The concept of ‘the generic’, which Badiou first deploys in Theory of the Subject in an essentially metaphorical reflection on the subjectivizing production of excess (271-4), comes into full philosophical force in Being and Event, where it is taken up to describe the ontological – set-theoretical – structure of a truth-procedure: the total multiplicity that will have been composed of all the elements in the situation that a faithful subject positively links to the name of an event (by way of a ‘fidelity operator’), from the perspective of this multiplicity’s always-futural and infinite completion, takes the form of a generic subset of the situation in which the subject of truth operates. As a consequence of their genericity, truth-procedures exhibit at least five critical traits: (1) their indiscernible, unpredictable and aleatory character; (2) their infinitude; (3) their excrescence relative to the situation; (4) their situatedness, and (5) their universality. The concept of a generic subset, itself, was first formulated by the mathematician, Paul J. Cohen, in his 1963 proofs of the independence of the Generalized Continuum Hypothesis (GCH) and the Axiom of Choice (AC) relative to the axioms of Zermelo-Fraenkel set theory (ZF). The problem Cohen faced was this: Kurt Gödel had already shown (in 1940) that both GCH and AC are consistent with ZF by showing that if ZF has a model, then a model can also be produced which satisfies ZF supplemented by GCH and AC. This means that one can never prove the negation of GCH or AC on the basis of ZF, but it does not imply that the statements themselves can be proven. To show that ZF is no more able to entail these theses than their negations, Cohen sought to construct a model in which AC and GCH fail to hold. This would show that they are independent – or undecideable – relative to ZF. Cohen’s strategy was to alter Gödel’s model S (in which GCH and AC do hold) by supplementing it with (i) a single element ♀ and (ii) everything that can be axiomatically constructed on its basis. The supplemented construction S(♀) must both be capable of satisfying the ZF axioms (and so remain a model of ZF), while encoding the information needed to falsify GCH or AC (information which can be extracted by the forcing procedure). The difficulty is this: though it suffices to encode the many ZF theorems concerning transfinite sets, ‘from the outside’ (when embedded in a sufficiently rich super-model, that is) Gödel’s model-structure appears to be countable (it can be placed in a one-to-one correspondence with the set of natural numbers). (The surprising fact that set theory has such models, if it has any at all, is guaranteed by the Löwenheim-Skolem Theorem.) Any supplement carrying that kind of information would spoil the structure’s claim to be a model of ZF, and so
♀ must have certain special properties if S(♀) is to be a model. Rather than describe it directly, it is better to examine the various properties of ♀ and determine which are desirable and which are not. The chief point is that we do not wish ♀ to contain ‘special’ information about S, which can only be seen from the outside […]. The ♀ which we construct will be referred to as a ‘generic’ set relative to S. The idea is that all the properties of ♀ must be ‘forced’ to hold merely on the basis that ♀ behaves like a ‘generic’ set in S. This concept of deciding when a statement about ♀ is ‘forced’ to hold is the key point of the construction.[i]
Leaving technical subtleties aside, the idea is to construct ♀ in such a way that for every predicate or ‘encyclopaedic determinant’ restricted to S (where ‘restricted’ means that its constants and quantified variables range only over elements of S), ♀ contains at least one element which fails to satisfy this predicate. This suffices to determine the generic: (1) as indiscernible, insofar as no predicate can separate it from the swarming multitudes of S, and for this reason the generic must present itself in time as unpredictable and aleatory, its lawless composition impossible to forecast; (2) as infinite, since it remains essentially possible to determine any finite multiplicity by means of a complex predicate, even if this is only a list of its constituents (the syntactic constraints of set theory, if nothing else, prevent us from ever writing an infinitely long formula); (3) as excrescent, meaning that it is a subset but not an element of the ‘situation’ (the model in which the generic is articulated), the reason for this being that if ♀ was an element of S, then the predicate ‘x Î ♀’ alone would be enough to capture it; (4) as situated or immanent, since genericity is by no means an absolute property, but one which is relative to the model in which it is articulated; (5) as universal, since the generic outstrips every mark of particularity to the extent that no element of the model is excluded from entering into a generic subset by reason of the predicates it bears. Finally, though it must be connected to an essentially non-mathematical (non-ontological) theory of the event in order to do so, genericity helps to capture the idea that truths are effected through the work of a subject whose existence precedes and outstrips its essence. The ‘existentialist’ resonance that the concept of genericity brings to the Badiousian theory of the subject must be taken seriously, for it bears directly on obstacles accompanying trait (2): insofar as every actual truth-procedure unfolds undeterministically in time, each procedure is at any actual moment, finite, and can lay claim to genericity only by projecting itself ahead of itself, by being the future it factically is not: the infinite truth-multiple that it seeks to complete but which it cannot fully determine in advance.
See also: MODEL, FORCING, ONTOLOGY, SET THEORY, MATHEMATICS, SUBJECT, TRUTH, EVENT.
BE: Meditations 26–36.
TS: Seminar of May 15, 1978, ‘Logic of the Excess’.
Paul Cohen (1966), Set Theory and the Continuum Hypothesis, New York, W.A. Benjamin.