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.
