Issue #42 July 2021

Alain Badiou – The Theory of Covering and the Ethics of the Idea

David Diao, "Iconic Crossing", 1985

The following translation is from Alain Badiou’s seminar from February 15, 2016. The video in French is here and the transcript here. The translator has added and occasionally rephrased Badiou’s extemporaneous remarks during the lecture for the sake of completion. The title ‘Ethics of the Idea’ and the divisions of text into four sections are also the translator’s, for ease of an organized reading.

General Introduction to Covering [recouvrement]

1. The fundamental idea we are working with for the moment, I remind you, is the idea that every figure of oppression comes down to an imprisonment in a finite figure of existence, right where an infinite perspective could have been upheld. In other words, we are transforming the problem of emancipation, or the process of liberating human possibilities. We are no longer treating it directly in the form of an explicit contradiction between disjointed and separated terms of the type ‘the oppressors’ and ‘the oppressed’. In fact, we suppose that what brings oppression upon oneself, oppression in all its different figures, is always the fear, the danger or risk, the possibility that something will emerge that would be radically in excess over this order guarded by the masters.

If the functionary order – without presupposing anything as its contrary – does not need to use specific methods of oppression, the order itself constitutes the oppressive figure. What we will discuss is this specific, identifiable figure of oppression.1placeholder

Our intuitive starting point is that oppression manifests itself whenever something dares to appear that could free itself from the order that detains or contains it. This is one possible meaning of the strong old revolutionary statement: “Where there is oppression, there is revolt.” Unfortunately, this is not quite true. It is not mechanically true. What is certain, on the contrary, is that where there is revolt, there is oppression. Wherever something that seems to disturb the general order rises up, that order will immediately, always, put in place precise and specific figures. That’s what will interest us here. How is an antagonistic possibility constrained?

What we will not directly examine is the general structure, the classically-analyzed figure of contradiction – social contradiction or contradiction of any sort, since it is not reserved to politics. We will take the following track. Covering is a general figure of oppression that can refer to any type of order – the academic order, the order of artistic surveillance, the political order of course, but also the romantic order once jealousy enters the mix (I will return to this; it is a particularly spectacular and violent example of finite covering), as well as the obscurantism that tries to ward off dangerous scientific innovations. Covering emerges – for one reason or another, and sometimes it is not even a reason, but more like a worry, a fear – when the ordinary functioning of the order is no longer enough to do what it would normally do in the course of its own movement. We must examine closely: what is the resource, the technique, the means whereby the order produces, with regard to this fear, the singular figure of its own duration, of the oppression it constitutes?

2. My hypothesis, which is ontological in nature, is that the dialectic adequate for thinking this, for thinking this in its being, is the dialectic of the finite and the infinite. A closed order, whatever its nature, has the ambition to perpetuate itself, that is, to maintain its closure as such, to prevent the manifestation of anything qualitatively foreign to this closure. This closed order can always be described as the maintenance of a certain type of finitude (what is finitude? we will discuss this shortly). Everything that appears to go beyond this finite closure and to overflow the very concept of finitude is a threat – for an order carries and transmits its own concept of finitude. That is why, in general, the order starts out by saying that it alone is possible. Thus it blocks the possible itself, at a precise point. Everything that seems to be in excess of it, to deregulate this closure, is perceived or received as a perilous in-finitization of the situation. And in particular, as the in-finitization of the possible. For the key to maintaining the order is to lock down what is possible. If something possible appears to go beyond or exceed the situation, one enters into a defense of finitude, a protection of finitude from the peril of infinitization of the situation. That is why, in the last resort, we are interested in the dialect of the finite and infinite: to enter into the heart of the figure of oppression – beyond the obvious description of oppression which is well known (the mechanisms of propaganda, of policing, of violence, etc.). What we are looking for is the deep underlying logic that is even more fundamental than the system of means for maintaining order, i.e., the means whereby the order seeks to actually crush that which appears to go beyond the norm and the rule.

So, there are different ways to approach the dialectic of the finite and the infinite, and there are different forms of the imposition of finitude.

My second hypothesis has to do with an extremely important procedure which I call covering. At the most general level, it is the attempt to neutralize the possible emergence of a new infinity by covering it with preexisting significations – significations that are already given in the situation and intend to forbid its development, on the one hand, but also its internal significance, the immanent sense of this infinite, of this excess, this new infinity. It is not a matter of crushing it from outside. It is not a matter of saying it did not happen or that nothing happened, but rather that this something does not have the signification it gives itself. It is possible, in fact, to analyze this situation in terms of oppression itself: it is to cover in some way, like putting a bag over it, the set of whatever is said and done in the name of this novelty. It is to cover it with significations that are old, generally stereotypical and internal to the situation, such that the very intelligibility of what has happened is covered, crumbled, annihilated. Such that even those who participate in it wind up no longer knowing if what they are doing is really what they say it is. For this procedure also aims at an intrinsic demoralization of the agents of the novelty, namely by convincing them, through numerous artifices, that what they believe is new is actually old, and not just old, but harmfully antiquated, etc. That is the general operation of covering. In order to cover something, one has to plaster preexisting significations over its advent, its upsurge, its embryonic instances. In that way one kills the pioneering intensity of the uprising or novel figure of infinity.

Now, I maintain that this business of covering is very important. I think it is the most continually exerted oppression, in the most lively fields of human creation. And I insist on the fact that it does not only want to make disappear the threatening being of the new infinity, novelty or thing-in-rupture. More profoundly, its goal is to render it definitively unintelligible, that is, to kill its very sense. By means of the covering, it intends to introduce a figure of non-sense or, paradoxically, of impossibility into that which had nonetheless appeared possible. The three political examples I provide below are typical: a parasitic signification was plastered over what happened, covering it in such a way that what was really at stake is no longer decipherable or legible – of course not in the future, but also for some of those who participated in the very thing thus covered. In these three cases, from point of view of the future, something obscure and ungraspable has been constituted by the covering. It has made it so that the thing is declared to be mixed up, covered, disintensified by disparate operations coming from the preexisting situation. The element of rupture, inauguration, novelty and creation that it seemed to bear is asphyxiated, annulled, disidentified – not by negating the fact that something happened, but by substituting something else for what did happen. What happened in the vital lives of its actors is not at all what happened from the point of view of the thing once it is covered.

Thus is created a dark and retrospective legend: at the very point where something like a hope or a novelty had seemed to infinitize the situation relative to the closed order, the covering makes appear a sort of formless stub, a thing that should not have been, or something insignificant or abominable – not because it is simply negated, but because it is covered. This operation is absolutely essential because, in a certain way, at every level of collective history, as well as personal, artistic, or scientific history, it organizes the battle of significations and not simply the battle of reality. It creates distinct memories. It creates historical legends. Ultimately, it qualifies, determines, or modifies the historical signification of what happened. From this point of view, covering is a long-term operation. It’s not like when you crush some present figure and afterwards everyone says, oh, it is crushed. Covering is a sort of poison injected into time. It does more than simply abolish what has taken place. It disfigures it such that it becomes unrecognizable. And this is even more intense, because then the disfigured thing is maintained as what actually took place. But what took place has been transformed into something totally different.

Some examples:

  • This is how a political event, potentially infinite in its possible consequences, is covered by negative platitudes. Already the living party of the French Revolution (1792-1794), which opened a real egalitarian process to infinity, was immediately covered up by commonplaces regarding the actions of that “cold monster” Robespierre, said to be embittered and bloodthirsty. The participants in the coup de force of Thermidor, the “thermidorians,” returned the dictatorship of property owners and the corrupt by passing through this covering (which is still wielded today by reactionaries on every side).
  • Similarly, the Cultural Revolution in China (1965-1968) was an unprecedented attempt to restart the real communist movement in the space of a socialist State that was on its way to ossification, involving the massive and direct intervention of students and workers. But the experts in finitude, Chinese as well as Western, rapidly qualified it as a desperate manipulation by Mao to restore himself to the power he had lost due to his own errors; and that he had unleashed unacceptable violence to do so.
  • One will say about May ’68 as well as its consequences in France: the biggest mass movement in Western Europe since the World War II, opening for the first time the possibility of a common political process for revolutionary students and workers on strike, was qualified, and often still is, as a tiny anarchistic tremor wrapping the “sexual liberation” in a perfectly fictitious revolutionary discourse.

One could find similar operations of annihilating an infinite potential with a finite covering in all the other truth-procedures: love, art, science. Proposed exercise: look for these examples in history, collective or personal!

To take up one variation on the theme, I would like to ask what the operations of covering look like in the domain of love. They consist in inhabiting love in such a way that it is permanently haunted by an uncertainty as to its very existence. The covering inhabits love from the inside out with the figure of an immanent darkness, which gradually invades it like a cancer from within. The figure of jealousy is the most complete figure of invasion. Proust described it admirably. He is a fabulous writer on this subject of interpersonal covering. He shows quite well how jealousy institutes a sort of surveillance grid that preexists the existence of the other. It cuts up time so that no continuity is possible anymore. Its suspicion is a permanent finitization of the general movement of love, which is thus blocked in a covering fragmentation (one of the authority measures of finitude). This is why one of the chapters of Proust’s book is called The Prisoner. Instead of love being the intense development of a new figure of existence à deux, it becomes a closure, a lock-up, a prison, where what ultimately matters – from the point of view of jealousy – is not the other but the other of the other. The obsession of jealousy is surveillance; but what threatens it at every moment, what puts love in peril, is the other of the other. Is there an other of the other? This concern becomes the top priority. It is this ungraspable alterity, which has become the fear of the other of the other, that blocks off love, suffocates it, or poisons it. You see how the covering is not exterior. It operates from the inside of the love, like a kind of splitting-up which it cannot help but impose.

David Diao, "Glissement", 1984

Finitude and the Constructible

Now we must ask ourselves: what is the logical substructure of these covering operations? What is a covering, beyond the metaphorical aspect? Under what conditions can one escape the covering? What conditions are necessary to oppose this form of finite dictatorship – which works by imposing a different signification, by plastering pieces of finitude over the infinite possibility that has arisen? How can one oppose it so that the emergence of a new, true infinity is allowed, authorized?

To begin with, we need to know exactly what will be understood by “finite,” since covering works by plastering finitude over something that could have eventually freed a new infinity. We must know what is “finite,” because intuition does not serve us very well here. As we will see, things that appear infinite may be finite from the point of view of the covering. It depends on its origin, on its meaning, on its situation, etc.

The starting point is as follows. Any set will be called finite when it has for elements only multiplicities which figure as definable parts in another preexisting set.

Let there be set E. Let there be a clearly-defined property P, in the sense that we know what it means for an element of E to possess property P. Then the set of elements of E that have property P constitute a definable part of E. This part is defined by property P.

The world, in its most abstract form, is composed of things and predicates about those things. There is an existing set of things (E), and then there is a predicate or property (P). ‘Definable’ applies to all the elements in the set which have that property. It is the subset defined by P, which has the property P. In this way, there is a definable correlation between the language of the situation in which one finds oneself and the elements in things which are correlated to this language by means of properties.

A definable part of a set is therefore a part submitted to the dominant language in the context of this set. This dominant language is made up of well-practiced properties which everyone knows, and which make up a part of the conservative world. A definable part is a part that is clearly constituted by a defined part, that is, by a set of elements which has property P.

Figure 1: The left side depicts constructible set E with the set of elements defined by property P. The centerline depicts the barrier of closed order, which the new infinity threatens to transgress. This revolt, creation, etc., is represented by the double arrow crossing the barrier. The new infinity is represented by a dotted-line circle to indicate its genericity and “non-constructibility.”

This operation which connects language to things – to things in the most abstract sense, to the multiplicity as such – is the basis of every operation of covering. Whatever risks crossing the barrier of the closed order [the double arrow] will be covered systematically by things like property P, that is, by things that are already knotted by language, things that are integrally known, completely banal, totally ordinary, simply because they have a property that the whole world knows.

The procedure functions like this. Suppose that something here is menacing – there is something that rises up to transgress and displace the barrier of the world, a new risk of infinity. The procedure of covering will consist in showing that this thing is not outside the world like it claims, because a covering composed by the definable part will be plastered over it. The alleged unknown will be covered by definable parts, and so it will be locked out completely, since the definable parts are obviously already known by everyone. For example, instead of seeing Robespierre as a revolutionary who tried to introduce figures of equality into the political system, you will see him as an “adventurist,” an “opportunist,” “bloodthirsty” and “bitter,” that is, determinations identified by everyone in the world such as it is.

Figure 2: The operation of covering. The set of P(x), the definable part of E, is mobilized to cover over the generic set of the emergent infinity. This depicts in the abstract how pre-existing significations are plastered over the novelty, covering it in the dominant language.

In every case, a set of elements with the well-known property P will be pasted over the threat of the undefinable ꝏ which crosses the barrier of the order.2placeholder That is the fundamental resource for the operation of covering. Behind the covering lies a theory that what exists submits completely to the documented language. The resource is the definable parts, the parts which have in the situation, in the order, their name, their properties, and nothing escapes these properties. This is called the ‘common’ of the situation, the consensual stuff, the stuff shared by everyone, like ‘French values’, things like that. These are the nominations pegged to the established language. This pegging of things to their name, to their properties constitutes the plaque of language, of actions, of determinations, of morsels of state power, etc., which is ready to be plastered over, to cover the transgressive movement of the novelty.

You can practice for yourselves finding examples where these procedures are constantly in use. In a dispute, for example, you are always in the process of imputing to the other that everything they say can be covered by what you deem a well-defined property, “you say that, but in reality I know very well…” The figure of oppression is constituted in just this way. That’s why so many commonplaces circulate in disputes, and why the chatter of the argument is a generally meaningless, long-haul chatter. Any inventiveness is crushed by the fact that one taps into what is aggressively constituted in the situation one shares. Covering is a procedure that is constantly available, because it suffices to belong to the situation! – the old situation, since at bottom the pieces used for covering are what circulates everywhere. These things can be evoked without saying much about what they are, because really all that matters is that they are saturated from the point of view of language. If you say ‘France’, everyone knows what you mean – except that not everyone does know what it means, and that today perhaps no one knows exactly what France is. But knowledge is not the real issue. The real issue is that, between the alleged thing, France, and the name, there is a lock-down [verrouillage] that does not propose to go any further. The knotting of the thing and its name will function all by itself. It will plaster itself over a situation that might have no relationship to it, a situation it is going to try to seal off or cover.

When we look closer, we see that it’s a little more complicated, for the word “property” is an equivocal word. It is not a very simple word. And so I gave you an example where, if we say that it is made of the definable, we end up with an entity that is indefinable and yet perfectly defined. It’s what one would call, in another context, an impossible point. An impossible point is something that is supposed to be fully definable, yet something else that is indefinable is stamped onto it. We should be suspicious here, if we want to avoid the traps of contradiction.

Even he who, wishing to oppress the people, covers what they are, what they recount, etc., even he tries to avoid explicit contradiction. He can’t always do it, sometimes he falls into it anyways. For example, I was very intrigued by the fact that Valls, the head of State, claimed that, “to begin understanding inevitably means to begin justifying.”3placeholder This in itself is a remarkable philosophical statement. What does this statement cover, or try to cover, if not the very understanding of what happened? It takes a floating cliché, namely: ‘when something horrible happens, the urgent thing to do is to not understand it’ (many people think that way, perhaps we all do in certain moments) and leads to a statement that borders on pure contradiction. If you try to oppose reason to fanaticism, secularism to religion, peace to violence, etc., you can’t base this opposition on the fact that one shouldn’t understand the opposite. On the contrary, the imperative of reason, including political reason, would be to understand what happened. You would have to assert a fully rational construction of things. The covering breaks down when one is caught red-handed wanting only to cover. If, on one side, you say that our values have been trampled by something abominable and, on the other side, you forbid people from comprehending it, you stand between what is definable and what is not. For how can you define what has happened, or declare it undefinable, if you don’t even know what it’s about?

These operations of covering, even in their propagandist vacuity, are taken to deploy a certain coherence. In every case, to comprehend them, we must try to see how to render the general concept of covering perfectly comprehensible. For that it is necessary to give a rigorous definition to the notion of ‘property’. To understand the operations of covering in their formal substructure, we must formalize our language. It is for this reason, and not because of some obsession, that the complete theory of covering is a mathematical theory, for it is fully focused on an elucidation, as rigorous as possible, of the notion of a property. The notion of a property is the notion of what is definable, and therefore the definition of what serves the covering, namely, the locking between things and language. If you want to understand the formal operations used by the adversary to try to cover that which you propose as potentially in excess of the situation, to understand it in its foundation, in the abstract, we must give a fixed meaning to the notion of property. Of course, no one wants to deal with it. We admit that it is extraordinarily tedious. But these are the dull things of thinking. In any case, we cannot do without knowing we must do them – that would be another semi-covering of the real activity…

We will assume that we have done this work, that we have a formally rigorous language at our disposal. We know what a property is, and we know what is definable, namely, a set constituted by elements given with well-defined properties. Starting from there, we will treat four points which I will now enumerate. They form a strategy:

  1. The definition of a “set marked by finitude” is a particular concept of the finite. In fact, the finite is a notion that is only intelligible in the type of procedure of finitude at stake in it. Finitude is not an objective given independent of a process. Here, finite means: that which can be used in a covering. It is finite in the sense that it serves to cover, disidentify, and finally render incomprehensible the emergence of a novelty on the side of the transgression of the order, the covering of something one could have imagined as infinite.
  2. The encounter of an alternative. This is very striking. We cannot avoid the fact that maybe everything is finite, meaning that the upholders of the order are right, and that whatever is not of the order assigned to finitude is in fact impossible, inexistent, even dangerous. It is not possible to prove that the doctrine of the oppressors is untenable. This means that, in a certain sense, we must stop thinking that everything’s going to collapse on its own. With Marx, there is certainly a hesitation on this point. Sometimes he gives the impression that History works in the right direction, towards the collapse of the system of domination; and at other times, particularly when he’s concerned with building the International, he sings a different tune. For it appears to be very difficult, it does not go without saying, but cuts across very complicated twists and turns, etc.

The important point is this. You cannot prove that the one who handles the covering, who says ‘I will cover this thing you think is a new infinity with what already exists, with what is drawn from my own order’ – you cannot necessarily prove him wrong and that his business won’t work. But it also can’t be proved that you’re wrong. If you suppose there is something which cannot be covered, an infinity, a true infinity that cannot be covered, this cannot be disproved.

I call this the discovery of a fundamental alternative. Because at bottom, your position vis-à-vis this problem cannot result from demonstrating the validity of your position. At some point, a choice must be made. You must choose. Every rational thought is inhabited by a fundamental choice. It cannot be shirked. You cannot say that you are convinced through and through by a rational demonstration that the position you are going to choose is true, appropriate, and finally victorious. It can’t be done, not at the level of extreme abstraction which is the theory of covering. But you can’t demonstrate the contrary either – that the position of the adversary will necessarily be victorious.

  1. Under what condition can you uphold, sustain the infinitizing position? Because it is not supported by a rigorous demonstration. You can choose it. But what does “choice” mean here?
  2. What I call fundamental ethics recapitulates all of this. It is the commitment to what must be undertaken in order to be on the side of the good, as I understand it – that is, on the side of the thesis according to which it is not true that everything can be covered.

I will take up these points one by one.

1. What is a set “marked by finitude”? It is a set that is constructed, that is fabricated exclusively with pieces that are the definable sets. It is a set that is constituted by pieces [morceaux] that are glued together, and all these pieces are in fact definable, in the strict sense ordered by the formalized language. A set “marked by finitude,” in the sense of covering, is not a quantitative notion. It is not a matter of it being small! What counts is its composition. What belongs to the constructible, what is the real material of every covering, is the multiplicity which is composed entirely by definable things, by things which in their very composition are intrinsically definable.

Now, the definable itself is defined as the parts of the set that are already there. Let’s see how a finite set, a “constructible set,” is constructed [Figure 3, left side]. This construction takes place in successive stages. Suppose you have a number of constructible sets. Inside these sets there is a definable part [the shaded-in circles]. Now, the new constructible set is going to be composed from the definable part of the previous sets. At every stage, the constructible set retains only the definable of the previous stage. It is composed of definable things and it does not contain anything that would not be definable.4placeholder

Figure 3: The two fundamental choices. On the left, accepting the axiom of constructability, ‘new’ sets are constructed from properties of previous sets, all is definable, etc. On the right, accepting the axiom of non-constructibility, the generic set is committed to as possible and the barrier of the closure of the world is crossed. The operation of covering continues to try to plaster pre-existing significations over the new infinity. The ethical choice and effort is to remain faithful to the relaunch point: there is something non-definable, not everything is subject to covering and the imprisonment in a finite figure. At a precise point the infinite set remains open to development.

This point about the definable is very interesting for a general analysis of our world. The constructible is an ordered, hierarchical process. In the most radical doctrine, the starting point is the void. Since there is nothing definable in the void (because nothing is there, nothing definable), you mark the set ‘void’. This ‘void’ is the only element of the set, namely, the one. Afterwards, as you progress, you will open a wider and wider range, but at each stage you will have new constructible sets composed entirely of things definable in a formal language which begins from the previous definable multiplicities. You can arrive by levels up to considerable complexities – and at a certain point, you may even arrive at an infinite constructible set. Hence ‘finite’ here is not defined by big or small, but by its internal structure: its submission to the current language. It is in this sense that one can say it is closed.

The constructible is a category that once would have been called “ideological,” for it admits as existing only things that are already submitted to the dominant language. In this affair, you do not accept that there is something undefinable. The finitude here is the finitude of submission. It is a limit where the undefinable is inacceptable. If you only admit what is definable, this means that, from a certain point of view, you only admit what the world already knows, what it has already named, structured, put in practice, etc. That is the structure of a dominant ideology as the general conservation of the system. It only admits operations upon its own definable, that is, in the language that it utilizes to name things and hierarchize them in the order of the definable. From this point of view, constructible sets are the general form of all the materials utilized in oppressive procedures, and singularly in the oppressive procedures of covering.

This is more sophisticated than the notion of dominant ideology, because it constitutes itself in networks that are capable of covering everything at the interior of the order – since inside the order, nothing will be admitted as really existing except things in the already-established, definable structure. This does not mean that new things won’t appear! You can always add a stage, but in the new stage there will only be something definable coming from the previous stage. This will be ‘new’ because it combines the definable in a different way, but it will not be new in the sense that it would not be of the definable. There will be a sort of automorphy of the dominant order which, from the point of view of the relation between multiplicities and the names given to them, will automatically maintain itself under the aegis of the dominant language – without ever letting anything show through that would not be reducible to this language. Put otherwise: there will be no unnameable (the unnameable which as you know is the title of a novel by Samuel Beckett).

The thesis supported here is that the world, in this dominant register which contains only constructible sets, which only knows the definable and hierarchized, is closed. This is not a quantitative notion. It is closed according to the internal composition of what composes it.


David Diao, "Skating on Thin Ice", 1985

Non-Constructible or Generic Sets

2. Is it possible to accept that only constructible sets exist? The greatest logician of the 20th century, Kurt Gödel, who invented the concept of the constructible, demonstrated with virtuosity that it was not contradictory to accept that all existing sets are constructible. (However, the fact that it is not contradictory is not the same thing as it being true.) So, when the masters of a situation say, ‘everything is constructible’, it holds water. You can’t just say to them, ‘No, no, that’s not true’, and then show them the ‘already present’ non-constructible! To refute the masters, you must create something. It is not a matter of looking for something in the situation that already exists under the law of the constructible. It is remarkable that Gödel demonstrated this in the mathematical theory of constructible sets, that is, sets submitted to the definable. If you add to general set theory the axiom, ‘everything is constructible’, it does not collapse. The masters who say that everything is constructible are not going to ruin the general system of possible thought. In fact, there are innumerable domains of concrete experience that are constructible – finite in a profound sense.

Gödel demonstrated this in a convincing way, but what happened was something quite interesting. The mathematics for which everything is constructible, the “easiest” from a certain point of view – there is no unnamable, nothing escapes to the edges, everything is classified, named, registered, constructible – well, no mathematician wanted it. It did not captivate them. Practically no one rushed into the paradise of the constructible. Instead, they posed the following question: since Gödel had demonstrated that it was not contradictory to admit that all existing sets are constructible, couldn’t one demonstrate that it is also not contradictory to admit that there is some inconstructible? This was a huge challenge. Because if you want to introduce some non-constructible, you are going to have to ‘construct’ something that is not definable, something that escapes the dominant system of language. Mathematics was haunted by this problem for decades: how is it possible to demonstrate that something could exist which, from the dominant point of view, is not constructible?

It would seem impossible: the imperative to construct the inconstrucible. In my view, this is the problem that confronts every creation, in a universal way: how to construct something that from the dominant point of view is not constructible? You can say this about a revolutionary party as well as an early cubist painting, the first dodecaphonic works of Schönberg, or Galois’ theory. In all these examples, and in many others, something is produced that, precisely from the point of view of the established order, is not constructible – that is, something that ultimately cannot be covered, that cannot be buried underneath the constructible because it is confirmed as non-constructible. These are the mysteries of creation. Creation always lurks in the vicinity of that which is not already there in the form of the constructible and definable. And at the same time, one works with what is already there. You can surpass the world, but you surpass it from inside this world. The procedures you invent necessarily borrow, whether they want to or not, from the ambient definable. This ambient definable will have to be twisted, maneuvered, in order to arrive at something it refuses. Every invention, from this point of view, is in some way refused by the world in which it is produced. It is not just difficult. It is refused, because the immanent law of the world is the imperium, the commandment of the constructible.

Finally Paul Cohen found the way. He demonstrated that, yes, one can admit sets that will not be reached through a constructible hierarchy – sets that are intrinsically non-constructible. And it is absolutely remarkable that he would name these sets generic. This word “generic” has a long history. In the Manuscripts of 1844, Marx uses it to designate the proletariat. He says that the proletariat is the representation of generic humanity, of humanity as such: the representation of that which, in humanity, falling exclusively under the heading of ‘humanity’, is not of the definable order imposed by a determined society. This point touches humanity as generic humanity, humanity as a genre of existence that is not coded or pre-coded in the figure that society always imposes on it. It is also the classless society, but this question of the generic in Marx shows that what he understood by the proletariat was the non-constructible point of bourgeois society: the existing non-constructible point. It exists, indeed, the people are there; but as a subjective capacity it was a point that the order not only could not construct but could not even imagine being constructed. I don’t know if there was a direct filiation – I don’t think so anyhow – but spontaneously, one could say, Paul Cohen rediscovered this old word ‘generic’ to designate, not the humanity immanent to the proletariat as that humanity stripped of properties from outside, but non-constructible sets.

And so we found ourselves in the following situation: it is possible to declare that everything is constructible; it is also possible to declare that, no, there is some inconstructible. What to do in such a situation? You must choose. There is nothing else to do. You cannot say both at the same time. If the mathematician does not assume the axiom of constructibility, it cannot be for reasons of coherence. To assume it is simpler and just as coherent. When he decides not to situate himself in the field of the constructible, it’s simply because he finds it more interesting to set up shop in the field of the non-constructible. More interesting, why? Because if you accept the non-constructible, you are going to accept that there exists something that cannot be covered. There is a non-covered core, generic in a certain way, which is radically exterior to the field of the constructible. It is not constructible, and so it is not composed of the definable. In one place, there is the undefinable. We can speak here of a relaunch [relance], for it will necessarily implicate a surpassing of the limit. It is stabilized because there is a point that in any case was discovered, manifested, a point that is in excess. It is a relaunch inasmuch as you can rely on this point to build something else, and also to modify the very definition of definability. Because you could say things are definable either in the old sense of the term, or in a new sense: the old sense of the term plus this point that cannot be covered! And so you are also in a position to construct a new universe of the definable – and not just to clash with the definable. It is enough to add new generic entities to it. They will work from the inside on the new definable through a sort of permanent instability.

This is exactly what Marx thought. The proletariat was the support of the revolution insofar as it was generic – insofar as it deployed, from within the society where everything is defined by proprieties, the social hierarchy, etc., something impossible to grasp from the point of view of the covering. This ‘something’ becomes the principle for reorganizing the whole of sociality around this point, which will itself diffuse and disappear in such a way that what ultimately exists is generic humanity. The proletariat will render the whole of humanity generic, completely leaving behind the previous system that defined the system of social positions.

We can retain this: there is a fundamental choice, though naturally it appears in concrete circumstances. It is a choice that you encounter each time you are confronted with the possibility, the uprising, of something other – and consequently when the logic of covering confronts you intensely, for X number of reasons. One aspect of what I call events is summed up by this point of view. One function of the event – an uprising, a creation, etc. – is to bring this choice to the light of day. An event can be defined by the necessary decision, with regard to what happens, either to remain within the constructible or to get out of it – that is, to expose oneself to the generic – even if it is something that, for quite a while, is undefined, poorly defined, impossible to grasp, because it is generic and cannot be classified, defined, systematized, etc.

This commitment as a coefficient of incertitude was a fundamental quality of the proletariat, though later on it lost it, to be sure. The proletariat was in fact simultaneously the promise of the future and also the phantom of society, unlike that which was constructible or definable by society or through it. It is necessary for politics to safeguard this, if it is to remain a politics of the generic. If it redefines everything, if it reclasses everything, it just substitutes one constructible order for another, what we call a relative constructibility. There is a general constructibility that starts with the empty set, and you can very well take the “proletariat” this way – as closed off in a unique and closed genericity – and then rebuild the definable universe starting with this closure. Then the State is the only new reality. The ‘proletarian State’ is largely the reconstitution of a new type of definability, of course, but it is also loses the generic’s character of being in movement and its capacity to dissolve, its capacity to spread throughout the entirety of humanity, where it dissolves the constructions of and subordinations to the dominant language.

This proves that an event can be correlated to something very simple but, in reality, very complex, though formally very simple: ultimately every fundamental choice is always a choice to accept a genericity, in one point. I believe this absolutely. It is to accept that something will escape the system of authority of the dominant language. But if you do this, it will also partly escape you, because you are also in the dominant language! And so there is an effort to ensure that the acceptance of the generic is a process, that one draws the consequences from it and it is not just undermined or anesthestized in relation to previous definitions.

I think that the antinomy Gödel-Cohen – which was not really an antinomy, since they were both in perfect agreement; neither of them really liked the constructible, and Gödel was very happy to see that another choice could be made – this antinomy is still an admirable formalization of the freedom of thought. It forces us to get right to the point: the choice is not prescribed, nor prescribable, because you cannot claim it’s coherent. You can say, “the constructible is still better, because it is clear and stable – language is in its element.” You can say, “the generic is tremendous, for it means adventure and trouble – it is what bursts the banks of definitions and language.” This is in reality a permanent debate, everywhere, affecting every level of human existence. The twists and turns of existence are such that you are often in one register on one side of things and wholly in another register on the other side; it’s not a global or systematic distribution. Mathematicians saw that in its profundity, and it is a major existential point, because at the end of the day, there is a fundamental choice: Am I going to take my place in the order of constructibility, that is, of the definable? Will I try to be placed? Because the restriction of the definable is that you will be placed somewhere; you are there, you have your attributes, your name… and likewise with everything surrounding you. Or am I going to assume something that cannot be placed? Because that’s what the generic is: something that is unstable as to its name, its disposition and even its phantom-like existence. Because, for example, two generic sets are obviously very hard to distinguish from each other, since they don’t have a stabilized definition. They are alike. And so we have to assume and undertake simultaneously the completely different and the completely identical. This is well known in fraternal political undertakings, amorous ventures, and similar initiatives. The whole intellectual history of the interpretation of love has gone back and forth over what is so tremendous about love: the fact that there were two [on était deux] and the fact that there was one [on était un]. We could say: love à la Gödel, love à la Cohen.


Fundamental Ethics of the Idea

3. Every thought contains a fundamental choice. Thinking is thus free in a profound sense – and not because it can say whatever it likes, which is the freedom of indifference and is of no interest, the freedom to say whatever stupidity. Suppose now that you have chosen to be on the side of the non-constructible, for one reason or another. What does that mean? What are the consequences? It is not inevitable, I insist, that you have a non-constructible set in hand, for a non-constructible set is not to be found just anywhere – unlike the constructible, which is easy to find. Your fundamental choice can take the form: there is some non-constructible. But what is going to materialize this? Because in fact the choice for the constructible is a clear choice: what does not conform to the dominant language doesn’t exist. Whereas for the generic, you affirm that it exists, but what does that mean? Because you are not going to just gossip about the subject, find some names, etc. It is simply the idea that there is some novelty that is not definable by the established order.

In reality, there you truly touch infinity. Ultimately, it must be affirmed that there is some infinity – infinity in the strict sense, that is, a non-constructible infinity. It must be affirmed and, afterwards, you will see – by working out the idea in the world – if it works, if it is verified, constructed, etc. Consequently, non-constructible subjectivity, generic subjectivity always assumes that you assume the existence of a non-constructible infinity, an assumption that you will have to work out in the real. It is not an inert assumption. If you assume it, that means you will watch over it carefully, you will defend it against any attempt to cover it, you will denounce the easily definable, you will track down the declarations of constructibility which are manifestly and specifically designed to maintain the order, and so forth. The whole thing is hard work, in every order of thought. But you will not do this work if you’ve not made the fundamental choice to do it – that is, if you have not in one way or another opted for the assumption that there really is some non-constructible infinity.

And that is what I call an Idea. Whatever sort of idea it may be, an Idea is always an infinite anticipation of the existence of a potentially generic universe. This type of infinity will help you testify to the fact that, in effect, something that transgresses the dominant order can exist. On this point, mathematicians have done terrific work. They have demonstrated that certain types of determined infinities exist which – if one accepts that they exist – attest by their very existence that the universe is not constructible. They have, in a way, demonstrated the power of the Idea. If you believe in this kind of infinity (since you cannot prove that a non-constructible infinity exists), it will testify that there is some non-constructible. It can be shown that the universe in its entirety cannot be reduced to the constructible by the sole fact that such an infinity exists somewhere. That has been proved. It is enough to affirm the existence of a certain type of determined infinity to make the universe completely tip over to the side of the possibility of the non-constructible.

I think this is an absolutely remarkable existential guideline. It means that if you have an Idea and you are in a state to really support it – which comes down to saying that you affirm its existence, that you manage to install this idea somewhere in a small fragment of the real – well, then you have a good chance that the world will tilt beyond the constructible. But if you have no Idea at all, it will not be easy to leave the constructible. That much is clear. The first form of this mathematical proof was discovered by the mathematician Scott in a theorem which says: if one “measurable set” exists (very poorly named, by the way), then the universe is not constructible, there is some non-constructible. It was upsetting, because we can see how the constructible and the non-constructible might appear to be a question about the entire universe; the question of whether everything is constructible or non-constructible seems to be about a characteristic of the entire universe. Whereas here it was enough for there to be one witness, if I may say so, one single very particular configuration, one set having certain properties, for the theory of the constructible to be rejected out of hand.

I think it is always that way when an invention, a novelty, a creation takes place in the real: it’s because someone, somewhere, has an Idea. They have an Idea in the strong sense, meaning: they affirm this idea existentially. They support it with their life, with their creations, with what they do, and they organize their existence around the fact that this thing, whose existence is affirmed by the Idea, can exist. In that case, they are in the situation of Scott’s theorem: if it exists enough for one to say that it exists, for others to mock the fact that it exists, that it has consequences, etc., well, it will mean that the universe was not as constructible as they said. And that therefore something of the order of covering, of domination, etc., was chipped, reduced.

It is utterly surprising, the symbolic connection between the formal theory of the constructible and everything we have just said about the obstacle of covering. Because if you hold firm to the Idea, to the kind of infinity it contains, this means that you have the means to oppose yourself to the covering, because the covering only survives on the assumption that, in the end, everything is covered by the constructible. To extract yourself from the covering, you must have an Idea, in the precise sense I give it: the possible recognition [reconnaissance] of a type of existence whose consequence would be that the universe is not constructible. This is something totally different than encountering by chance something that would not be constructible! It is a hard labor which reworks our perception of the world in such a way that you find pathways that install little by little the coherence of your Idea. Because we know it is coherent, Cohen has shown it. You will not be contradicted in this affirmation. You assume it exists, it exists. Perhaps you will find nothing. But the theorem says that normally you should find something. Because if the correlate to your Idea exists, then the universe is not constructible, and there are limits to covering.

4. From all this, we can draw a fundamental ethics, which will serve as our conclusion: you must always take responsibility for an Idea [assumer une Idée]; participate in the discovery; free yourself in this way from finitude; and open thought to the real infinite.

‘You must assume an Idea,” that is, you must oppose yourself to the fundamental thesis of our contemporary world: the imperative, “Live without an Idea!”, which has a whole doctrine of covering behind it. This is what is uniquely hidden behind the apparently anodyne, even progressive motif of the ‘death of ideologies’. The death of ideologies means: “Enjoy (if you can), live without an Idea. Enjoyment is a sufficient norm. Stand before the great planetary market, buy something if you can, it’ll be great, otherwise don’t embarrass our spirit with your ideas.” The imperative of the contemporary world to live without Idea has a whole doctrine of covering behind it, because every Idea from the past was covered by the method I explained here. So the first point is to hold on to an Idea: a political Idea, obviously, but it is bigger than that. It means: place your existence as such under the sign that you will not give up on your Idea, that is, a type of infinity.5placeholder And so you will act in such a way that, ultimately, the encounter with the generic will have taken place. The fact that the universe will no longer be, for you and those around you, constructible, will happen. That is what I call: always assume the Idea.

“Participate in the discovery.” Obviously, armed with the Idea, you can intervene and undo the coverings. Coverings are precarious from the moment someone has an Idea, that much is sure. There are many very concrete examples of this, even in ordinary life. If you have an Idea of what life can be, it will not let itself be easily covered by mortifying debris, by the miserable bits.

Between “free yourself from finitude” and the Idea, there is a path, clearly. It is the path of the first fundamental choice (“I am Cohen”).

“Open thought to the real infinite”: that synthesizes it all. The synthesis will retrospectively prove that infinity is always relative to a given constructivism. Infinite is always to have the strength to not let oneself be covered; it is always to hold on to the Idea; it is always to free oneself in this way from finitude (partially, never totally); and consequently, it is to make a creation out of one part of one’s existence. If it were absolutized, it would be a chimera. But out of part of one’s existence, surely a creation can be made. A creation that will not be covered –when you will guard in such a way that it will not be covered.

Timothy Lavenz is an independent writer, translator, and researcher who spends his time between Santa Fe and Baltimore. Many of his writings can be found at His current creative focus is poetry – to innerve the ghosts of concepts and set the ear of language aflame. He has published articles in Kunst und Kirche, Oraxiom, and Epoché Magazine, poems in Sequestrum. His co-translation of Pascal Quignard’s The Response to Lord Chandos won the French Voices Award and is forthcoming. He is also a dedicated contemplative with abiding interest in Christian theology, Vedanta, and all things mystery. He curates the YouTube channels Experimentum Vocis (poems and poetics), Marian Weigh (Catholicism), and Our Golden Change (the integral yoga of Sri Aurobindo). His latest book of poems is Chipper Rabbit.

Instead of focusing on finite methods of oppression, Badiou is going to focus on the quasi-automatic ‘defensiveness’ of the order of finitude itself. This ‘defensiveness’ and self-maintenance of the closed order will be treated under the heading of covering. Covering is needed whenever the finite methods of oppression are not enough to maintain the order. The covering imprisons in finitude: it covers over the emergent infinity, treating it as if it does not and cannot exist. That is the master idea of ‘oppression’ Badiou will focus on here.
Note that in this part the translation only seeks to reproduce Badiou’s unscripted teaching at the blackboard. It is not an exact transcript and certain repetitions have been omitted.
Valls’ remark was about Islamic terrorism. He implies that to understand the reasons for X is necessarily to excuse or justify X.
This will be the case for any ‘new’ constructible set, which is represented by the [etc.] in Figure 3.
“Placer son existence sous le signe de ce qu’on ne cédera pas sur une Idée.” Note the deliberate contrast Badiou is drawing between his ethics and Lacan’s, which the latter articulated in Seminar 7 thus: ne pas céder sur son désir, do not give up on your desire. See Badiou’s La Métaphysique de bonheur (in English, Happiness) to understand his view on the link between assuming an Idea and joy.

July 2021


Alain Badiou – The Theory of Covering and the Ethics of the Idea

Translation by Timothy Lavenz

Nineteenth Century Interpretations of Thought and the Claims of Twenty-First Century Science

by GW Middleton

Views of Reality from Stagecraft, Science, and ‘Nowhere’

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