# How does Badiou analyze natural situations?

I'm having trouble applying Badiou's method of looking at situations as sets (EDIT: specifically sets in a model of ZFC). The following example was in the introduction to one of his books, Infinite Thought, and I don't see how we can interpret it in a set-theoretic way.

Take, for an example of a natural situation, the ecosystem of a pond. The multiples which it presents include individual fish, tadpoles, reeds and stones. Each of these elements is also represented at the level of the state of the situation... Each element of an ecosystem is also one of the ecosystem's subsets, because each of their elements also belong, in turn, to the ecosystem; for example each fish's eating and breeding habits belong to the ecosystem as well as to each fish. These elements are thus normal multiples... The test of whether a situation is natural or not is whether there is any element of the situation whose content is not also part of the situation.

Infinite Thought, Introduction by Oliver Feltham and Justin Clemens, p 25.

There's a bit of a mistake here, because (p 24, itself citing Being and Event 146)

[A]ll of [a natural situation's] multiples are either normal or excrescent, and each normal element in turn has normal elements.

So the pond could, but does not need to, have excrescent multiples. However they make a good case that every element of the pond is a normal multiple, and you could probably make the same case for any other natural situation. (EDIT: according to these two books from Google, and also Mauro ALLEGRANZA, natural situations only have normal multiples. So the definition provided above is false.)

The crucial part is "each normal element in turn has normal elements." If this is true, at least one element of the pond is the null set,* but that doesn't make a lot of sense.

Regarding the pond, or really any other natural situation: Which, if any, of its elements are excrescent multiples? What does the null set mean in the context of a natural situation?

*Or else the set wouldn't be well-founded. The proof I sketched was easy enough and tedious enough that I didn't include it here. I think it's correct but if I am wrong about my math please tell me.

• "... or else the set wouldn't be well-founded". Is Badiou's set-theory using Regularity Axiom ? There are Not-wellfounded set-theories. – Mauro ALLEGRANZA Dec 13 '17 at 7:43
• Badiou does use regularity. He's very keen on ZFC, though I haven't found a good explanation for why, other than its popularity. – Canyon Dec 13 '17 at 7:45
• It occurs to me now that there might not be any natural situations, which would be kind of funny, but I don't think he's taking it in that particular direction. – Canyon Dec 13 '17 at 7:46
• According to Badiou's definition, a system is natural if all its terms are normal, i.e. it has no excrescent terms: terms that are included into the situation but do not belong to it. – Mauro ALLEGRANZA Dec 13 '17 at 9:38
• Maybe Badiou consider atoms (or urelements) i.e. objects that are not sets (thet have no elements) and thus they acts like the empty set. – Mauro ALLEGRANZA Dec 13 '17 at 9:49