"[After much technical research] I came to think that it was necessary to
shift ground and formulate a radical thesis concerning mathematics… I came to
the conclusion that the sole manner in which intelligible figures could be
found within [the paradoxes of set theory] was if one first accepted that the Multiple, for mathematics, was not
a (formal) concept, transparent and constructed, but a real whose internal gap, and impasse, were deployed by the theory…
The entire history of rational thought appeared to me to be illuminated once
one assumed the hypothesis that mathematics, far from being a game without
object, draws the exceptional severity of its law from being bound to support
the discourse of ontology.”
A set is a multiple.
It really is as simple as that. Any set is a multiplicity, and each of the multiples "contained" within that set may themselves be a multiplicity. The natural question to ask next is: "a set is a multiple of what, exactly?"
The beauty of set theory, as we shall see, is that a set is composed of elements that are undefined by the theory itself. Set theory provides the properties for sets in general, without specifying the properties of a set's members. A set may practically be composed of "all fruits in this basket", or "all patients in this hospital with insurance", or "all citizens of the USA", but the theory does not depend on these individual properties or determinations. This is set theory's advantage as an ontology: what "is"? Sets. Multiplicities.
What is a number?
We use numbers all the time, but what are they exactly? From the outset of his Number and Numbers, Alain Badiou proposes, “A paradox: …we have at our disposal no recent, active idea of what number is… We know very well what numbers are for: they serve, strictly speaking, for everything, they provide a norm for All. But we still don’t know what they are…” [italics mine]. Lamenting the modern day role of numbers as that which are relegated to pure arithmetic or statistics, not to mention polling numbers and market exchange values, Badiou proposes a journey through the genealogy of the concept of number in history. His goal is to rescue the concept of Number from banal empirical definitions as well as outlandish mystical musings.
While Badiou's exposition of the history of constructing numbers from the Greeks through the great mathematicians of Frege, Dedekind, and Peano is quite fascinating in its own right, we shall concentrate on the particular contributions of Georg Cantor to this history. The fundamental question of concern to all these thinkers was the following: Is it possible to think of numbers neither as empirical (as observed in the world), nor as transcendental (as beyond our comprehension), but as a production of thought itself? That is, can we construct numbers on the basis of thought alone, axiomatically, and not rely on limited conscious observations on the one hand, nor the rule of God on the other? To be able to achieve such a feat would allow for a unique realm of thought untainted by the complications of perception and the aporias of mysticism.
Georg Cantor
Set theory proper begins in the 1880's with Georg Cantor. Cantor provided us with his novel "set-theoretical" approach to constructing numbers. As noted above, within the development of this set theory it is crucial to note that there is no explicit definition of what a set is. And therein lies the beauty of the conceptual edifice-- it does not rely on an a priori description of a set and its elements via empirical predicates. While it is intuitive to think of sets as "collections of objects", it is not necessary to define what these objects are beforehand, and these objects (or elements) may be sets themselves. All that is required is the concept of "belonging" [1]. We should also note that “sets” may also be referred to as “multiples”, and that Badiou prefers to use this convention. Set theory thereby seeks to provide the rules for any set whatsoever.
The power set
Constructing natural numbers
To construct the natural numbers, one starts with the "empty set", or the set that contains no elements. This empty set is marked ∅. Next, one can construct a set that contains the empty set, by taking the power set of the empty set: [∅]; this set containing a single element of the empty set. This process can be continued ad infinitum [3]. The perceptive reader will notice that these enumerations correspond to what we know as the "ordinal" (or "natural") numbers: ∅ = 0, [∅] = 1, [∅, {∅}] = 2, [∅, {∅}, {∅,{∅}} ] = 3 , and so forth.
Badiou spends many chapters of Being and Event (as well as other essays) speaking on the philosophical significance of building fundamental sets on "nothing" (the empty set). His discussion is situated within the historical philosophical debate over the One versus the Many. For our purposes, it will suffice to understand that the significance of founding set theory (the very foundation of mathematics) on the empty set enables thought to construct mathematics without a "privileged" element: mathematics starts with nothing (0) and not something (1).
This is pretty remarkable, right? We effectively just formed our familiar counting numbers from nothing and a rule of succession. Ordinals are composed by, or "contain", all the ordinals that precede them. Intuitively, we could say 3 is represented by the set [0, 1, 2]. Ontologically, the Being of 3 is the equivalent set composed of empty sets: [ ∅, {∅}, {∅, {∅}} ]. This is the case for the Being of any natural number, and all that is needed is the concept of an empty set and the operator of "belonging" to compose a set.
Now, is there a limit to these natural numbers? Stay tuned!
- Dr. G
Next: Infinity
Notes
[0] Editor’s Note: So, Donald Trump just won the US election. Kind of a nightmare scenario for someone like Alain Badiou, amirite? Or maybe just the spark for a powder keg of revolutionary possibilities… A coming event waiting to be intervened on…
I considered moving up discussion of Badiou’s The Rebirth of History: Times of Riots and Uprisings in response to this, where Badiou lays out his theory of riots and the profound political truths and transformations that can result from them. However, once again, this theory is best clarified after an understanding of, at least, Being and Event and his Ethics. So I will hold off for now, and will intend to return to it at a more appropriate time. Or, sooner if current events necessitate unpacking its major concepts and ideas in the service of great revolt.
[1] “...set theory distinguishes two possible relations between multiples. There is the originary relation, belonging, written ∈, which indicates that a multiple is counted as element in the presentation of another multiple”. So if A = [x,y,{z,w}], then x∈A, y∈A, and {z,w}∈A. “But there is also the relation of inclusion, written ⊂, which indicates that a multiple is a sub-multiple of another multiple... the writing B⊂a, which reads B is included in a, or B is a subset of a, signifies that every multiple which belongs to B also belongs to a: (∀y)[(y ∈ B) → (y ∈ a)]”. So if B = [{x,y,z}, h], then [{x,y,z}]⊂B, [h]⊂B, and [{x,y,z},h]⊂B. Note that inclusion can be reduced to the primary relation of belonging. [Alain Badiou, Being and Event, 81]
[2] Power set Axiom (or Subset Axiom): “If a is a set and F(x) is any well-formed expression in the language of ZF [axiomatic set theory] with a single free variable, then there is a set b whose elements are those elements of a for which F(a) is true. ∀x∃y ∀z [z ∈ y ⟷ z ∈ x & F(z)]” [Mary Tiles, The Philosophy of Set Theory, 122]. Note that the power set is simply the group of inclusions or subsets of the original set, as defined in [2].
[3] If A = ∅, then B = p(A) = [∅]. C = p(B) = [∅, {∅}]. D = p(C) = [ ∅, {∅}, {∅, {∅}} ]. And so forth. [Mary Tiles, The Philosophy of Set Theory, 125 and 134]
-
Alain Badiou, Being and Event pg. 5 (italics
mine)
A set is a multiple.
It really is as simple as that. Any set is a multiplicity, and each of the multiples "contained" within that set may themselves be a multiplicity. The natural question to ask next is: "a set is a multiple of what, exactly?"
The beauty of set theory, as we shall see, is that a set is composed of elements that are undefined by the theory itself. Set theory provides the properties for sets in general, without specifying the properties of a set's members. A set may practically be composed of "all fruits in this basket", or "all patients in this hospital with insurance", or "all citizens of the USA", but the theory does not depend on these individual properties or determinations. This is set theory's advantage as an ontology: what "is"? Sets. Multiplicities.
What is a number?
We use numbers all the time, but what are they exactly? From the outset of his Number and Numbers, Alain Badiou proposes, “A paradox: …we have at our disposal no recent, active idea of what number is… We know very well what numbers are for: they serve, strictly speaking, for everything, they provide a norm for All. But we still don’t know what they are…” [italics mine]. Lamenting the modern day role of numbers as that which are relegated to pure arithmetic or statistics, not to mention polling numbers and market exchange values, Badiou proposes a journey through the genealogy of the concept of number in history. His goal is to rescue the concept of Number from banal empirical definitions as well as outlandish mystical musings.
While Badiou's exposition of the history of constructing numbers from the Greeks through the great mathematicians of Frege, Dedekind, and Peano is quite fascinating in its own right, we shall concentrate on the particular contributions of Georg Cantor to this history. The fundamental question of concern to all these thinkers was the following: Is it possible to think of numbers neither as empirical (as observed in the world), nor as transcendental (as beyond our comprehension), but as a production of thought itself? That is, can we construct numbers on the basis of thought alone, axiomatically, and not rely on limited conscious observations on the one hand, nor the rule of God on the other? To be able to achieve such a feat would allow for a unique realm of thought untainted by the complications of perception and the aporias of mysticism.
Georg Cantor
Set theory proper begins in the 1880's with Georg Cantor. Cantor provided us with his novel "set-theoretical" approach to constructing numbers. As noted above, within the development of this set theory it is crucial to note that there is no explicit definition of what a set is. And therein lies the beauty of the conceptual edifice-- it does not rely on an a priori description of a set and its elements via empirical predicates. While it is intuitive to think of sets as "collections of objects", it is not necessary to define what these objects are beforehand, and these objects (or elements) may be sets themselves. All that is required is the concept of "belonging" [1]. We should also note that “sets” may also be referred to as “multiples”, and that Badiou prefers to use this convention. Set theory thereby seeks to provide the rules for any set whatsoever.
The power set
A fundamental function one can perform over any
set is that of creating its “power set”, or the set made up of all the
“inclusions” or “parts” of the initial set. For example, if A = {x,y,z}, the
power set p(A) = {∅, x, y, z, {x,y}, {y,z}, {x,z}, {x,y,z}}, keeping in
mind that an empty set is an implicit element of every set (see below).
For all finite sets A of size n, the number of elements in p(A) will equal 2n.
The power set is a general feature of all sets [2].
As a visual aid (from the Cohen article
attached to the previous post) power sets of finite sets are elaborated as
follows:
Constructing natural numbers
To construct the natural numbers, one starts with the "empty set", or the set that contains no elements. This empty set is marked ∅. Next, one can construct a set that contains the empty set, by taking the power set of the empty set: [∅]; this set containing a single element of the empty set. This process can be continued ad infinitum [3]. The perceptive reader will notice that these enumerations correspond to what we know as the "ordinal" (or "natural") numbers: ∅ = 0, [∅] = 1, [∅, {∅}] = 2, [∅, {∅}, {∅,{∅}} ] = 3 , and so forth.
Badiou spends many chapters of Being and Event (as well as other essays) speaking on the philosophical significance of building fundamental sets on "nothing" (the empty set). His discussion is situated within the historical philosophical debate over the One versus the Many. For our purposes, it will suffice to understand that the significance of founding set theory (the very foundation of mathematics) on the empty set enables thought to construct mathematics without a "privileged" element: mathematics starts with nothing (0) and not something (1).
This is pretty remarkable, right? We effectively just formed our familiar counting numbers from nothing and a rule of succession. Ordinals are composed by, or "contain", all the ordinals that precede them. Intuitively, we could say 3 is represented by the set [0, 1, 2]. Ontologically, the Being of 3 is the equivalent set composed of empty sets: [ ∅, {∅}, {∅, {∅}} ]. This is the case for the Being of any natural number, and all that is needed is the concept of an empty set and the operator of "belonging" to compose a set.
Now, is there a limit to these natural numbers? Stay tuned!
- Dr. G
Next: Infinity
Notes
[0] Editor’s Note: So, Donald Trump just won the US election. Kind of a nightmare scenario for someone like Alain Badiou, amirite? Or maybe just the spark for a powder keg of revolutionary possibilities… A coming event waiting to be intervened on…
I considered moving up discussion of Badiou’s The Rebirth of History: Times of Riots and Uprisings in response to this, where Badiou lays out his theory of riots and the profound political truths and transformations that can result from them. However, once again, this theory is best clarified after an understanding of, at least, Being and Event and his Ethics. So I will hold off for now, and will intend to return to it at a more appropriate time. Or, sooner if current events necessitate unpacking its major concepts and ideas in the service of great revolt.
[1] “...set theory distinguishes two possible relations between multiples. There is the originary relation, belonging, written ∈, which indicates that a multiple is counted as element in the presentation of another multiple”. So if A = [x,y,{z,w}], then x∈A, y∈A, and {z,w}∈A. “But there is also the relation of inclusion, written ⊂, which indicates that a multiple is a sub-multiple of another multiple... the writing B⊂a, which reads B is included in a, or B is a subset of a, signifies that every multiple which belongs to B also belongs to a: (∀y)[(y ∈ B) → (y ∈ a)]”. So if B = [{x,y,z}, h], then [{x,y,z}]⊂B, [h]⊂B, and [{x,y,z},h]⊂B. Note that inclusion can be reduced to the primary relation of belonging. [Alain Badiou, Being and Event, 81]
[2] Power set Axiom (or Subset Axiom): “If a is a set and F(x) is any well-formed expression in the language of ZF [axiomatic set theory] with a single free variable, then there is a set b whose elements are those elements of a for which F(a) is true. ∀x∃y ∀z [z ∈ y ⟷ z ∈ x & F(z)]” [Mary Tiles, The Philosophy of Set Theory, 122]. Note that the power set is simply the group of inclusions or subsets of the original set, as defined in [2].
[3] If A = ∅, then B = p(A) = [∅]. C = p(B) = [∅, {∅}]. D = p(C) = [ ∅, {∅}, {∅, {∅}} ]. And so forth. [Mary Tiles, The Philosophy of Set Theory, 125 and 134]
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