Reading Summary of Meditation 3 - paradoxes and critical decision
As it has been established in Meditation One, Badiou has affirmed the multiple as that which presents itself in presentation and the one as operation only in making consist what in Cantor's words are "absolutely infinite multiplicities, or inconsistent [multiples]" (p41). Badiou has recalled Plato (Meditation Two) as first thinking inconsistent multiplicity, though in its Platonic expression relegated to a speculative dream. In this Meditation—giving voice to the mathematicians Cantor, Frege, Russell, Zermelo, and Fraenkel—Badiou sketches a conceptual framework for unpacking Plato’s dream via the mathematical theory of the pure multiple known as “set theory.” The creation of Georg Cantor, "set theory" has to do with inclusion and exclusion--the relationships of membership--among objects or numbers taken as a unit or "set." This forming into a set should be understood as the product of the count-as-one. Christopher Norris, in his reader Badiou's Being and Event, notes the implications: "Thus sets are defined as products of the count-as-one, that is, the classificatory procedure that consists in grouping together a certain range of such entities and treating then as co-members of a single assemblage whatever their otherwise diverse natures or properties. The latter point is crucial in mathematical terms but also for Badiou's socio-political thinking since it allows the set theorist--or anyone who has truly absorbed its implications--to ignore any merely contingent or localized differences between such entities and accord them strictly equal status as regards their membership of any given set." Thus, any ontology (the science of being qua being) of the multiple must concede that it operates in structure, that is, via the count-as-one (the formation into a set). One then begins to see the direction Badiou is heading, to correlate the axiomatic rules (Zermelo-Fraenkel) of set theory to the non-being-one's constant manifestation as a count-as-one. To understand axiomatic set theory is to understand the folly of any attempt to totalize or comprehensively unify structure, or rather, the count-as-one.
Interestingly, consider Cantor's definition: "'By set what is understood is the grouping into a totality of quite distinct objects of our intuition or our thought'"(p38). From what has been said in the paragraph above it will become clear that Badiou does not accept Cantor's definition as such. He writes, "Without exaggeration, Cantor assembles in this definition every single concept whose decompostion is brought about by set theory: the concept of totality, of the object, of distinction, and that of intuition. What makes up a set is not a totalization, nor are its elements objects, nor may distinctions be made in some infinite collections of sets (without special axioms), nor can one possess the slightest intuition of each supposed element of a modestly 'large' set. 'Thought alone is adequate to the task, such that what remains of the Cantorian 'definition' basically takes us back...to Parmenides' aphorism: 'The same, itself, is both thinking and being'" (p38).
As Cantor's theory developed, "What was presented as an 'intuition of objects' was recast such that it could only be thought as the extension of a concept, or a of a property, itself expressible in a partially (or indeed completely, as in the work of Frege and Russell) formalized language" (p39). This language enabled advances in at least two distinct directions:
1. "It became possible to rigorously specify the notion of property, to formalize it by reducing it--for example--to the notion of a predicate in a first-order logical calculus, or to a formula with a free variable in a language with fixed constants. I can thus avoid, by means of restrictive constraints, the ambiguities in validation which ensue from the blurred borders of natural language" (p39).
2. "...it became legitimate to allow that for any formula with a free variable there corresponds the set of terms which validate it....Such security amounts to the following: control of language (of writing) equals control of the multiple....The speculative presupposition is that nothing of the multiple can occur in excess of a well-constructed language, as the referent-multiple of a a property, cannot cause a breakdown in the architecture of this language if the latter has been rigorously constructed. The master of words is also the master of the multiple" (p39-40).
Unfortunately, the two-fold thesis above is false. In fact, Russell's own paradox cut at the very core of set theory, though he himself is criticized for finding a rather self-serving "solution." To better understand the general context for these paradoxes in the development of set theory consider the reading: Christopher Norris On Formalized Language and Set Theory. Here it will be adequate to note Badiou’s critique: “It so happens that a multiplicity (a set) can only correspond to certain properties and certain formulas at the price of the destruction (the incoherency) of the very language in which these formulas are inscribed….In other words: the multiple does not allow its being to be prescribed from the standpoint of language alone. Or, to be more precise: I do not have the power to count as one, to count as ‘set’, everything which is subsumable by a property. It is not correct that for any formula λ(a) there is a corresponding one-set of terms for which λ(a) is true or demonstrable” (p40). In fact, this marks the second ruining of the definition of a set (both Cantor’s and Frege-Russell’s).
Considering its importance to Badiou’s schema, one would be wise to conceptually grasp Russell’s famous paradox (p40-1):
*Consider also the counter-examples such as “the set of everything I manage to define in less than twenty words,” which is found to be self-satisfying—“But it feels a bit like a joke."
** “Thus, forming a set out of all the sets a for which ~(a ∈ a) is true seems perfectly reasonable." And say that p (for paradoxical) is this set.
*** p = {a / ~(a ∈ a)}, or, all a’s such that a is not an element of itself. “But what can be said about his p?”
**** “If it contains itself as an element, p ∈ p, then it must have the property which defines its elements; that is, ~( p ∈ p)."
***** If it does not contain itself as an element, ~( p ∈ p), then it has the property which defines its elements: therefore, it is an element of itself: p ∈ p.
****** Finally, we have: ( p ∈ p) → ~( p ∈ p). “This equivalence of a statement and its negation annihilates the logical consistency of the language."
What we see here is essentially that set p is in excess of the "formal and deductive resources of language" (p41). What Badiou finds striking is that it is at such an impasse that Cantor "forces a way through with his doctrine of the absolute" (p41). Badiou writes, "If some multiplicities cannot be totalized, or 'concieved as a unity' without contradiction, [Cantor] declares, it is because they are absolutely infinite rather than transfinite (mathematical). Cantor does not step back from associating the absolute and inconsistency. There where the count-as-one fails, stands God" (p41). Badiou continues, "Cantor, essentially a theologian, therein ties the absoluteness of being not to the (consistent) presentation of the multiple, but to the transcendence through which a divine infinity in-consists, as one, gathering together and numbering any multiple whatsoever" (p42). Such is the decisive decision of ontologies of presence mentioned in Meditation One.
The real effects of the paradoxes are immediately of two orders:
2. "It is necessary to prohibit paradoxical multiple, which is to say the non-being whose ontological inconsistency has as its sign the ruin of language...a criterion of the set, which is excluded--is not correlate to formulas such as ~(a ∈ a), formulas which induce incoherency" (p43).
"Between 1908 and 1940 this double task was taken in hand by Zermelo and completed by Fraenkel, von Neumann and Godel. It was accomplished in the shape of the formal axiom system, the sytem in which, in a first order logic, the pure doctrine of the multiple is presented, such that it can still be used today to set out every branch of mathematics" (p43). Furthermore, this formal axiom system should not be seen as merely an "artifice of exposition" but an "intrinsic necessity" to set theory (p43). "Axiomatization is required such that the multiple, left to the implicitness of its counting rule, be delivered without concept, that is, whithout implying the being-of-the-one.
Badiou identifies three major characteristics of the Zermelo-Fraenkel axiomatization system (p44-6):
1. It's lexicon contains simply one relation, ∈ (belonging). "In particular this system excludes any construction of a symbol whose sense would be 'to be a set'." Moreover, the sign ∈, "unbeing of any one, determines, in a uniform manner, the presentation of 'something' as indexed to the multiple."
2. The Zermelo-Fraenkel system "immediately revokes a being, strictly speaking, a 'something' which is thereby disposed according to its multiple presentation. Furthermore, "the ZF system postulates that there is only one type of presentation of being: the multiple. The theory does not distinguish between 'objects' and 'sets.' That there is only one type of variable means: all is multiple, everything is a set."
3. "...a property only determines a multiple under the supposition that there is already a presented multiple...the axiom of separation (or of comprehension, or of sub-sets) provides for this. Consider further The Axiom of Separation.
As it has been established in Meditation One, Badiou has affirmed the multiple as that which presents itself in presentation and the one as operation only in making consist what in Cantor's words are "absolutely infinite multiplicities, or inconsistent [multiples]" (p41). Badiou has recalled Plato (Meditation Two) as first thinking inconsistent multiplicity, though in its Platonic expression relegated to a speculative dream. In this Meditation—giving voice to the mathematicians Cantor, Frege, Russell, Zermelo, and Fraenkel—Badiou sketches a conceptual framework for unpacking Plato’s dream via the mathematical theory of the pure multiple known as “set theory.” The creation of Georg Cantor, "set theory" has to do with inclusion and exclusion--the relationships of membership--among objects or numbers taken as a unit or "set." This forming into a set should be understood as the product of the count-as-one. Christopher Norris, in his reader Badiou's Being and Event, notes the implications: "Thus sets are defined as products of the count-as-one, that is, the classificatory procedure that consists in grouping together a certain range of such entities and treating then as co-members of a single assemblage whatever their otherwise diverse natures or properties. The latter point is crucial in mathematical terms but also for Badiou's socio-political thinking since it allows the set theorist--or anyone who has truly absorbed its implications--to ignore any merely contingent or localized differences between such entities and accord them strictly equal status as regards their membership of any given set." Thus, any ontology (the science of being qua being) of the multiple must concede that it operates in structure, that is, via the count-as-one (the formation into a set). One then begins to see the direction Badiou is heading, to correlate the axiomatic rules (Zermelo-Fraenkel) of set theory to the non-being-one's constant manifestation as a count-as-one. To understand axiomatic set theory is to understand the folly of any attempt to totalize or comprehensively unify structure, or rather, the count-as-one.
Interestingly, consider Cantor's definition: "'By set what is understood is the grouping into a totality of quite distinct objects of our intuition or our thought'"(p38). From what has been said in the paragraph above it will become clear that Badiou does not accept Cantor's definition as such. He writes, "Without exaggeration, Cantor assembles in this definition every single concept whose decompostion is brought about by set theory: the concept of totality, of the object, of distinction, and that of intuition. What makes up a set is not a totalization, nor are its elements objects, nor may distinctions be made in some infinite collections of sets (without special axioms), nor can one possess the slightest intuition of each supposed element of a modestly 'large' set. 'Thought alone is adequate to the task, such that what remains of the Cantorian 'definition' basically takes us back...to Parmenides' aphorism: 'The same, itself, is both thinking and being'" (p38).
As Cantor's theory developed, "What was presented as an 'intuition of objects' was recast such that it could only be thought as the extension of a concept, or a of a property, itself expressible in a partially (or indeed completely, as in the work of Frege and Russell) formalized language" (p39). This language enabled advances in at least two distinct directions:
1. "It became possible to rigorously specify the notion of property, to formalize it by reducing it--for example--to the notion of a predicate in a first-order logical calculus, or to a formula with a free variable in a language with fixed constants. I can thus avoid, by means of restrictive constraints, the ambiguities in validation which ensue from the blurred borders of natural language" (p39).
2. "...it became legitimate to allow that for any formula with a free variable there corresponds the set of terms which validate it....Such security amounts to the following: control of language (of writing) equals control of the multiple....The speculative presupposition is that nothing of the multiple can occur in excess of a well-constructed language, as the referent-multiple of a a property, cannot cause a breakdown in the architecture of this language if the latter has been rigorously constructed. The master of words is also the master of the multiple" (p39-40).
Unfortunately, the two-fold thesis above is false. In fact, Russell's own paradox cut at the very core of set theory, though he himself is criticized for finding a rather self-serving "solution." To better understand the general context for these paradoxes in the development of set theory consider the reading: Christopher Norris On Formalized Language and Set Theory. Here it will be adequate to note Badiou’s critique: “It so happens that a multiplicity (a set) can only correspond to certain properties and certain formulas at the price of the destruction (the incoherency) of the very language in which these formulas are inscribed….In other words: the multiple does not allow its being to be prescribed from the standpoint of language alone. Or, to be more precise: I do not have the power to count as one, to count as ‘set’, everything which is subsumable by a property. It is not correct that for any formula λ(a) there is a corresponding one-set of terms for which λ(a) is true or demonstrable” (p40). In fact, this marks the second ruining of the definition of a set (both Cantor’s and Frege-Russell’s).
Considering its importance to Badiou’s schema, one would be wise to conceptually grasp Russell’s famous paradox (p40-1):
*Consider ~(a ∈ a), or, a is a set which is not an element of itself. Such as “the set of all whole numbers is not itself a whole number."
*Consider also the counter-examples such as “the set of everything I manage to define in less than twenty words,” which is found to be self-satisfying—“But it feels a bit like a joke."
** “Thus, forming a set out of all the sets a for which ~(a ∈ a) is true seems perfectly reasonable." And say that p (for paradoxical) is this set.
*** p = {a / ~(a ∈ a)}, or, all a’s such that a is not an element of itself. “But what can be said about his p?”
**** “If it contains itself as an element, p ∈ p, then it must have the property which defines its elements; that is, ~( p ∈ p)."
***** If it does not contain itself as an element, ~( p ∈ p), then it has the property which defines its elements: therefore, it is an element of itself: p ∈ p.
****** Finally, we have: ( p ∈ p) → ~( p ∈ p). “This equivalence of a statement and its negation annihilates the logical consistency of the language."
What we see here is essentially that set p is in excess of the "formal and deductive resources of language" (p41). What Badiou finds striking is that it is at such an impasse that Cantor "forces a way through with his doctrine of the absolute" (p41). Badiou writes, "If some multiplicities cannot be totalized, or 'concieved as a unity' without contradiction, [Cantor] declares, it is because they are absolutely infinite rather than transfinite (mathematical). Cantor does not step back from associating the absolute and inconsistency. There where the count-as-one fails, stands God" (p41). Badiou continues, "Cantor, essentially a theologian, therein ties the absoluteness of being not to the (consistent) presentation of the multiple, but to the transcendence through which a divine infinity in-consists, as one, gathering together and numbering any multiple whatsoever" (p42). Such is the decisive decision of ontologies of presence mentioned in Meditation One.
The real effects of the paradoxes are immediately of two orders:
1. "It is necessary to abandon all hope of explicitly defining the notion of set. Neither intuition nor language are capable of supporting the pure multiple...these multiplicities are deployed in an axiom-system in which the property 'to be a set' does not figure" (p43).
2. "It is necessary to prohibit paradoxical multiple, which is to say the non-being whose ontological inconsistency has as its sign the ruin of language...a criterion of the set, which is excluded--is not correlate to formulas such as ~(a ∈ a), formulas which induce incoherency" (p43).
"Between 1908 and 1940 this double task was taken in hand by Zermelo and completed by Fraenkel, von Neumann and Godel. It was accomplished in the shape of the formal axiom system, the sytem in which, in a first order logic, the pure doctrine of the multiple is presented, such that it can still be used today to set out every branch of mathematics" (p43). Furthermore, this formal axiom system should not be seen as merely an "artifice of exposition" but an "intrinsic necessity" to set theory (p43). "Axiomatization is required such that the multiple, left to the implicitness of its counting rule, be delivered without concept, that is, whithout implying the being-of-the-one.
Badiou identifies three major characteristics of the Zermelo-Fraenkel axiomatization system (p44-6):
1. It's lexicon contains simply one relation, ∈ (belonging). "In particular this system excludes any construction of a symbol whose sense would be 'to be a set'." Moreover, the sign ∈, "unbeing of any one, determines, in a uniform manner, the presentation of 'something' as indexed to the multiple."
2. The Zermelo-Fraenkel system "immediately revokes a being, strictly speaking, a 'something' which is thereby disposed according to its multiple presentation. Furthermore, "the ZF system postulates that there is only one type of presentation of being: the multiple. The theory does not distinguish between 'objects' and 'sets.' That there is only one type of variable means: all is multiple, everything is a set."
3. "...a property only determines a multiple under the supposition that there is already a presented multiple...the axiom of separation (or of comprehension, or of sub-sets) provides for this. Consider further The Axiom of Separation.
"A crucial question remains unanswered....where is the absolutely initial point of being? Which initial multiple has its existence ensured such that the separating function of language can operate therein?" (p48). Thus Badiou concludes, "This is the whole problem of the subtractive suture of set theory to being qua being. It is a problem that language cannot avoid, and to which it leads us by foundering upon its paradoxical dissolution, the result of its own excess. Language--which provides for separations and compositions--cannot, alone, institue the existence of the pure multiple; it cannot ensure that what the theory presents is indeed presentation" (p48).
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