Let us examine the argument whereby Badiou seeks to establish thought’s suture to the unpresentable.
The axiom of separation stipulates that every assertion of existence concerning a set necessarily presupposes a pre-existing set. Thus there must be an originary set whose existence provides the precondition for every subsequent set.
This is the emptyset:
the set to which nothing belongs. Ontological discourse, which is the presentation of presentation, the presentation of counting-as-one qua belonging, affirms no existence, which is to say, no belonging prior to that to which nothing belongs. It affirms the being-nothing of belonging as that to which no belonging belongs.
Consequently, it is not belonging (consistency) whose existence is originally declared, but non-belonging (inconsistency) as that which is already presupposed by every subsequent belonging. In fact, ontological discourse begins with a negation of belonging, rather than an affirmation of non-belonging; but a negation whose very act is equivalent to an affirmation, or as Badiou puts it, an ‘existence which does not exist’ (Badiou 1988: 81, 2006a: 68 tm).
Thus the ontological axiomatic accomplishes what Badiou refers to as ‘thought’s subtractive suture to being’ by declaring the existence of an inexistence. It asserts the being of the unpresentable through a negation of presentation.
François Wahl raises a particularly interesting objection here: the argument that enjoins us to deduce the existence of non-belonging from the negation of belonging merely reiterates the ontological argument.
For even if, as Badiou claims, the axiom of the empty-set affirms an inexistence rather than an existence, since inexistence is no more of an ‘index of existence’ than was perfection in the classical version of the ontological argument, then why should the inexistence of belonging enjoy any more right to existence than any other inexistence – such as that of the One for instance? (Wahl 2002: 177) How does the negation of belonging which establishes the existence of the void’s ‘being-nothing’ differ from the negation of unity through which Badiou asserts the ‘non-being’ of the One? However, Wahl’s objection misrepresents Badiou’s argument. The logical import of the axiom of theempty-set is neither that ‘non-belonging exists’, nor that ‘the unpresentable’ exists, nor even that ‘inexistence exists’.
It does not predicate existence of any concept, whether it be that of ‘non-belonging’ or ‘inexistence’. Rather, the axiom asserts that ‘there exists a set _ such that no set _ belongs to it’.
The existential quantifier does not attribute a property to a concept. Its import is that, even in order to deny belonging, it is at least necessary to affirm the existence of a mark of belonging.
It is this ‘at least’ that is singularized by the assertion of the existential quantifier. And it is by negating this mark of belonging that the axiom affirms the existence of a mark of non-belonging.
This is also why the being-nothing of the void cannot be formally conflated with the non-being of the One. The former is an assertion of negation, while the latter is neither an assertion nor a negation since the unification of multiplicity carried out through the operation of belonging has no ontological status. Belonging is an operation, not an entity; an operation whose inexistence is converted into existence when it in turn is counted as belonging to another set. It is this operation whose negation is asserted by the axiom of the empty set, which is effectively the only belonging that exists.
Set-theory begins by asserting the negation of belonging, a negation already presupposed by every subsequent operation of belonging through which multiplicity is counted-as-one. Thus the inexistence of the void, the assertion of the negation of belonging, is both the precondition for the operation of the One qua belonging, since all belonging presupposes the existence of (the name of ) non-belonging; and that which ensures that the existence of belonging per se is never affirmed.
Set-theory begins by declaring that nonbelonging exists, a non-belonging which authorizes all subsequent belonging, but the theory neither asserts nor presupposes the existence of belonging.
The axiom of the empty-set asserts that the name of the unpresentable is presented; or that there exists a name of inexistence. This nuance is crucial: asserting the existence of a name in discourse is quite different from asserting the existence of an extra-discursive concept.
For it is through this nomination that presentation is able to suture itself to the unpresentable without presenting it. Thus, Badiou writes, ‘the inaugural advent’ of the unpresentable consists in ‘a pure act of nomination’ (Badiou 1988: 72, 2006a: 59) which ‘since it is a-specific […] consumes itself, thereby indicating nothing but the unpresentable as such’ (Badiou 1988: 72, 2006a: 59 tm). This nomination neither marks the return of the One, since it does not make anything consist, nor does it index a multiplicity, since what it presents is strictly nothing.