Reading Summary of Meditation 5 -
It should be fairly clear by now that Badiou is arguing that in the "execution" of any ontology--or rather, in the "mathematical theory of the multiple, or set theory"--the "grand Ideas" of the multiple must conform to a "system of axioms" which excludes "any explicit definition of the multiple" (p60). In fact, nine axioms make up this "economy of presentation" which allows Badiou to examine the first principles of being itself. Here in Meditation Five, Badiou introduces five additional axioms which have a "strictly operational value" so as to "think the singularity of this existential statement on the void" (p60). Because these axioms will be utilized in recurring fashion, I shall be brief in introducing the axioms of Extensionality, Powerset, Union, Separation, and Replacement before turning again to the Axiom of the Void-Set.
1. The Axiom of Extensionality
"The axiom of extensionality posits that two sets are equal (identical) if the multiples of which they are multiple, the multiples whose set-theoretical count as one they ensure, are 'the same'....The axiom...thus amounts to saying: if every multiple presented in the presentation of α is presented in that of β, and the inverse, then these two multiples, α and β, are the same....The logical architecture of the axiom....indicates....The identity of multiples is founded on the indifference of belonging" (p60-61).
This is written: (∀γ) [(γ∈α)↔(γ∈β)]→(α=β), which reads: for every multiple γ, it is equivalent and thus indifferent to belong to α or to belong to β, then α and β are indistinguishable and interchangeable.
This does not constrain us from saying there is something "different", α and β have differential markings, thus α and β must "depend on what belongs to their presentations" (p61). Badiou writes: "What possible source could there be for the existence of difference, if not that of a multiple lacking from a multiple? No particular quality can be of use to us to mark difference here, not even that the one can be distinguished from the multiple, because the one is not. What the axiom of extension does is reduce the same and the other to the strict rigour of the count such that it structures the presentation of presentation. The same is the same of the count of multiples from which all multiples are composed, once counted as one...." (p 61). This axiom of the same and the different, or the same and the other, doesn't tell us if anything exists, it simply establishes for any existent multiple, "the canonical rule of its differentiation" (p61).
2. The Axiom of the Powerset (the set of subsets)
"This axiom affirms that given a set, the subsets of that set can be counted-as-one: they are a set....[Therefore,] the logical structure of this axiom is not one of equivalence but of implication" (p62). Considering set B is a subset of α:
It is written: β is a subset of α, or β is included in α; which is an abbreviation for: (∀γ) [(γ∈β)→(γ∈α)], which reads, for all γ, if γ belongs to β, then γ belongs to α. This defines the relation of inclusion between β and α.
Much more will be said about belonging and inclusion in Meditation Seven and Meditation Eight.
3. The Axiom of Union
This axiom seeks to answer the question: "Does the count-as-one extend to decompositions? Is there an axiom of dissemination just as there is one of composition [Axiom of the Powerset]?" (p63). Badiou notes, "Since a multiple is a multiple of multiples, it is legitimate to ask if the power of the count via which a multiple is presented also extends to the unfolded presentation of the multiples which compose it, grasped in turn as multiples of multiples. Can one internally disseminate the multiples out of which a multiple makes the one of the result?" (p63). The Axiom of Union therefore states "that each step of the dissemination is counted as one" (p63).
It is written: (∀α)(∃β)[(δ∈β)↔(∃γ)[(γ∈α)& (δ∈γ)]] which reads, “…if α is presented, a certain β is also presented to which all the δ’s belong which also belong to some y which belongs to α” (p64). Furthermore, the "multiple β gathers together the first dissemination of α, that obtained by decomposing into multiple the multiples which belong to it, thus by un-counting α.
“Given α, the set β whose existence is affirmed here will be written ∪α (union of α)” (p64). Badiou concludes, "The fundamental homogeneity of being is supposed henceforth on the basis that ∪α, which disseminates the initial one-multiple and then counts as one what is thereby disseminated, is no more or less a multiple itself than the initial set. Just like the powerset, the union set does not in any way remove us from the concept-less reign of the multiple....Ontology announces herein neither One, nor All, nor Atom, solely the uniform axiomatic count-as-one of multiples" (p64).
4. The Axiom of Separation
See, Axiom of Separation.
5. The Axiom of Replacement
This axiom is really quite simple. "The idea--singular, profound--is the following: if the count-as-one operates by giving consistency of being-one multiple to some multiples, it will also operate if these multiples are replaced, term by term, by others. This is equivalent to saying that the consistency of a multiple does not depend upon the particular multiples whose multiple it is. Change the multiples and the one-consistency--which is a result--remains, as long as you operate, however, your substitution multiple for multiple" (p65). Moreover, "the axiom of replacement is suited--even to the point of over-indicating it--to the mathematical situation being presentation of the pure presentative form in which being occurs that-which-is" (p65).
6. The Axiom of the Void-Set, Subtractive Suture to Being
"...the axiom says: the unpresentable is presented, as a subtractive term of the presentation of presentation. Or, a multiple is, which is not under the Idea of the multiple. Or: being lets itself be named, within the ontological situation, as that from which existence does not exist....the axiom of the void-set will begin with an existential quantifier...." (p67-68).
It is written: (∃β)[~(∃α)(α∈β)] which reads, there exists β such that there does not exist any α which belongs to it.
"...the unicity of the void-set is immediate because nothing differentiates it, not because its difference can be attested. An irremediable unicity based on in-difference is herein substituted for unicity based on difference....But, the one is not, and thus I cannot assume that being-void is a property, a species, or a common name. 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...it is because the one is not that the void is unique" (p68-69). It is at this point we merely ascribe a proper name to the void, that is ∅, an old Scandinavian letter, "zero affected by the barring of sense" (p69).
It should be fairly clear by now that Badiou is arguing that in the "execution" of any ontology--or rather, in the "mathematical theory of the multiple, or set theory"--the "grand Ideas" of the multiple must conform to a "system of axioms" which excludes "any explicit definition of the multiple" (p60). In fact, nine axioms make up this "economy of presentation" which allows Badiou to examine the first principles of being itself. Here in Meditation Five, Badiou introduces five additional axioms which have a "strictly operational value" so as to "think the singularity of this existential statement on the void" (p60). Because these axioms will be utilized in recurring fashion, I shall be brief in introducing the axioms of Extensionality, Powerset, Union, Separation, and Replacement before turning again to the Axiom of the Void-Set.
1. The Axiom of Extensionality
"The axiom of extensionality posits that two sets are equal (identical) if the multiples of which they are multiple, the multiples whose set-theoretical count as one they ensure, are 'the same'....The axiom...thus amounts to saying: if every multiple presented in the presentation of α is presented in that of β, and the inverse, then these two multiples, α and β, are the same....The logical architecture of the axiom....indicates....The identity of multiples is founded on the indifference of belonging" (p60-61).
This is written: (∀γ) [(γ∈α)↔(γ∈β)]→(α=β), which reads: for every multiple γ, it is equivalent and thus indifferent to belong to α or to belong to β, then α and β are indistinguishable and interchangeable.
This does not constrain us from saying there is something "different", α and β have differential markings, thus α and β must "depend on what belongs to their presentations" (p61). Badiou writes: "What possible source could there be for the existence of difference, if not that of a multiple lacking from a multiple? No particular quality can be of use to us to mark difference here, not even that the one can be distinguished from the multiple, because the one is not. What the axiom of extension does is reduce the same and the other to the strict rigour of the count such that it structures the presentation of presentation. The same is the same of the count of multiples from which all multiples are composed, once counted as one...." (p 61). This axiom of the same and the different, or the same and the other, doesn't tell us if anything exists, it simply establishes for any existent multiple, "the canonical rule of its differentiation" (p61).
***A note about the axioms of Powerset, Union, Separation, and Replacement***
These axioms are of the form: "Take any set a which is supposed existent. There then exists a second set B, constructed on the basis of a, in such a manner", or rather, by "this or that rule" (p62). To be more clear, "for all a, there exists B such that it has a defined relation to a" (p62). Badiou concludes, "These four axioms...concern guarantees of existence for constructions of multiples on the basis of certain internal characteristics of supposed existent multiples" (p62).
2. The Axiom of the Powerset (the set of subsets)
"This axiom affirms that given a set, the subsets of that set can be counted-as-one: they are a set....[Therefore,] the logical structure of this axiom is not one of equivalence but of implication" (p62). Considering set B is a subset of α:
It is written: β is a subset of α, or β is included in α; which is an abbreviation for: (∀γ) [(γ∈β)→(γ∈α)], which reads, for all γ, if γ belongs to β, then γ belongs to α. This defines the relation of inclusion between β and α.
Much more will be said about belonging and inclusion in Meditation Seven and Meditation Eight.
3. The Axiom of Union
This axiom seeks to answer the question: "Does the count-as-one extend to decompositions? Is there an axiom of dissemination just as there is one of composition [Axiom of the Powerset]?" (p63). Badiou notes, "Since a multiple is a multiple of multiples, it is legitimate to ask if the power of the count via which a multiple is presented also extends to the unfolded presentation of the multiples which compose it, grasped in turn as multiples of multiples. Can one internally disseminate the multiples out of which a multiple makes the one of the result?" (p63). The Axiom of Union therefore states "that each step of the dissemination is counted as one" (p63).
It is written: (∀α)(∃β)[(δ∈β)↔(∃γ)[(γ∈α)& (δ∈γ)]] which reads, “…if α is presented, a certain β is also presented to which all the δ’s belong which also belong to some y which belongs to α” (p64). Furthermore, the "multiple β gathers together the first dissemination of α, that obtained by decomposing into multiple the multiples which belong to it, thus by un-counting α.
“Given α, the set β whose existence is affirmed here will be written ∪α (union of α)” (p64). Badiou concludes, "The fundamental homogeneity of being is supposed henceforth on the basis that ∪α, which disseminates the initial one-multiple and then counts as one what is thereby disseminated, is no more or less a multiple itself than the initial set. Just like the powerset, the union set does not in any way remove us from the concept-less reign of the multiple....Ontology announces herein neither One, nor All, nor Atom, solely the uniform axiomatic count-as-one of multiples" (p64).
4. The Axiom of Separation
See, Axiom of Separation.
5. The Axiom of Replacement
This axiom is really quite simple. "The idea--singular, profound--is the following: if the count-as-one operates by giving consistency of being-one multiple to some multiples, it will also operate if these multiples are replaced, term by term, by others. This is equivalent to saying that the consistency of a multiple does not depend upon the particular multiples whose multiple it is. Change the multiples and the one-consistency--which is a result--remains, as long as you operate, however, your substitution multiple for multiple" (p65). Moreover, "the axiom of replacement is suited--even to the point of over-indicating it--to the mathematical situation being presentation of the pure presentative form in which being occurs that-which-is" (p65).
6. The Axiom of the Void-Set, Subtractive Suture to Being
"...the axiom says: the unpresentable is presented, as a subtractive term of the presentation of presentation. Or, a multiple is, which is not under the Idea of the multiple. Or: being lets itself be named, within the ontological situation, as that from which existence does not exist....the axiom of the void-set will begin with an existential quantifier...." (p67-68).
It is written: (∃β)[~(∃α)(α∈β)] which reads, there exists β such that there does not exist any α which belongs to it.
"...the unicity of the void-set is immediate because nothing differentiates it, not because its difference can be attested. An irremediable unicity based on in-difference is herein substituted for unicity based on difference....But, the one is not, and thus I cannot assume that being-void is a property, a species, or a common name. 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...it is because the one is not that the void is unique" (p68-69). It is at this point we merely ascribe a proper name to the void, that is ∅, an old Scandinavian letter, "zero affected by the barring of sense" (p69).
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