This note responds to

Mathematical Interlude (1/n)

The infinite by immanent dimensioning of parts (excerpt from Immanence of Truths)

July 16, 2020

[Note: The following is a translation of the first 8 sections of chapter 14 of Immanence of Truths, where Alain Badiou introduces the concept of "non-principal ultrafilter k-complete", arguably the central idea in the whole book. We have selected these sections because they offer an important point of contact between the theory of topological coverings, at stake in Logics of Worlds, and Badiou's set theoretic framework for discussing generic procedures. Furthermore, it helps us to reinforce the crucial place of a "scalar metaphor" or point of view in Badiou's system, as the ultrafilter is an instrument to immanently assess the size of very large cardinals — while also connecting set theory to complexity theory in an interesting and fruitful way]

Chapter 14
The infinite by immanent dimensioning of parts

(L’Immanence des Vérites, p. 317-351)

1. Where are we now?

In chapters C12 and C13, and of course in suites S12 and S13, we have made good progress on the first two paths to infinity that we defined in chapter C11, that is, the negative way of absolute transcendence and the way of resistance to divisions.

The first path is that of Plato's speculations about the One — where one cannot affirm neither that it is nor that it is not — and of negative theology, which seeks to think of the infinite being of God as that which excludes any finite determination. This conception is found in theories which postulate the existence of an active ideality capable to overcome the limits of the established world. For example, infinity of communism with regard to the world governed by private property, infinity non-figurative painting with regard to the aesthetics regulated by imitation, infinity of the Galilean physical universe with regard to the closure of the ancient cosmos, infinity of the amorous promise with regard to existence regulated by work and family. All of this remains inaccessible ("Utopian", "without form", "sacrilegious" or "romantic"), if we try to think about it using only the parameters of the existing world and if we assume its laws to be intangible — or if we exclude the illegal power of events.

The second route is opened by the difficult questions concerning the resistance of the One-infinite to divisions, and here the simplest theological paradigm is the case of the Christian God, of the partition of God between the three persons of the Father, the Son and the Spirit. But it is also found in the classic observation of the maneuvers of division by which all power and all oppressions try to break the efforts of a popular revolt's coalition, or to secure their hold on new territories. The maxim of “divide and conquer” means: the reign of finitude requires that all signs of a new infinity be destroyed by the divisions inflicted upon its support. Conversely, that a type of active infinity manages to invent its own persistence beyond divisions attests to the real of this infinity.

In order to clarify the other two ways – the immanent power of “large parts” and the increasing proximity to the absolute referent —  we have to familiarize ourselves with operational concepts that are very exciting both in their strict formalism as well as because they demonstrate the subtleties of the ontology of the multiple without-one. The crucial concept, which we have already discussed in S10, received, as I said, the name of ultrafilter. We will try throughout of this book to give this term the status of a philosophical category, rather than substituting it for a more classical philosophical name.

2. Informal approach to the question of the "large" parts of a multiplicity

Given any existing multiplicity, we can always maintain - it's either a tautology or a petition of principles - that it is "large" insofar as it includes "large" parts. It is like those very large and extremely populated cities that give us the feeling of being in a powerful country. Or the way the breadth of actual knowledge and the complex singularity of lived experience of an individual, already seems to destine her to the infinity of the Subject that she is capable of becoming. If the proper characters of a given multiplicity impose, from its very interiority, that this multiplicity is infinite, we will say that the infinity in question is immanently apprehensible through its very composition — through, in sum, its parts, the very large parts which only an infinite power could bind in the form of a unique set.

Note that this immanence, this interior force which constraints a multiplicity to be infinite because of its internal composition, is not at stake in the first two access routes to infinity. Indeed, both of those paths remain extrinsic. The path of negative transcendence starts from operations available "outside infinity" and shows that they fail to think a multiplicity which will then be said to be infinite by negation: in-finite. The second route, the path of resistance to divisions, requires an outside agent who somehow tries to reduce infinity by inflicting a cutting, a segmentation that will ruin its own power. Again, it is a negative resource, namely, the ability to resist an external operation, which characterizes this type of infinity.

With immanent infinity - and also, as we will see, with the infinite in proximity to the absolute referent - we enter into properly affirmative characterizations of types of infinity. The instrumental concept here is not that of the negation of resources or of external constraints but of that which gives us the means to positively think what an immanent power may be, the positive immanence of an infinite constraint. The simplest idea seems the following: we will say that a multiplicity is infinite if we manage to clarify what it means for it to contain a very large number of very large parts. "Ultrafilter" is really only the name of a mental apparatus of "measurement" of avery large set of large parts of a given set.

The difficulty of this undertaking is well known to philosophers since Antiquity: the notions of "big" and "small" are all quite relative and don't seem to be able to give rise to a stable concept. Perception is misleading here, because what is big from a perspective might be small when compared to something else — which means that perception is incapable of clearly distinguishing the large and the small. Socrates sums it up admirably in The Republic (I give here my version of the text):

“We said that the sight of the large and the small does not really separate them, but conjoins them. To shed some light on this, pure thought is forced to conceive of the big and the small as disjoint and not as inseparable, and therefore to contradict what we actually see. Here we have a obvious contradiction between seeing and conceiving. It is this contradiction which pushes us to seek what the the large and the small are, in their being”

The whole problem is then to determine how we might conceive what is large “in itself”. Or more precisely, in the case that occupies us here, what a large part of a given multiplicity is, and, more precisely, what a set of large parts, itself large, can be. It is indeed the inner thrust of the countless large multiplicities contained in this set which will determine, from its interior, immanently, its infinite extension.

Experience shows that it is indeed the localized system of zones of power, their eventual entanglement, which characterize an effective infinity — and not the intrinsic definition of an isolated quantity. Creativity — be it collective or individual — always realizes itself as a complex of works, a network of localized differentiations, a power of germinal of variations, in short, multiplicities in the making. The "greatness" of a realization of truth, political, existential, artistic, ultimately lies in the relationship between its constituent parts rather than in an isolated whole. I have already proposed to say that the real Subject, the one who unfolds the consequences of an event, must be called a "configuration" rather than a whole, and that this "configuration” is here the name of an activity. Let's say a action of truth configures in the situation a complex of "large" parts which expose, immanently to this situation, the power of a new infinity.

This is why, at the ontological level of the question, the rational approach is not and cannot be to say what exactly what "a" large part of a whole really is. We would inevitably be committed to the inseparable dialectic of the big and the small, which Plato demands we move away from. We will rather propose the concept of a network of parts, where the links they have with each other, as well as the connections they exclude or compel, attest that in any case most of them are so "large" that we might say they are "almost" equivalent to the initial situation, to the set of which they are a part. Just like the complex of revolutionary organizations, when they are in the historical moment of a seizing of power, penetrates into every corner of the evolving situation, accumulating, entangling, under the sign of the Idea, the organizations of women, young people, factories, neighborhoods, artists, whose ultimately greatness comes from their organic connection and the active sharing of the ideas that move them. Or as in an inventive process, the individual unfolds in him resources of an intellectual, emotional, mnemonic and physical type whose entanglements alone cause to arise in him powers unknown to himself. We only need to consider, in the process of invention, the accumulation of disparate powers into a single gesture of the painter to understand that the measurement of a new infinity exists in the local convergence of heterogeneous multiplicities.

The concept of ultrafilter and its variants organize the positive response to the following question: given a multiplicity, does it contain a large set of distinct parts that can be roughly said to be all truly "large", that is to say, of comparable extension to the Whole of which they are the parts, that is the initial set? "Large” is taken here in the sense of: having a power comparable to that of the totality being considered, a power which is, in sum, comparable to that of the overall situation.

3. The concept of filter and the notion of "almost everything"

The concept of "almost" will find surprising applications throughout the rest of this book. Let’s dwell a bit on its seemingly evasive nature. What might it mean, precisely, for a part of a whole to be “almost” like the whole itself. The problem is that we cannot use the precise measurement of infinite sets, namely their cardinality, to define the large parts, the average ones and the small ones. Why ? Because a part of a given infinite whole obviously cannot have a cardinality greater than the whole. At most, it can have a cardinality equal to that of the whole. But is that enough for us to speak of a "large part"? For example, the multiples of the number 123,456,789,321,654,987 constitute a part of ω which is of cardinality ω. Intuitively, however, one has the impression that this part is far from being "almost" the same size than ω, since the whole numbers which are not multiples of 123,456,789,321,654,987 appear to be part of ω that is much larger than that made up of these multiples ... This feeling is produced by the intuition that what remains of the whole, when we remove a really big part of it, must be quite small. The greatness of a part of a whole is intuitively measured by the smallness of that which is left of the whole when we remove this large part of it.

There is an ontological reason for this: if no part can be of a higher cardinality than that the whole, on the other hand a small part may be of lower cardinality. So it's very easy to give some really compelling examples of what is a small part of a set of cardinality κ. We know that a part whose cardinality is less than κ is certainly small. For example, a finite collection of whole numbers is a very small part of ω. It is even easy to "classify" small parts as follows: a group of 2000 whole numbers is a part of ω of cardinality "two thousand", it is finite, and therefore very small compared to ω, but it remains clearly larger than a part composed of 2 numbers…

Finally, the notion of "part almost as large as the whole" is much easier to specify by going through the complementary parts: it is a part of which the "rest" is really small. So the part of ω defined by what remains when we remove part of 2000 numbers is certainly great, because the cardinal "two thousand" is much smaller than the cardinal at all, ω, which is the first infinite cardinal.

Having said that, what we generally want is a procedure that gives us an infinite family of “large parts”. We will not get to them one by one! We will often be satisfied with obtaining (and this is the goal of the operator we are looking for) a family where "almost" all the elements are very large. It's like fishing: a net whose “medium-size” meshes will retain all the great beautiful fishes, but perhaps also some average-sized fishes, which we will then return to the water.

How to design such a net with regard to pure multiples, to sets? We are going to talk, not about direct measurement of the “size" of parts - no more than measuring fish one by one before they jump into the net - but about operations that engage indirectly the greatness (or smallness) that we are looking for in the said parts of the set. The first operation, very intuitive, is as follows: it will be said that a very large part A is such that it takes up a lot of space in a set E, so that it doesn’t leave much room for another large part B. This requires that many elements of B, the second supposedly large part, "encroach" on the first, which is A, and that therefore the number of elements common to two very large parts be itself very large. We call the "intersection" of A and B the elements common to A and B. The idea which here, according to Plato, "separates" the big from the small is, as you would expect, not a definition of the big (that would be the method of Aristotle), but an axiom - therefore an active relationship - that says: the intersection of two large parts is a large part. Down below, there is the mediation of the "small": if what remains by removing part A is small, and if what is left by removing part B is also small, then we can be sure that what remains by removing all that is common to A and B will also be small. The underlying idea, mathematically simple, is that what remains by removing the common part to A and to B is simply the addition of the two remains obtained in removing A (all by itself), and removing B (all by itself). These remains are small, if A and B are large. And the addition of two little things cannot be that large…

Another such relation is equally evident. If you know that A is a large part, and that another part B contains all of A, it is certain that B, being larger than the already large part A, is also a big part. We will simply say this, which is like a second axiom of the large: what includes the great is great.

Two other observations are ontologically trivial. On the one hand, the totality, that is the initial set E, being, so to speak, the largest part of itself, is certainly a large part, the whole-part, in fact. And on the contrary, the nothing, the void, the part that has no element, is the very norm of the small, if one considers that all being is multiplicity.

The axiomatic determination of the great is then suspended between the nothing and the whole, so that the intersection of two large parts is also large, and that a large part which is greater than another large part is itself large. Such is the extremely simple content of the concept of filter formally introduced by Henri Cartan in 1931, and the use of which in set theory, and therefore ultimately in philosophy, is and will prove to be considerable.

The word "filter" here conforms to its concrete origin: a filter "retains" what is large and lets slide what is small.

Let's recap. Given a set E, we call filter on E a set F of parts of E which has the following properties:

1. The set E is in F.
2. The empty set is not in F.
3. If two parts are in F, their intersection is also there.
4. If a part contains a part which is in F, it is there as well.

4. The concept of ultrafilter

Perhaps the most basic concept in all of the theory of infinite multiplicities is that of ultrafilter, which we will now define. An ultrafilter adds to the four axioms of the filter a property of exhaustion with regard to the selective action of the filter: given any part of E, either it is in the ultrafilter, or its complementary part is there.  What is here a “complementary part”? Quite simply, if A is a part of E, the part complementary to A is the set of elements of E which are not in A. We note the complement of part A of a set E: CE (A).

Note in passing that the notion of the complementary introduces here a use of negation. Indeed, the complement of A is that which, from E is not in A. In this sense, the notion of ultrafilter dialectizes that of filter, because a filter makes no use of the notion of complement. It may very well be the case that part A is not in a filter and that its complementary part CE (A) is not there either. What an ultrafilter introduces, via its “completeness” condition, is that if A is not in the filter, then CE (A), which in a sense is the negation (in E) of A — everything that of E is not of A — must be there. In short, given any part of E, it is "ultrafiltered", in the following sense: either it is in the ultrafilter, or its negation is there. We could say analogically that it is a filtering appropriate to a situation organized by an antagonistic contradiction: a subset of the situation is summoned to be in the ultrafilter, if it is not, then its "negation" (its complement) will be there. The ultrafilter is also a set theoretic expression of the principle of the excluded middle. Given a part X of a set E and an ultrafilter on E, either the part X belongs to the ultrafilter, or the complementary CE (X) of this part belongs to it. These two possibilities cannot be combined. Indeed, X and CE (X) cannot be in the ultrafilter at the same time, otherwise their intersection should be there. Since there is nothing common to a part and its complement, their intersection is empty. But the vacuum is ontologically small, and cannot be included in a filter. However, there is no third possibility. Which simply means that, unlike to what happens at the level of appearing - which I systematically studied in Logics of the Worlds - where there is blur, where an element can be "more or less" in a world of appearances, veiled, or disappearing, in the pure ontology of multiples, there are only two possibilities and it is mandatory that one of them be realized: to be or not to be in a given whole.

The ultrafilter systematizes this point by stating that either a part is in the ultrafilter, or the negation of this part in E – its complement - is there. An ultrafilter excludes (filters) any third possibility, it prohibits a part that is only partially in it, or that is itself not there nor its complement.

Let's recap once more. An ultrafilter is a set ULT of parts of a set E such that:

1. The set E is in ULT.
2. The empty set is not there.
3. If two parts are in ULT, their intersection is there too.
4. If a part contains a part which is in ULT, it is there as well.
5. Given part of E, either it is in ULT, or its complementary part in E is there.

5. Maximum ultrafilters

The fact that an ultrafilter on E is a privileged structure among the filters can also said in the following way — itself quite homogeneous to the virtues that we recognize in this concept: an ultrafilter is a maximum filter. This means that an ultrafilter on a set E is not included, or is not a strict part, of any other filter on E. We will note here the correlation between exhaustion (any part of E is "treated" by ultrafiltration, which retains either this part or the complementary part) and maximality (no filter envelops an ultrafilter). The demonstration of this point, very simple, is given in a next section. This maximality already points to infinity in a sense that is not purely quantitative. We will show it using a crucial distinction between "principal" ultrafilters, namely, those built around a fixed part of the base set E – a fixed part that is like the “principle” of the ultrafilter — and “non-principal” ultrafilters, which are not built around a fixed element and, in that sense, are “wandering” ultrafilters.  With these “non-principal” ultrafilters we can say we got a maximality without the One. That is, an incomparable infinite intensity which, contrary to the historical thesis of the infinite uniqueness of God, is not separated from the multiple.

6. Principal filters and ultrafilters

The question of the One is introduced into the theory of filters as the condition that there be a single “support” for all the elements of the filter. There are indeed particular filters: those that are composed of all the parts of E that contain a given non-empty part of E. These are filters, for the reason that the parts that contain a fixed part A of the whole of E form a collection which obeys to the four axioms of the filters. Let's call FA all the parts of E which include part A. We have:

1. Let B and C be two parts of E which both contain the fixed part A. What is common to B and C, their intersection, also contains A, since B and C both contain it. So this intersection is also part of FA.

2. If B contains A, and if B is an element of FA, and C contains B, then C necessarily contains A, and therefore C is an element of FA.

3. Given a fixed part of E, it is clear that E (the whole) is an element of FA.

4. Since the vacuum cannot contain anything, it does not contain A, and is therefore not an element of FA.

A filter thus obtained is said to be “principal”. The principality has relation to the One in the sense that the filter is defined by a determined part of the initial set. In other words, the "greatness" of the parts which are in the filter is originally fixed by the size of this determinate part. Here we find the difficulty raised by Plato: the big is not really separate from small, and its definition is both relative and extrinsic. Indeed, if the fixed part A can be determined as small, then many elements of the filter, starting with itself, will also be small. Of course, if A is large, then all the elements of the filter will be large. But it's the singularity of part A which guarantees this property, and not the axioms which define the filter. In other words, it is the One of the common part to all the elements of the filter which gives the measure of the magnitude of these elements, not the filtering as such. A principal ultrafilter may even contain a very small part, for example a part which has only one element, a part of type {a}, a singleton. To accomplish this, consider the filter F{a}, composed of all the parts of which “a” is an element.

Note in passing that a singleton is, as its name suggests, a true allegory of the One. It isolates as part of the whole a single element of this totality. I showed in Being and Event  that the concept of singleton is fundamental to understanding the relationship of the State to individuals under its power. It's not indeed the individual as a potentially infinite multiplicity, capable of truth, participating in an event, irreducible and creative, that the State considers, but precisely the singleton of this individual. This is not the actual individual Ahmed who is incorporated by the State as "citizen", but indeed {Ahmed}. This is what clarifies the general identity of all individuals with regard to the State, their essential anonymity. They are all singletons, therefore units taken not for what they are, but for minimal parts of the State.  

A principal filter, which accepts the singleton, cannot really inspire much confidence. And in fact, it can perfectly be a very bad “fishing net” if one does not wish to retain tiny fishes. And this because one set - without any other guarantee - gives the measurement of the filter. We have here a kind of ontological theorem, typically a-thesist, which we will meet in various refined forms as we go along: A given type of infinity is all the more intense and immanent the more it is removed from the power of the One.

How then to define a principal ultrafilter? An ultrafilter, remember, has the property of completeness of filtering: if it does not contain a part, it contains the complement in E of this part. Suppose the ultrafilter is principal. All its elements contain a given part A of the initial set E. Suppose that this part A has two elements, a1 and a2. It follows that all elements of the ultrafilter must contain these two elements of E. Let us then consider the part of E which has a single element, namely element a2. It is {a2}, the singleton of a2. This singleton {a2} does not belong to the principal ultrafilter, since in order to be there it needs to contain the part with two elements {a1, a2}, which is obviously not the case of the singleton in question. But the complement of the singleton also cannot belong to the ultrafilter, because this complement cannot contain anything that is in the singleton {a2}, which means that it does not contain a2, and most certainly not {a1, a2} either, and it is therefore not in the principal ultrafilter. What can we  conclude from this? Well, that our supposed principal ultrafilter, defined by a fixed set of two elements, is not an ultrafilter, it is a vulgar filter, since it is lacks the power of completeness: neither {a2} nor C ({a2}) are elements of this alleged ultrafilter.

If we carefully consider the above reasoning (that will be more fully formalized below), we can also conclude something else. For an ultrafilter to be principal, the fixed part that defines it cannot have two different elements. But a part which does not have two different elements and is not empty — since the empty set is not an element of any filter — is a singleton. A principal ultrafilter is an ultrafilter whose defining fixed part is a singleton. It is therefore doubly attached to the One: only one part defines it, and this part has only one element. And this is why we must be doubly wary of it.

7. Why the principal ultrafilters are of little use if we want to enlighten and strengthen the function of infinity in thought

From the point of view of thought, it is extremely interesting to understand why the concept of principal ultrafilter is only a didactic transition between the principal filters, which organize all forms of domination, and non-principal ultrafilters, which open to the types of infinity that underlie truth procedures, and in particular emancipation procedures, i.e. procedures of political truth.

A principal filter is nothing but a particular exercise of uniqueness: there is a part X of E which is common to all the elements of the filter. It's like the filter has some kind of stable root, which makes its efflorescence, however extensive, remain attached to the One

that grounds it. That's how modern reactionaries interpret what they say is the magnitude, and even the universality, of the shimmering diversity of opinions, of the unlimited bloom of satisfied desires — in short, of imperial democracy. All of this requires, they believe, a big strong root sunk into the historic land – namely, economic liberalism and imperialism with its necessary condition, namely, the private appropriation of all that exists – which is supposed to be absolutely stable — and its eternal perpetuation. In that case, the word "principal" acquires a clear meaning: we will filter the thoughts and actions of men, and it ultimately does not matter whether they accept or not that their personal happiness depends on the "ingrained" strength of private property, since it is unimaginable and impossible to pluck this root out. And therefore it is absolutely necessary to rally in defense – even if that requires invasions, state conspiracies and drones - of this root and of all that, subjectively, depends on it: the blind adherence to the “values” of what calls itself today by  names like "West" or "International Community”. In short, this is the true "principality" of the contemporary filter: it is the instrument for filtering beliefs, so that all consolidate the modern form - globalized liberal capitalism - of domination by the finitude of the One.

We could as well say that when it comes to the arts, academicism — in whatever epoch — spells out the same type of submission to the One. We will admit everything, we will be enchanted by everything, we will proclaim everything, as long as this “everything” preserves the roots held to be essential to the system of art, which makes this "all" admissible. At a given historical moment, the X which makes the One for the filter of tastes may be figuration in painting, tonality in music, realism in romanesque prose or versification in poetry. But it can obviously change, and even take an opposite perspective, for example, that the only rule is the absence of rules, that the only important thing is that the artist “expresses himself”, that the fragment, the subjective confession, the exposed sexuality and anarchic cutting of sentences prevail over totalizing construction, over the collective fresco, over courteous love or rhymed Alexandrians. In all cases, oppression by finiteness is that a latent figure of the One, a system of received ideas, a common opinion, that organizes the filter of tastes.

Thinking in terms of the principal ultrafilter takes us even further in this same direction: it perceives that the One of the principal filtering is just a brute rooting system of all things in the plethoric X of the powers of privatization, mental nullity and violent looting that characterizes modern capitalism, just as in the formal prejudices that govern the "inventions" of the dominant aesthetic. If it is a principal ultrafilter which organizes thought, however, then the term common to the dynamics of all the "big" parts of the situation, their root, needs to be clearly identifiable by all, not as a massive and complex system of more or less clandestine abject constraints - somehow hidden behind words like "freedom individual ”and“ representative democracy ”, or“ free evolution of the artistic judgment "- but as a One which is the One of a one, that is, a singleton. A principal ultrafilter is the instance which, radicalizing the — eventually transitory — characteristic of "principality" of filters, that is, their subordination to the One, reducing this common element to the bone. The One must present himself as one.

The singleton is here the radical highlighting of the "principal" essence ultrafiltration of all existing things. That's what Marx means, in his famous Manifesto, when he declares that all the communist principles boil down ultimately to one: the abolition of private property. Private ownership of the means of production is the root through which all social relations and all historical becoming are, in the capitalist context, ultrafiltered — and so this is what the communist revolution must unroot.

However, the example of Marx must point us here towards a complex idea: for the revolution to be possible, we must be able to identify the root of oppression. We must see that society is constituted as a principal filter which organizes the grouping of all things around the dominant ideology (the necessary inequalities, the virtue of the rich, the glorification of competition, etc.). But materialism consists in understanding that this ideological filter is a principal ultrafilter, and that the complexity of the dynamics of social domination and its ideology, the filter of domination, conceals an essential simple element, namely, the private ownership of the means of production, aptly named, in its contemporary form, as Capital.

Communism, which must get to the bottom of things, cannot stop, in its theoretical analysis of social forms and of the ideology which corresponds to them, at the denounciation of the filtering of the dominant opinions. It must, above all, assure the analytical and intellectual promotion of the principal ultrafilter — Capital — beyond the simple apparent filter which constitutes bourgeois "modernity". It is in this sense that “radicals” are those who consider the “root” of things. Marx’s and Engels’ Manifesto is largely concerned with the shift from the analysis in terms of the principal filter (the villanies and lies of the dominant ideology and the concrete forms of bourgeois sociality) to an analysis in terms of the principal ultrafilter: Capital, its growth, its concentration and its circulation as the unified core of an ultrafiltration of the whole society.

And here we encounter an inevitable risk: that communist action will itself constitute, after all, as an alternative principal ultrafilter, a principal ultrafilter whose core One, formally centered in a singleton, even if totally different from the one that organizes capitalist hegemony. We now know that this alternative principal ultrafilter was called the Party-State, and that its simplicity, its uniqueness, its singleton, took the form of a proper name, whether Lenin, Stalin, Mao, Castro, Tito, Ho Chi Minh or Enver Hoxha.

The whole contemporary political problem is then to seek how it is possible to overcome this centering on the One of the State and the chief-of-State, heading towards what is the actual measure of the Communism, namely an egalitarian society untied from the power of the One. What must then support this political analysis, whether it knows or not, is inevitably a non-principal ultrafilter.

The peculiarity of the cultural revolution in China has no other abstract, almost ontological, definition than this: trying to organize, in and through mass political action, the de-centering of the communist process, so that its new infinity might be defined not by a simple change of the One which defines a principal ultrafilter - abolition of private property under the law of an omnipotent state whose symbol is the absolute leader - but through the promotion a non-principal ultrafilter, finally detached from the finitude, in that circumstance, of the power of the One. That is, a society aiming at the autonomy of the collective and the decline of the State.

Liberal ideology believes that it can get rid of communist revolutions and their leaders by declaring that these were only dictatorships and dictators. Eh yes ! All those whose names we have just mentioned have been the one of this transient One, the revolution and the seizure of power organized a principal ultrafilter, which they hoped, and us with them, to be a transition to that which alone can really liberate the emancipatory powers of infinity, namely non-principal ultrafilters. They were the dictatorial geniuses of politics. Evil geniuses? That is another question. As we will see in Sections VI and VII, their colleagues from other truth procedures, the founding artists, the prodigious scholars, or the Heloise and Abelard’s of love, have often had to act as transitory symbols of the One, and therefore as the principal character of the mental ultrafilter which governed their decisive inventions.

You could say that against tyranny without truth that apparently formalizes a principal filter, with its mixture of authority of a vague set X and of dissolution of all antagonism of type A/ C(A), between a part A and its complement, the revolutionary transitions, in all orders of creation capable of truths, favored constructions whose primary form is that of a principal ultrafilter, symbolized by a proper name, and returning to all the levels of collective life the explicit work of contradiction.

We note - this is the most important lesson of the twentieth century - that the passage of a complex filter, invisibly regulated by private property and its principal power, to a non-principal, egalitarian ultrafilter, withdrawn from the power of the One, has so far been unable to go beyond this first step. A stage which consisted of the exercise, supposedly transient,of a completely new principal ultrafilter, which has broken the rules of law as well as those of the old State apparatus. It did so by an inevitable creative violence whose singleton is a proper name, which signified for everyone a politics of emancipation. Far from being a barbarous deviation, the emergence in the labour of emancipatory politics, of the power of the proper name — and even before the modern communists we had Spartacus, Robespierre, Toussaint-Louverture, and many others - looks like a necessity which we are about to assign a certain ontological status. Precisely so we might then destitute its indefinite necessary character.

We will return to these analyzes in detail when it is a matter, in the final sections of this book, of analyzing the different infinities created by different truth procedures. But let us take note of something that, once again, is a symptom of the fact that mathematics — focused as it is on the being qua being of all that is, and leaving aside the peculiarities of the world which is that of our living animality, —sees more clearly the most complex problems well before they emerge in other disciplines of thought.

In set theory, in fact, the main instruments concerning the question of infinity are clearly the non-principal ultrafilters, while the principal filters and ultrafilters only play the role of second-class stooges in all of this. You could say, metaphorically, that, in its field, mathematics has already arrived at a sort of mental communism. This is why, in the rest of this book, I will follow the example of my  mathematical masters on this matter — Kunen, Jech and Kanamori — and, unless otherwise noted, "ultrafilter" will from now on mean "non-principal ultrafilter".

You will find a precise definition of the non-principal ultrafilter below, in the second half of this chapter, which is partially devoted to the formal review of filters and ultrafilters. We just need, for now, to say that a non-principal ultrafilter is an ultrafilter subtracted from the power of the One. In other words, there does not exist, as in the case of the principal ultrafilter, a "piece" - in fact, as we saw above, a singleton - which is common to all elements of the ultrafilter. This absence of a common root frees up, in a way, the ultrafilter from any final stowage.

8. Connection of the concept of ultrafilter with the question of the great infinities: infinity of the number very large parts that make up the ultrafilter

If an infinity is "very large", there may be space in it for a very large number of very large parts to co-exist. By this I mean filters and ultrafilters, which, as required by an axiom of filters, admit not only that the intersection of two large parts is still large, but also that the intersection of an extremely large number of parts, for example, an infinite number of parts, and even a very large infinite number of parts, is also in the ultrafilter.

We are still in the same terrain as our previous remarks, for an ultrafilter is the emblem of a kind of equality between terms and their connections. This point is completely foreign to finitude, accustomed to the notion that the connection of two terms, and even the connection of "n" terms, can be reduced to a kind of minimum which may be far removed from the power of the larger terms. Obviously, if you're looking for the elements common to a set A which has a billion elements and to a set B which has three elements, it is clear that there cannot be more than three. In general, the intersection of a large number of finite sets, in the ordinary sense, that is, of sets all smaller than ω, cannot have more elements than the smallest of them. The infinite arranges things in a completely different way. So the infinite set of even numbers is as large as the set of all numbers, of which it is a part. If you take the set of even numbers and the set of multiples of three, both of which are as large as the total set of numbers, their intersection, what they have in common, consists of all the numbers which are multiples of six, and this set is again as large as the total of whole numbers. In any case, with sets defined as "even numbers", "multiples of three "," multiples of six", we are considering very large parts of the basic set which is the total of all natural whole numbers, in other words the cardinal ω. So we are at the level of an ultrafilter, since we handle very large parts. The space of infinitude teaches us, through non-principal ultrafilters, that a subset of a situation can combine with an infinity of other subsets, so that this combination is itself in a situation of equality, not only with the other subsets at stake in the combination, but even, eventually, with the whole big picture.

This point agrees with the leitmotif of this whole book: infinity is the generic name of that which institutes, at a given point of the situation - in fact, at a post-evental point - a power simultaneously egalitarian and commensurable with the whole situation. And this opposes the thesis of finitude, which objects that any maximum power is tyrannical, or, conversely, that only powerlessness can be equally shared. It is fundamental to think about the necessary link between equality – which is a general maxim, in all areas where thinking leads to creations of universal value - and infinity, and specially the infinity that is compatible with a non-principal ultrafiltering of its parts. We have here the dialectical reverse of the most important oppressive thesis, which can be stated as: since we are naturally finite, we are unequal in nature. The thought of infinity is the inevitable ontological condition for a thought of equality. And today, for us, the major operator of this thinking is that of a non-principal ultrafilter, a filter which is not enslaved to the One, which filters very large parts in such a way that the number of these parts is itself infinite.

To clarify this point, we need to introduce an elementary formalism which one can follow, if one so desires - and I hope we do desire it!–,  in the last sections of this chapter. I remind you that in everything that follows, except where explicitly mentioned, "ultrafilter" means "non-principal ultrafilter”.

What does it mean to say that an ultrafilter ULT on a set E has a very large number of very large parts? This means, operationally, that the intersection of a very large number of the parts of E which are in the ultrafilter is also in the ultrafilter. Which means that E is so large that even if you take, in E, an enormous mass of large different parts, what these parts have in common remains extremely large, to the point of being positively ultrafiltered.

We can see here an abstract matrix of the egalitarian power of a collective. Not only does it bring together a huge amount of different people, which already gives it tremendous power, but what these different people have in common - for example the Idea they all share and which in fact has brought them together - is itself of an intensity comparable to that of their objective number. This is what Mao meant when he wrote that a true idea, when it takes hold of the masses, is like a "spiritual atomic bomb". This "atomic bomb" is neither reducible to the number represented by the masses, nor to the Idea as such, because it equalizes their power in a movement that merges both and which is in fact the only true power in history.

We must therefore bring into play, in addition to the ultrafilter composed of parts of E, the infinity of the "common" of these parts. It's not that complicated, and a minimal formalism further simplifies the definition of what we will call the completeness of the ultrafilter.

Let κ be an infinite. We say that an ultrafilter on a set E is κ-complete if the intersection of λ elements of the ultrafilter (therefore of λ large parts of E), however smaller λ is than κ, is itself an element of the ultrafilter.

Suppose we have on E a non-principal ultrafilter ULT. Let's say that λ is an infinite number. Finally, let μ be a infinite number smaller than λ. So what is common to μ parts of E which are in ULT, that is the intersection of these μ parts, is still so big that it still belongs to ULT. We say in these conditions that ULT is λ-complete. The number λ here expresses a measure of ULT's ability to hold in itself what is common to an infinity of large parts, if this infinity remains less than infinity λ.

It is intuitively clear that the larger the infinite number λ is, the more the fact that ULT is λ-complete is a "strong" property, a property that says something about the power of the set E on which this ultrafilter exists. Indeed, the more the "common" part of large parts from E is itself large, the more that means that there is in E not only room for differences (since there is already a very large number of very large separate parts), but also of the place for the common, for what is identical in these different parts, which turns out to be very large. Metaphorically, we will say that the more the gigantic masses are capable of holding something powerful in common — in terms of visions of the future and ideas — the more the “spiritual atomic bomb” is capable of changing the world.  Likewise, the greater the infinity of a love,  such that the shared experience of the world is ultrafiltered by love — the part that becomes a common experience of a Two of love — the more this love will be a power capable of an entire existence’s upheaval.

We actually know a limit for λ, as the value of the "common" of large parts, which remains so large as to still be in the ultrafilter of the large parts of set E. This limit is none other than the measure of E itself, its own infinite magnitude, namely the cardinality of E, let's say κ. This is not surprising. The possible common power of the separate parts of E cannot surpass the power of the situation, of the world, where these parts are deployed, therefore the power of the Whole - here measured by κ - of which these are the parts. We will see later that the demonstration of this point leads to a very interesting idea. In fact, if an ultrafilter is complete for an infinite value greater than κ, for example for the infinite number that comes just after κ, that is, the successor cardinal of κ, which we denote κ +, then the ultrafilter is necessarily a principal ultrafilter. Which means that it contains at least one singleton, and that therefore it is again enslaved to the One. It is a kind of ontological theorem.

If you claim that the power of the common can exceed that of the situation where this commonality brings together many parts — if in short you say: my idea goes far beyond everything that makes up the reality of the world where I proclaim it, you abandon equality in favor of the One, simply because you have given up the materialism of the Idea, which is that the becoming of its existence is immanent to the world. Indeed, insofar as a non-principal ultrafilter in an world M - in other words, the operations of a truth’s unfolding in this world - cannot be complete beyond the power of M, we can assume, as a theorem of the immanence truths, the following critical implication: if a real historical process claims to be complete beyond the overall power of the situation of the world where it exists, then this process, far from working in a materialist way towards the emancipation of humanity, is enslaved to the ideality of the One.

Positively, the fact remains that the maximum power of a non-principal ultrafilter is precisely that it is maximally complete, that is to say, what it can "bear" in common with large parts of the world is commensurable with the world itself. If therefore κ is the infinite number that measures the power of the world - here reduced to a set E -, the concept of κ-complete ultrafilter means that infinity κ is so large that there is room for the intersection of parts of E as numerous as itself, namely a number of parts limited only by κ, still remains in him. We can affirm that such an ultrafilter is maximal, because the combinatorial "greatness" achieved by the intersection of these large parts is at the level of the total power of the situation, being limited only by it.

It is easy to imagine that the internal power exerted by such large parts requires that the extension of the initial situation be of a considerable infinity. Indeed, an infinity κ which admits within it the existence of a κ-complete ultrafilter is truly gigantic.

We will study this point in detail later. But we can already remark something. We can easily demonstrate that it is inaccessible, and a little less easily, that he subsumes κ inaccessible infinities smaller than itself. We can also demonstrate that it resists all the partitions that we have examined, and that it is therefore compact, and even that it subsumes κ smaller compact infinities than itself, and we can also show that it is a Ramsey cardinal.

We will gladly maintain that any real creation, any new subjectivity, is actually supported by an infinity of this kind. It indeed organizes within itself areas of greatness (of intensity) whose combinations remains of the same order of complexity as the whole. The slightest meeting of an emancipation political cell can absorb in itself the whole greatness of the Communist Idea, just like the complex of preparatory drawings for a painting is of the same type of infinity as the finished work. We will say in all cases that a truth procedure admits a full ultrafilter of its own infinity, and that's why the network of its local efficacies is commensurate with the entire procedure.

Other Notes