[Badiou and Science] 1.1 Sets and Multiplicities

“[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.” — Alain Badiou, Being and Event pg. 5 (italics mine)

As we begin talking about sets and set theory, we will inevitably encounter the various notations and formalisms used to describe the theory and its logic in its most abstract sense. While it is not completely necessary to be able to read these formalisms to understand the radical consequences of set theory (as many of its major conclusions can be related in natural language) we will maintain that the ability to do so greatly enhances Badiou’s case for positioning set theory as a fundamental language, and that learning to read set theory’s symbols only requires a brief primer (for example, see this useful cheat sheet). For this post, I will spare the body of the formalism details, but will ask the interested reader to refer to the end notes for further explanation.

A Set is a Multiple
It is both convenient and intuitive to think of a set as a “collection of objects”, and one is certainly not prohibited from doing so. However, the beauty of set theory is that it defines a set as composed of objects (“elements”) that are undefined by the theory itself. So set theory provides the properties for sets in general, without specifying the actual 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. Every set is a “multiplicity” of objects, and each of the objects “contained” within a set may themselves be another multiplicity (e.g. the set “all patients in this hospital” is composed of the set “all patients with insurance” and the set “all patients without insurance”). This is set theory’s advantage as an ontology: what “is”? Sets. Multiplicities.

The origins of set theory begin with treating numbers as their primary object of inquiry. But what are numbers?

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…” [pg. 2, italics mine].

Lamenting the modern-day role of numbers as that which is 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 as it was explored historically. His goal in Number and Numbers is to rescue the concept of Number from both banal empirical definitions and outlandish mystical musings, and to instead deliver it unto a conceptually solid ground apprehended by pure thought alone.

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, without having to rely on limited conscious observations on the one hand, nor the rule of God on the other? Is thought even capable of such a task? 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’s Theory of Sets
Set theory proper begins in the 1880’s with Georg Cantor. Cantor’s work provides us with his novel “set-theoretical” approach to constructing numbers, and it is deceivingly simple: All that is required is the concept of “belonging”, marked “∈” [1]. All sets “belong” to another set.

For example, we could say that the set [shirt, pants] and the set [socks] belong to set [clothing]: so [shirt, pants] ∈ [clothing] and [socks] ∈ [clothing]. We note that each of these sets may further consist of sub-elements or sub-sets (e.g. [socks] may consist of [left socks, right socks]).

The objects we chose for the above example may seem arbitrary. However, within the development of set theory it is crucial to note that there is no explicit definition of what a set is, and therein lies the beauty of its conceptual edifice — set theory 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. Set theory thereby seeks to provide the abstract rules for any set whatsoever.

We will also note that “sets” may also be referred to as “multiples”, and that Badiou prefers to use this convention. So, Sets == Multiples.

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 subset of every set (see below). For all finite sets A of size n, the number of elements in p(A) will always equal 2ⁿ (so, in the latter example, as A has 3 elements, its power set will necessarily have 2³= 8 elements). The power set is a general feature of all sets [2].

As a visual example (from the Cohen article attached to the previous post) the power set of a finite set can be elaborated as follows:

Constructing Natural Numbers
We now have enough tools to start producing our basic counting numbers. To construct the natural numbers, one starts with the “empty set”, or the set that contains no elements. This empty set is marked ∅ or { }. Next, one can construct the set that contains the empty set, the “set of the empty set”: [∅]; this set containing the single element of the empty set. Then, one can construct the set containing this set and the empty set: [∅, {∅}]. If one takes the set containing the latter set and the two former sets, we construct: [∅, {∅}, {∅,{∅}} ].

The perceptive reader will notice that these enumerations correspond to what we know as the “ordinal” (or “natural” or “counting”) numbers:

∅ = 0

[∅] = 1

[∅, {∅}] = 2

[∅, {∅}, {∅,{∅}} ] = 3

[∅, {∅}, {∅,{∅}}, {∅, {∅}, {∅,{∅}}} ] = 4

… and so forth.

This is pretty remarkable, right? We effectively just formed our familiar counting numbers from “nothing” and a rule of succession (the “adding” of a set to itself). As one can clearly see, these ordinals are merely composed of — or “contain” — all those ordinals that preceded them [3].

Intuitively, we can say that “3” is represented by the set [0, 1, 2]. Ontologically, we can say that the Being of “3” is the equivalent set composed of empty sets: [ ∅, {∅}, {∅, {∅}} ]. Such is the case for the Being of any natural number, and all that we need is the concept of an empty set and the relation of “belonging” to define these ordinals as such.

Badiou spends many chapters of Being and Event (as well as other essays) speaking on the philosophical significance of building fundamental sets and numbers on “nothing” (the empty set). His discussion is situated within the historical philosophical debate over the One versus the Many, and which of these concepts is more fundamental than the other. 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 a pre-existing something [“1”].

Now, is there a limit to these natural numbers? Stay tuned!

- Dr. G

Next: Infinity

Notes

[0] An earlier version of this post promulgated the power set function as the rule of succession for constructing the ordinals. This was incorrect — for years I erroneously believed this to be this case as you can demonstrate it to be so in passing from 0 to 1 to 2 to 3 (but, as it turns out, no further than that). I have since updated the demonstration of ordinal succession with its standard treatment, and apologize for any confusion this may have caused.

[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: “If a is a set, then there is a set P(a) whose elements are all subsets of a. ∀x ∃y ∀z [z ∈ y ↔ ∀w (w ∈ z → w ∈ x)]” [Mary Tiles, The Philosophy of Set Theory, 123]. Note that the power set is simply the group of inclusions or subsets of the original set, as defined in [1].

[3] We shall see later that the axioms of set theory will properly formalize these rules of succession. More precisely, using the axiom of union (or the pair-set), if A = ∅ and B = [∅], we can produce a C = A U B = ∅ U [∅] = [∅, {∅}], then a D = [∅, {∅}] U [{∅, {∅}}] = [∅, {∅}, {∅, {∅}}], and so forth. [Mary Tiles, The Philosophy of Set Theory, 125 and 134]