Since Aristotle ontology has been a privileged sub-discipline of philosophy; otherwise known as the discourse of being. Badiou puts forward a radical thesis: if being is inconsistent multiplicity, then the only suitable discourse for talking about it is no longer philosophy but mathematics. For Badiou mathematics is ontology … This thesis enables Badiou to reformulate the classical language of ontology—being, relations, qualities—in mathematical terms: more specifically, those of set theory, because it is one of the foundational disciplines of contemporary mathematics; any mathematical proposition can be rewritten in the language of set theory. [Feltham and Clemens, ‘An Introduction to Alain Badiou’s Philosophy’, 13]Set theory ‘is the formal theory of non-unified multiplicities’. One of the reasons Badiou is drawn to it because it avoids the need to ground his ontology on any absolute definitional ‘foundation’.
There is neither definition nor concept of a set in set theory. What there is in its place is a fundamental relation—‘belonging’—as well as a series of variables and logical operators, and nine axioms stating how they may be used together. Sets emerge from operations which follow these rules. Or again:
Sets are made up of elements. The elements of a set have no distinguishing quality save that of belonging to it. … The relation of belonging is the basic relation of set theory: it is written α ε β; α belongs to β, or, α is an element of the set β. There is another relation in set theory, terms inclusion, which is based entirely on belonging. Sets have ‘subsets’, that are included in the sets. I appreciate, of course, that this language is adopted straightforwardly from mathematics, where these terms (‘belonging’, ‘inclusion’) are deployed in a neutral sense. But when mathematics is recast as ontology, and ontology then determines a set of specifically political, psychoanalytic and aesthetic philosophical engagements with the world, I wonder if things don’t get problematic.
I understand (I think) the appeal of Badiou’s sets to an anti-foundationalist mindset; an anti-foundationalist is a very desirable one to possess. Badiou’s thinking in terms of sets avoids essentialism (what some people call humanism) in ontology; it formulates ontology in a rigorous and schematic way that nevertheless retains the invigorating fluidity and relativism that so intoxicates postmodernists and hard-line deconstuctionists—for, in a mathematical sense, belonging to any given set does not fix an element. In fact, any given element belongs to an infinite number of sets that are continually overlapping and reconfiguring the relational possibilities of the element itself depending on context (the fact that 9 belongs to the set of odd numbers doesn’t prevent it belonging to the set of square numbers and so on.) Nor is Badiou, of course, suggesting anything so crude as mapping set theory directly onto the world in which we live. In Being and Event he says: ‘we are trying to think multiple presentation regardless of time (which is founded by intervention) and space (which is a singular construction, relative to certain types of presentation) . If I understand it correctly, Badiou’s ontology concerns the structure of situations, rather than situations themselves.
Nevertheless, this emphasis on ‘belonging’ keeps knocking a heavy knout against the back of my head. Belonging? Is there a less ideological neutral word in any language? It’s the core strategy of oppression; the identification of ‘one of us’ and its inevitable correlative ‘not one of us’. To think of a political-historical structure that enacted belonging as its root determinant is to think (for me) of the old Apartheid South Africa. What I mean by this is that, more than enacting set-theory in a crude literal sense (parceling human beings into the sets ‘white’, ‘black’, ‘coloured’ and so on) the old, unmissed South African regime elaborated a politics grounded in an ontology of belonging.
More, this Apartheid set-theory didn’t pretend absolutely to fix its elements, just as set-theory doesn’t absolutely fix its elements. What I mean by this is that including a certain human being in the set ‘black’ does not prevent them from also being included in lots of other sets (‘workers’, ‘church-goers’, ‘people who must pay their electricity bills’ and so on). Of course, including 9 in the set ‘odd numbers’ means, even though 9 can be included in an infinite number of other sets, nevertheless 9 is stuck in that set—it cannot appeal against its inclusion. It is always an odd number for ever, without right of dissent.
Now, you might say: but 9 is an odd number. (A proponent of Apartheid might say: but Nelson Mandela is black). Or, you might say: people are not numbers. But if set theory is to become ontology, then people (amongst other things) are elements, determined ontologically by the various facts of their ‘belonging’ and ‘not-belonging’.
‘Belonging’ has its neutral flavours, of course; but in terms of human relations it connotes slavery (this person belongs to me, the wife belongs to the husband) and capitalism (my money, my possessions, my property, my belongings). If a set is constructed, it is surely worth asking: who has determined that this is a set? To what end? And more importantly: how can it be contested and challenged?’
One of the dangers of introducing mathematics into philosophy is that it tends to suggest that this latter question must be answered: it cannot. That’s just the way it is. Dostoevsky, in Notes from the Underground talks about this when he discusses the appalling tyranny of ‘two plus two equals four’. But to imagine an ontology of political action, rather than of ‘belonging’, is to require us to refuse all tyrannies.
In ‘Philosophy and Politics’ Badiou defines ‘political justice’ in terms of egalitarianism. But ‘equality’
does not refer to anything objective. It is not a question of equality of status, of income, of function, and even less of the supposedly egalitarian dynamics of contracts or reforms. Equality is subjective … is in no way a social programme. Moreover it has nothing to do with the social. It is a political maxim, a prescription. Politcal equality is not what we want or plan, it is what we declare under fire of the event, here and now, as what is and not what should be. Really?
There’s a kind of romantic individualism at the heart of this (‘in matters of justice … it is true, as true as a truth can be, that it all depends on you’). Certainly ‘the State’ has no part to play in Badiou’s understanding of political justice:
Any definitional or programmatic approach to justice turns it into a dimension of the action of the State. But the State has nothing to do with justice. … The modern State aims solely at fulfilling certain functions, or at crafting a consensus of opinion. Its sole subjective dimension is that of transforming economic necessity—that is, the objective logic of Capital—into resignation of resentment. 
But isn’t the objective logic of Capital precisely belonging (‘this belongs to me’) Imagine a world in which ‘belonging’ is actually just another word for ownership (which is to say, possession); and possession is the opposite of freedom. Sets are defined as much by what they exclude as what they include; and the dominant experience of humanity in the world is exclusionary. The desire—political, cultural, racial, even philosophical—to stress inclusion, the comfort of belonging, the ‘we’ that provides solace, necessarily involves the possession of x by y.
Those people who derive meaning and comfort from belonging to the sets Aryan, German, wealthy orient a larger number of people via un-belonging, the malign-belonging, of poor, exterminatable and so on. Philosophy’s duty is to what the set excludes, not to belonging as such.