conquering flesh said:
my simplistic question is: why does badiou base his philosophy in set theory & 'memberships'?
My simplistic answer is: he doesn't. Or more precisely – the foundational metaphor at work here is misleading when it comes to situating the position of set theory within Badiou's theoretical edifice.
The important point is Badiou's equation ontology = mathematics. This is a strictly philosophical proposition, and a thoroughly original one, though Badiou argues that it continues a materialist line of thought that runs through Galileo.
Galileo argued that physics fell into the remit of scientists rather than philosophers. Badiou extends this to metaphysics. Philosophers have failed to answer the ontological question because they have hitherto failed to grasp that it is mathematics that articulates what is expressible about being qua being.
It follows from Badiou's position that philosophers should not compare mathematical concepts to their ready made philosophical ones (eg interminable analytic philosophy debates about the "correct" definition of necessity, possibility, conditionality), but rather "close read" mathematics and derive/extract "metaontological" concepts from it. Badiou's exegesis of
apparentenance (belonging, or "membership" – though Badiou avoids this interpretation) is a case in point. It is a metaontological extraction of what's at stake in the epsilon relation.
The most important example of this in L'Etre et l'evenment concerns infinity. Badiou argues, compellingly in my view, that the Cantorian conception of transfinite numbers (aka the aleph hierarchy) represents a thoroughly novel – and superior – understanding of infinity, incommensurable to previous philosophical approaches to the question (all of which, in Badiou's view, treat infinity as the horizon of the finite, and are fatally compromised by theology). Mathematics achieves something that rationalist materialist philosophy has always striven for, namely the desacralisation of infinity.
Now, all the debates about ZFC, hypersets, topoi etc occur
within this framework. One can quite legitimately question Badiou's focus on ZFC in L'Etre et l'evenment. That book was written 20 years ago and concerns itself with mathematics that were developed 20 years before that. An awful lot has happened since then, not least the spectacular advances in category theory and the renewed question marks over foundation (AFA, hypersets etc).
But I'd argue that
(i) none of these developments has yet entirely displaced ZFC's position within mathematics. If ontology = mathematics, then the peculiar fact that all of mathematics can be rendered in ZFC has to be acknowledged, and one should expect metaontological analysis of ZFC to produce something fruitful.
(ii) even if many of the formulations in L'Etre et l'evenment do need to be drastically revised in the wake of new mathematical discoveries (and for my money there is something dodgy in Badiou's treatment of foundation), this does not weaken Badiou's fundamental proposition that mathematics = ontology – in fact, it can even be said to develop it further. A case in point is Logique du Monde, in which Badiou reworks much of his previous system in the light of topos theory (and this does not involve "ditching" ZFC as borderpatrol states, but rather developing a complex dialectic between ONTO-logy (the science of being) and onto-LOGY (the science of being-there)).