Reading Summary of Meditation 7 -
This pregnant meditation is perhaps the most revealing of those which make up the fist half of Badiou's magnum opus. For the theologically minded reader, the logical-mathematical framework can be grasped with some work and repetition of thought. However, it will be absolutely necessary to grasp the extra-mathematical themes in this meditation--which will, in fact, illuminate the framework of Badiou's socio-political thought (moreover, his entire direction of thought). From Badiou's reliance on the terms of set-theoretical axiomatic ontology, emerges a precise and compelling conceptual framework about the way in which human "sets" or "multiples" are related.
1. Belonging and Inclusion: "...set theory distinguishes two possible relations between multiples. There is the originary relation, belonging, written ∈, which indicates that a multiple is counted as element in the presentation of another multiple. But there is also the relation of inclusion, written ⊂, which indicates that a multiple is a sub-multiple of another multiple: we made reference to this relation [Meditation Five] in regard to the power-set axiom. To recap, the writing of β ⊂ α, which reads β is included in α, or β is a subset of α, signifies that every multiple which belongs to β also belongs to α: (∀γ) [(γ∈β)→(γ∈α)]" (p81-82). It is the distinction between belonging and inclusion which is paramount for understanding the "gap"--which is the "impasse" of being.
If β belongs to α then β is termed an element of α. If γ is included in α then γ is termed a subset of α. "But these determinations--element and subset--do not allow one to think anything intrinsic. In every case, the element β and the subset γ are pure multiples" (p82). Again, it is the power-set axiom which "helps to clarify the ontological neutrality of the distinction between belonging and inclusion" (p82).
Axiomatic set theory allows one to imagine belonging and inclusion to represent a separate and distinct count-as-one. For instance, take set α {1,2,3}, then it is the case that the singletons {1}, {2}, and {3} belong (or are elements) of set α. However, the power set axiom decides a recount (a count of all possible subsets) of the original set α, this recount will be named p(α). The set p(α) would read {{∅}, α itself, {1,2}, {1,3}, {2,3}}, therefore if we term {1,2} as γ, one may say γ is included in α. Notice the two distinct counts. The first count-as-one, congruent with structure itself, is the set α {1,2,3}. The second count-as-one, which gathers all inclusions, is the set p(α) or, {{∅}, α itself, {1,2}, {1,3}, {2,3}}. In fact, α and p(α) are distinct multiples. "...this second count, despite being related to α, is absolutely distinct from α itself. It is therefore a metastructure, another count, which 'completes' the first in that it gathers together all sub-compositions of internal multiples, all the inclusions. The power-set axiom posits that this second count, this metastructure, always exists if the first count, or presentative structure, exists" (p83). "However, there is an immediate consequence of this decision: the gap between structure and metastructure between element and subset, between belonging and inclusion, is a permanent question for thought, an intellectual provocation of being" (p84). Therefore, "belonging and inclusion, in the order of being-existent, are irreducibly disjunct" (p84).
2. The Theorem of the Point of Excess: One conceptual problem which emerges from the power-set axiom is the excess of p(α) over α itself, p(α) is "larger than the initial multiple" (p84). Another way of saying this is that p(α) is "an operation in absolute excess of the situation itself" (p84). The theorem of the point of excess establishes that at least one "multiple" or "set" is included in p(α) but does not belong to α. Again, one is to think the gap between presentation (elements counted-as-one) and representation (the gathering of all inclusions into a count-as-one).
3. The Void and the Excess: Suppose we examine {∅} rather than α. How might one think the subset of the empty-set? Badiou argues that the void maintains "with the concept of inclusion two relations that are essentially new with respect to the nullity of its relation with belonging...
A. "the void is a subset of any set; it is universally included..."
B. "the void possesses a subset, which is the void itself" (p86)
The first of these two relations has been substantiated above and in Meditation Four and Meditation Five. It amounts to saying that ∅ is included in every set. Note the inclusion of ∅ in our hypothetical set p(α) above.
The second of these relations appears slightly more enigmatic. "This is because intuitively, and guided by the deficient vocabulary which shoddily distinguishes, via the vague image of 'being-inside', between belonging and inclusion, its seems as though we have, by this inclusion, 'filled' the void with something. But this is not the case. Only belonging, ∈, the unique and supreme Idea of the presented-multiple, 'fills' presentation. Moreover, it would indeed be absurd to imagine that the void can belong to itself--which would be written ∅ ∈ ∅--because nothing belongs to it. But in reality the statement ∅ ⊂ ∅ solely announces that everything which is presented, including the proper name of the unpresentable, forms a subset of itself, the 'maximal' subset" (p88).
4. One, Count-As-One, Unicity and Forming-Into-One: As you will recall from Meditation One, the one is not. The count-as-one, or rather, the result of any structured presentation is merely a "nominal seal of the multiple" (p90). "As for unicity, it is not a being, but a predicate of the multiple. It belongs to the regime of the same and the other, such as its law is instituted by any structure. A multiple is unique inasmuch as it its other than any other. The theologians, besides, already knew that the thesis 'God is One' is quite different from the thesis 'God is unique.' In Christian theology, for example, the triplicity of the person of God is internal to the dialectic of the One, but it never affects his unicity (mono-theism)" (p90). The fourth, phantom meaning, of the one is termed a forming-into-one. It is not that dissimilar to the count-as-one. It may be helpful to draw once again from the power-set axiom. As one will recall, there is a second count, a count of the count which gathers all inclusions into the one-result--noted above as p(α). In all cases the one is a "retroactive fiction" of the count (p91).
This pregnant meditation is perhaps the most revealing of those which make up the fist half of Badiou's magnum opus. For the theologically minded reader, the logical-mathematical framework can be grasped with some work and repetition of thought. However, it will be absolutely necessary to grasp the extra-mathematical themes in this meditation--which will, in fact, illuminate the framework of Badiou's socio-political thought (moreover, his entire direction of thought). From Badiou's reliance on the terms of set-theoretical axiomatic ontology, emerges a precise and compelling conceptual framework about the way in which human "sets" or "multiples" are related.
1. Belonging and Inclusion: "...set theory distinguishes two possible relations between multiples. There is the originary relation, belonging, written ∈, which indicates that a multiple is counted as element in the presentation of another multiple. But there is also the relation of inclusion, written ⊂, which indicates that a multiple is a sub-multiple of another multiple: we made reference to this relation [Meditation Five] in regard to the power-set axiom. To recap, the writing of β ⊂ α, which reads β is included in α, or β is a subset of α, signifies that every multiple which belongs to β also belongs to α: (∀γ) [(γ∈β)→(γ∈α)]" (p81-82). It is the distinction between belonging and inclusion which is paramount for understanding the "gap"--which is the "impasse" of being.
If β belongs to α then β is termed an element of α. If γ is included in α then γ is termed a subset of α. "But these determinations--element and subset--do not allow one to think anything intrinsic. In every case, the element β and the subset γ are pure multiples" (p82). Again, it is the power-set axiom which "helps to clarify the ontological neutrality of the distinction between belonging and inclusion" (p82).
Axiomatic set theory allows one to imagine belonging and inclusion to represent a separate and distinct count-as-one. For instance, take set α {1,2,3}, then it is the case that the singletons {1}, {2}, and {3} belong (or are elements) of set α. However, the power set axiom decides a recount (a count of all possible subsets) of the original set α, this recount will be named p(α). The set p(α) would read {{∅}, α itself, {1,2}, {1,3}, {2,3}}, therefore if we term {1,2} as γ, one may say γ is included in α. Notice the two distinct counts. The first count-as-one, congruent with structure itself, is the set α {1,2,3}. The second count-as-one, which gathers all inclusions, is the set p(α) or, {{∅}, α itself, {1,2}, {1,3}, {2,3}}. In fact, α and p(α) are distinct multiples. "...this second count, despite being related to α, is absolutely distinct from α itself. It is therefore a metastructure, another count, which 'completes' the first in that it gathers together all sub-compositions of internal multiples, all the inclusions. The power-set axiom posits that this second count, this metastructure, always exists if the first count, or presentative structure, exists" (p83). "However, there is an immediate consequence of this decision: the gap between structure and metastructure between element and subset, between belonging and inclusion, is a permanent question for thought, an intellectual provocation of being" (p84). Therefore, "belonging and inclusion, in the order of being-existent, are irreducibly disjunct" (p84).
2. The Theorem of the Point of Excess: One conceptual problem which emerges from the power-set axiom is the excess of p(α) over α itself, p(α) is "larger than the initial multiple" (p84). Another way of saying this is that p(α) is "an operation in absolute excess of the situation itself" (p84). The theorem of the point of excess establishes that at least one "multiple" or "set" is included in p(α) but does not belong to α. Again, one is to think the gap between presentation (elements counted-as-one) and representation (the gathering of all inclusions into a count-as-one).
3. The Void and the Excess: Suppose we examine {∅} rather than α. How might one think the subset of the empty-set? Badiou argues that the void maintains "with the concept of inclusion two relations that are essentially new with respect to the nullity of its relation with belonging...
A. "the void is a subset of any set; it is universally included..."
B. "the void possesses a subset, which is the void itself" (p86)
The first of these two relations has been substantiated above and in Meditation Four and Meditation Five. It amounts to saying that ∅ is included in every set. Note the inclusion of ∅ in our hypothetical set p(α) above.
The second of these relations appears slightly more enigmatic. "This is because intuitively, and guided by the deficient vocabulary which shoddily distinguishes, via the vague image of 'being-inside', between belonging and inclusion, its seems as though we have, by this inclusion, 'filled' the void with something. But this is not the case. Only belonging, ∈, the unique and supreme Idea of the presented-multiple, 'fills' presentation. Moreover, it would indeed be absurd to imagine that the void can belong to itself--which would be written ∅ ∈ ∅--because nothing belongs to it. But in reality the statement ∅ ⊂ ∅ solely announces that everything which is presented, including the proper name of the unpresentable, forms a subset of itself, the 'maximal' subset" (p88).
4. One, Count-As-One, Unicity and Forming-Into-One: As you will recall from Meditation One, the one is not. The count-as-one, or rather, the result of any structured presentation is merely a "nominal seal of the multiple" (p90). "As for unicity, it is not a being, but a predicate of the multiple. It belongs to the regime of the same and the other, such as its law is instituted by any structure. A multiple is unique inasmuch as it its other than any other. The theologians, besides, already knew that the thesis 'God is One' is quite different from the thesis 'God is unique.' In Christian theology, for example, the triplicity of the person of God is internal to the dialectic of the One, but it never affects his unicity (mono-theism)" (p90). The fourth, phantom meaning, of the one is termed a forming-into-one. It is not that dissimilar to the count-as-one. It may be helpful to draw once again from the power-set axiom. As one will recall, there is a second count, a count of the count which gathers all inclusions into the one-result--noted above as p(α). In all cases the one is a "retroactive fiction" of the count (p91).
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