That said, here's the entry: last but least, the VOID...
The word ‘void’ is surprisingly equivocal in Badiou’s writings. Leaving aside the non-ontological, ‘operational’ notion of the void that Badiou discusses in his first Manifesto (where ‘the void’ figures as the empty place in which a philosophy receives the truths of its time), we can discern, in Badiou’s work, at least four distinct senses of ‘void’:
1. The void as the ultimate ground of ontological identity.
2. The void as pure, non-self-identical, inconsistent multiplicity.
3. The void as the emptiness of the count-as-one, itself.
4. The void as the ‘gap’ between presentation and what-is-presented.
These senses are not always easy to untangle. Sense 1 is the one most readily divined from the empty set (Ø) of ZF. The identity between any two sets is determined by the axiom of extensionality, which simply states that ‘two’ sets are identical if they have all the same elements. Since a set is never anything but a set of sets, this criterion implies a regress: before the identity of a set can be established, we must first establish the identity of its elements by looking at its elements, and so on. The only stopping point to this regress is Ø, whose existence is asserted by the axiom of the empty set, and which, alone, is immediately self-identical. The void, in this sense, figures as the ‘prime matter’ from which presentation is composed; every presentational multiplicity is conceived as ‘a modality-according-to-the-one [selon-de-l’un] of the void itself’ (BE 57).
According to Sense 2, the void is thought as a sheer chaos of self-differentiation – the plethos of being-without-unity that appears in the ‘dream’ evoked at the end of the Parmenides. The void is that aspect of every situation that is still-uncounted, and which inheres in every presentation as the invisible but ineffaceable residuum of inconsistency, a ‘yet-to-be-counted, which causes the structured presentation to waver towards the phantom of inconsistency’ (BE 66). This sense, too, resonates with the set-theoretic Ø – as witnessed by the theorem that Ø is a subset of every set – but at the price of a slight ambiguity: Ø is, mathematically speaking, a perfectly consistent set, neither eluding identification nor threatening the stability of the sets in which it inheres. And so it is that, ‘with the inconsistency (of the void), we are at the point where it is equivocally consistent and inconsistent […] the question of knowing whether it consists or not is split by the pure mark (Ø).’[i]
While these first two senses of ‘void’ seem capable of enjoying a strained harmony as interpretations of the empty set, Sense 3 seems to speak of something else altogether. There is no straightforward set-theoretic formulation of the notion of ‘the emptiness of set-formation’. If there is dissonance here, it is unheard by Badiou, for whom
it comes down to exactly the same thing to say that the nothing [the void] is the operation of the count – which, as source of the one, is not itself counted [Sense 3] – and to say that the nothing is the pure multiple upon which the count operates – which ‘in itself’, as non-counted, is quite distinct from how it turns out according to the count [Sense 2].
The nothing [the void] names that undecidable of presentation which is its unpresentable, distributed between the pure inertia of the domain of the multiple, and the pure transparency of the operation thanks to which there is oneness [d’où procède qu’il y ait de l’un]. The nothing is as much that of structure, thus of consistency, as that of the pure multiple, thus of inconsistency. (BE 55; emphasis added)
Here, yet another thought of the void emerges, playing on the equivocity of ‘between’ (which can indicate either a distribution or an interval): The void is, now, ‘the imperceptible gap, cancelled then renewed, between presentation as structure and presentation as structured-presentation, between the one as result and the one as operation, between presented consistency and inconsistency as what-will-have-been-presented’ (BE 54; trans. modified): this is Sense 4 in our list. It is not at all clear how this conception of the void is related to the empty set (unless we imagine it to mark a suture in Miller’s sense). Here, Badiou seems to be naming something like the sheer moment of differentiation that must be supposed to hold sway between a set and its elements—a moment which, structurally, recalls nothing so much as the nothingness that, to Sartre’s eyes, interposes itself between impersonal consciousness and its objects, and which arises as an ‘impalpable fissure’ that arises in the heart of consciousness in the event of its reflexive presentation.[ii]
I would argue that it is this notion of the void that allows us to make productive and non-trivial sense of Badiou’s thesis that
for the void to become localisable at the level of presentation, and thus for a certain type of intrasituational assumption of being qua being to occur, there must be a dysfunction of the count, resulting from an excess-of-one. The event will be this ultra-one of chance, on the basis of which the void of a situation may be retroactively discerned (BE 56; trans. modified).
One way of explaining this thesis would be to show how an ‘event’, a moment when presentation folds back on itself, breaking with the axiom of foundation and foiling the extensional regime of ontological identity, formally replicates what Sartre called ‘the immediate structures of the for-itself’.[iii] The void—the ‘nothingness’—which erupts in such circumstances is not described by a set theoretical ontology, but a Sartrean one—one capable of systematically articulating the relation between the ‘void’ exposed by an event and the form of subjectivity to which an event gives rise (including its dimensions of temporality, possibility, normativity, and liberty). What remains lacking is a univocal concept through which these various senses of ‘void’ can be synthesized.
See also: EVENT, ONTOLOGY, ONE, SUBJECT, STRUCTURE, SET THEORY, SUTURE, SARTRE
[i] Badiou & Tzuchien Tho, ‘The Concept of Model, Forty Years Later: An Interview with Alain Badiou’, in CM, p. 99.
[ii] Jean-Paul Sartre (1956), Being and Nothingness: An Essay in Phenomenological Ontology, trans. Hazel E. Barnes, New York, Philosophical Library, pp.77-8
[iii] Cf. Sartre, Being and Nothingness, esp. Part II, Chapter I, Section I.