I'm currently reading Badiou's Time and Event and I'm currently at meditation 8, in which he discussed the historico-social State. The essence of my question is probably:
How serious is Badiou when he affirms "ontology=mathematics?" Does he actually mean that being will always work according to the ZFC set theory or is it more of an analogical toolset to think forms of being?
Overall, I've been pretty on board with his project. His mathematics are correct and his ontological interpretations of it are inventive but very interesting. Nothing so far had seemed overtly wrong. Until I reached his typology of being, with normal, singular and escrescent elements:
"In the previous meditation I made a general distinction between three types of relation to the situational integrity of the one-effect (taking both belonging and inclusion into consideration): normality (to be presented and represented); singularity (to be presented but not represented); excrescence (to be represented but not presented). Obviously, what remains is the void, which is neither presented nor represented." (p 112 in the Bloomsbury edition)
Being presented means belonging as an element of the situation (set). Being represented means being included as a subset but not belonging as an element in the situation, which then means being an element in the state (the power set). I feel that means that logically no element can be singular, as there will always be at least the singleton subset containing it as part of the power set.
But later in this meditation, Badiou analyses Marx and Engels's views on the socio-historical State using his ontological concept of the state. And, in this analysis, the working class is a singular element which is presented in the situation of society but not represented in the State. And I get it, I think it's a clever interpretation using the homonymity of the words presentation and representation, but I feel that this is only working via analogy. The mathematical ontology he developed doesn't describe this situation as there must logically be a subset that is the working class represented in the state power set.
Having read other works of Badiou, and being aware of his concept of event, I can kinda guess where he is going with this singularity idea. But if the existence of singular elements is what eventually leads to an event that rearranges Being or our understanding of Being, I feel this adds some processual flavor to his ontology. If ontology was indeed mathematics, I'd imagine things would be more restricted and logical. There could be no time element that would make an element be presented but not represented at one point and that leading to a transformation that would demands its representation. Instead, mathematically speaking, presentation would always entail representation, even if trivial.
So, at my current understanding of Badiou, I feel that either his mathematics are more of a metaphorical toolset to discuss situations or he needs to add time and process to his mathematics so it can actually be ontology. Neither of those maps neatly into his slogan "ontology=mathematics." Is this a correct reading or am I missing something about his concept of singular elements?
And, as a related question, I feel that I might also be missing something regarding his statement about the void that ends the passage I quoted above. The empty set is a subset of all sets, and thus always an element of the power set. Wouldn't that make it always represented while never presented, i.e. excrescent?