Alain Badiou - Number and Numbers - Reviewed by John Kadvany, Policy & Decision Science/johnkadvany.com - Philosophical Reviews - University of Notre Dame
Like many philosophers, Alain Badiou relies on technical systems of mathematical logic as a foundation for philosophical exploration. Donald Davidson used Tarski's theory of truth for formal languages to ground his approach to natural language semantics. Modal logic is frequently used to discuss problems of necessity, time, or belief. This review follows those two themes. Badiou's vision in brief: he despairs the lack of objectivity and relativism implicit in the "linguistic turn" -- whether of Lacan, Foucault, and Derrida, but probably equally much Rorty or Searle -- and so seeks a directly ontological alternative, somehow avoiding constructivist methods. Enter modern set theory, particularly Zermelo-Fraenkel (ZF) set theory, already essential to Being and Event. The real numbers return us to Badiou. To motivate his project, Badiou's book begins with a history of number concepts in the late 19th century, a critical time in the foundations of mathematics.
Set Theory > Basic Set Theory
The following basic facts are excerpted from “Introduction to Set Theory,” Third Edition, by Karel Hrbacek and Thomas Jech, published by Marcel Dekker, Inc., New York 1999. 1. Ordered Pairs We begin by introducing the notion of the ordered pair. If a and b are sets, then the unordered pair {a, b} is a set whose elements are exactly a and b. As any object of our study, the ordered pair has to be a set. Definition. If a ≠ b, (a, b) has two elements, a singleton {a} and an unordered pair {a, b}. Theorem. Proof. With ordered pairs at our disposal, we can define ordered triples (a, b, c) = ((a, b), c), ordered quadruples (a, b, c, d) = ((a, b, c), d), and so on. (a) = a. 2. A binary relation is determined by specifying all ordered pairs of objects in that relation; it does not matter by what property the set of these ordered pairs is described. Definition. It is customary to write xRy instead of (x, y) ∈ R. 3. Definition. The Axiom of Extensionality can be applied to functions as follows. Lemma.
Badiou: A Philosophy of the New