what I alluded to in my last lecture at the Panthéon, it is not on the same level that the idea of the similar is introduced. The sameness of the elements of the set is as such counted as playing its role in the parts of the set.

This certainly has its importance for us, since what is at stake at the level of analytic theory? Analytic theory sees the One arise at two of its levels. The One is the One that repeats. It is at the foundation of this major incidence in the speech of the analysand, which signals a certain repetition, with regard to—what?—a signifying structure.

What is, moreover… considering the diagram I have given of analytic discourse …what results from the positioning of the subject at the level of “the jouissance of speaking”?

What is produced, and what I designate at the level called surplus-jouissance, is S1, that is, a signifying production which I propose… at the risk of giving myself the task of making you feel its incidence …which I propose to recognize in what exactly?

– What is “the sameness of difference”? – What does it mean that something we designate in the signifier with different letters, they are the same? – What could “the same” mean, if not precisely that it is unique, from the very hypothesis with which, in set theory, the function of the element begins?

The One in question… the one the subject produces, let us say “ideal point” in analysis …is very precisely, in contrast to what is at stake in repetition, – the One as only one, – the One inasmuch as—whatever difference may exist—all differences are equivalent: there is only one, it is difference.

This is what I wanted to conclude with this evening, besides the fact that time and my fatigue urge me incidentally. The illustration of this function of S1 as I have set it in the defining formula of analytic discourse, I will give in the sessions to come.