[Badiou and Science] 1.6 Axioms of Foundation and Choice

“The remarkable conceptual connection affirmed here [by the axiom of foundation] is that of the Other and foundation. This new Idea of the multiple stipulates that a non-void set is founded inasmuch as a multiple always belongs to it which is Other than it. Being Other than it, such a multiple guarantees the set’s immanent foundation, since “underneath” this foundational multiple, there is nothing which belongs to the initial set…

The hypothesis I advance is the following: within ontology, the axiom of choice formalizes the predicates of intervention. It is a question of thinking intervention in its being… Ontology declares that intervention is, and names this being ‘choice’ (and the selection, which is significant, of the word ‘choice’ was entirely rational)… The axiom of choice subsequently commands strategically important results of ontology, or mathematics: such is the exercise of deductive fidelity to the interventional form fixed to the generality of its being.” — Alain Badiou, Being and Event, pgs. 186, 227-228

As alluded to in the previous section, some of the set theory axioms were specifically designed to address paradoxes or difficulties that had come up in the course of the theory’s development. We shall briefly go over two such axioms, whose development will have decisive consequences for understanding Badiou’s concepts of revolutionary sites and interventions.

The Axiom of Foundation and Russell’s Paradox
In its simplest terms, the Axiom of Foundation states that a set cannot belong to itself; that is, a set must always contain an element such that the contents of this element are not also elements of the initial set. Or, for a set a where set b ∈ a, there must be a b where b ꓵ a = Ø (a and b have no shared element in common) [1]. a ∈ a is therefore barred from being a valid set. This is intuitive, as such a set would be problematic due to the “infinite regress” it would cause:

Historically, the Axiom of Foundation came in the wake of Bertrand Russell’s notorious paradox of self-referencing sets. There are several famous formulations of this paradox, and I encourage the reader to view this delightful video that visually details the paradox, and which also efficiently covers the philosophical complexities involved in its discovery. (Of note, the video references poor Gottlob Frege’s mental breakdown that occurred when he was presented with the paradox by Bertrand Russell, in a manner much similar to Georg Cantor’s depression in the face of his aforementioned Continuum Problem. Alas, mathematics appears to have driven many a great thinker to madness…)

Here is one formulation of the paradox that I find to be particularly effective:

Consider a set that does not contain itself as “normal” (for example, the set of all circles is not itself a circle). Now consider a set that contains itself as “abnormal” (for example, the set of all non-circles is itself a non-circle). Finally, consider the set of all normal sets — is this set normal or abnormal? It cannot be normal, or it would contain itself and be abnormal. It cannot be abnormal, or it would not contain itself and be normal. This is Russell’s paradox. [Adapted from Howard DeLong, A Profile of Mathematical Logic, pgs. 81–82]

This type of paradox threatened to render set theory inconsistent, being formulated using well-formed expressions produced by the theory itself. In order to avert such an devastating possibility, the Axiom of Foundation was constructed to effectively outlaw any sets belonging to themselves.

The consequences of requiring a set to contain a distinct set with which it shares no common element is straightforward when it comes to numbers: as all numbers are constructed from the ordinals, which are themselves all composed of sets based on the empty set Ø, the set with which any number has no shared elements will always be the empty set (recall, the empty set contains no elements of its own). As Badiou likes to say, all natural numbers are “founded on the void” Ø . As expected, this simply means that no number “contains” itself. This is intuitive since all ordinals only contain as elements those ordinals that precede them, and by definition not themselves — for example, 3 = [2 1 0] = [ {∅,{∅}}, {∅}, ∅].

However, when it comes to sets of physical objects the consequences are a bit more profound. We can use an example akin to one Badiou uses to demonstrate the Axiom of Foundation in Number and Numbers: Take the set of “living things”. Among the sets contained in this concept will be “humans”, “animals”, and “plants”. So [animals] ∈ [living things]. Now, it would be trivial to say that [animals] will also be composed of [organs], [tissues], [cells], [molecules], [atoms], etc. What will be recognized it that several of these objects also belong to the set [living things]; that is, while [organs] ∈ [animals] and [tissues] ∈ [organs] (and [tissues] ∈ [animals]), both [organs] ∈ [living things] and [tissues] ∈ [living things]. “Cells” clearly compose these latter three concepts, and therefore belong to them as well. But that which compose cells — the elements of the set [cells] — do not belong to [living things]. In this way, while say [living things] ꓵ [organs] = { [tissues] [cells] }, [living things] ꓵ [cells] = Ø — they share no sets in common. As such, [cells] grounds the set [living things]: there is nothing (Ø) “below” [cells] from the perspective of [living things]. Cells, in a very literal sense, supply the “foundation” of living things [2].

The Axiom of Foundation therefore requires that every non-empty set be disjoint from at least one of its elements. This prohibits infinite regress by guaranteeing a “halting-point” to the set, or a “∈-minimal element”.

It will be of interest that in aiming to prevent self-belonging sets, the Axiom of Foundation had to require that all sets contain an element essentially possessing material “foreign” to itself. This injunction, at the very heart of set theory, creates the quite real possibility of “indiscernible” elements from such an x that belongs to A: elements that are not recognized by the language of A, but are yet real at point x. It is through this axiom that “unknown”, “novel”, or “repressed” elements can come to transform a situation by being incorporated into it from the “foundation”.

We shall take leave of these mysterious (yet tantalizing) remarks for now, but shall unravel them further when we discuss Badiou’s concept of revolutionary (or “evental”) sites. Associated with this concept are the formal powers inaugurated by the Axiom of Choice.

The Axiom of Choice
The development of the Axiom of Choice also has a fascinating and storied history. It is the only one of the nine axioms that is considered “optional” (it is what distinguishes traditional ZF set theory from ZFC set theory — ZF “with the Axiom of Choice”), primarily because its incorporation results in paradoxes that, while perfectly coherent on the mathematical plane, are inconsistent in the physical world (such as the famed Banach-Tarski Paradox). Despite this, its stature amongst the other axioms has been widely accepted due to its necessity in proving several key mathematical theorems.

The Axiom of Choice posits a “choice function” that uses elements of existing sets to produce a new set. The axiom essentially acts as a “collecting” function: it states that given a group of non-empty sets, a set exists made up of one element from each of these initial non-empty sets [3].

Such an operation is trivial to visualize for finite sets. For example, take the following sets of [shirts] and [pants], from which we can then “choose” sets of [outfits]:

There are two important points to note about the Axiom of Choice, related to both to the choice function inaugurated by the axiom itself, and the set that results from the actions of the choice function:

• The choice function is anonymous. The axiom does not dictate the particular collection that can result from the choice function; that is, the specifics of the choice function are not specified by the axiom (for example, which outfit is chosen by this choice function, above? All we can say is that it is an outfit, but which outfit is not explicitly defined).
• The outcome of the choice function is (potentially) illegal. The resultant collection “exists”, insofar as it is a set in congruence with the laws of set theory, but this set may not be consistent with the laws regulating the sub-model in which it is developed (for example, the outfit [Shirt3 Pant1], while a potential existent, may not be acceptable attire within its local world).

Discussing the Axiom of Choice when it comes to finite sets is relatively straightforward. But what about infinite sets? For infinite sets the same rational procedure of collection holds, but representing the resultant sets becomes a complex endeavor. As stated, the choice function is not unique, and the choices possible will be “infinite”.

What becomes increasingly transparent when applying the Axiom of Choice to infinite sets is the anarchic quality of the axiom — again, which elements are chosen to forge the resultant set are not prescribed or pre-determined by the axiom, and this new set is the result of infinite possibilities. The Axiom of Choice solely postulates that the result does indeed exist, without specifying any particular individual realization.

Recognizing the Axiom of Choice as embodying properties that make it illegal and anonymous with respect to a given universe of sets is an inspired move by Badiou, who acutely observes:

“In the first place, given that the assertion of the existence of the function of choice is not accompanied by any procedure which allows, in general, the actual exhibition of one such function, what is at stake is a declaration of the existence of representatives — a delegation — without any law of representation. In this sense, the function of choice is essentially illegal in regard to what prescribes whether a multiple can be declared existent…

Second: what is chosen by a function of choice remains unnamable. We know that for every non-void multiple B presented by a multiple a the function selects a representative… But the ineffectual character of the choice — the fact that one cannot in general construct and name the multiple which the function of choice is — prohibits the donation of any singularity whatsoever to the representative f(B) [the resultant set B of the choice function f ]. There is a representative, but it is impossible to know which one it is; to the point that this representative has no other identity than that of having ti represent the multiple to which it belongs. Insofar as it is illegal, the function of choice is also anonymous. No proper name isolates the representative selected by the function from amongst the other presented multiples. [Being and Event, pg. 229]

Again, tantalizing remarks that will be interrogated further, and become more clear, once we discuss Badiou’s concepts of Situations and Events. Ultimately, the importance of the Axiom of Choice to Badiou’s work will be in its unconstrained organizing function — its ability for Subjects to create “novel” sets from the “pieces” of already existent sets, resulting in sets that may be incongruous and threatening to the dominant situation and world in which they are collected. The Axiom of Choice thereby generates pure “possibility” within the matheme, and the formal potential for revolutionary intervention.

’Til next time!
- Dr. G

Next: Summary of Section 1

Notes

[1] “Any non-void [non-empty] set posses as least one element whose intersection with the initial set is void; that is, an element whose elements are not elements of the initial set. One has β ∈ α but β ⋂ α = ∅. Therefore, if γ ∈ β, we are sure that ~( γ ∈ α). It is said that β founds α, or is on the edge of the void [∅] in α” [Alain Badiou, Being and Event, pg. 500].

[2] Of course, one could trivially argue whether “cells” really are the minimal living object (an argument which really amounts debating the social construction and agreement over the concept “living thing”), but that does not change the general point that we are trying to illustrate with this simplified example: all sets have a foundational “stopping point”.

[3] “…the axiom of choice could be formalized in the following manner: (∀a)(∃f)[(∀B) [B ∈ a & B ~= ∅) → f(B) ∈ B] ]. The writing set out in this formula would only require in addition that one stipulate that f is the particular type of multiple termed a function; this does not pose any problem.” [Alain Badiou, Being and Event, pg. 226]