**Badiou’s Ontology**

The only possible ontology of the One, Badiou maintains, is theology. The only legitimately posttheological ontological attribute, by implication, is multiplicity. If God is dead, it follows that the “central problem” of philosophy today is the articulation of “thought immanent to the multiple” (D, 12). Each of the truly inventive strands of contemporary philosophy—Badiou mentions Deleuze and Lyotard in particular, along with Derrida’s “dissemination” and Lacan’s “dispersive punctuality of the real”—have thus presumed the “radical originality of the multiple, ” meaning pure or inconsistent multiplicity, multiplicity that is ontologically withdrawn from or inaccessible to every process of unification, every counting-as-one. 1 For Lyotard and Deleuze, of course, such multiplicity is caught up with (the neo-Aristotelian) substantial or intensive connotations of difference, fragmentation, and incommensurability. We know that Badiou’s innovation is to subtract the concept of multiplicity per se from any such reference, however implicit, to the notion of substantial differences between multiples, indeed from the very medium of the “between.” Instead, “what comes to ontological thought is the multiple without any other predicate other than its multiplicity. Without any other concept than itself, and without anything to guarantee its consistency” (CT, 29).

Since the concept of the multiple is subtracted from any constituent reference to unity or units, its only conceivable foundational point must be void pure and simple, a none rather than a one. The multiple must have literally no limit, or, to put it another way, its limit must be void from the beginning. This is precisely the step that Badiou—unlike Bergson or Deleuze, for instance—has been only too happy to take. Only on this condition, only as founded on nothing or nothing but itself, can the concept of multiplicity be made properly absolute. Were the multiple to be founded on something (else)—an élan vital, a primordial agonism, a Creative or chaotic principle, an elementary unit or “atom”—its multiplicity would to some degree be constrained by this thing beyond its immanent logic. The multiplicity of elements in our physical universe, for example, however vast, is certainly constrained in a number of ways, not least by its origin in an inaugural Bang.

At both ends of the scale, then, Badiou’s pure multiplicity must have no limit to its extension, neither intrinsic nor extrinsic, neither from above nor from below. Any such limit would reintroduce a kind of One beyond the multiple or reduce the sphere of the multiple itself to a kind of bounded unity. Pure multiplicity must not itself be made to consist. Badiou needs, in short, a theory that both confirms the multiple as unlimited self-difference and “bases” it only on the absence of a limit, that is, on the sole basis of an original nothing or void. Both requirements are fulfilled, very neatly, by contemporary set theory. As prescribed by set theory, the multiple is neither cobbled together from more elementary particles nor derived from a (divisible) totality, but multiplies (itself) in pure “superabundance.” 2 Even a relatively dry textbook on the history of mathematics enthuses about set theory as “indescribably fascinating, ” and no one has made more of the theory’s philosophical potential than Alain Badiou.

The Elements of Set Theory

Badiou himself, to be sure, is a philosopher rather than a mathematician, and L’Etre et l’événement is a work of philosophy rather than of mathematics. If every new piece of mathematical research makes a direct contribution to the extraordinarily ramified discourse of ontology, Badiou’s own philosophical project, though conditioned by this research, is concerned with the properly “metaontological” task—that is, the active identification of mathematics as ontology (since there is nothing within the discipline of mathematics itself that affirms this identity) and the elucidation of those properly fundamental principles that shape the general “site” of every work of ontological research (since most such research takes these principles for granted). Badiou is happy to admit that set theory is now far from the cutting edge of most truly inventive ontological work, but it retains an exemplary philosophical or metaontological value as that branch of mathematics which the nature of its objects and terms, that is, what they are or how they are made. A survey of set theory is absolutely essential to any discussion of Badiou’s work, and in working through it we shall be only reconstructing the first, elementary, stages of the abstract argument of L’Etre et l’événement itself. Badiou himself provides the patient reader with all of the technical knowledge required, and the initiated can certainly skip the following outline. Here, for the mathematically illiterate—those readers like myself, whose mathematical education ended in secondary school—I summarize the most basic aspects of the theory in deliberately analogical style. It is a risky technique and is certain to annoy (or worse) those familiar with the pure mathematics involved. We already know that all analogies with substantial objects or situations are in a very real sense wholly inappropriate, and can serve only to convey the basic gist of the logic involved. The analogies presented here are intended as strictly disposable pedagogical aids. What set theory itself provides is precisely a way of describing terms whose only distinguishing principle is distinction itself—the distinction inscribed by an arbitrary letter or proper name (EE, 36).

Badiou sees in set theory’s nine canonical axioms nothing less than “the greatest effort of thought ever yet accomplished by humanity” (EE, 536). These axioms—of extensionality, of subsets, of union, of separation, of replacement, of the void or empty set, of foundation, of the infinite, and of choice— postulate, by clearly defined steps, the existence of an actually infinite multiplicity of distinct numerical elements. 6 Any particular set, finite or infinite, is then to be considered as a selection made from this endless expanse. At its most basic level, the modern exercise of “mathematical thought requires the presumed infinity of its place” (C, 162; cf. EE, 59). If we accept the coherence of this presumption—and this coherence is what the axioms are designed to establish, in purely immanent fashion—what a set is is a collection of these previously given elements, considered as a completed whole. 7 As one textbook puts it, “In set theory, there is really only one fundamental notion: the ability to regard any collection of objects as a single entity (i.e., a set).” 8 The precise number of elements involved in any such collection is strictly irrelevant to the definition, and infinite sets actually figure here as much simpler than large finite ones. 9 The elements thus collected are always themselves sets, however far we go down the scale toward the infinitely small. The sole limit or stopping point of such regression is what is defined as a purely memberless term (or “urelement”). In the strictly ontological, set-theoretic, situation, the only such term is the void or empty set, whose own existence is postulated pure and simple. 10 That the void is alone foundational means there is no elementary mathematical particle, no indivisible or “smallest possible” number: the empty set is never reached by a process of division. 11

If all elements are sets and thus equally multiple in their being (i.e., are multiple in the “stuff” that their elements are made of, the stuff that these elements count as one), what distinguish different sorts of elements or sets are only the sets to which they in turn belong. In ontological terms, we can declare the existence of a multiple only insofar as it belongs to another multiple: “To exist as a multiple is always to belong to a multiplicity. To exist is to be an element of. There is no other possible predicate of existence as such.” 12 As a rough analogy, consider the set of all galaxies, with its many millions of elements. Each galaxy may be said to exist as an element of this set, that is, it counts as one member of the set. “In itself, ” of course, what makes up a galaxy is a very large set of physical components: stars, planets, parts of planets, and so on, down to subatomic collections of electrons and quarks. But as far as set theory is concerned, such substantial realities are of no consequence: “There is only one kind of variable …: everything is a multiple, everything is a set” (EE, 55). Questions of scale do not apply: “Neither from below nor from on high, neither through dispersion nor by integration, the theory will never have to encounter a ‘something’ heterogeneous to the purely multiple” (EE, 77–78). In terms of their organization as a set, a set of galaxies, a set of nations, a set of algebraic letters or of molecules will be treated in exactly the same way. This is how set theory meets the ontological requirement reviewed in the last chapter, that the unity or oneness of an element be considered not an intrinsic attribute of that element but a result, the result of its belonging to a particular set. To pursue our analogy: a galaxy exists as a galaxy (as distinct from a mere group of stars) to the degree that it belongs to what we define as the set of galaxies, to the degree that it fits the rules by which that set counts or recognizes its elements. 13 Remember that what is is inconsistently multiple and eludes all presentation, but whatever can be presented consists as counted as one in some set (or sets).

The elements of set theory as such are not galaxies or anything else, but pure bundles of multiplicity, distinguished only by arbitrary notations. Given a set called S made up of three elements called x, y, and z, for the purposes of set theory the only thing that separates these elements is the literal difference of the letters themselves. What the letters might represent is of no consequence whatsoever. They represent, very literally, nothing at all. There is no relation of any kind between the one produced by a counting for one in a set and the “intrinsic” qualities of such a one—this very distinction has no meaning here, since the word “’element’ designates nothing intrinsic” (EE, 74). Consequently, “the consistency of a multiple does not depend on the particular multiples it is the multiple of. Change them, and the oneconsistency, which is a result, stays the same” (EE, 78). Take, for instance, the set defined by a national population whose elements include all of those counted for one in a particular census. The fact that an individual belongs to this set has strictly nothing to do with the particular nature or idiosyncratic experiences of the individual as such. The individual citizen belongs to the nation precisely as an arbitrary number (on an identity card, on an electoral roll, etc.). In other words, the only form of predication involved here is belonging itself. A given element either belongs or does not belong to a given set. There can be no partial or qualified belonging, and since to exist is to belong to or be presented by another set, it is impossible for an element to present itself, that is, to belong to itself. Belonging (written ∈) is the sole ontological action or verb (EE, 56).

Now then, although to belong (to a set) is the only form of predication, it is immediately obvious that those elements which belong to a set can, if we so choose, be variously grouped into distinct parts (or subsets) of that set, that is, distinct groups of belongings. A subset p of a set q is a set whose own element(s) all belong to q. A part or subset is said to be included in its set. The distinction between belonging and inclusion—and thus between member and part, element and subset—is crucial to Badiou’s whole enterprise. Elements or members belong to a set; subsets or parts are included in it. (The most inclusive of these subsets—the whole part, so to speak—clearly coincides with the set itself.) For example, it is possible to establish, within the set of galaxies, an altogether astronomical number of subsets or parts of this set: for example, galaxies grouped according to shape, number of stars, age of stars, the presence of life forms, and so on. The elements of a national set can be distinguished, in the same way, according to the subsets of taxpayers or prison inmates, social security recipients or registered voters, and so on. The elements of these subsets all belong to the national set, and in their “substance” remain indifferent to the count effected by any particular subset. To belong to the subset of French taxpayers has nothing to do with the substantial complexity of any individual taxpayer as a living, thinking person. Such elemental complexity is always held to be infinitely multiple, nothing more or less.

The whole of Badiou’s admittedly complex ontology is based upon this simple foundation. Before reviewing his terminology in more detail, however, it is worth pointing provisionally to three especially important theoretical consequences, which concern matters of selection, of foundation, and of excess.

In the first place, axiomatic set theory decides the basic ontological question “What is a set?” in terms of a strictly extensional (rather than what used to be called intensional) principle of selection. This was once a matter of some debate. An intensional notion of set presumes that a set is the collection of objects that are comprehended by a certain concept. The sets of prime numbers, of red things, of people living in London, are intensional in this sense. Versions of intensionality were defended by Frege and Russell. 14 In today’s standard version of set theory, however, “the guiding idea is that the members of a set enjoy a kind of logical priority over the set itself. They exist ‘first.’” 15 The first and most widely endorsed of the theory’s axioms, the axiom of extensionality, simply declares that “a set is determined solely by its members.” Under the axiom of extensionality, sets y and z are the same if they have the same elements, regardless of how these elements might be related or arranged. 16 (By the same token, every difference between two beings is “indicated in one point …: every difference proposes a localization of the differing.” 17 ) Relations between elements have no place in the set-theoretic universe as such. Considered as a set, the set {a, b, c} is exactly the same as the set {b, a, c}. As Cantor points out, we can begin talking about the mathematically relevant features of a set M, such as its size or cardinality, only “when we make abstraction from the nature of its various elements m and of the order in which they are given.” 18

This extensional or combinatorial conception of set ensures the entirely open-ended character of the set-theoretic universe. Since the only requirement for the construction of a set or collection is the presumed “priority” of collectible elements, as Penelope Maddy confirms, “every possible collection can be formed, regardless of whether there is a rule for determining which previously given items are members and which are not.” 19 Sets are determined solely by their elements. Just how these elements are brought together, in the extensional conception of set, is a perfectly open question: the possibilities can include, as a matter of principle, every conceivable intensional selection, as well as purely haphazard selections made without reference to any concept at all. The extensional selection may conform to a property or may be determined by a completely random choice. 20 Mary Tiles suggests a useful illustration: “Faced with a page of print one cannot say how many objects there are on it. One needs to know whether to count letters, words, sentences, lines, etc.” 21 An intensional approach to the enumeration of the sets included on such a page would seek to specify the (vast) range of definitions distinguishing letters, words, sentences, and lines, before “counting” the elements that fall under each definition—say the number of words beginning with e, the number with three letters, the number with Latin etymologies, and so on. An extensional approach would accept the validity of any sort of “combinatorial” approach to collections—every possible intensional selection,

expressly considers as well as purely arbitrary collections, such as the set of words enclosed by a rough circle drawn on the page.

Relative tolerance of such an open conception of “being-with” is a characteristic indication of the differences between Badiou’s classical approach to mathematics and the intuitionist approach he so staunchly opposes. Intuitionists refuse to accept a purely extensional understanding of infinite sets, just as they deny many applications of the axiom of choice. 22 An intuitionist conception of set requires a well-defined principle of construction, that is, clear criteria for members’ belonging. But intuitionism is resisted (for compelling reasons) by the great majority of working mathematicians, and it is Badiou’s conviction that people, like numbers, are not constructible in the intuitionist sense. Seen through Badiou’s ontological lens, the human universe is one where absolutely no criteria of membership or belonging apply. Badiou’s ontology recognizes no constraints (social, cultural, psychological, biological, or other) as to how people are grouped together. It remains the case, clearly, that at this particular moment in history our dominant groupings are indeed national, religious, ethnic, or otherwise communal, but—from the ontological point of view—this dominance is strictly contingent. There is nothing about people, Badiou presumes, to suggest that they should be grouped in one way rather than another. As a rule, the most truly “human” groupings (i.e., those most appropriate to a purely generic understanding of humanity) are those made in the strict absence of such communal or social criteria.

In the second place, for any particular set to be “founded” means, in settheoretical terms, that it has “at least one element which presents nothing of which it itself presents” (NN, 93). The foundational term of a situation is that element to which, as seen from within the situation, nothing belongs (i.e., has no members in common with the situation). As an apparently nondecomposable term, this term figures as the most elementary or basic element of the situation, the term upon which all recognizable or situated belonging is based: as far as the situation is concerned, it cannot be broken up into still more fundamental constituent parts. In a situation made up of sets of books, a single book would serve as this foundational term: so would a single musical note in situations made up of sequences of notes. The set of living things offers another example. This set includes elements on several levels of complexity, from ecosystems and species to organisms, to the organs and cells of the organism, and perhaps to certain components of the cell (mitochondria and so on), but at a certain point of cellular organization there are elements (mitochondria, say) whose own elements (proteins, membranes, biochemical structures) are not themselves elements of the set of living things. Such biochemical structures are fundamental to the set of living things—they are that upon which living elements are built, but are not themselves living (NN, 92–93). In the case of mathematical entities, only the postulated empty set can play this foundational role. Since the empty set has no elements by definition, any set that includes the empty set is founded in this sense: it includes something with which it has nothing (no elements) in common.

A direct ontological consequence of this principle is that no (founded) set can belong to itself. The set of whole numbers, for instance—which is certainly well founded—cannot itself be a whole number (EE, 51, 59). A “normal” or ontologically acceptable set cannot be self-founding. 23 (As we shall see in the next chapter, what this ontological prohibition makes illegal is nothing other than an event as such.) To put things a little more formally, what is known as the axiom of foundation (addressed in mediation 18 of L’Etre et l’événement) states that, given a set x, there is always an element y of x, such that y has no elements in common with x. This means that, starting out from a given collection of members, we are blocked from counting indefinitely down from that set to a member of the set and then to a member of that member. Eventually, we must reach something that belongs to the set but that itself has no members that can be discerned from within that set—the empty set, or “urelement.” In the metamathematical applications of ontology, this member or urelement can be anything at all, so long as it is defined as having no members in common with the set. Elementary particles might act as urelements in certain physical situations; so could an individual phoneme, say, in linguistic situations made up of sets of phonemes.

Finally, the number of possible ways of grouping together the elements of a set—the number of parts, or subsets of the set—is obviously larger than the number of elements themselves. As a consequence, and no matter what kind of situation we might consider, “it is formally impossible that everything which is included in it (all subsets) belongs to the situation. There is an irremediable excess of subsets over elements.” 24 It would be hard to exaggerate the importance of this excess in Badiou’s philosophy. There must always be more subsets than elements, because these subsets include not only each individual element, considered as the sole element or “singleton” of its own private subset, but also every possible combination of two or more elements—say, to stick with our national analogy, elements combined according to civil status, tax rates, criminality, levels of education or salary, or indeed according to any arbitrary criterion (“everyone with black hair, ” “everyone living east of the Seine, ” etc.). The combination of all these parts—the set made up of the subsets of a set—is in massive excess of the

set itself. More precisely, given a finite set with n elements, the number of its subsets or parts is 2 to the power of n. For example, the set α with three elements, x, y, and z, has eight parts (23), as follows: {x}, {y}, {z}, {x, y}, {x, z}, {z, y}, {x, y, z}, and {Ø} (this last, empty, subset {Ø}, for reasons I will explain in a moment, is universally included in all sets). A set with nine members has 512 (i.e., 29) parts.

But with an infinite set—and all human situations are infinite—the excess of parts over elements is, thanks to the undecidability of the continuum hypothesis, properly immeasurable. So are the ontological consequences Badiou draws from this excess. This gap between α (a set that counts as one its members or elements) and the set of its subsets p(α) (a set that counts as one its included parts or subsets) indicates precisely “the point at which lies the impasse of being” (EE, 97; cf. 469). This point, whose measurement or specification is ontologically impossible, is thus the real of being-as-being. 25

To give a rough sense of the kind of excess involved here, consider the set made up of the letters of the English alphabet, a set with twenty-six elements. Excluding repetitions, we know that these letters can be arranged in 226 different ways. Allowing for repetition, they can be arranged in any number of “words, ” a small portion of which are listed in the most comprehensive English dictionaries available. Moving from the combination of letters in words to the combination of words in sentences, we move into a still vaster combinatorial range, a tiny fraction of which is covered by the history of the English language and the various ways it has been used. The sort of overabundance Badiou has in mind here is a bit like that of the excess, over the relatively small collection of letters at our disposal, of all that has ever been or could have been said. Thanks to this immanent excess of parts over elements, Badiou—unlike Bergson or Deleuze—has no need to invoke a cosmic or chaotic vitalism in order to secure “the principle of an excess over itself of pure multiplicity, ” nor does he need to explore the virtual dimensions of an “indetermination or undecidability that affects all actualisation. For it is in actuality that every multiple is haunted by an excess of power that nothing can measure, other than … a decision.” 26

Precisely this excess of parts over members locates the place of ideology in Žižek’s clear and compelling sense: “At its most elementary level, ideology exploits the minimal distance between a simple collection of elements and the different sets one can form out of this collection.” 27 More specifically, the ideology of a situation is what organizes its parts in such a way as to guarantee the structural repression of that part which has no recognizable place in the situation—that part which, having no discernible members of its own, is effectively “void” in the situation. Such, Žižek continues, is “the basic paradox of the Lacanian logic of pas-tout: in order to transform a collection of particular elements into a consistent totality, one has to add (or to subtract, which amounts to the same thing: posit as an exception) a paradoxical element which, in its very particularity, embodies the universality of the genus in the form of its opposite.” 28 This element is what Žižek calls the symptom of the situation. 29 For example, Hegel’s rational-constitutional state requires the irrational exception of the proletarian “rabble” as “an element within civil society which negates its universal principle.” Likewise, the anti-Semitic situation requires for its coherence the phantasmatic figure of the Jew as its intolerable Other, just as the contemporary liberal-capitalist consensus is built on the marginalization of the variously “excluded” (the unemployed, the homeless, the undocumented, and so on). Such symptomal elements—rabble, Jew, immigrant—are perceived within the situation as “absence embodied.” 30 Hence Žižek’s most concise “definition of ideology: a symbolic field which contains such a filler holding the place of some structural impossibility, while simultaneously disavowing this impossibility.” 31

What Badiou calls the “site” of an event plays almost exactly the same role in his own system, and Žižek’s terminology fits it nicely: the site, or symptomal real, is both that around which a particular situation is structured (i.e., its foundational term), and “the internal stumbling block on account of which the symbolic system can never ‘become itself,’ achieve its ‘self-identity.’ [1]

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[1] Peter Hallward, *Badiou: A Subject to Truth* (Minneapolis: University of Minnesota Press, 2003), null4, http://www.questia.com/PM.qst?a=o&d=109746408.