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 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.
