Being as Pure Multiplicity

Badiou's ontology begins with a rejection of the primacy of the One. The philosophical tradition from Plato and Aristotle through Leibniz and Hegel has tended to think being in terms of unity — the one substance, the one God, the one concept in which all difference is reconciled. Badiou inverts this: being is not one, it is multiple — pure, inconsistent, infinite multiplicity. Unity is not a feature of being but an operation: the counting-for-one, the act of a situation in which a multiplicity is presented as a unified element. The One is not; it results. Ontology must begin not with unity but with the void and with set-theoretic structure.

The claim that mathematics is ontology is not the claim that the physical world is made of mathematical objects. It is the claim that the discourse adequate to pure being — being as being, without qualification or predication — is mathematical, specifically set-theoretic. When set theory deploys the axiom of the void, the axiom of extensionality, the axiom of the power set, it is saying something true about being as such. Physics says something true about physical beings; biology about living beings; but axiomatic set theory, in its formal purity, says something about being qua being. The identification is exact: Badiou's ontology is not a metaphor drawn from mathematics but a literal identification.

The empty set — the set that has no elements — is the unique primitive of set theory. Every set is ultimately built from the empty set through successive operations: the singleton of the empty set, the set containing that singleton and the empty set, and so on. In ontological terms, the void is the ground of all being: every presentation is ultimately a counting from the void. This has political as well as metaphysical implications: the void is what every situation includes but refuses to count; it is the unrepresented, the excluded, the "part that has no part" — and the event always arises from the site of the void, making visible what the situation's structure has hidden.

The identification of mathematics with ontology is the thesis of the Introduction and Meditations 1–5 of Being and Event. It draws on Cantor's transfinite set theory and the Zermelo-Fraenkel axiom system. The philosophical tradition Badiou argues against — the primacy of the One — is traced through Parmenides, Spinoza, Hegel, and Heidegger.