Alain Badiou: Two Names for Infinity

What follows is a hyper-transcription of a lecture that Alain Badiou gave at the European Graduate School in 2010, called “Infinity and Set Theory: Repetition and Succession.” Until today, I have produced dozens of hyper-transcriptions, some of which are below.

Badiou begins his lecture with a diagram which looks something like the following:

Nothingness | 0, 1, 2, 3, … | Infinity

We can read the diagram from left to right in the following way: nothingness comes before the series of finite numbers which comes before infinity. At this stage of our understanding, that which is finite is always sandwiched between two negative forms of being (nothingness and infinity). On the one side there is nothingness. Nothingness can be thought of as that which is purely negative. On the other hand, infinity can be thought of as that which is without limit. In any case, there is currently, within this traditional understanding, no affirmative or positive definition of infinity. And, finally, we have the finite in the middle. The entire lecture is dedicated to exploring this understanding of numbers. If we like, we can also draw the following diagram:

Nothingness | The Finite (0, 1, 2, 3 …) | Infinity

The finite is a form of positive existence which is composed through the name of nothingness itself. To begin with, we know that 1 is not the same number as 2, and 2 is not the same number as 3. There are real differences between each unique number: there is something within the number 2 which is not within the number 1, there is something within the number 3 which is not within the number 2, and so on. This all concerns the realm of the finite, which is also the realm of differences and movement. We can describe the realm of the finite as the realm of differences and movement because what we notice is that the passage from 1 to 2 and from 2 to 3, and so on, implies that there is a continuation. This continuation manifests itself as a repetition which is without limit.

The infinite is not something which can be affirmed, rather it is the absence of limits upon something else. So this is how we can come to understand the negativity of infinity. Infinity itself has no being. It is not being, but limitation. It is not being itself but that which is the absence of a limit for a process. The infinite is the very possibility of the continuation of a process, the infinite ensures that the process continues without interruption. This is why we can refer to it as a negative determination. So, to revise our main argument: the finite is always between the negation of being (nothingness) and the negation of the limit (the infinite).

This leaves unanswered the question, how is it that finite numbers continue along their process? What is the process of this continuation? How is it that finite numbers can continue along their process without stopping? How is there an absence of limits? In other words, how is it that there is an absence of limits on the continuation of the process of finite numbers? To make sense of this, we must begin with a formalization of the question: how do we pass from the number, N, to the number, N+1?

Let be the number of terms. For example, = {0, 1, 2, 3}. Here, N = 4 because we have four numbers (0, 1, 2, and 3). A successor of is N1:

N = { 0, 1, 2, 3 }

N1 = { 0, 1, 2, 3, N }

When we want the successor of as N1 then we must take the entire set of elements from and add the previous name of that set itself. Thus, to pass from the number to the number after, N1, we must take the contents of the first as well as the name of the predecessor (where is the name of the first set of numbers). So, in the final instance, we always add the name to that which came before. For this reason, our writing is composed uniquely of names. We introduce a new unique name (N1) as a substitute for the old name (N). The operation for passing from one unique name of a number to another unique name of number is very simple: you place one element after all of the others within the new set which is uniquely the name of the set which came before.

And what is this name, in the first instance? It is a name of nothingness. In the end, the only material that we have at our disposal is a list of names. We have as our first name the proper name of nothingness and after that we have a composition of new names. For example we can decide that the following is the name of nothingness:

Ø

And we can give this a new name:

1 = { Ø }

 Here, the new name is 1. After that we can construct another new name as follows:

2 = {Ø, { Ø }}

Here, the new name is 2. Finally, we arrive at the point: only names exist, as well as nothingness. Only names and nothingness exist. The previous example looks more like this then:

Name “Two” equals Nothingness and One as the Name of Nothingness

Arithmetics is perhaps the complete understanding of the world: the world is composed of only names and nothingness. We must not forget that we give the name one only because the one is composed of nothingness {Ø}, and that we give the name two as the composition of the name of nothingness plus nothingness, and so on. So, when we pass from one number to another number we by necessity take the first number (which is a collection of names from 0 to N+1) and we put it inside of the new numbers as one of its elements. The successor is thus always, by definition, one element more. 

We can summarize this in two ways: first, we have a “primitive name” such that a name is absolutely primitive if it is the name of nothingness itself (but there is nothing before the name), and; second we have an operation (which is to succeed).

  • Primitive Name
  • Operation, S.

The operation which concerns us is an operation which moves the continuation or the process along. It is identified as an ‘S‘ because it ‘succeeds’ the line of numbers, it ensures that something comes after. So we have:

{Ø} and S

With both of these we have all of the processes required for the construction of the finite. The successor of a number is composed exactly of the contents of that number plus the previous number: S(n) = … ].

All of this is possible because the name of ‘something’ is not the same ‘thing’ as the ‘something’ itself. This is a point of contingency. For example, the name of the void is not identical to the void. If the name of the void were identical to the void then the thing would be nothing rather than something which exists. The name of a number is not identical to the number itself. The name, N, for N = { 0, 1, 2, 3 }, which is a set of names, is different from the set itself. We know this because the set does not include itself. For example, where 4 = { 0, 1, 2, 3 }, there is no number 4 within the set at all. So we must always have a name for the number which is not within the set itself. Interestingly, at this point Badiou draws a diagonal line across the right side of N, as if to “bar” the name from the set of elements inside of it.

Finally, the name introduces something new. This is an important point because it answers the question: why is the set of finite numbers not a pure repetition? The name of a set is a creation which results from an operation and introduces something new into the chain of numbers. The number two is not reducible to the number one even though the number one is within the set of the number two. And the number one is not within the set of the number one.

When we place the name of something inside of its own set of elements we are performing the operation of succession. By doing so we produce something new which is the successor of the set. It is what comes after the set: what succeeds the set and what names the new set.

Counting is a basic operation of thinking and yet we are generally not reflexive about how the operation functions. Only a philosopher is truly reflexive about all of this. We always function by putting the name of something inside of a set in order to construct something new. For example, when we write a novel we often decide on the name of a character by included all of those characteristics of this character (his hair color, eye color, personality, etc) under the novelty of a name. It is thus the collection of all of the characteristics of the person as well as his name.

So, we have the following as the name of the void:

Ø

And we have the process of succession as follows:

 Ø, S(Ø, S(Ø,S(Ø)), ….

This can be read in the following way: the name of the void, the successor of the name of the void, and the successor of the successor of the name of the void, and so on. For those who are interested, you can find a full explanation of all of this on page 160 of the newest English edition of Being & Event.

All of this is really a presentation of numbers via the operation of succession:

Ø, S(Ø), S,(S(Ø)) = 0, 1, 2

What is two? Two is the operation to ‘succeed’ twice. There are two successors for the number two. A number is therefore always the result of a repetition. But it is not the result of a pure repetition of the number itself. Rather, it is the repetition of the operation of succession. It is the successor which repeats. Three is always three times the same operation, it is three successions: the name of the void, the successor of the name of the void, the successor of the successor of the name of the void, and the successor of the successor of the successor of the name of the void. If we do this operation five times then you will have the number five as a result.

creative repetition is always a succession of numbers within the finite. It is a paradox of sorts. The new number is really different from the number which came before it – we have proof that 4 is not 3, and so on – but it is also another composition of the void which gives it a unique name. The number or the name of the number itself changes even where the operation remains the same. You go from one to two to three to four, and so on, using the exact same operation. This is why it can be annoying to count because even though the numbers change you inevitably get bored of the repetition of the operation itself.

We have a new definition: the finite is a mixture between novelty and repetition. This is why we can describe succession as creative repetition. But, we should also note that the finite is the insistence of succession. It is the succession, without limit, of repetition. The numbers always continue and this is the insistence of the repetition. As such, the finite is under the law of repetition: ‘one more, once more again, again once more again, etc’.

In French we describe this using the word Encore, which means, to succeed. This is also the title of one of Lacan’s seminars, seminar XX. In fact, within that seminar there is an entire meditation concerning the very concept of repetition. What we notice is that within concrete life, repetition exists. And this is why we are finite beings: we are always in the field of repetition. Very often, we must do the same things. There is an insistence, which is not always part of our conscious agency, to repeat. And so the subjective repetition is very often also a creative repetition. It is the same with numbers. There is something profoundly similar with respect to subjectivity and number.

Perhaps we can approach a definition of the subject in this way: the subject is always known by his repetitions. We know somebody by noting their repetitions. When we claim to know somebody we often make that claim in full awareness that there will be some creative repetitions. A person seems to always have the same behaviors or opinions, even if there is a creative element to those behaviors or opinions – even if they change a little bit here or there. The operation nonetheless insists. Finally, this finite structure can produce many differences, novelties, surprises, complexities, within the entire world. This is precisely why we can claim that the finite is under the law of encore … continue, continue…

We approach our first definition of the infinite: the infinite is the space of repetition itself. In this respect, it is logical to conclude that the space of repetition must be infinite if we can continue within it without limit. If we can endlessly repeat the operation of succession then we can say that the space of the infinite is without limits. In other words, if we know that repetition is without limits then there must be a space for that repetition which is itself without limits. And so we can not know the creative nature of repetition without claiming that there is something which is without limits. If repetition were within limits then we would think of it like a circle, or a loop. In fact, it seems to me that this was Lacan’s conception of the repetition of the drive:

Badiou claims that our understanding of the finite realm necessitates a thinking which does not close in upon itself like a boomerang. We do not always return to the same. Conversely, with number, we always have a linear continuation. We have an obligation to assume that there is something without limits, without closure, within the realm of repetition itself. This is qualified in a very precise sense by the operation of succession. The space of repetition, then, is infinite, and it is linear. It is not, like the traditional understanding of the Lacanian drive, closed in upon itself like a boomerang and circular. If there is no limit point to numbers then there is something infinite which keeps moving as if in a chain and it is this chain which is the result of a creative repetition. We never arrive at the limit point precisely because this negative notion of infinity is that there is an absence of limits. And it is this absence of limits which permits the repetition of succession to continue, infinitely.

The first name of infinity is thus virtual infinity. Virtual infinity names the position that the infinite does not exist except as an absence of limits to the finite. So the infinite exists alongside the possibility for the continuation of the finite. What exists is always finite and so the infinite, if it were to exist, would fall into the finite realm of numbers. What exists is the unlimited process of the finite. The point is that virtual infinity reduces the infinite to the absence of limits for the finite. So, finally, the infinite is always at the service of the finite and is not an affirmation of something which exists itself.

Most mathematicians do not admit that there is something like an actual or real infinite. They typically only admit the virtual infinite as that which has no limit and which is never finished within the finite realm. Badiou describes the position of virtual infinity as a compromise situation: it is a compromise between the finite and the infinite. This compromise operates in the service of the finite precisely because the infinite does not exist as a separate being. Rather, the infinite exists inside the finite as its negative necessity via the law of succession.

The infinite is an internal law of the finite.

According to this view the infinite is itself a part of nothingness. Whereas the finite is sandwiched between two forms of nothingness we can claim that the first form is pure nothingness (nothing at all) and the other form is infinity as that which does not exist. Thus, the infinite is really “No Thing” because it is not a Thing. So we have Nothing and No-Thing. The infinite is a virtual law of the finite but it is No-Thing.

If we decide that we want to go beyond this understanding of infinity then we must allow ourselves to realize that the infinite comes also in the form of a thing. There is no other possibility to overcome the problem. We must affirm the existence of something infinite and not only the infinite as the pure absence of limits in the service of the insistence of repetition within the finite realm. And so the infinite must therefore exist as a point which is beyond repetition itself – it can not be inside of the repetition. It is the repetition which is without limit. If we want to go beyond virtual infinity then we must propose something which is not a form of a law which exists inside of the repetition. The new infinity must itself be infinite. But how can we propose something like that?

We had the following:

Ø, S(Ø), S,S(Ø)), etc

0, 1, 2, N+1 …

We must begin by affirming the existence of a term which does not succeed and which is therefore not inscribed within the repetition. This term must exist after one term, like all whole numbers (N+1), but it must also exist after all numbers. But we immediately run into the problem of thinking a set of all numbers.

If we were to think about all numbers then it can not be as a consequence of the operation of succession. We can not simply at N+1 to the chain of finite numbers. There is no point where the succession produces something which is beyond succession. And so this is the paradox. We can not produce a term for infinity by using the internal law of succession because this would immediately place us back within the finite realm. So, we can not produce something infinite using succession as such.

Repetition as such can not produce any term which is infinite. That is impossible. Consequently, we must affirm the existence of something completely new: a term which does not succeed. A term which is outside the scope of the repetition. Badiou proposes the following matheme for the affirmation of a term for infinity:

¬ E(x) [(Infinite = S(x)]

There does not exist one x where the infinite is the successor of x. In other words, the infinite does not succeed at all. We shall name the infinite, “omega” using the following symbol: ω.

¬ E(x) [ω = S(x)]

Not Exist X, omega is the successor of x. This is our first possible concept of a positive infinite. However, by the use of the symbol of negation (¬) placed as it is on the existential quantifier we note immediately that this is another negative definition of infinity. However, this time our negation is not the negation of a limit. What we are dealing with this time is the negation of succession itself – a negation of the operation which sustains the creative repetition.

There is no x for which omega is the successor of x.

So it is not the without limit of the space of the succession, of the space of the repetition. But it is the negation of the repetition itself. We can claim that if something infinite exists then the first infinite – or the beginning of the infinite – is always in the form of an interruption of repetition. It is a rejection of continuation and therefore finds itself radically outside of the continuation of the operation of succession. This is omega, then: the first infinite outside of repetition. What we have with our matheme is a reversal of the first negative definition of infinity – it is a reversal of virtual infinity.

  1. Definition One: NO LIMITS AT ALL
  2. Definition To: NO SUCCESSION AT ALL

And we have the following:

  1. Virtual infinity is the name of the strength of repetition: repetition can continue without limits. It is the strength of the finite, then. The finite is stronger precisely because it can continue as virtual infinity.
  2. Real infinity is not inside of repetition and is not the result of succession. It is the weakness of the finite. It is beyond the possibility of the finite.

The point is now to examine our decision for real infinite: can we rationally accept the existence of something which is beyond the repetition? Something which is beyond the successive construction of the finite numbers? How can we do that? How rationally can we do something like that? Cantor’s greatness was to answer this question. Cantor is the father of the modern conception of the infinite. Before Cantor we only had access to the virtual form of infinity, it was the dominant conception. Cantor affirmed the existence of something entirely new: the omega which exists beyond all finite numbers, ω. How can we be beyond all finite numbers? Cantor’s idea was to claim that we can take all finite numbers and presume that they are a set (as if the repetition was finished). The idea is that we can put the set of that which is without limits as a new limitation. 

So we are between two names:

Ø & ω

We can write that omega is the name for the space of the repetition as such. The space where all numbers are defined by successive repetition.

ω = { 0, 1, 2, 3, … }

There is in fact a limit, and this limit is omega itself as the complete recollection as the total process of the collection of numbers. We have here a very obvious problem which is the notion that somehow we can close the set to produce a set of all sets. But we must understand that when we speak of the closure of the set in this instance we are actually speaking about the closure of a set which remains unlimited in terms of the operation of succession. The closure we are discussion is therefore something which is perhaps better described as a pure interruption of the operation of succession. We are simply affirming something which does not succeed. And so there is no clear contradiction then between the non-limit of succession and the name of the set which is outside the scope of the succession.

A succession operates on the name which came before it but omega deals with a ‘before’ which is not in the same sense as the ‘before’ of succession. For succession, two is before three. We know exactly what is before 3, it is 2. But with omega what comes before is the operation itself. 

For Lacan, feminine jouissance is something which is infinite. Why is it that the infinite is so often associated with the feminine? Often, woman is represented as the point where man does not understand his own limitations. Woman, according to Lacan’s graph of sexuation, is represented as being without limits of the male process. Classically, the man is on the side of the process of numbers – there is a succession. The male is the being of succession. And, classically, woman is represented as without limits according to the total space of repetition. She is an interruption of repetition itself and exists outside of the succession of numbers. This is why we sometimes claim that man is a finite number and woman is an infinite number, or that man is quantitative and woman is qualitative.

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