Structuralism’s Prohibition of the Event
Perhaps something has occured in the history of the concept of structure that could be called an ‘event’, if this loaded word did not entail a meaning which it is precisely the function of structural – or structuralist – thought to reduce or to suspect. Let us speak of an ‘event’, nevertheless, and let us use quotation marks to serve as a precaution. What would this event be then? Its exterior form would be that of a rupture and a redoubling.
Derrida, J., Structure, sign and play in the discourse of the human sciences
Set-theory notation does not use inverted commas to bracket what the count-as-one counts as one. Instead it uses what computer programmers know as curly braces: {}. In programming languages with a curly-braced syntax, the braces delimit the scope within which names are bound to variables, expressing for example the distinction between local and global variables. They demarcate a context, and their syntactic nesting (e.g. {{}}) enables contexts to be enclosed within each other in a hierarchy.
What Badiou calls the operation of the count, which retroactively establishes the one-ness of the counted ensemble, sets a limit and gathers everything that falls within that limit together in a regime of co-presentation. The count-as-one states only that the things counted are presented together, are as one: it is a gesture of at-onement.
It does not, in itself, prescribe any but the most minimal structure: that of co-presentness, simultaneity. It turns out that via the nesting of scopes and judicious invocation of the Void ({} or Ø, the empty set), one can start to number things, place entities in order, index them or make them the subject of statements describing their relationships. Minimal structure plus syntactic recursion gives you all you need to say almost anything, albeit at considerable length.
Now, I have reached a passage in Being and Event that is both evidently very significant and bewilderingly opaque: it is the pair of meditations in which Badiou first introduces “the matheme of the Event” (which is that of a recursively self-belonging set, ex = {x ε X, ex}), and then introduces the axiom of Foundation (which rules that sets cannot recursively belong to themselves in this way), in order to show that “mathematical ontology” can only recognise “the Event” as that which is otherwise-than-being. The “matheme of the event” describes a kind of hyper-set, which the axiom of Foundation (which Badiou calls a “metaontological thesis of ontology”) defines as unfounded, and hence ab-surd for the purposes of the axiom system to which it belongs.
Reading around and ahead (Wikipedia is an ever present help in trouble here), I find that this has something to do with the axiom of Choice, and thus presumably with the discussion of forcing and subjectivity that Badiou is going to get into later. Clever things are, clearly, afoot: one needs to bite one’s tongue and forge ahead. All the same: up until this point, Badiou’s correlation of mathematical with ontological terminology has generally had a certain intuition on its side. It’s not hard to accept “multiple” as a synonym for “set ” (or “ensemble”), or to follow the mapping between the nesting of sets and the relationship between “presentation” (the initial count-as-one”) and “re-presentation” (the count-of-the-count). But it far from obvious why eventhood, the “rupture and redoubling” Derrida speaks of, should correlate with the particular matheme Badiou puts forward for it. In spite of Badiou’s brief discussion of the significance of name and Idea of “the Revolution” for the French Revolution, it’s still hard to grasp what self-belonging in a set-theoretical context has to do with the kind of rupture that structuralism was allegedly destined to reduce…except insofar as such self-belonging violates a relatively-dispensible axiom of axiomatic set theory.
I think the generous thing to do at this point is to allow an element of intrigue to enter the picture, and read on in the hope that the plot will eventually coalesce. But it may also be worth revisiting this moment of uncertainty, to discover whether it doesn’t after all contain some stumbling-block…
May 15th, 2006 at 7:08 am
“I find that this has something to do with the axiom of Choice” – hmm, suspect you’re being led up the garden path here. The matheme of the event is dodgy, IMO, though I think the basic insight that eventality = ill-foundedness is correct.
May 15th, 2006 at 8:13 am
wikipedia says:
Which suggests that ZFC implies the ill-foundedness of the event, whereas ZF/C may not…
May 15th, 2006 at 10:28 am
Ah right, see where you’re coming from now. This is a technical point – to prove (AF) => (no infinite descending epsilon chains) does not need choice, but to reverse the implication you need a very weak form of choice (countable dependent choices, aka DC): assume ~AF, pick a set with no founding element, pick an element of that, pick an element of that – hence construct an infinite descending epsilon chain… clearly you need a smidgin of AC to do all the successive pickings.
cf proofs that “infinite” “Dedekind infinite” for similar applications of DC
The point here is that you need DC to prove AF is equivalent to the (marginally more intuitive formulation) that there are no infinite descending epsilon chains (let’s call it AF*). In a set theory where AC fails and AF is weakened to AF* there is the possibility of a set that has no foundational element (so is “really” illfounded), but nevertheless slips through the net because the epsilon chain cannot be gathered into a set.
Hence we have something that should be “evental” that creeps into the realm of “being”, but only because our police/bureaucracy isn’t powerful enough to discern its eventality.
Not sure what this amounts to, tho.
May 15th, 2006 at 10:29 am
sorry that should read: “infinite” = “Dedekind infinite”