[Badiou and Science] 1.5 Axioms and Predicates
“A great theory, which had to show itself capable of providing a universal language for all branches of mathematics, was born… It became possible to rigorously specify the notion of property, to formalize it by reducing it — for example — to the notion of a predicate in a first-order logical calculus, or to a formula with a free variable in a language with fixed constants… axiomatization is not an artifice of exposition, but an intrinsic necessity… Axiomatization is required such that the multiple, left to the implicitness of its counting rule, be delivered without concept…” — Alain Badiou, Being and Event, pgs. 38–39, 43
Axioms traditionally allow us to bring structure and order to an area of thought. A stronger argument, and one that Badiou ascribes to, is that axioms enunciate fundamental and basic information about a universe of objects. As such, axioms outline the proposed “reality” of these objects in the broadest sense, and particularly in areas of thought suspended from physical evaluation (c.f. Euclid’s five axioms of geometry). We shall discuss the importance of axiomatizing set theory for these purposes, and then begin to explore the consequences of these axioms, starting with predicates and propositions.
The Axioms of Set Theory
Set theory was formalized into axioms in the early 1900's by Ernst Zermelo and Abaraham Fraenkel, forming the ZFC axiom system (“ZF” representing the original axioms, with “ZFC” representing the ZF axioms appended with the Axiom of Choice).
The historical impetus behind axiomatizing set theory was not only to provide a rigorous and clear formulation of the theory, which existed in scattered pieces up until that point, but to additionally address the paradoxes and contradictions within set theory that had appeared since its inception. We will investigate some of these paradoxes in the next post — as it helps to have an understanding of predicates and the role of language in the theory first — but shall nevertheless present the axioms that attempt to “solve” these paradoxes here in anticipation of that discussion.
We will recall that all sets are purely defined by the undefined relation of “belonging” (∈). What is added to this primary structure in order to construct the axiomatic are “logical” operators, such as the existential quantifier ∃ (“there exists”), the universal quantifier ∀ (“for all”), implication → (“if-then”), biconditional ↔ (“if and only if” or “iff”), and conjunction V (“aVb is true iff a is true and b is true”). This architecture allows the axiom system to exist as a “first-order predicate logic” that can then exert a binary logic upon statements created within it (see “Predicates” below).
With just the resulting nine axioms, the entire universe of numbers (and mathematics itself) can be enumerated. We have briefly presented some of these axioms already, particularly those inaugurating the empty set, the power set, and the infinite set. We reproduce the full set of axioms in the below Appendix for clarity, and attach their logical formalizations. We will reiterate that one does not necessarily have to understand how to read these formulations in order to understand Badiou’s arguments, but being able to do so only requires a brief lesson (see here) and will further enrich an understanding of Badiou’s own later formulations. (See Appendix: The Axioms of Set Theory, below).
Again, these axioms regulate the form and function of any and all sets. Note their elegance and relative compactness — through just nine basic axioms the whole known mathematical universe could now be derived, independent of the exploration of the physical world. As Peter Hallward, one of Badiou’s most ardent acolytes, keenly puts it:
“The axiomatization of set theory as the foundation for mathematics completed the process begun by Descartes and the arithmetization of geometry, namely, the liberation of mathematics from all spatial or sensory intuition. Numbers and relations between numbers no longer need to be considered in terms of more primitive intuitive experiences (of objects, of nature) or logical concepts. The whole of mathematics could now be thought to rest on a foundation of its own making, grounded on its own internally consistent assertion” [Badiou: A Subject to Truth, pg. 340, italics mine].
Badiou treats the axioms that deploy set theory as a language capable of describing the primary structure of what “is” in the most general sense — again, mathematics as an ontology (see Section 1.0). The ultimate power of the axioms lies in their being able to construct the idea of a set, but without having to specify what any set actually contains (aside from other sets). These axioms thereby form the rules of composition for any “thing” that can be considered as a set, and while various theorems can be applied to create “sub-models” of this set-theoretical universe, every universe of sets is bound to these axioms.
The overarching way I picture these effects is the following:
Predicates and Propositions
The practical beauty of the set theorem axioms is that while they are primarily meant to describe the mathematical universe, their structuring and logic can theoretically apply to anything that can be conceived of as sets. So while mathematicians primarily works on sets composed of number objects, the theory itself is agnostic to the specific objects of any set. Therefore, one can adapt the domain of set theoretical discourse to apply to any practical group of objects.
This ability can first be seen in the Axiom of Separation, which allows for the use of “well formed expressions”(F(x)) to extract elements with certain properties out of sets. The corresponding notion of predicates and propositions allows one to “separate” out the objects of our experience — as sets — through the use of language. These predicates may then be marked “true”or “false” in their consistency with the rest of the dominating theorems and statements of the system.
For example, if a = [8 .33 -5], we can arbitrarily form the expression F(x): x is an integer within domain a. Then we plainly observe that F(8) is true while F(.33) is false.
As a physical example, consider the following socks:
A predicate (essentially a variable function) becomes a proposition (true or false) for a fixed value of a specified domain. A short video summarizing this use of predicates can be seen here.
Mathematician Howard DeLong broadly defines predicates and propositions in following way, incorporating the use of natural objects as potential domains of discourse:
“The central notion to be explained is that of a propositional function (or predicate, as it will also be called). Let there be fixed some nonempty domain of discourse, that is, some set of objects which our logic will be about. Examples may include the set of physical objects, the set of living animals, or the set of natural numbers. The members of the domain of discourse will be called individuals. An n-place propositional function (or n-place predicate) is a function of n individual variables, where the domain of definition is the domain of discourse and the domain of values is a set of propositions. Hence, when each variable in a propositional function (predicate) has assigned to it an individual, the result is a proposition.” [Howard DeLong, A Profile of Mathematical Logic, pg. 111].
Therefore, having the axiomatization of set-theory be equipped with logical operators creates a “first-order predicate calculus” or “first-order logic”, whereby predicates or statements generated by various theorems can be tested for their veracity in a binary fashion — propositions deemed as “true” or “false” when processed through logic of the mathematical universe under consideration.
Being able to apply these concepts of predicate and proposition outside of pure mathematics and onto the empirical world, while still conforming to the regulations of the axiomatic, will form the crux of Badiou’s philosophical project. But before we explore this further, two other axioms — the Axiom of Foundation and the Axiom of Choice — deserve further mention.
’Til next time!
- Dr. G
Next: Foundation and Choice
Appendix: The Axioms of Set Theory
[Adapted from Mary Tiles, The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise]
For sets a, b, c… and free variables … x, y, z:
Axiom of Extensionality: if two sets have the same elements, they are identical.
(Or, the equality of two sets is determined by the elements they contain.)
∀x ∀y ∀z [z ∈ x ↔ z ∈ y ] → x = y ]
Axiom of the Empty Set: there is an empty set containing no elements (Ø)
∃x ∀y ~( y ∈ x )
Axiom of Pairing: if there are two sets a and b, there is also a set [a,b]
∀x ∀y ∃z ∀w [w ∈ z ↔ w = x V w = y ]
Axiom of Union: if a is a set, then there is a set of all unique elements of a, written Ua.
(For example, if a = { [1 2] [2 3] }, then Ua = [1 2 3])
∀x ∃y ∀z [z ∈ y ↔ ∃w (w ∈ a V z ∈ w ) ]
Axiom of Infinity: there is a set that contains the empty set, and is such that if a belongs to it then so does U{a, [a]}.
(Equivalently put, there is a set that consists of the empty set and all “successors” of element a)
∃x [Ø ∈ x & ∀y ( y ∈ x → ∃z (z ∈ x & ∀w (w ∈ z ↔ w ∈ y V w = y) ) )
Axiom of Foundation: if a is a non-empty set, there is an element b of a such that there is a set which does not belong to both b and a.
(We will discuss this axiom later in more detail…)
∀x [~(x = Ø) → ∃y (y ∈ x & ∀z (z ∈ x → ~ (z ∈ y ) ) ) ]
Axiom of Separation: If a is a set and F(x) is any well-formed expression in the language of ZF, there is a set b whose elements are those of a and for which F(a) is true.
(This allows subsets of a set to be “separated” out by predicates; examples below…)
∀x ∃y ∀z [z ∈ y ↔ z ∈ x & F(z) ]
Axiom of the Power Set: 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 ]
Axiom of Choice: if a is a set, all of whose elements are non-empty sets where no two of which have any elements in common, there is a set c which has precisely one element in common with each element of a.
(We will discuss this axiom later in more detail…)
[The Axiom of Choice’s formulation is quite long and complex; we shall pass over it’s detailed formalization for now.]
