143
* University of Bristol
Filozofski vestnik | Volume XLI | Number 2 | 2020 | 143–178 | doi: 10.3986/fv.41.2.07
Tzuchien Tho*
Sets, Set Sizes, and Infinity in Badiou’s
Being and Event
Introduction
In 2006, Lucca Fraser published the article “The Law of the Subject: Alain Ba-
diou, Luitzen Brouwer and the Kripkean Analysis of Forcing and the Heyting
Calculus.”
1
Still one of the best Anglophone commentaries on Badiou’s L ’Être et
l’Événement, she argued that Kripke forcing was sufficient to render the theory
of the subject in Badiou’s L’Être et l’événement within an intuitionistic frame -
work. This pluralization of logical frameworks allowed us some insight into the
possibilities of mathematical ontology of which Badiou was not aware. That is,
it forced us, and eventually Badiou, to recognize that his commitment to clas -
sical logic, where the principle of excluded middle was true, was unnecessary
to the claims about the subject within his project. In turn, Badiou agreed and
acknowledged this point but argued that this only applied to the subjective di-
mensions of mathematical ontology.
2
Classical logic was still needed to express
the core of the ontological analysis, based on Zermelo-Frankel set theory with
the axiom of choice (hereafter ZFC).
3
The paper here follows in Fraser’s example. Yet instead of dealing with the the -
ory of the subject in L ’Être et l’événement, I turn to the theory of the “count”, the
fundamental aspect of Badiou’s theory of Being and beings in L’Être et l’événe -
ment. Further, I am not interested here to challenge the classical logical frame -
work upon which the work is couched, although it is far from unassailable. What
is in question here is instead the relation between the count and multiplicity. In
particular, I argue that Cantorian transfinite cardinality, a method of reckoning
1 Zachary Luke Fraser (Lucca Fraser), “The Law of the Subject: Alain Badiou, Luitzen Brou -
wer and the Kripkean Analyses of Forcing and the Heyting Calculus”, Cosmos and History:
The Journal of Natural and Social Philosophy 2 (1-2/2006), pp. 94–133.
2 Alain Badiou, “New Horizons in Mathematics as a Philosophical Condition: An Interview
with Alain Badiou [with Tzuchien Tho]”, Parrhesia 3 (2007), pp. 1–11.
3 Ibid., p. 7.
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tzuchien tho
the measure of the sizes of sets, is not a necessary feature of the coherence of the
project of mathematical ontology in L ’Être et l’événement. The implication of this
for Badiou’s project is that the “subtraction of the one” also implies a pluralism
of the one. That is, the pluralization of how unity is constituted, located, and
operational. While subtraction is a rejection of the givenness of the one, the re -
jection of a view entrenched in traditional metaphysics, it retains the form of the
one as the result of the count. Here the one is not a given but a result. From this
count, the set theorical universe is populated by the entities generated by the
count. However, these entities need not be measured in the standard Cantorian
way and can be subject to different forms of measurement.
As will be argued, the implication of the pluralism of the one is that the count-
as-one of finite and transfinite sets is indifferent to Cantorian cardinality. This
indifference is crucial because the count is responsible for the count of the in-
consistent multiple at the very heart of the project. This initial count is one that
formally introduces the set (within the discourse of ZFC) as the basic term of
Badiou’s mathematical ontology. This renders questions of measure and, in par -
ticular, the measure of the infinite orthogonal to this theory. Badiou emphasizes
this in his work, noting that the infinite has no intra-mathematical meaning.
Badiou’s ontological project is thus not only a subtraction from the metaphysics
of the “one” but also a subtraction from the metaphysics of infinity, which is
dependent on it. What results instead is a pluralism of these terms based on the
more fundamental distinction, as Badiou himself puts it, between concept and
existence.
4
A pluralism with respect to the unfolding of the ontological concept
is thus no challenge to dimension of existence. This orthogonality reveals that
Badiou’s ontological project is free to embrace a pluralism about unity and in-
finity unanticipated by Badiou and the founders of set theory. This pluralism of
the one reveals new contours of theoretical possibility previously unexamined
by Badiou and his commentators.
Counting and the inscription of the void
The central goal of Badiou’s L’Être et l’événement is to employ the structures
provided by set theory (especially ZFC) to analyze existence. What this means
is not at all straightforward. Badiou argues neither for a Pythagorean-Platonic
ontology in the vulgar sense, where existence reduces to sets, nor mathematical
4 Alain Badiou, Being and Event, trans. O. Feltham, Continuum, London 2005, p. 159.
145
sets, set sizes, and infinity in badiou’s being and event
reality as ultimate reality, nor for set theory as a model for existence. Rather, no
structure, be it mathematical, or something else, could serve as a direct map of
ultimate reality since any ontology must first reckon with the fact that Being is
itself unstructured. Any ontology must first force Being to relate to a structure
that would be alien to it. Hence, for Badiou, since Being is itself unstructured, in
order for ontology to be possible, it must be drawn into structure by the means
of the count. Hence, this first step into the possibility of a mathematical ontol -
ogy cannot be given by set theory itself. This “drawing in” is operated by “in-
scription”, the localization of unstructured being into the structure of set theo -
ry. Basically, the unstructured is inscribed within structure by the count of the
void, since whatever is not already within the structure must appear as, or be a
presentation of, a “nothing”.
The basic picture then is that Badiou makes the link between set theory and un-
structured Being by counting it (Being) as a set containing the void. The positivi-
ty of the void in set theory therefore stands in for what is “underneath” structure
but nonetheless localized within the set theoretical discourse. This bridge be -
tween unstructured Being and the structures of set theory rejects the two tradi-
tional tendencies in ontology. The first tendency is that ontology or metaphysics
in general is a rational description of the fundamental structures of being. This
presumes that Being has an inherent structure that is knowable and reducible
to some basic form. We can set this aside because Badiou maintains the view,
shared by a lineage of thinkers since the Platonists that Being is inaccessible
to us through the means of the categories we apply to beings or the senses. The
second notion is that ontology plays the role of drawing out the immanent but
obscure qualities of Being-as-such. Badiou similarly rejects this path of analyz -
ing the “deep” allure of the Being by cutting off our relation to the “presence” of
Being in favor of reducing it to a count of the void. In short, all that is “above”
and “beyond” cannot appear as a “one” (i.e. an entity), therefore it is counted as
a “void”. This act, which we may term “subtraction”, produces nothing myste -
rious. It is simply the count of the void. This is the null set {} or Ø, the void after
it has been counted, that forms the basis of the arithmetic counting of pure sets.
From this Ø, we count its successor {Ø}, and its successor’s successor {Ø, {Ø}},
and {Ø, {Ø}, {Ø, {Ø}}}, and so on. The success of this process allows Badiou to
provide a key distinction in the mathematical ontology of L’Être et l’événement.
That is, the arborescent branching of the count of the null set (the count of the
void), allows us to retroactively designate Being, seen through the lens of con-
146
tzuchien tho
sistent structure, as “inconsistent”. Hence, instead of subordinating Being to
the logic of beings (the unit entity), or searching for a non-rational presence of
the Being behind and beyond being, unstructured Being is taken up into ontol -
ogy (the realm of consistent multiplicity) as “inconsistent multiplicity”. This is
wholly dependent on the existence of a structurally coherent (and well-ordered)
consistent multiplicity.
Badiou’s act of ontological subtraction may be seen as deflationary to any met -
aphysics. It is certainly fair to judge it so from the perspective of the traditional
goals of ontology. Yet, it is not deflationary of ontology insofar as the subtraction
does not restrict ontology to consistent multiplicity. Instead, the distinction be -
tween consistent and inconsistent multiplicity provides the central materials for
analysis in L’Être et l’événement. Ontology, in Badiou’s sense, has therefore the
task of unfolding the unexpected relations between consistent and inconsistent
multiplicity. More importantly, the fundamental aim of the book, the claim that
truths and events exist, is the claim that, because of the incompleteness to any
sufficiently strong set theory, there are sets that can be proven to exist but can-
not be constructed (non-constructible sets). Hence, the inscription of Being into
set theory through the count of the void will entail the unfolding, albeit incom-
plete, of the inconsistent multiple within the structure of consistent multiplicity.
From the point of the inscription of Being, qua inconsistent multiplicity, the op -
eration of the count is crucial in distinguishing between ordered and structured
presentation (sets in set theory), and inconsistent multiplicity (the inscribed
void). In this way, structured presentation, what Badiou calls “consistent mul -
tiplicity” is given by the ordered universe of pure sets. With the inscription of
the count, further counts produce a structure of sets that are recursive counts
of the inscription. We can illustrate this with the sequence of natural numbers
(0,1,2,3, etc.).
We can note that this counting structure maps succession order with size. How -
ever, this correspondence between order and size will not be so obvious when
it comes to counts that go beyond the finite numbers. What is conspicuous here
is that Badiou’s ontology in L’Être et l’événement relies on the count and there -
fore ordinality. Cardinality plays a much less important role. Why does this dif -
ference matter? It matters insofar as the count cannot be subordinated to the
difference between the finite and the non-finite (or infinite). The oneness of
147
sets, set sizes, and infinity in badiou’s being and event
the set is indifferent to whether the set is the void qua inconsistent multiple, a
defined quantity (such as the keys on a qwerty keyboard), or the uncountably
many points on a continuum. Badiou’s use of set theory to anchor mathematical
ontology relies fundamentally on ordinality. We shall underline this claim and
its stakes further.
What is the difference between cardinality and ordinality?
Now, the two concepts of cardinality and ordinality are not determined by the
axioms of ZFC. The axioms indicate and restrict the kinds of sets that can exist.
Cardinality and ordinality are instead structures we use to analyze these sets
in terms of “how many” and “which one”, respectively, of a given set. In other
words, they measure and count sets, respectively. Cardinality can be determined
by matching or mapping (measurement) a set onto another pre-given set (i.e.
numbers) or itself while ordinality is determined by ordering. Under normal fini-
tary circumstances, cardinality and ordinality correspond neatly to each other,
a set of five marbles has a cardinality of five because the five marbles can be
mapped to a set of five entities in a pre-given set or to the first five natural num-
bers. On the other hand, putting the marbles in order will get us to the “fifth”
marble, thereby completing the ordinal count of the marbles.
5
Given the set of
finite natural numbers, which does not terminate, any arbitrarily large number
will be finite and the ordinal that counts the sequence that allows us to arrive
at that number will also tell us the size (cardinal) of the set that is given. The
situation is slightly different for non-finite sets.
For finite cases, we can arrange the natural numbers in order and map them to
their subsets (e.g., the squares, cubes and fourths, etc.). This one-to-one map -
ping is the structure of cardinality that allows us to see that these sets are the
same size.
Naturals n 1 2 3 4 5 …
Squares n
2
1 4 9 16 25 …
5 There are ways to deliberately distinguish cardinality and ordinality in finite sets, exploit -
ing the fact that ordinalities are pertinent to ordering while cardinalities are indifferent to
order as we shall later examine.
148
tzuchien tho
Cubes n
3
1 8 36 64 125 …
Fourths n
4
1 16 81 256 625 …
This is a principle known at least since Thābit ibn Qurra but better known
through Galileo who puts forth this mapping in the first day of the Discours -
es and Mathematical Demonstrations Relating to Two New Sciences.
6
The point
here is that, through mapping, we can clearly see that the measure of these dif -
ferent sets of numbers are equivalent even though square, cubes, and fourths
are subsets of the set of natural numbers. In other words, they share the same
cardinality.
Now it is possible to demonstrate the difference between ordinality and cardi-
nality in finite sets as well. If we take two subsets of the natural numbers, say,
odds and evens, we can make a new set of the union of the two. This maps to the
natural numbers and thus allows us to claim equal cardinality. However, since
we remain within the context of the finite, if we start with the odds, there will be
no way to “reach” the evens.
[1, 3, 5, 7, 9 … 2, 4, 6, 8…]
Regardless of any assumptions about the infinite, there would be no determi-
nate “which one” except for the odds since the ordering does not allow access to
the evens in a finite number of steps (i.e. the order means that we would never
access the evens since the first ellipsis implies a non-terminating sequence). In
other words, we will not know where the number “2” arises in the sequence.
This problem accentuates the conceptual difference between ordinality and car -
dinality in order to underline why size and order are distinct structures even
6
For a reconstruction of the “alternative” history of the infinite, involving figures likes Ibn
Qurra, please see Mancosu’s 2016 essay collection. Paolo Mancosu, “Measuring the size
of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevi-
table?”, Abstraction and Infinity, Oxford University Press, Oxford 2015, pp. 116–153. This
paper was earlier published as Paolo Mancosu, “Measuring the size of infinite collections
of natural numbers: Was Cantor’s theory of infinite number inevitable?”, Review of Sym-
bolic Logic 2 (4/2009):612-646. For Galileo’s demonstration see Galileo Galilei, Discourses
and Mathematical Demonstrations Relating to Two New Sciences, trans. H. Crew and A. de
Salvio, Macmillan, New York 1914, pp. 31–37.
149
sets, set sizes, and infinity in badiou’s being and event
within the finite cases. In non-finite cases, this difference is accentuated follow -
ing a similar kind of reasoning.
Without giving its demonstration, let us assume that there are non-finite cardi-
nalities and ordinalities. For the cardinal, we designate א
0
as the size of the finite
ordinals collected as a set. This is not a finite number because if it were finite,
it would be part of the set being measured. It is non-finite. It is transfinite and
infinite in technical and common parlance respectively. However, we hesitate
in using the term “infinite” because it is significantly different from traditional
notions of infinity as the completion of a non-terminating sequence. By “tradi-
tional” I mean here the wavering notions of the infinite as either the “potential”
infinite of an unending sequence of successive finite terms, or the “number of
numbers”, the “actual” totality of numbers qua termination of the succession of
finite terms. Here, the traditional ideas concerning the infinite is indeed a family
of ideas and intuitions that runs the spectrum from the negative imagination of
the “very large” but only ever finite, to an infinite totality that, at the pain of con-
tradiction, must invoke a transcendence over numbers themselves. Variations
across this spectrum run the gamut in the history of the infinite since antiquity.
The origins of set theory offers an alternative. This alternative breaks with the
traditional dialectic of the infinite with the one: either the infinite is not one,
and therefore a negative entity, or the infinite is a one, and therefore a totality.
As neither a potential infinite nor a totality of finite terms, the modern infinite
affirms the non-termination of the count and therefore its non-totality.
By the time of the formalization of ZFC, this break with the traditional dialectic
of the one and the infinite has already been accomplished by the innovations of
Cantor and Dedekind. However, it is clarifying to see its canonical expression in
the ZFC axioms. The axiom of infinity in ZFC specifically bars us from this archa-
ic attachment of the infinite to the one.
The axiom of infinity
7
states:
The axiom states that there is an infinite set in the sense that there is a set with
Ø as its element, and also the successor of that set which is the union of that set
7 Kenneth Kunen, Set Theory, revised edition, College Publications, London 2011, p. 17.
150
tzuchien tho
and its subset. Therefore, there is at least one infinite (or non-finite) set in the
sense that it collects the set of the null set and all its successors. This expresses
infinity neither as termination nor even transcendence of a non-terminating se -
quence. Rather it expresses a limit. Regardless of the size of the set offered, there
is a set to which it and its successor belongs.
While being one of two of the ZFC axioms that assert the existence of a particu -
lar kind of set, the other one being the axiom of the void or null set, it does not
assert that the infinite set terminates some non-terminating sequence (i.e. there
is no “last” finite number). In the ZFC axiom of infinity, there is no “last one”
that counts the series for the series to be measured. In fact, the very existence
of a “last one” indicates a finite, rather than an infinite set according to this
very axiom.
Traditionally, the infinite could only be defined against the backdrop of its rela-
tionship to the “one”. This dialectic of the one and the infinite pushed the tradi-
tional notion to polarize between a potential infinite (an indefinite, non-termi-
nating term), and a contradictory notion of a “number of numbers” that counts
all the numbers. That is, either a negation of the one-total (hence potential), or
the embrace of the one-total (hence contradictory). In this modern ZFC context,
this dialectic fails to hold. There is neither a “greatest” finite number that counts
all the finite, nor is there a completion of the non-terminating series of finite
numbers. That is, it elides both the actual and potential infinity. The cardinal
א
0
is instead a measure of that non-terminating series of finites. Hence, the nat -
ural numbers, the squares, cubes, etc., are all measured by the same cardinal.
Adding a finite number to the cardinal א
0
returns the same cardinal, moreover,
adding squares, cubes, fourths (though there are some shared numbers), each
measuring the cardinal א
0
also returns the same cardinal. This feature of the
cardinal matches up with intuitions of inexhaustibility traditionally associated
with the infinite, where the subtraction of a finite from an infinite returns an
infinite, and the addition of an infinite returns an infinite. This was Galileo’s
observation in the text related to the principle noted above:
Salviati. So far as I see we can only infer the totality of all numbers in infinite,
that the number of squares is infinite, and that the number of the roots is infinite;
neither is the number of squares less than the totality of all the numbers, nor the
151
sets, set sizes, and infinity in badiou’s being and event
latter greater than the former; and finally that the attributes “equal,” “greater,”
and “less,” are not applicable to the infinite, but only to finite quantities.
8
However, whereas Galileo presented this feature of the infinite as a kind of puz -
zlement, Cantor and his faction took it up as a positive property of the infinite.
Dedekind, the other founder of modern set theoretically based mathematics (ar -
riving at results independently from Cantor), took the Galilean puzzle instead as
the positive definition of the infinite.
A system S is said to be infinite when it is similar to a proper part of itself; in the
contrary case S is said to be a finite system […] My own realm of thoughts, the
totality S of things, which can be objects of my thought is infinite. For if s signifies
an element of S, then the thought s’ , that s can be object of my thought, is itself
an element of S.
9
For Dedekind, the system S was his terminology for a set S, “similar” here means
“equal in size”. The infinite set he denotes here is a set where the proper subset
is equal in size (ie. “similar”) to itself. In turn, the set that Dedekind gives as
an example is the set of his “thoughts”. Since every thought can be thought of,
thoughts of thoughts provide a simple way to generate a subset that is equinu -
merous to the original set. The thought of a thought is a thought which already
belongs to the original set. Hence, any set that expresses this same feature can
be considered infinite. What is crucial here is that the finite sets are determined
negatively only as those that are contrary to this case. It is worth remarking that
the finite here is the negative case of the prior definition of the infinite set. The
ground has shifted from a constructive move from the finite to the infinite to that
of a prior infinite multiplicity.
10
Let us designate this shifting of ground by naming this kind of measurement
as “Cantor-Dedekind measure”. This can be contrasted to “Euclidian measure”
which designates that proper subsets must always be smaller than the origi-
8 Galileo, Discourses and Mathematical Demonstrations Relating to Two New Sciences, pp.
32–33.
9
Richard Dedekind, Essays on the Theory of Numbers, Dover Publications, Mineola 1963,
pp. 63–64.
10
Cf. Alain Badiou, Number and Numbers , trans. R. Mackay, Polity Press, Boston 2008, pp.
38–54.
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tzuchien tho
nal set. We recognize that Euclidian measure applies for finite sets. The set of
odd numbers in a finite set will be strictly less than the original set (composed
of elements other than odd numbers) because it is a proper subset. In turn, as
Dedekind argues, the property of infinite sets is to have proper subsets that
measure up to the original set.
11
Ordering and transfinitude
Now, we can shift focus away from measurement and turn to ordering. Ordering
or ordinality identifies a very different structure. If given a set, say the empty set,
we can always move up in order by counting that set to produce a new set that
is a successor of that set. By doing this, we can model the ordering of numbers.
Number Von Neumann ordinal
0={} Ø
1={0} {Ø}
2={0,1} {Ø, {Ø}}
3={0,1,2} {Ø, {Ø},{Ø, {Ø}}}
… …
The ellipsis at the bottom of the chart indicates that this constructive process of
the ordinal “count” is non-terminating and therefore opens up into the transfi-
nite sphere as a well.
11
Cantor argues roughly the same thing in Mitteilungen zur Lehre vom Transfiniten. Georg
Cantor, “Mitteilungen zur Lehre vom Transfiniten”, Zeitschrift für Philosophie und philos -
ophische Kritik 91 (1887–1888), pp. 81–125, 240–265. Quoted in Mancosu, Abstraction and
Infinity, p. 132.
153
sets, set sizes, and infinity in badiou’s being and event
This hierarchy produced by the ever-widening elements contained in each suc -
cessor. The “V” shape is due to this. Hence, it is proper to ask the measure of
each of these sets that identify the ordinality of the set. In this way, measure and
order inform each other. For instance, the first number 0 contains no elements
while the third number contains three elements constructed from previous
counts of the set with no elements. However, the difference here is that ordinal -
ity requires a correspondence to a well-ordering of the elements, a property in-
dependent from measurement. Measurement, whether in the Cantor-Dedekind
sense or in the Euclidian sense, requires no appeal to the “least” element. How -
ever, for ordinality to make any sense, it must begin with a least element. In
the example above drawn from the natural numbers, this least element is 0,
denoted by the empty set. In other words, in order to “arrive” somewhere in
the sequence, it must “begin” somewhere. This difference of structure means
that the transfinite ordinals, denoted by ω
n
, function in a different way to the
transfinite cardinals א
n
. The main difference can be reflected by the properties
of the traditional infinite mentioned earlier. If we add a finite to the infinite, we
get back the infinite. This principle holds in the domain of cardinality. Adding
something finite to a set with a transfinite measure does not change its measure.
However, adding something finite, like 1, to a transfinite ordinal, can indeed
make a difference. The point here is rather simple. The first transfinite ordinality
is a border between two orders of succession. Whatever comes before it can be
subsumed under it, whatever comes after it moves beyond the border. This is
simply what follows from the property of well-ordering between orders. Hence,
the addition of an element at the start of the sequence is not the same as the
addition of an element at the end of the sequence.
z+ω = ω < ω + z
Though ordinality and cardinality differ in structure, they are not opposed.
They are tools that capture different structures in the analysis of sets, especially
non-finite sets, which cannot be collapsed into one another. However, insofar as
Badiou privileges the count over measure, it is ordinality, not cardinality, that is
the guiding structure of his mathematical ontology in L’Être et l’événement. The
count is the count of succession rather than that of measurement. Hence, al -
though Badiou accepts the entire machinery of ZFC set theory, the work in L ’Être
et l’événement nonetheless selects different aspects of the theory to demonstrate
certain ontological claims. What is clearly primary is the structure of order, the
154
tzuchien tho
feature of ZFC that grants the “count” of the inconsistent multiplicity entry into
the status of being an “ontology”. What is secondary is the analysis of their
measure, which pertains only to the representations of this primary structure.
The key indication of this is the obsolescence of the notion of “infinity” in the
text. While we have already hinted at this, we shall turn to make this point more
concrete in what follows.
The obsolescence of infinity
The aim of distinguishing between cardinality and ordinality as structures in
the previous section aims at pluralizing the notions of the non-finite (or infinity)
that results from the advent of the Cantorian revolution. The goal of this section
is to draw out how this implies an obsolescence of the “infinite”. This will in
turn highlight the pluralization of the “one” in Badiou’s ontological project in
L ’Être et l’événement.
What are the ontological implications for Badiou in distinguishing cardinality
and ordinality? First, this distinction between measure and order indicates two
different kinds of structure. Cardinality is operated by the one-to-one mapping
of one set to another or itself. In non-finite cases, a set is equal in measure to
at least one of its proper subsets. Second, ordinality requires a least element,
the constructive hierarchy of sets admits to no termination and, more impor -
tantly, is indifferent to the properties of traditional infinity. Badiou identifies
the structure of ordinals (from finite to the transfinite), and not cardinals, as
the arborescent domain of ontology. The parallel but independent structure of
cardinality is not rejected but taken up in a secondary role. This point is best
viewed through what I shall call the obsolescence of infinity at work in L’Être et
l’événement.
The argument here is simple at first glance. Cardinality and ordinality deal with
infinity in very different ways. A non-finite or transfinite cardinal is one where
a proper subset is equal measure to the original set. A non-finite or transfinite
ordinal marks the border between the finite and non-finite in a non-terminat -
ing sequence. However, the addition of an element from the “left” of the sum
returns the same ordinal but an addition to the right does not. This is also true
of non-terminating sequences in the finite case. For ordinals, the “count” or the
ordering of succession determines the order of transfinitude to which a given
155
sets, set sizes, and infinity in badiou’s being and event
set corresponds. The count, as succession, is entirely indifferent to whether it is
progressing in the finite or non-finite domain.
There are two crucial implications here. First, the count is indifferent to the
distinction of the finite and non- or trans- finite. The count, in Badiou’s sense,
is not restricted to the countable (denumerable) sets. This means that such a
distinction, between the denumerable and indenumerable is only a secondary
analysis made upon the count. We shall examine this later on. The second im-
plication is that the traditional dialectic between the one and the infinite loses
all significance in this context. What we get in return is a theory of the “count”
that, through “counting-as-one”, is neither finite nor infinite in the traditional
sense. This installs a reckoning of ontological consistency subtracted from both
the notion of the one-consistent as finite and the many-inconsistent as infinite.
Badiou’s act of “subtraction of the one”, as he calls it, is not only a move to priv -
ilege the multiple against “the one” but also to thoroughly withdraw the relation
of the one and the infinite. We shall later examine what this obsolescence of the
infinite means for infinity.
Let us treat the first implication first. The traditional paradoxes of infinity con-
cern the one and the many. The infinite arises in cases where the many cannot
be reduced to the one or some rule of the one. Hence, traditionally, the infinite
cannot be real because it cannot be reduced to a sum of entities (definite parts)
that are actual. More importantly, it cannot be a totality except in a non-quan-
titative sense. Therefore, appeals to actual infinity from Antiquity until the late
modern period tend to be what the Scholastic medievals called “hypercategore -
matical”. The hypercategorematical may be some ultimate Being (God) or abso -
lute reality that outstrips the discursive resources of the very notion of quantity.
Alternatively, to speak of an infinite or infinitesimal quantity in the early mod -
ern period, theorists like Leibniz used a “syncategorematical” notion of infinity
which corresponded to a “fictional” or “manner of speaking” to designate ob -
jectivity to the infinite within a restricted domain of discourse (ie. a differential
ratio). This allows for the engagement with the structures available through a
commitment to the infinite or infinitesimal without a commitment to its reality.
12
12
Richard T.W. Arthur has argued for Leibniz’s commitment to the actuality of syncategore -
matical infinities. However, the point here is simply that the syncategorematical use of in-
finity does not commit us to its actuality. Richard T. W . Arthur, “Leibniz’s Syncategorematic
156
tzuchien tho
In brief, the primacy of the one is the reason why the infinite cannot be actual.
The infinite is neither reducible to “ones” nor can the “ones” form a totality (a
whole) in any quantitative sense. The implication to be drawn here is that the
irreducibility of the infinite to the one is what forbids its actuality. In turn, the
difference between finite and infinite is simply marked by this property of the
quantitative reducibility to the one either as part or totality. The limits of this
discourse to the reducibility to the one is precisely what is rejected in L’Être et
l’événement. This is of course what is meant by Badiou’s notion of the “subtrac -
tion of the one”. However, this means the subtraction of the infinite as well.
Since the concept of the infinite is grasped as the transcendence from the do -
main of the one, the subtraction of the one implies the obsolescence of the in-
finite as well. From this, Badiou’s ontological use of the count is also subtracted
from the one and the infinite. The count-as-one is therefore the designation of
an entity (i.e. a set) that does not fall into the traditional dialectic of the one and
the infinite (many). The further implication is that Badiou’s count does not re -
spect the border between the quantities made up of ones (the realm of finitude)
and the counts of non-finite multiplicities.
This indifference of the count to the finite and non-finite is subject to possible
misunderstandings. If the count is understood under the aegis of the traditional
relation between the one and the infinite, the distinction between the countable
and uncountable (denumerable and indenumerable) quantities would present
a challenge to Badiou’s project. If this were the case, the count could not be
universally or generically applied across finite and non-finite cases. That is, the
count would itself be subject to the dialectic of the one and the infinite. Instead
what Badiou introduces with the count is the creation of the border between the
consistent and the inconsistent. The count is therefore orthogonal to the distinc -
tion between the finite and non-finite (i.e. the transfinite) and produces, rather
than be produced by, the distinction between the consistent and inconsistent.
Actual Infinite”, in O. Nachtomy, R. Winegar (eds.), Infinity in Early Modern Philosophy,
Springer, Cham 2018, pp. 155–179. Richard T.W. Arthur, “Leibniz’s syncategorematic infin-
itesimals”, Archive for History of Exact Sciences 67 (5/2013), pp. 553–593. David Rabouin
and Richard T.W. Arthur, “Leibniz’s syncategorematic infinitesimals II: their existence,
their use and their role in the justification of the differential calculus”, Archive for History
of Exact Sciences 74 (5/2020), pp. 401–443.
157
sets, set sizes, and infinity in badiou’s being and event
What this implies is that the count is the border between inconsistency and con-
sistency, rather than the distinction between the finite and non-finite. This fact
indicates that the count privileges ordinality as its primary functional role. Just
to be explicit, if, for the sake of argument, we take Badiou’s count to be cardinal
measure, we would have to apply some measure to the inconsistent multiple.
This is impossible since the inconsistent multiple cannot be measured unless it
is first counted-as-one. As such, the inconsistent multiple would have the car -
dinality of the void. The void stands as what is “before” any count and there -
fore any measure. Insofar as the inconsistent is neither finite nor non-finite, this
does not identify any relation between the finite and non-finite.
13
The ordinal structure through which the count functions, on the other hand,
operates on succession. For the domain of pure sets, the count, outside of the
“first” count of the void, operates only by the recursive counting of the prede -
cessor. Hence, nothing about the inconsistent multiple has to be assumed for
the count to function. As such, the inconsistent multiple can be taken as a void
or really any other kind of ground suitable for an ur-set. There is hence no need
for the assumption of any distinction between the finitary status of the count it -
self. Further, as the count moves up the hierarchy of successors, the count itself
does not distinguish between finite and transfinite. That is, while it is true that
ω
0
must be handled in a different way than a finite ordinal (i.e. it does not com-
mute), the count itself, the process of succession operates indifferently between
the finite and infinite.
From this we must assert the fact that the count-as-one does not distinguish
between the finite and infinite. This is crucial because the basis of Badiou’s on-
tological project in the L ’Être et l’événement does not begin from finite cases that
slowly build towards transfinite cases. The count-as-one does not imply finitude
or denumerability for either what is counted or what results from the count.
Most importantly, it does not imply the difference between denumerability and
indenumerability.
13
It should be noted that Badiou has, in L’Immanence des vérités (2018) introduced another
approach to inconsistent multiplicity. Here, he uses the Von Neumann universe of sets (V),
which is not itself a set but a class, to handle the relation between inconsistent multiplicity
and its relationship to sets. The later work certainly implies a more committed relationship
to the positivity to the infinite. I reserve my analysis only to the structure worked out in
L ’Être et l’événement. Alain Badiou, L ’Immanence des vérités, Fayard, Paris 2018.
158
tzuchien tho
This indifference of the count to denumerability and indenumerability of the
one lead us to the second implication here, the obsolescence of the dialectic
between the one and the infinite. The traditional dialectic rejects any “actual” or
“determinate” infinity because it cannot be made a one-totality. In its tradition-
al Pre-Cantorian form, the infinite can only be hypercategorematical, a “one”
transcending quantity, or syncategorematical, a stipulated infinity (ie. a fiction)
based on a restricted domain. Of course, from this traditional perspective, the
post-Cantorian transfinite would not qualify as “infinite”. However, the trans -
finite can be considered “infinite” in the sense that the modern view remains
the (very useful) distinction between the denumerable and the indenumerable.
Insofar as the cardinal א
0
and the ordinal ω
0
identify the measure and count of
the denumerables (respectively), thereby forming the limit between the denu -
merable and indenumerable, the transfinites can be considered “infinite” in a
meaningful way. This is however gained only by correlating the transfinites with
some pre-given (and familiar) sequence of numbers (the rationals vs. the reals)
and with geometrical properties (e.g., the continuum). These are legitimate ap -
plications of ZFC but correlations of this sort are not intrinsic to it. The axiom
of infinity describes a limit through the operator of belonging and the principle
of succession. Denumerability and non-denumerability are applications of this
construction.
The main implication here for the purposes of understanding Badiou’s count-
as-one from the axiom of infinity is that the oneness of any set is granted regard -
less of its finiteness or non-finiteness. From this, unity and infinity do not form
a determinate negation. Transfinitude or the form of non-termination is a robust
form of the count-as-one rather than its exception.
Given that the Cantor-Dedekind definition of infinity is encoded in a less quan-
titative way in the axiom of infinity in ZFC, the analysis of sets moves away from
a reliance on number fields as a form of correlation. Hence, the move away from
the traditional dialectic between unity and infinity does not take place in the
form of a mere rejection of this dialectic but rather as an orthogonal side-step -
ping of this relation. Within the framework, the one and the infinite are not
opposed. Instead, transfinite orders are successive orders of limits which cor -
respond neither to the concept of the one nor the infinite (in the traditional
sense). We hence assert the obsolescence of infinity through its replacement by
a non-terminable but well-founded hierarchy of successive sets.
159
sets, set sizes, and infinity in badiou’s being and event
If we take this view retrospectively back into the history of mathematics, it
seems that this obsolescence of infinity does not really come at any cost to math-
ematics. The idea of an absolute infinite never had any real mathematical con-
tent (even while sustaining a metaphysical and theological importance). The
infinite either remained thoroughly potential or had to be, for the most part,
fictionalized.
14
Instead, where this obsolescence of infinity matters most is in
the transformed status of the “one”. It is unfortunate that the Cantorian trans -
formation of mathematics is sometimes reduced to a reform of the concept of
the infinite and a declaration of its actuality. The “one”, through the count qua
succession is formally retained. However, the one no longer plays the role of
the distinguishing mark of the finite, and thus no longer separates the finite
from and non-finite. Instead it plays the role of marking consistent multiplicity
across these traditional distinctions. With this subterranean transformation of
the “one”, we make a more thorough reckoning with this guiding concept of
the “subtraction of the one”. That is, as Badiou argues elsewhere, the act of
subtraction is characterized by the retaining of the positivity of a negation.
15
In
the case of ontology, the subtraction of the one is not a rejection but rather the
retaining of its positive role in ontology through the rejection of its traditional
dialectic with the infinite. This notion, in parallel to the obsolescence of the
infinite does play an important mathematical role. The most crucial of these is
the availability of the transfinite ordinals, the use of unity without totality in
treating non-finite terms. What this implies is the pluralization of the notion of
the “one”. We shall examine this further on.
Cardinality as representation
Before turning to the pluralization of the one, we turn briefly to the positive role
played by cardinality in L ’Être et l’événement.
If, as we have seen, the ontological project in L’Être et l’événement proceeds
through the structure carved out by ordinality, it is also important to indicate
what remains of cardinality in the project. As has been emphasized multiple
14
Mancosu points to some exceptions to this mainstream history of the infinite. Cf. Manco -
su, “Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of
infinite number inevitable?”, pp. 117–119.
15
See Alain Badiou, “Destruction, Negation, Subtraction – On Pier Paolo Pasolini”, Lacan-
ian Ink, https://www.lacan.com/badpas.htm, accessed Aug. 2020, Los Angeles 2007.
160
tzuchien tho
times, Badiou does not reject cardinality, but treats it as a secondary structure.
In fact, for Badiou the structure of cardinality is appropriate for the analysis of
representation.
For L ’Être et l’événement, presentation and representation differ in their grafting
onto two kinds of operations we can make on sets. Presentation is associated
with “belonging” (∈). Therefore, whatever is presented in a situation are the
elements of the set.
16
Representation, on the other hand, is associated with “in-
clusion” (either as subset ⊆ or proper subset ⊂). This inclusion is an operation
that recognizes subsets of the situation (situation qua set). Of course, all the
elements of a set can be considered as what “is included” in the set but the
subsets also involve all possible combination of those elements. This concept
of representation involves mere subsets up to the set of all the subsets of the
situation. This is the set of all subsets produced by the operation of inclusion is
the powerset.
This takes us to Cantor’s theorem. The theorem states that given a set A, its pow -
erset P(A) is of a strictly higher cardinality than A. For finite cases, this is obvi-
ous. A set of three elements {x, y, z} will have subsets {Ø}, {x},{y},{z}, {x,y},{x -
,z},{y,z}, {x,y,z}. Hence, a set of three elements, a cardinality of 3, will have a
powerset made up of 8 sets, hence a cardinality of 8. The basic reckoning here
is that the powerset will have a cardinality of 2
n
, where n is the cardinality of
the original set. For non-finite sets, Cantor argued, the same follows. Hence the
powerset of all finite numbers, א
0
, is 2
א0
. This powerset of א
0
is therefore also of a
strictly greater cardinal. The implication here is that there are cardinalities high-
er than the set of all infinite numbers that can be indicated by the application
of the powerset operation P(A). Furthermore, for any set, finite or nonfinite, the
application of the powerset will render a set of strictly higher cardinality.
This feature of the main development in set theory is interpreted by Badiou as
the difference between presentation and representation. Badiou refers to the
powerset as the “state”. One could find reasons to quarrel about this description
16
Although Badiou credits Lyotard for the terminology of “situation”, he cites Barwise and
Perry on treating a set as a situation. Cf. Jon Barwise, “Situations, Sets and the Axiom of
Foundation”, Logic Colloquium 1984, J.B. Paris, A.J. Wilkie, G.M. Wilmers (eds.), Elsevier
Science Publishers, Amsterdam 1986, pp. 21–36. Cited in Badiou, Being and Event, p. 484.
Thanks to Julian Rohrhuber who carefully pointed out this passage recently.
161
sets, set sizes, and infinity in badiou’s being and event
of the state but we take it up here only as a technical term. Given a situation,
there is a state of the situation. This is the difference between a set and its pow -
erset. This also lines up with presentation and representation. On this account,
Badiou offers two analyses. The first is that given the operation of the power -
set, the sequence of cardinal numbers is interminable. The second is that the
general application of the powerset results in an outstripping, not only of the
quantity of the sets involved in the operation, but the outstripping of the scale of
measurement. That is, insofar as one moves upwards in cardinality by the pow -
erset of all the sets in that cardinality, each new powerset installs a new scale of
quantity (the hierarchy of transfinites) that is irreducible to the “how many” of
the lower scale. Here, Badiou notes that, “the natural measuring scale for mul -
tiple-presentations is not appropriate for representations. It is not appropriate
for them, despite the fact that they are certainly located upon it. The problem
is, they are unlocalizable upon it.”
17
The point here is that for non-denumerable
sets (non-finite cases) cardinality locates lower sets within them but cannot lo -
calize them due to the expansion in cardinality. Reinterpreting Galileo’s demon-
stration, all sets of natural and rational numbers are of the same cardinality א
0
.
In the same way, any mapping of transfinite cardinalities to their powersets will
result in this same “unlocalizability”. The distinctions of a lower dimension are
“lost” in the higher dimension. The “state” of a situation, whose constituents
are subsets of the situation is therefore always at a higher cardinality than the
set of the constituents itself.
The resources of cardinality are employed in L’Être et l’événement. However,
they are used in order to address the difference between presentation (governed
by belonging), and representation (governed by inclusion). This gap, for Badi-
ou highlights what he calls the “impasse of ontology”. This is undergirded by
Easton’s theorem which roughly states that the cardinality of the powerset 2
א0
of
א
0
is arbitrarily greater than א
0
.
18
Essentially, this means that the cardinality of
the given powerset of a cardinal א
0
and greater is any cardinal arbitrarily greater
than א
0
. Badiou puts it in the following way:
17
Badiou, Being and Event, p. 278.
18
A few conditions apply here. Crucially, the set in question is regular (ie., obeying the axi-
om of regularity/foundation).
162
tzuchien tho
This theorem roughly says the following: given a cardinal λ, which is either ω
0
or
a successor cardinal, it is coherent with the Ideas of the multiple to choose, as the
value of │p(λ)│ ─ that is, as quantity for the state whose situation is the multiple ─
any cardinal π, provided that it is superior to λ and that it is a successor cardinal.
19
Easton’s theorem here is a deepening of the Galilean “paradox”. It turns out that
the immeasurability of quantities beyond the finite is a general condition that
goes beyond the gap between the denumerable and the indenumerable. This is
what Badiou will call the “quasi-total errancy”. The scales of cardinal infinites
interplay with an uncontrollable degree of arbitrariness.
To what degree is this an impasse of ontology then? Badiou argues that,
Consequently, Easton’s theorem establishes the quasi-total errancy of the excess
of the state over the situation. It is as though, between the structure in which the
immediacy of belonging is delivered, and the metastructure which counts as one
the parts and regulates the inclusions, a chasm opens, whose filling in depends
solely upon a conceptless choice.
20
The “quasi-total errancy” that Badiou refers to here is precisely the role of car -
dinality in L’Être et l’événement. That is, given well-ordered sets and consistent
multiplicities beyond the finites, the application of cardinality reveals a field of
arbitrariness. In other words, all questions of quantity and measure fall into a
state of underdetermination where measure is stipulated (chosen) rather than
deduced. From this Badiou offers an interpretation of this arbitrariness of the
domain of the infinite consistent with our pluralist argument:
Being, as pronounceable, is unfaithful to itself, to the point that it is no longer
possible to deduce the value, in infinite extension, of the care put into every pres -
entation in the counting as one of its parts. The un-measure of the state causes
an errancy in quantity on the part of the very instance from which we expect -
ed-precisely-the guarantee and fixity of situations. The operator of the banish-
ment of the void: we find it here letting the void reappear at the very jointure be -
tween itself (the capture of parts) and the situation. That it is necessary to tolerate
19
Badiou, Being and Event, p. 279.
20
Ibid., p. 280.
163
sets, set sizes, and infinity in badiou’s being and event
the almost complete arbitrariness of a choice, that quantity, the very paradigm
of objectivity, leads to pure subjectivity; such is what I would willingly call the
Cantor-Gödel-Cohen-Easton symptom. Ontology unveils in its impasse a point at
which thought-unconscious that it is being itself which convokes it therein-has
always had to divide itself.
21
The pluralization of the “one”
Through examining the difference between cardinality and ordinality, we saw
that Badiou’s count-as-one is expressed in a transformed concept of the “one”. It
no longer forms part of the dialectic between the one and the infinite, where the
infinite, if it is in any sense actual, must be presented as a totality. What results
is a pluralization of the “one”.
The argument so far has been the following. The Cantor-Dedekind revolution,
while traditionally interpreted as the actualization of infinite sums, should in-
stead be understood as the obsolescence of infinity. The key reason for this is
that the “one” as repetition and totality, within this new domain, ceases to play
the role of distinguishing between the finite and the infinite. From a cardinality
perspective, although denumerability and non-denumerability sustains the tra-
ditional border between the finite and infinite as measure, the “infinite” in this
sense fails to correspond to the traditional notions of the infinite. Even if we take
the infinite to be non-denumerability, we see the problems of the indefinite re -
produced via Easton’s theorem. Hence the “infinite” in this case, or the realm of
the “infinite”, fails to be “actual” if we understand this actuality as conditioned,
at least, by determination. If we take this question from the perspective of ordi-
nality, we do recover the border between denumerability and indenumerability.
The transfinite ordinal as a limit ordinal is logically prior to transfinite cardinal -
ities.
22
But here, as we have argued, the structure of ordinality does not require
21
Badiou, Being and Event, p. 280.
22
It should be noted here that Cantor saw ordinality and its well-ordering to be more funda-
mental than cardinality. Georg Cantor, “Über unendliche, lineare Punktmannichfaltigkeit -
en”, in E. Zermelo (ed.), Gesammelte Abhandlungen mathematischen und philosophischen
Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel, J.
Springer, Berlin 1936; reprinted Olms, Hildesheim 1966, pp. 165–209, p. 169. Although the
advent of the cardinal approach to infinity was historically prior, the priority of the ordinal
is asserted from a logical perspective. Cf. Hans Niels Jahnke, “Cantor’s cardinal and ordi-
164
tzuchien tho
the infinite in order to stipulate a transfinite limit ordinal. In turn the primacy of
ordinality in Badiou’s ontological project in L’Être et l’événement, based on the
count-as-one ordinality, renders infinity an obsolete concept for ontology. The
claim here is to focus instead on the pluralization of the “one”.
The pluralization of the notion of the “one” is simply rendered by the notion of
the set. The set is a “one” in that it is counted in the ordinal sense. It therefore
matters that this count is not confused with measurement. Sets that are built
out of the ZFC axioms are placed within a family of branching sets that fulfils
the condition of well-ordering.
23
The basic structure that this ordinality analyses
is therefore the structure of succession (i.e. for any set, say a pure set, there is a
successor set that counts that set). However, this only gets at the structure and
not the existence of sets, the notion of a “one” produced by counting (succes -
sion). What then accounts for this parallel structure of existence?
For the project in L’Être et l’événement, Badiou asserts throughout that it is the
count that produces the one, eschewing the notion of the “one” as a given.
Hence it is the count, an act of counting, that introduces the existence at the ba-
sis of the analysis. To stave off misunderstanding, mathematical ontology does
not go around counting what exists. Instead it is a rational and logically classi-
cal analysis of the result of the act of the count, the unfolding relation between
inconsistent multiplicity, what is counted as void, and consistent multiplicity,
the aggregation of counts following on the count of the void and its successors
(ordinals). What constitutes the grounds of this ontological project theorized
by the two “seals” [sceaux] that Badiou introduces in Meditation 14 of L ’Être
et l’événement. Badiou’s approach pluralizes the “one” treated as the product
of the count and thereby concretizes the subtractive aspect of the ontology by
revising its traditional basis.
The two “existential seals” occur as limits. The first limit is that which seals off
inconsistent multiplicity from consistent multiplicity. Hence, this is the count
of the void. Whatever is or is in inconsistent multiplicity is excluded from the
nal infinities: An epistemological and didactic view”, Educational Studies in Mathematics
(48/2001), pp. 175–197.
23
Note that while not all sets per se can be well-ordered, in ZFC, Zermelo’s theorem shows
the equivalence between the axiom of choice and the well-ordering theorem.
165
sets, set sizes, and infinity in badiou’s being and event
domain of set theoretical entities by inscribing the inconsistent as the void of
the consistent structure carved out by sets in their ordinal arrangement. From
this, as we have already discussed, sets branch off in ordinals by counting the
void. This results in a well-ordered tree of sets grounded in this first seal. Now,
the second seal is the limit between the finite ordinals, the first transfinite ordi-
nal, and the transfinite orders that follow after the limit as successors. Since the
structure of ordinality is constructed by succession, nothing within this struc -
ture offers a distinction between the finite and transfinite. The first tranfinite
ordinal is stipulated rather than given from the background structure. This bor -
der between the finite and transfinite is thus the limit of the finite series and the
start of the transfinite series, a second existential seal.
We have discussed the first seal previously and thus set it aside. The second
existential seal has also been briefly addressed. However, it is important here
to underline that although, under a traditional reading, this limit ordinal intro -
duces an “actual” infinite in the practice of set theory, what it actually provides,
in Badiou’s reading, is a further application of the subtraction of the one. Here,
what is crucial is that the transfinite ordinal is not a new “one” that counts the
finite series in completeness. Rather, it is an inscription of a consistent multi-
plicity beyond finitude. In this sense, the infinite and the transfinite limit do not
coincide. As Badiou argues:
[W]e have not yet defined infinity. A limit ordinal exists; that much is given. Even
so, we cannot make the concept of infinity and that of a limit ordinal coincide;
consequently, nor can we identify the concept of finitude with that of a successor
ordinal. If α is a limit ordinal, then S(α), its successor, is ‘larger’ than it, since αÎ
S(α). This finite successor — if we pose the equation successor=finite — would
therefore be larger than its infinite predecessor — if we pose that limit = infinite —
however, this is unacceptable for thought, and it suppresses the irreversibility of
the ‘passage to infinity’. If the decision concerning the infinity of natural being
does bear upon the limit ordinal, then the definition supported by this decision
is necessarily quite different. A further proof that the real, which is to say the
obstacle, of thought is rarely that of finding a correct definition; the latter rather
follows from the singular and eccentric point at which it became necessary to wa-
ger upon sense, even when its direct link to the initial problem was not apparent.
166
tzuchien tho
The law of the hazardous detour thereby summons the subject to a strictly incal -
culable distance from its object. This is why there is no Method.
24
Badiou moves on from this to define infinity according to this limit ordinal. Any
set where the limit ordinal belongs is infinite, any set belonging to the limit is
finite. The ontological significance here is not captured by this distinction. What
this second seal indicates is that it replays the distinction between the inconsist -
ent and consistent multiples by treating the finites as the inconsistent (because
interminable) multiples. That is, just as nothing within consistent finitude can
generate the limit transfinite ordinal, nothing in inconsistent multiplicity can
generate the count-as-one of the void. This leads us to the key distinction of our
investigation. Here Badiou argues,
In the order of existence the finite is primary, since our initial existent is Ø, from
which we draw {Ø}, S{Ø}, etc., all of them ‘finite’. However, in the order of the
concept, the finite is secondary. It is solely under the retroactive effect of the
existence of the limit ordinal ω
0
that we qualify the sets Ø, {Ø}, etc., as finite;
otherwise, the latter would have no other attribute than that of being existent
one-multiples.
25
What Badiou goes on to develop here, at the end of Meditation 14, is the concep -
tual effect of the Cantorian revolution. This is the distinction between existence
and concept. According to the criterion of the order of existence, there is no
relation between the finite and infinite, since, qua ordinals, the sets succeed
each other. If {α} is finite, so is its successor S{α}. It is only with the limit ordinal
ω
0
that the limit between the finite and non-finite is marked. Yet, as Badiou ar -
gues, this limit is stipulated, and it is only through this stipulation that the finite
and infinite can be distinguished. As Badiou argues here in the passage above,
what this reveals is not so much a realm beyond the finite, that is, the infinite,
but rather a region of the infinite classified as the “finite”. This second seal is
therefore a repetition of the first seal. Just as the inconsistent multiple can be
recognized as inconsistent only after the first seal which designates consistent
multiplicity, finitude can only be recognized (and defined) by the designation of
the transfinite limit ordinal.
24
Badiou, Being and Event, p. 157.
25
Ibid., p. 159.
167
sets, set sizes, and infinity in badiou’s being and event
Badiou’s argument here reflects onto our earlier discussion of Dedekind’s argu -
ment for the non-finite by means of infinite cardinality. In both cases, the finite
is treated as a special case for the ordinary “infinitude” of sets. Hence, finitude is
only available as a concept with respect to the definition of the infinite. Here, we
can quibble with Badiou on the elaboration of this point. Just as the inconsistent
multiple is not inconsistent before the forming of consistent multiplicity, the
existence of sets and their successors are not finite before the transfinite limit
ordinal. It is therefore incorrect to treat the finitude as primary in the order of
existence. The crucial difference here is that the count, the structure analyzed
by ordinality, is fundamentally indifferent to the distinction between finite and
infinite whereas cardinality is fundamentally sensitive to this difference.
Badiou here reinforces the subtraction of the one by drawing out the further
implication that the one, the counting-as-one (and forming-into-one) as a result
of an act of counting rather than a given. The one, the concept through which
being must be analyzed, will always be alien to what is analyzed. This holds in
the case of the original count (the count of the void) as well as within consistent
multiplicity. That is, the one, within the post-Cantorian context, is released from
its association with finitude. The forming-into-one of a set is shown to be finite
only in specific contexts but generically non-finite. The limit ordinal demon-
strates this in Badiou’s interpretation because it shows finitude to itself be a
result of a “one” that is formally outside of it and thus capable of “counting” it
(qua ordinal) and standing as a limit, a “one” that stands beyond the “ones” that
populate finitude.
The new “one” within the post-Cantorian context is thus completely trans -
formed. Another way of saying the same thing is to reaffirm that the dialectical
relationship between the one and the infinite no longer holds. The one is no
longer distinguished from the infinite but rather part and parcel of it. Just as the
infinite is no longer the “beyond” of finitude, the one is no longer the obstacle
of infinity.
The reasoning here concerning the one indicates a deep rift between treating
ontology by means of ordinality or cardinality. Cardinality relies on the one-to-
one mapping between sets (or a set to itself) and therefore relies on a traditional
notion of the one as a pre-given unit-entity. Here, the infinite or the transfinite
involves the equal cardinality of wholes and parts, a suspension of the identity
168
tzuchien tho
between oneness and wholeness (or totality). Hence despite the general coher -
ence in set theory between ordinal and cardinal treatments of the transfinite,
Badiou’s emphasis on ordinality, via the structure of the count, implies the ac -
commodation of different means of reckoning the measure (i.e. cardinality) of
any given set. Hence Badiou’s ontological interpretation of ordinality and the
limit ordinal implies a pluralization of the one in the sense that the ordinal
structure can accommodate different theories of cardinality.
In what follows, we will examine an alternative non-Cantorian treatment of car -
dinality and revisit the primacy of ordinality in the mathematical ontology of
L ’Être et l’événement. What we aim to underline is the pluralization of the “one”
as the consequence of the ontological project of L ’Être et l’événement.
Numerosity and ordinality
If we go back to the traditional problems of infinity and the Cantorian revolu -
tion, we find that the hallmark of the new “infinity” (in cardinal terms) was
the conceptual break with Euclidian measurement. Recall that for Dedekind,
an infinitely sized set is one where at least one subset is equal in size to the
original set. Cantor’s transfinite cardinal also has this property. Also recall that
this is logically independent of transfinite ordinality, a question that is relevant
but not determinant in questions of measurement per se. However, an alterna-
tive approach to infinity has always existed in the sub-currents of mathematics
and its philosophical expositors even in the Cantorian age. This is an alternative
approach that rejects the relinquishing of traditional Euclidian measurement.
Hence, to deepen our inquiry into the pluralization of the one, we should exam-
ine the recent resurgence of this Euclidian approach. The refusal to relinquish
Euclidian part-whole relations is to express a commitment to the notion of the
oneness as wholeness. Its friction with the canonical Cantorian view will allow
us to grasp what the pluralism of one offers us.
In what its proponents call “numerosity theory”, the Euclidian principle, which
maintains that the part is always lesser than the whole, is maintained. Although
Bolzano, a senior contemporary of Cantor and Dedekind, developed some fea-
tures of a concept of the infinite that maintains the Euclidian principle in his 1851
Paradoxes of the Infinite [Paradoxien des Unendlichen], numerosity theory is a
169
sets, set sizes, and infinity in badiou’s being and event
distinct recent development.
26
This emerged in a series of papers since 2003 by
Benci and Di Nasso around “Numerosities of labelled sets: a new way of count -
ing.”
27
Since 2003, the research program has grown significantly to include sev -
eral aspects of mathematics including probability. It has also sparked harsh
criticism in the philosophy of mathematics notably by Parker.
28
Significantly,
it has also been taken up philosophically and historically by Mancosu, moving
beyond the narrow domain of non-standard analysis and set theory.
Numerosity theory, from Benci and Di Nasso’s 2003 paper, extends the Euclidi-
an principle standard from finite cases to infinite cases. Citing from Mancosu’s
reconstruction of the paper
29
, we take Benci and Di Nasso’s theory as operating
from the maintenance of three principles:
1. if there is a bijection between A and B then v (A) = v (B)
2. if A ⊂ B then v (A) < v (B)
3. If v (A) = v (A’) and v (B) = v (B’) then the corresponding disjoint unions (∇)
and cartesian products (x) satisfy:
v ( A ∇ B) = v(A’ ∇ B’)
v (A x B) = v ( A’ x B’)
The first is simply the definition of equivalence from bijection, a standard prin-
ciple of cardinality. The third is a definition of sums and products in numerosity
also standard to sets of this kind. What is distinctive is the second principle that
maintains the strictly “lesser than” difference between a set and its proper sub -
set. This is what is under contention.
26
Bernard Bolzano, Paradoxien des Unendlichen, C. H. Reclam, Leipzig 1851.
27
Vieri Benci and Mauro Di Nasso, “Numerosities of labeled sets: A new way of counting”,
Advances in Mathematics, 173(2003), pp. 50–67. Cf. V. Benci, M. Di Nasso, and Marco For -
ti, “An Aristotelian notion of size”, Annals of Pure and Applied Logic, 143 (1–3/2006), pp.
43–53.
28
Matthew Parker, “Set size and part-whole principle”, The Review of Symbolic Logic, 6
(2013), pp. 589–612.
29
Mancosu, “Measuring the size of infinite collections of natural numbers: Was Cantor’s
theory of infinite number inevitable?”, p. 139–145.
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tzuchien tho
How does numerosity work? Intuitively, what the theory does is provide labels
to sets such that subsets can be indexed (labelled) together in finitary way. Man-
cosu uses, as an illustration, the Italian game of Tombola, a version of Bingo.
On a master board with numbers 1–90, pegs are placed when a number is drawn
from a well at random. If a called number corresponds to a number on one’s
own board (an arbitrary rearrangement of 1–90), one pins the number down.
The goal is to cover the board with pegs corresponding to the master board. In
the middle of the game, how does one check one’s progress in the game? There
are three options.
First, one could ignore the order of the numbers and simply check if the num-
ber of pegs on one’s board corresponds with the number of pegs on the master
board. This bijective function would be equivalent to the Cantorian approach
to cardinality. Second, one could list the pegs in order such that the numbers
1-90 are listed in order. This corresponds to an ordinal approach. Finally, one
could label the pegs by designating the numbers by decades: 1-10, 11-20, 21-30,
etc. Each label would be partial, given we are in the middle of the game. Under
practical circumstances, this would indeed be an odd way to see how one is pro -
gressing in the game of Tombola. Nonetheless, what this does is the breakdown
of a potentially linear sequence into incomplete partial sums.
The third approach is thus an intuitive image of numerosity. The goal, as Manco -
su argues, “is to split a set of objects into boxes each one containing only finitely
many objects. The metaphor of putting things in box number 10, 20, and so forth
will be captured by the idea of a labelled set. From now on we deal only with
countable sets.”
30
A potentially confusing point here is the fact that in a game, each of the boxes
(ie. labels), will be incomplete as we have not finished the game. The point is
that each of these partial sums constitute an approximation of the cardinality
marked by each label. The whole sum will be the sequence of these approxima-
tions. The formal definition of numerosity will then rely on how one is to operate
these sequences of approximation. Here, I will again cite Mancosu’s simplified
outline of a “calculus” for numerosities.
30
Ibid., p. 139.
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sets, set sizes, and infinity in badiou’s being and event
Def. 3.1
The sum of two labeled sets A, B is
A ⊕ B = where l
A
⊕ l
B
(x) = l
A
(x) if x is in A and l
B
(x) if x is in B.
[Caveat: take disjoint unions of A and B only]
Def. 3.2
The product of two labeled sets A, B is
A ⊗ B = where l
A
⊗ l
B
(x,y) = max { l
A
(x) ; l
B
(y)}.
Def. 4 Definition of numerosity
A numerosity function for the class L of all countable labeled sets is a map num:
L -> N onto a linearly ordered set < N, ≤ > such that the following properties are
satisfied
31
:
(1) If #A
n
≤#B
n
for all n, then num(A) ≤ (B)
(2) x