I just finished meditation 6, and my worst fears each time appear closer to becoming true. Following his formulation of the axiom of union, which states that:
"for every set, there exists the set of the elements of the elements of that set. That is, if α is presented, a certain β is also presented to which all the δ’s belong which also belong to some γ which belong to α. In other words: if γ∈α and δ∈γ, there exists a β such that δ∈β. The Multiple β gathers together the first dissemination of α, that obtained by decomposing into multiples of multiples which belong to it, thus un-counting α:
(Vα)(Eβ)[(δcβ) ↔ (Eγ)[(γcα) & (δcγ)]]"
This must follow to guarantee the consistency of the ZF system's condition that the relation of belonging to a set does not take place between elements qua individuals and sets; it takes place as the relation between sets and sets. What this implies is that for any set its members as such may be shown to be multiples themselves with other multiples as their own members. Consequentially, it is implied by the irreducibility of the multiple that for every set, another set may be formulated that includes the members of that first set as being themselves sets of another set of members, thus including the members of what the first set took as self-standing elements. We thus avoid the concentration of ontology in 'the one' multiplicity which can be counted as not being itself a set, thereby avoiding the taking of a set as being composed of elements (what Badiou thus terms 'containing the dissemination of the first set). However, this apparent constriction has its obverse offering in the axiom of the void set. At the very least, now everything turns on how Badiou will respond the second part of what he calls the ‘double-question’:
“(b) Is there a halting point- given that the process of dissemination, as we have just seen, appears to continue to infinity.”
Perhaps this knot devolves in what Alexei meant in his short note by calling attention to Badiou’s reluctance to bite the Fichtean bullet; that of asserting an infinite task. His preliminary offering of the ‘axiom of the void’ seems to be the ‘ad-hoc’ operator which can initiate the prevention of that consequence. And the way this is introduced seems too rushed, and suspiciously so:
“The solution to the problem is quite striking: maintain the position that nothing is delivered by the law of the ideas, but make this nothing be through the assumption of a proper name. In other words, verify via the exendrary choice of a proper name, the unpresentable alone as existent; on its basis the ideas will subsequently cause all forms of presentation to proceed.”
Now, perhaps I’m missing something crucial, but isn’t this merely saying ‘the axiomatic system to follow will rest by the implicit reference to a nothing, explicited only through a proper name; that is to say, with no positive account of its contents.” Of course, since presumably there is no ‘content’ to the pure multiple, in the sense that it could be accounted for explicitly by a consistent multiplicity. That is to say, the void set functions precisely in that empty term which exists only as that which both (a) doesn't belong to any set and (b) has no members and thus sets as members of its own. In what sense, should we then, take this set to exist? It exists in the sense that the name itself cannot stand without itself being quantified existentially as excluding any relation to belonging. In other words, it cannot stand as a free variable being occurring freely in a formula in which the other terms are quantified, and thus cannot be be assigned a property, i.e. it cannot function as an operator for the count-as-one. This is why the void set is defined thus:
(Eβ)[ ¬(Eα)(α c β)]
But this is approaching operational dogmatism: the axiomatic rule must be followed on the basis of a term for which not only we cannot account for- but that even attempting to account to for it becomes explicitly prohibited by its own principle. The interesting result is that this is meant to prevent the boring pseudo-Kantian impasse that we have no ‘access to the thing itself’; or put in Badiou’s own language, a consistent multiple that can serve, in its count as one (and thus affirmed consistency), as primary with respect to all other multiplicities.
Yet this seems like a transcendental copout, having designated this void as an empty term to which no multiple belongs (thus pure difference, indifferent to content). How is this anything but a formalized paraphrase of ‘that which transcends the phenomenal’ being, by definition, non-graspable by thought/language? One could without much trouble read this like the strict Kantian definition of finitude as transcendental horizon (as Heidegger does).
“There exists that to which no existence can be said to belong”, “the unpresentable is presented, as a subtractive term of the presentation of presentation”, or “a multiple exists which is subtracted from the primitive idea of the multiple”.
But then this multiple which does not conform to the idea of the multiple has the operational content of deferring its belonging to multitudes; to sets as such. This is too much- since it guarantees that as soon as we try to even utter something about the void set we already violate its own law. This is a fantastic way to use the ‘proper name’ as a philosophical deity, in the strict sense of ontotheological. Of course, the system might thereby show its consistency pretty faultlessly afterwards: that assumption in place, the rest becomes almost uninterestingly valid.
The void as that which doesn’t belong, which subsists without the quality of belonging only proper to the multiple makes this a term which hushes any objector in situ. Not only we cannot define the void, but that we can all of a sudden use it as the backbone for our entire axiomatic system without ever calling into question its consistency (for, of course, it has none).
In short, I think this (again) is a blend of the two fantasies of philosophy: the scientific rigor of inductive principles and the space for an ‘unaccountable’ term which would put an end to all pretensions of essentialism (differance, ontological difference, the inconsistent multiple). I’m not sure what to make of this, but I will nonetheless see what possibilities this offers structurally. It might not be the most interesting approach I’ve read (it’s not hard to see why Dreyfus would want to tear this guy apart) but he’s make a good job of making himself noticeable by pissing off everyone off. The analytics will abhor this intrusion of nothingness as worthy of consideration (and consider it ad hoc); continentals, especially of Heideggerean/Derridean tendency, will be appalled at pretences of the axiomatic structure that will follow. This is very nicely identifiable in the following passage:
“There are not ’several’ voids, there is only one void, rather than signifying the presentation of the one, this signifies the unicity of the unpresentable such as marked within presentation”.
This has been appropriated by Zizek through his own notion of the ‘parallax’ as designating either ‘the empty place without content’ or ‘the excess of content for which no place occurs’. Either we take this ‘void’ as an empty term for which no content can be attributed, or we take it as that which cannot be captured by the structure of (consistent) multiplicities and thus ontology; as an excess. Of course, in strict Badiouean nomenclature, saying this much would already be too much. Perhaps this is why the privilege granted to the ZF system appears suspicious from the start. Guess we'll have to see...
"for every set, there exists the set of the elements of the elements of that set. That is, if α is presented, a certain β is also presented to which all the δ’s belong which also belong to some γ which belong to α. In other words: if γ∈α and δ∈γ, there exists a β such that δ∈β. The Multiple β gathers together the first dissemination of α, that obtained by decomposing into multiples of multiples which belong to it, thus un-counting α:
(Vα)(Eβ)[(δcβ) ↔ (Eγ)[(γcα) & (δcγ)]]"
This must follow to guarantee the consistency of the ZF system's condition that the relation of belonging to a set does not take place between elements qua individuals and sets; it takes place as the relation between sets and sets. What this implies is that for any set its members as such may be shown to be multiples themselves with other multiples as their own members. Consequentially, it is implied by the irreducibility of the multiple that for every set, another set may be formulated that includes the members of that first set as being themselves sets of another set of members, thus including the members of what the first set took as self-standing elements. We thus avoid the concentration of ontology in 'the one' multiplicity which can be counted as not being itself a set, thereby avoiding the taking of a set as being composed of elements (what Badiou thus terms 'containing the dissemination of the first set). However, this apparent constriction has its obverse offering in the axiom of the void set. At the very least, now everything turns on how Badiou will respond the second part of what he calls the ‘double-question’:
“(b) Is there a halting point- given that the process of dissemination, as we have just seen, appears to continue to infinity.”
Perhaps this knot devolves in what Alexei meant in his short note by calling attention to Badiou’s reluctance to bite the Fichtean bullet; that of asserting an infinite task. His preliminary offering of the ‘axiom of the void’ seems to be the ‘ad-hoc’ operator which can initiate the prevention of that consequence. And the way this is introduced seems too rushed, and suspiciously so:
“The solution to the problem is quite striking: maintain the position that nothing is delivered by the law of the ideas, but make this nothing be through the assumption of a proper name. In other words, verify via the exendrary choice of a proper name, the unpresentable alone as existent; on its basis the ideas will subsequently cause all forms of presentation to proceed.”
Now, perhaps I’m missing something crucial, but isn’t this merely saying ‘the axiomatic system to follow will rest by the implicit reference to a nothing, explicited only through a proper name; that is to say, with no positive account of its contents.” Of course, since presumably there is no ‘content’ to the pure multiple, in the sense that it could be accounted for explicitly by a consistent multiplicity. That is to say, the void set functions precisely in that empty term which exists only as that which both (a) doesn't belong to any set and (b) has no members and thus sets as members of its own. In what sense, should we then, take this set to exist? It exists in the sense that the name itself cannot stand without itself being quantified existentially as excluding any relation to belonging. In other words, it cannot stand as a free variable being occurring freely in a formula in which the other terms are quantified, and thus cannot be be assigned a property, i.e. it cannot function as an operator for the count-as-one. This is why the void set is defined thus:
(Eβ)[ ¬(Eα)(α c β)]
But this is approaching operational dogmatism: the axiomatic rule must be followed on the basis of a term for which not only we cannot account for- but that even attempting to account to for it becomes explicitly prohibited by its own principle. The interesting result is that this is meant to prevent the boring pseudo-Kantian impasse that we have no ‘access to the thing itself’; or put in Badiou’s own language, a consistent multiple that can serve, in its count as one (and thus affirmed consistency), as primary with respect to all other multiplicities.
Yet this seems like a transcendental copout, having designated this void as an empty term to which no multiple belongs (thus pure difference, indifferent to content). How is this anything but a formalized paraphrase of ‘that which transcends the phenomenal’ being, by definition, non-graspable by thought/language? One could without much trouble read this like the strict Kantian definition of finitude as transcendental horizon (as Heidegger does).
“There exists that to which no existence can be said to belong”, “the unpresentable is presented, as a subtractive term of the presentation of presentation”, or “a multiple exists which is subtracted from the primitive idea of the multiple”.
But then this multiple which does not conform to the idea of the multiple has the operational content of deferring its belonging to multitudes; to sets as such. This is too much- since it guarantees that as soon as we try to even utter something about the void set we already violate its own law. This is a fantastic way to use the ‘proper name’ as a philosophical deity, in the strict sense of ontotheological. Of course, the system might thereby show its consistency pretty faultlessly afterwards: that assumption in place, the rest becomes almost uninterestingly valid.
The void as that which doesn’t belong, which subsists without the quality of belonging only proper to the multiple makes this a term which hushes any objector in situ. Not only we cannot define the void, but that we can all of a sudden use it as the backbone for our entire axiomatic system without ever calling into question its consistency (for, of course, it has none).
In short, I think this (again) is a blend of the two fantasies of philosophy: the scientific rigor of inductive principles and the space for an ‘unaccountable’ term which would put an end to all pretensions of essentialism (differance, ontological difference, the inconsistent multiple). I’m not sure what to make of this, but I will nonetheless see what possibilities this offers structurally. It might not be the most interesting approach I’ve read (it’s not hard to see why Dreyfus would want to tear this guy apart) but he’s make a good job of making himself noticeable by pissing off everyone off. The analytics will abhor this intrusion of nothingness as worthy of consideration (and consider it ad hoc); continentals, especially of Heideggerean/Derridean tendency, will be appalled at pretences of the axiomatic structure that will follow. This is very nicely identifiable in the following passage:
“There are not ’several’ voids, there is only one void, rather than signifying the presentation of the one, this signifies the unicity of the unpresentable such as marked within presentation”.
This has been appropriated by Zizek through his own notion of the ‘parallax’ as designating either ‘the empty place without content’ or ‘the excess of content for which no place occurs’. Either we take this ‘void’ as an empty term for which no content can be attributed, or we take it as that which cannot be captured by the structure of (consistent) multiplicities and thus ontology; as an excess. Of course, in strict Badiouean nomenclature, saying this much would already be too much. Perhaps this is why the privilege granted to the ZF system appears suspicious from the start. Guess we'll have to see...
