"As the name implies, a subtractive ontology is to be distinguished from a discourse which pretends to convey being as something present and substantial, something accessible to a sort of direct experience or articulation. By subtractive ontology Badiou means a discourse which accepts that its referent is not accessible in this sense. As Badiou conceives it, being is not something that shows itself in a sort of primordial revelation; still less is it the object of some divine or quasi-divine act of creation.... The ontologist knows that the ground of being eludes direct articulation, that it is thinkable only as the non-being upon which pivots the whole discourse on being....in mathematical set theory (the theory of consistent multiplicity) the ultimate ‘stuff’ presumed and manipulated by the theory is itself, as we shall see in a moment, inconsistent – it can be presented only as no-thing. In other words, ontology does not speak being or participate in its revelation; it articulates, on the basis of a conceptual framework indifferent to poetry or intuition, the precise way in which being is withdrawn or subtracted from articulation."
The above is not like Hegelian types of 'consistent' set theory which cannot handle void sets, the withdrawn or dialetheisms of the kind Morton describes in this post and following (using Graham and Priest), more typical of real reason.
I'm also reminded of this post from Women, Fire and Dangerous Things on non-Badiouian set theory:
"The classical theory of categories provides a link between objectivist metaphysics and and set-theoretical models.... Objectivist metaphysics goes beyond the metaphysics of basic realism...[which] merely assumes that there is a reality of some sort.... It additionally assumes that reality is correctly and completely structured in a way that can be modeled by set-theoretic models" (159).
He argues that this arises from the correspondence-representation model.