As far as the “restricted action”, I am not aware of this particular phrase coming up in anything that I have read, though it is very possible that I passed over it without noticing. It does make sense if one were to think about it in relation to Badiou’s “truth procedure”, or his version of a ‘militant engagement’. Every event, in order to be revealed (though nothing is really ‘revealed’) in a situation required the fidelity of a subject, and likewise the careful, calculated, bit-by-bit piecing together of the consequences of a truth. I would understand that to be a sort of “restricted action”, since, as such, it would not be reactionary. It is also important to note that ’situation’ has a very different meaning for Badiou than it would for say, the Situationist.
Happy Holidays as well,
Keith

One question: In a bit of Infinite Thought I read today there’s a reference to Mallarme and the phrase “restrained action”. This same phrase appears in a text that I really love, the manifesto of the Malgre Tout collective, a Franco-Argentine group connected with someone named Miguel Benasayag. Friends of mine translated it and are trying to find a home for it, they translate this phrase as “restricted action”, I’m going to write them to say perhaps “restrained” is a better phrasing. Anyway, does Badiou use this term elsewhere? The term ’situation’ also occurs in a number of excellent contemporary Argentine left-theory works, that’s one of the main reasons I’ve started reading Badiou (that and some other friends of mine).
happy holidays,
Nate

A little of both, actually. Somewhere in Theoretical Writings he says that all philosophers should stop what they are doing to become mathematicians – though I am sure he doesn’t say this without a bit of a grin and some silent laughter. I am still working with most of this myself , often forgetting, often misreading, but will hazard some input.

You can go to irrational numbers to find pdf’s of Badiou’s Number and Numbers ,where he goes through his own genealogy of set theory, and a transcription of a recent talk given at Birkbeck, where he discusses some of the political importance for applications of Cohen’s generic sets. You are right to point out his use of set theory in the context of his polemic with Deleuze – which is also part of the reason for his insistence that mathematics is an ontology (or rather, simply just is ontology). Badiou considers it a task of any modern philosophy to engage in a thought of multiplicity, which is precisely what Deleuze set out to do. However, Deleuze was to mistake sets as always being numerical, which they are not. Badiou’s own ‘taste’, I suppose, led him via set theory, to reject any form of ‘vitalist’ philosophy or the empirical givenness of sets. It is also in line with his desire to take philosophy and ontology away from ‘approximations’, ‘post-modernism’, and what he terms ‘idealinguistry’. Multiplicity happens to be the only thing that set theory is able to ‘write’ (and there are here some dense Lacanian references for Badiou that at present I do not adequately understand). Set theory therefore deals with being as pure multiplicity, as the question it always answers but never asks. The following I have taken from a letter to Dedekind written by Cantor that appears in From Frege to Godel - definitely not a book to consider as any sort of introduction to set theory and was personally the cause of quite a few headaches:

“If we start from the notion of a definite multiplicity (a system, a totality) of things, it is necessary, as I discovered, to distinguish two kinds of multiplicities (by this I always mean definite multiplicities).

For a multiplicity can be such that the assumption that all of its elements “are together” leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as “one finished thing”. Such multiplicities I call absolutely infinite or inconsistent multiplicities.

As we can readily see, the “totality of everything thinkable”, for example, is such a multiplicity; […]

If on the other hand the totality of the elements of a multiplicity can be thought of without contradiction as “being together”, so that they can be gathered together into “one thing”, I call it a consistent multiplicity or a “set”.”

There is also something crucial missing from your post relating to Badiou’s use of set theory that has to do with how he conceptualizes (and insits on the reality of) truth(s). I don’t want to take up too much space with this so I’ll just skip through some things. The null set (zero, the void, nothingness) is universally included in every set, such that the number three would be written: (0,(0), (0,(0))). Zero and the infinite - neither of which are deducible and remain axiomatic decisions - are synonymous here (at least, that is, in the sense of their having neither inside nor outside). A truth, or event (which does require a subject), reveals that which is the void or infinity of every situation – as that which ‘departs from the set’ thus rendering it ‘momentarily’ inconsitent. It’s that ‘something which was not previously counted in a situation comes to be counted’, i.e. named. I suppose it is something like a ‘potential’ event though Badiou rejects any existence of the virtual. In set theory there is always an excess of how a set can be represented vs. what it presents, which is an actual excess. Badiou always likes to give the example of an amorous encounter: if someone doesn’t notice it, doesn’t name it, then it doesn’t happen and was not an event. This would not, however, prevent it from existing in the situation, as a possibility I suppose. If it is named (the ‘I love you’), then the two faithful subjects go about constructing their world via a ‘truth procedure’ which changes the knowledge of the situation, retroactively. That’s all I can think to write at the moment. Fortunately, the ever-slow translation of Being and Event is near publication, and I suspect a lot of these questions will find answers therein.

The set of DVDs I own that are movies starring Jim Carrey = {}

The set of books by Ayn Rand that I own = {}

Note that this means the books by Ayn Rand that I own are the same as the movies with Jim Carrey that I own.

I don’t know if the good old empty set can illuminate the philosophical nothing (Heidegger’s Das nicht, is that what he called it?) at all. The empty set is…well, something. It’s a set. What’s a set? A collection of things—most of the time. There are formalizations that go below this level, but I never learned them.

Tzuchien writes: The set theoretic apporach to the question of being and nothing is a unique one in that it doesn’t really bother itself with being a language which conforms with our usual semantic and conceptual difficulties with “being and nothing” and such… rather it forces practioners to speak its language. I’ll buy that, at least the last sentence. It’s hard for me to reflect on what these things might be, but I know what I was taught they could do—at least with respect to each other.

Finally, as regards a preview button, I don’t WordPress, so I couldn’t say. I’m surprised it doesn’t have it, though.

I think Badiou’s interests span this and more, focusing at times on the axioms of set theory itself… esp. on the two of them: the axiom of choice and the continuum hypothesis. Therein lie the formalization of “being” and “event” between the “being” of mathematics and the “event” of construction.
The set theoretic apporach to the question of being and nothing is a unique one in that it doesn’t really bother itself with being a language which conforms with our usual semantic and conceptual difficulties with “being and nothing” and such… rather it forces practioners to speak its language. There are real problems when the null set is excluded as a set. Things either get really complicated or … well… inconsistent. Frege famously used the null set to set the parameters on what the number “1″ means…
The beautiful thing about set theory is that the “sets” aren’t really objects at all, they define an indefinite parameters of possible objects. So, all the things that could ever become associated with the arrangement of “1″ or “73″ or “331″: trees, giraffes, icecream cones, etc. So it rather transforms/rejects “predicate” oriented forms of thinking about so-called “properties.” You might consider the ontology of “possible worlds” as a radical version of this rejection. Thus, in a way, Badiou opens a place to talk about things without “predicates” but rather “names,” with an organization of singularities rather than an organ of interrelated predicate relations. “Formalization” is not a language at all, mathematics, in the tradition of set-theory seeks to get away from the hegemony of discursive units and discouse in general.

My basic question with all this re: Badiou, particularly the null set and attendant categories (void, nothing), is what this stuff means/is used for outside set theory/math/formal languages. In some sense, I’m not sure I follow what an empty set could mean outside a formal language. As I remember this is one way to describe one of Schelling’s disagreements w/ Hegel, about being and nothing. As I remember it from a course I had in 1999, the argument was presented, basically, as a question of whether being and nothing are predicates like all others or special predicates (the conditions of predication). I think this turned into a question on what one means by nothing, what a thought of nothing might actually be - to my mind, it can’t be, by definition: to say “nothing is…”, by use of the word ‘is’ doesn’t get outside of being and into nothing. The closest there could be would something like “an object which is black and white in all the same places at exactly the same time”, a sort of null set in the sense of one just can’t think it and sort of stops thinking (rather like what I’m told koans are/were supposed to do in zen).

A preview button is a great idea. I’m not very tech savvy. Do you have any idea how I could put one in, or is there a resource you could direct me to to find out? I already pester my computer-knowing friends too much as it is.

happy holidays,
Nate

A = {1, 2, 3}
B = {3, 4, 5}
C = {4, 5, 6}

A ∪ B = {1, 2, 3, 4, 5}
A ∩ B = {3}
A ∪ C = {1, 2, 3, 4, 5, 6}
A ∩ C = {}

I hope that looks better (is there a way to put a preview button here?) Anyway, the null set is not some claim about the world, but a possible collection—specifically, the one with nothing in it. And yes, it is different from all sets with elements, but that’s nothing to get worked up over.

The null set is the basis of the formal construction of the natural numbers N = {0, 1, 2, 3, 4, 5, …} If numbers aren’t primitive objects how do you build them from sets? Let’s build a set of certain oddball sets. We do this by saying that the null set is in the set and that all the other members can be generated by repeated applications of the formula (I know that “repeated applications of the formula” is vauge and unformalized, but trust me, it can be expressed in terms of sets) S(X) = X ∪ {X} So
S({}) = {{}}
S ({{}}) = {{{}}, {{{}}}}}
S ({{{}}, {{{}}}}} = {{{{}}, {{{}}}}}, {{{{}}, {{{}}}}}}}
I realize that looks like gibberish (and I may have unbalanced brackets there), but the Set N of {} and other sets generated by repeated applications of the rule S(X) = X ∪ {X} has some interesting properties: in particular, certain set opeations and relations on it look exactly like the basic operations and relations of arithmatic. For example, if j and k are both in N then either j ∈ k, j = k, or j ∋ k But it gets better: if j ∈ k then k is the results of some applications of S(j). That means for elements of N ∈ behaves just like < for the natural numbers as we know and love them….

Suddenly, I think I may be causing more harm than good. Email me if this is of any interest. Otherwise, apologies.