Badiou asserts that mathematics, in particular set theory, is
ontology. In his view philosophy in its current state is unable to
discuss being adequately. As a result, the important position that
ontology has held in philosophy since Aristotle is not only under
threat, it is generally agreed to have become untenable, and there are
few serious and convincing attempts to reattain it. With the aid of set
theory, Badiou intends to change all that.
To support this assessment of set theory's suitability, Badiou sets out to explain what would be required of a language that can talk about being. As we saw in part 1, being is considered an inconsistent multiplicity. The proposed language must therefore be able to show a multiplicity as non-unified (inconsistent). The following three requirements have to be met: (1) multiples cannot be collections of individual things (of individual count-for-ones), instead they must be shown to be multiples of multiples; (2) multiples cannot be unified into a One at the level of the universe, instead they must be limitless; (3) multiplicities cannot be identified by one particular concept, or they would be unified.
Set theory, Badiou then goes on to demonstrate, is able to satisfy these requirements in the following ways: (1) Although sets are composed of multiple elements, they are themselves sets as there is no fundamental difference between a set and an element; (2) as a result of Russell's paradox, an axiom of set theory states that no set can include itself, and therefore there is no set that includes all sets; (3) there are rules about how sets operate, but no unifying definition of what a set is, they simply emerge from the operations performed.
Nevertheless, the inconsistent multiplicity is only one of two doctrines that Badiou employs to link set theory's infinity of sets, and his concept of the multiplicities of situations. The second is the doctrine of the void, and we need now turn our attention to that.
The idea of the void might seem strange at first, because it isn't simply nothing, as the name might suggest. To understand it we could contrast it with what it is not. Whatever is counted-for-one in a situation is “something”. The converse then, is that “nothing” goes uncounted. We know that there are indeed uncountables in a situation, for instance the inconsistent multiple prior to the count-for-one, and the operation of the count-for-one. Although they are uncountable, they are still necessary to the existence of a presented situation, but they cannot be presented inside the situation because they constitute the situation as a situation. They, then, are the void of the situation.
The void is considered to be subtractive in the sense that it is subtracted from presentation (it is not presented), and also because it does not engage with the particularities of the situation.
How does this connect to set theory? Set theory asserts that an initially set exists, namely the empty set or null set. It is from this set that an infinity of other sets emerge. The inconsistent multiples of a situation can be said to be linked to set theory by being constructed out of the null set, which is set theory's presentation of the void.
That accounts for the general connection between situations, and set theory's infinity of sets. But that is not all, the structure of specific situations can also be transcribed as particular types of sets.
The elements of a set have no other distinguishing feature other than the fact of belonging to the set. As a result, elements are indicated by variables. The set unifies those elements, but each element could also belong many different sets or subset. The particular structure of a set might prevent or limit this, but that is another matter.
Axioms, for their part, signify decisions in thought. They are neither pure nor without the potential of being reformed, as they have been reformulated a few times when logical inconsistencies came to light (Russell's paradox is a revolutionary case in point). Badiou has settled on the standard Zermelo-Fraenkel set theory (commonly abbreviated ZFC). For a discussion see Wikipedia's entry. Infinite Thought lists the nine axioms as: Extensionality, Separation, Power-Set, Union, Empty Set, Infinity, Foundation, Replacement and Choice. (p. 14)
It is now time to turn our attention to Russell's important insight. Gottlob Frege's formulation of set theory can be considered a logical foundation, and it was in reference to Frege's thought that Russell was able to discover the paradox. Frege held that in first order logic, for every well formed formula that defines a concept, there exists a set of elements that satisfied the formula. This would seem true most of the time. For instance, the set of all purple oranges would include all the available purple oranges, even if there were none (an empty set).
Bertrand Russell noticed, however, that not all well formed formulae would be satisfiable. In particular, he noticed that the formula: the set that includes as its elements every set that is not also a set of itself involves a fundamental contradiction, for if the overall set does not also include itself, then it should include itself, but once it includes itself it should no longer be included. This is called Russell's Paradox.
The axiom of separation was developed in order to avoid this contradiction. So, whereas Frege's formulation described the new set directly into existence, Russell's modification requires there be an existing set for the new set to exist. In other words, the new set is separated out.
This has a direct bearing on Badiou's understanding of the relation between language and being. Being is always in excess of language, just as one can only separate out a new set through a formula (language) when a larger set already exists (undefined being).
Discourses such as chemistry, biology, or fine art would have something to say about beings themselves, about their features and identity. Ontology makes no claims about those, instead it talks about the structure of what is presented in a situation.
In part 3 we will take a look at more complex applications of Badiou's adoption of set theory in the language of being, in particular the three types of situations as composed of different types of multiples, and the concept of indiscernibility, which is a challenging one but which I hope to clarify with the aid of the trustworthy text Infinite Thought.
To support this assessment of set theory's suitability, Badiou sets out to explain what would be required of a language that can talk about being. As we saw in part 1, being is considered an inconsistent multiplicity. The proposed language must therefore be able to show a multiplicity as non-unified (inconsistent). The following three requirements have to be met: (1) multiples cannot be collections of individual things (of individual count-for-ones), instead they must be shown to be multiples of multiples; (2) multiples cannot be unified into a One at the level of the universe, instead they must be limitless; (3) multiplicities cannot be identified by one particular concept, or they would be unified.
Set theory, Badiou then goes on to demonstrate, is able to satisfy these requirements in the following ways: (1) Although sets are composed of multiple elements, they are themselves sets as there is no fundamental difference between a set and an element; (2) as a result of Russell's paradox, an axiom of set theory states that no set can include itself, and therefore there is no set that includes all sets; (3) there are rules about how sets operate, but no unifying definition of what a set is, they simply emerge from the operations performed.
Nevertheless, the inconsistent multiplicity is only one of two doctrines that Badiou employs to link set theory's infinity of sets, and his concept of the multiplicities of situations. The second is the doctrine of the void, and we need now turn our attention to that.
The idea of the void might seem strange at first, because it isn't simply nothing, as the name might suggest. To understand it we could contrast it with what it is not. Whatever is counted-for-one in a situation is “something”. The converse then, is that “nothing” goes uncounted. We know that there are indeed uncountables in a situation, for instance the inconsistent multiple prior to the count-for-one, and the operation of the count-for-one. Although they are uncountable, they are still necessary to the existence of a presented situation, but they cannot be presented inside the situation because they constitute the situation as a situation. They, then, are the void of the situation.
The void is the 'suture' of being to presentation because it is the point through which a situation comes to be – the count-for-one – yet by which being – as inconsistent multiplicity – is foreclosed from presentation ... [sic] ... The void of a situation is simply what is not there, but what is necessary for anything to be there. - p. 12
The void is considered to be subtractive in the sense that it is subtracted from presentation (it is not presented), and also because it does not engage with the particularities of the situation.
How does this connect to set theory? Set theory asserts that an initially set exists, namely the empty set or null set. It is from this set that an infinity of other sets emerge. The inconsistent multiples of a situation can be said to be linked to set theory by being constructed out of the null set, which is set theory's presentation of the void.
That accounts for the general connection between situations, and set theory's infinity of sets. But that is not all, the structure of specific situations can also be transcribed as particular types of sets.
The elements of a set have no other distinguishing feature other than the fact of belonging to the set. As a result, elements are indicated by variables. The set unifies those elements, but each element could also belong many different sets or subset. The particular structure of a set might prevent or limit this, but that is another matter.
Axioms, for their part, signify decisions in thought. They are neither pure nor without the potential of being reformed, as they have been reformulated a few times when logical inconsistencies came to light (Russell's paradox is a revolutionary case in point). Badiou has settled on the standard Zermelo-Fraenkel set theory (commonly abbreviated ZFC). For a discussion see Wikipedia's entry. Infinite Thought lists the nine axioms as: Extensionality, Separation, Power-Set, Union, Empty Set, Infinity, Foundation, Replacement and Choice. (p. 14)
It is now time to turn our attention to Russell's important insight. Gottlob Frege's formulation of set theory can be considered a logical foundation, and it was in reference to Frege's thought that Russell was able to discover the paradox. Frege held that in first order logic, for every well formed formula that defines a concept, there exists a set of elements that satisfied the formula. This would seem true most of the time. For instance, the set of all purple oranges would include all the available purple oranges, even if there were none (an empty set).
Bertrand Russell noticed, however, that not all well formed formulae would be satisfiable. In particular, he noticed that the formula: the set that includes as its elements every set that is not also a set of itself involves a fundamental contradiction, for if the overall set does not also include itself, then it should include itself, but once it includes itself it should no longer be included. This is called Russell's Paradox.
The axiom of separation was developed in order to avoid this contradiction. So, whereas Frege's formulation described the new set directly into existence, Russell's modification requires there be an existing set for the new set to exist. In other words, the new set is separated out.
This has a direct bearing on Badiou's understanding of the relation between language and being. Being is always in excess of language, just as one can only separate out a new set through a formula (language) when a larger set already exists (undefined being).
the axiom of separation states that an undefined existence must always be assumed in any definition of a type of multiple. - p. 17
Discourses such as chemistry, biology, or fine art would have something to say about beings themselves, about their features and identity. Ontology makes no claims about those, instead it talks about the structure of what is presented in a situation.
unlike Plato and Aristotle's ontologies, there is neither cosmos nor phenomena, neither cause nor substance. Set theory ontology does not propose a description of “the furniture of the world”, nor does it concern itself with “carving reality at the joints”. Its own ontological claim simply amounts to saying there is a multiplicity of multiplicities. - p.17
In part 3 we will take a look at more complex applications of Badiou's adoption of set theory in the language of being, in particular the three types of situations as composed of different types of multiples, and the concept of indiscernibility, which is a challenging one but which I hope to clarify with the aid of the trustworthy text Infinite Thought.
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