Yet another note on Badiou and set-theoretic ontology…Compared to “the ontologist”, who has full freedom of movement in the set-theoretic universe, the “inhabitant of the situation”, whose analysis of multiples is restricted to the predicates that can be drawn up using the language of the situation, is missing a trick or two. In particular, he cannot make use of the axiom of union to “see inside” the elements of the situation. This inability is one index of the gap between the power of the encyclopaedia of the situation, which presides over representations (predicatively discriminated subsets of the situation), and the full structural effectiveness of the axioms governing presentation in general.

There now follows some reasonably unfancy maths. Take a fairly dull finite set, S1={a, b, c}. The axiom of union says that it is possible to form (or, in truth, that “there exists”) a new set on the basis of this set, which contains all of the elements that are present in either a, b or c. So, in the event that a={1, 2}, b={2, 3} and c={3, 4}, the union of S1={a, b, c} would be S2={1, 2, 3, 4}. This new set S2 “opens out” the sets in the first set S1, exposing their elements. Note that none of the elements in the resulting set S2 bears any trace of the element to which it belonged in the first set S1 - there is no indication that 2 was in a and b, or that 4 was only in c. However, all of the elements of our original set S1 are re-presented in the set of parts of S2, the powerset P(S2).

An interesting feature of P(S2) is that it contains not only a, b and c, but also every possible hybrid (e.g. {1, 3}, or {2, 3, 4}) that can be made out of the unions of selected parts of each. Any set whatsoever can be considered as a part amongst others of the powerset of the union of its elements. We are thus able to form new composites out of the parts of elements of a given set. However, we cannot necessarily say, for any given hybrid drawn from P(S2), which particular sets from S1 participate in its composition. For instance, the hybrid set {2, 4} may be supposed to be a hybrid either of a={1,2} and c={3, 4}, or of b={2, 3} and c. The axiom of union effectively anonymises the material it exposes for recomposition.

(This structural hybridisation, incidentally, is one good reason why the event cannot be “illegal” merely from the standpoint of the conceptual mappings assembled in the encyclopaedia of the situation: not only are conceptually “anomalous” multiples not prohibited by the axiom system governing presentation, their existence is actually legislated by it!)

Early on in Being and Event, Badiou distinguishes between ontological and non-ontological situations, saying that only the ontological situation is permitted to present the empty set, {}. Suppose however that, given a non-ontological situation, we were repeatedly to decompose its presented multiples into their constituent matter, using the axiom of union to form a new set that could be likewise decomposed in its turn, and so on. Would we not sooner or later expect to arrive at a set that contained the empty set - or, to put it another way, would we not finally expect the consistency of presented multiples to yield to inconsistency, to present that presentation is ultimately the presentation of nothing and that being itself is pure inconsistent multiplicity? This is to my mind a similar question to the one I asked about the infinite descent of nested “black boxes” in Graham Harman’s Latourian ontology: clearly there are no ur-elements in this ontology either, but wouldn’t one finally expect the law of structure organising this hierarchy to expose its own contingency?

It is here perhaps that the burden of Badiou’s insistence that every real situation is “infinite” may be weighed: there is no finite path to the ontological situation from any real situation. But the constructivist orientation would insist that there is nevertheless always a path, that there is no multiple whatsoever that does not have its place in the botanico-mathematical hierarchy of constructions that flower forth from the void. The question therefore remains: is there ultimately a chain, be it an infinite chain, linking any real situation to the ontological situation?