Livingston, Paul :Being and Event: Forcing and the Generic

Being and Event: Forcing and the Generic

As we saw in Chapter 1 , the most central burden of Badiou’s argument in Being and Event is to demonstrate the genuine possibility of the advent of radical novelty beyond being itself in what he calls the “event,” given the ontological theory of being as codified in the axioms of standard, ZF set theory. The undertaking involves Badiou’s exposition in the rarified and complex results of set theory’s investigation of the nature and relations of the immense variety of infinite sets, the “paradise” of infinities to which Cantor first showed the way. An infinite set, according to the definition Cantor drew from Dedekind, is any set whose elements can be put into a one-to-one correspondence with the elements of a proper subset of itself. Thus, for instance, the set of all natural numbers is an infinite one , since it bears a one-to-one relationship to the set of all even natural numbers (to match them up, we just pair each number with its double). The set containing all of the natural numbers, ω (or, as it is also sometimes called, ω0), is then the first infinite set. Its size or “cardinality” is designated א 0. 21 As Cantor already argued, however, there are many more. Recall that, by Cantor’s theorem, the power set of a set is always cardinally bigger (that is, it contains more elements) than the set itself . Thus it is certain that the power set of ω 0 is strictly “bigger” than ω0 itself; this power set essentially exceeds the cardinality of ω0 and cannot be put into one-to-one correspondence with it. The power set of ω 0 can also be identified with the set of all real numbers, or points on a continuous line . The question that then leads to the most complex developments of set theory is one that Cantor also already posed: how much bigger is this power set, the set of points on a continuum, than ω0 itself? Cantor formulated the question as a hypothesis, the so-called “continuum hypothesis,” which he struggled in vain through the last years of his life to prove or disprove . The hypothesis asserts that the cardinality or size of the power set of ω 0 is equal to א 1 , the first cardinal larger than א 0. 22 If the hypothesis holds, there is no third cardinal between the size of ω0 and the size of its power set; if it fails to hold, there may be one such, or infinitely many such cardinals. In its more general form, the hypothesis holds that the cardinality of the power set of any infinite set is equal to the very next cardinality (that, for instance, the cardinality of p(ω1) is א 2 , the cardinality of p(ω2) is א 3 , and so on). The continuum hypothesis may at first seem to represent only a very specialized problem in the development of the peculiar theory of transfinite cardinals, but given Badiou’s assumptions and terminology, it actually marks a question that is essential to the success of his doctrine of the event. Remember that the power set of any set is, for Badiou, the “state” representation of what is presented in the original set. Given this, and if, as seems plausible , the sets of interest to ontology are uniformly infinite, then the continuum hypothesis in its general form, if it holds, establishes that the gap between a situation and its state, in Badiou’s sense, can always be regulated by a uniform system of measure. In particular, if the hypothesis holds, the size of the state is always greater than the size of the original set, but the extent to which it is greater is strictly measurable and controllable through the regular succession of cardinals: א ,0 א ,1 א ,2 א 3, etc. If the continuum hypothesis turns out to be true, therefore, there will always be what Badiou terms a “measure of the state’s excess”; it will always be possible to determine how much “more” a representation contains than what is initially presented, how much novelty it is possible to add to the situation. 23 If it does not , on the other hand, this “state excess” will be unmeasurable, allowing the event full range to “wander” and “err,” introducing its radical consequences in an essentially unpredictable way throughout the situation in which it intervenes. We now know that the continuum hypothesis is neither provable nor refutable from the standard ZF axioms of set theory. That one can neither demonstrate the continuum hypothesis nor its negation means, for Badiou, that although there is no way to prove the doctrine of the event within ontology, there is no way that ontology can rule it out either. Nothing in being necessitates the event, but nothing shows that it cannot take place. And the detailed derivation of this result, Badiou argues, shows a great deal about the conditions under which it is possible to think, or assert, the event. It is to the examination of these conditions that Badiou now turns. Badiou thus takes the set-theoretical result that the continuum hypothesis is neither provable nor refutable from the axioms to have a profound ontological as well as political significance. It was Gödel himself who demonstrated the second half of this result, that it is impossible to refute (or prove the negation of) the continuum hypothesis within the standard axioms of set theory. His method was to exhibit a restricted model of the standard axioms in which, as he demonstrated, the continuum hypothesis in fact holds. 24 In doing so, he made use of a formalized notion of constructability, which is in fact the formal basis of the “constructivism” that Badiou cites as the greatest threat to his own doctrine. The condition of constructability places a restriction, much in the spirit of Russell’s theory of types, on the sets that can exist. In particular, it holds that a set exists only if it can be “constructed” by taking all and only elements of some already existing set that have some particular (first-order specifiable) property, P, which is itself definable solely in terms of the already existing set. That is, P must be such that it is possible to determine its extension solely by considering the elements of this existing set and asking whether or not they belong to this extension; if P satisfies this condition, it is said to predicatively define this extension. 25 For instance (to adapt an example given by Cohen), given the set of all finite integers, ω, it is possible predicatively to define the set of all finite integers having a specific numerical property (such as being even or odd), but it is not possible predicatively to define the set P which contains all n such that there is a partition of ω into n disjoint sets of a certain sort. 26 This is because , in considering whether a particular number, say 5, belongs to P or not, we must consider all possible partitions of ω into 5 sets. The definition thus requires us implicitly to run through the entire set of all sets of integers (including , possibly, the set P itself), which cannot be said to “exist” yet, simply given the existence of ω. It is thus termed an “impredicative” definition and the set P is said to be non-constructible. The restriction to constructible sets yields a hierarchy of sets, the so-called “constructible universe ,” that, although perhaps somewhat restricted with respect to the universe of sets overall, nevertheless contains many (if not all) of the transfinite cardinals and can, as Gödel showed, serve as a model for ZF (that is, all the axioms hold for this “restricted” universe). 27 Moreover , because of the restriction of constructability, the sets within the constructible universe are strictly orderable into a unified and unequivocal hierarchy. It follows that, as Gödel showed, if we assume the constructible universe is the (whole) universe of sets, the cardinality of p(ωx) = the cardinality of (ω x +1); that is, within the constructible universe, the continuum hypothesis in its general form is provably true. 28 The limitation to the constructible universe formulates the natural-seeming thought that a new set can only be said to exist if we can define it predicatively: that is, only if we can say, in terms of “already existing” sets, what defines it. The assumption of the constructible universe thus amounts to a restriction of the axiom schema of separation to allow only properties that are “predicative” in this sense to define a set. Introducing the limitation also introduces a strict measure for the “excess,” in Badiou’s terms, of the state over the situation. Other consequences of significance follow as well. For instance, if we stay within the constructible universe, the axiom of foundation does not have to be held as an axiom, since it now follows directly from the other axioms of set theory; the effect of the restriction to constructability is thus also to require that all sets be well-founded (that is, that their decomposition halts somewhere in a basic, founding element). 29 By demonstrating one model of the ZF axioms (the constructible universe) in which the continuum hypothesis holds true, Gödel thus demonstrated that it is impossible, in the ZF axioms in general, to prove its negation; it is thus impossible to prove that the continuum hypothesis does not hold for ZF set theory in general. The other half of the result, that it is impossible to prove the continuum hypothesis in ZF, was demonstrated by P. J. Cohen in 1963. The complex technique of “forcing” that he used is robust in its formal apparatus and subtle in its conceptual implications. For Badiou, it is significant most of all in that the demonstration that it is impossible to prove the continuum hypothesis shows also that it is impossible to prohibit the event in ontology, and indeed helps to demonstrate how it might, paradoxically, appear there by “subtracting” itself from what ontology can discern. Cohen’s general method, once again, was to construct a model; this time, however, the aim is to develop a model in which the continuum hypothesis is definitely not true . If there is such a model, it will follow that the hypothesis definitely cannot be proven in ZF. The details of the actual construction that Cohen used are complex. I shall therefore try to convey only a sense for the general strategy, pausing on the parts of it that are of particular interest to Badiou.

The intuitive idea is to construct a certain kind of model of ZF and show that within this model, we can make the cardinality of p(ω0) arbitrarily high (i.e., much higher than א 1 if we wish, making the continuum hypothesis false). In order to do so, we must begin with a certain kind of set of cardinality א 0 , the so-called “quasi-complete” set or situation. 30 The strategy will then be to add to such a set a “generic” or “indiscernible” extension; if we can do so, it will be possible to show that we can (essentially by stipulation) make the cardinality of the continuum, or p(ω0), as high as we like. A set is called “discernible” if there is some property specifiable only in terms of existing sets that discerns it; in other words, if a set is discernible within a larger set S , then there is some property definable in terms ranging only over members of S that picks out all, and only, the things in S that are in that set. 31 In this sense , the discernible sets will be all the sets that an “inhabitant” of S (who is restricted to considering only elements of S in defining his terms) can talk about, or have any knowledge about. Now, the demonstration that the continuum hypothesis can fail depends on our demonstrating the existence of an indiscernible (or non-constructible) set, a set that, although real, is definitely not nameable in a language thus restricted, or discerned by any property it can name (Badiou symbolizes the indiscernible set: ‘♀’). We can then add this indiscernible set to an existing quasi-complete situation to produce a “generic extension” of the original set and we will subsequently be able to demonstrate the falsehood of the continuum hypothesis with respect to the thus extended situation. 32 Cohen’s technique for generating the indiscernible set, and subsequently demonstrating its existence, is a complex piece of formalism. Intuitively, however, the idea behind it is this. We construct ♀ by “running through” all the possible properties λ that discern sets. For each discernible property λ, however, we include in ♀ one element that has that property. Once we’ve run through all the properties in this way, we know that the set we’ve created has “a little bit of everything”; since it has one element of each discernible property, there is no discernible property that discerns this set itself. (This is, yet again, an instance of the general “technique” of diagonalization.) 33 Thus we definitely have an indiscernible set. This set will exist, but it will not have any possible determinant (for we have built it in such a way, by running through all the specific determinants, that no one specific determinant can determine it). It is in this sense that it is the “anonymous representative” of the whole range of discernible subsets of the original situation. 34 Its appearance in ontology, according to Badiou, marks the free and immanently indeterminable circulation of the errant consequences of the event. Developing the implications of the formal argument , Badiou draws out the consequences he sees in it for the theory of the subject and the possibility of truth. Art , science, politics, and love are “generic procedures”; their pursuit, by analogy with the construction of a generic extension, progressively adds to the existing situation the indiscernible set of consequences of an event. 35 This addition is conceived as connecting the generic set to the event by means of what Badiou calls an “enquiry”; each member of the existing situation which is “investigated” is indexed positively or negatively as belonging or not belonging to the generic extension, and it is of the essence of the enquiry that it can traverse an infinite number of elements. 36 Such progressive addition, at its infinite limit, constitutes the addition of a “truth” to the existing situation; it is to be strictly distinguished from the discernment within a situation, by means of properties, of what (is not necessarily true) but merely “veridical” in it. 37 In intervening, a subject “forces” a new situation which, like Cohen’s “generic extension,” adds to the original situation a set of consequences which are indiscernible by any concepts or properties formulable within it. 38 Because they are collectively indiscernible, these consequences cannot be picked out by any term of an “encyclopedia” or schematization of possible knowledge accessible from within the situation; the consequences of an event are in this sense “subtracted” from positive knowledge. 39 Nevertheless , as Cohen demonstrated, it is possible to “force” them by successively considering conditional statements about the membership of certain elements in the generic set . Though it is not possible for the inhabitant of the initial situation to determine whether a given element is an element of the generic set, he can say (by means of forcing) that if the element is in the generic set, such -and-such statement about that set will be true (in the extended situation to be created). The elements that are considered as possible elements of the indiscernible generic set are thus treated as “conditions” and these conditions determine , by way of the forcing relation, statements that will be true of the new model created by adding the generic set to the existing one. It is possible in this way to build up a series of consistent conditions such that the entire infinite series of conditions, if thought of as complete, determines a set that cannot be specified by any positive predicate but, in that it contains at least one element discerned by each possible predicate, is “typical” or “generic” of the initial set as a whole. 40 By running through the series of conditions and their consequences, we are in a sense “reading out” the generic set, term by term, in such a way as to preserve its genericity , or its indiscernibility by any internally definable predicate . 41 In so doing, though we are not directly in a position to determine which elements belong to the generic set, we are in a position to determine what statement will hold true of the generic set if a certain element belongs to it (and if it is indeed generic). At the infinite end of the process, we will have the complete specification of a set that is indeed generic and cannot be determined by any internally definable predicate of the language. With the addition of the generic set, various statements that were not formerly accurate or “veridical” in the initial, unextended situation will be so in the new, extended one. It is thus that the subject, as Badiou says, “forces a veracity, according to the suspense of a truth.” 42 The centerpiece of Cohen’s own demonstration is the proof that it is possible with this method to force the truth of a (more or less) arbitrary statement about the cardinality of the power set of ω0; for instance, we can force the truth of p(ω0) = א 35 or p(ω0) = א 117 or whatever we like. This is accomplished by considering an arbitrarily high cardinal ( א 35 or א 117 or whatever) and showing by means of the construction of series of conditions that it is possible to distinguish, in the extended situation formed by adding the generic set, at least as many different subsets of ω0 as there are elements of that (arbitrarily high) cardinal . 43 Thus, in the extended model formed by adding the generic set (which is itself composed entirely of subsets of the unextended situation, though these subsets were initially indiscernible in the unextended situation), these statements become true and the continuum hypothesis fails. From the perspective of the initial situation and its state, these consequences of the addition of the generic set remain random; only the generic procedure itself, in “fidelity” to the event, picks them out. The subject is then definable as anything that can practice this fidelity; the result— and with it Badiou closes the book— is an updated, “post-Cartesian,” and even “post-Lacanian” doctrine of the subject. On this doctrine, the subject is not a thinking substance; it is equally not (in the manner of Lacan) a void point, or (in the manner of Kant) a transcendental function. 44 It is the “faithful operator” of the connection between the event and its infinite consequences, the generic procedure of truth in its coming-to-be.