Saturday, February 18, 2006
(8:30 AM) | Anonymous:
The laicity of axioms
Anthony raises two good questions: 1) whether use of mathematics as a discourse about ontology is similar to phenomenology, and 2) regarding the status of axioms in Badiou's system. I'll only take up the second point.Badiou admits only two points of pure decision: 1) the claim that 'maths is ontology' (although there are arguments and historical references made here in support, so maybe it's just an induction?); 2) the new occurs in being as an event - which amounts to assigning temporality and causality to the jumble of conclusions reached about sets (inconsistency, infinity, the possibility of forcing etc). Once induction #1 is made, and once set theory is selected as the mathematical language, axioms follow promptly - and not for any reason other than those that led Zermelo down that path in the first place.
Naive, intuitive - that is to say non-axiomatic - set theory, Cantor's original paradise, entered its post-lapsarian phase with Russell's paradox. Although he never stated them, all of Cantor's operations are said to be derivable from three axioms: the axiom of extensionality, the axiom of abstraction, and the axiom of choice. (Suppes 1960) The problem is with the second: simply put, it was too broad, too powerful. Instead of asserting the existence of sets unconditionally, Zermelo's axiom schema of separation makes the assertion of a particular subset completely conditional. Consider:
- (Ey)(Ax)(x e y iff p(x)) (axiom of abstraction)
- (Ey)(Ax)[x e y iff x e z & p(x)] (axiom schema of separation)
- (Ey)(Ax)[x e y iff -(x e x)]
- [y e y iff -(y e y) (taking x = y and instantiating)
- [y e y & -(y e y)], which is logically equivalent to (2), and contradictory
- (Ey)(Ax)[x e y iff x e z & -(x e x)]
- [y e y iff y e z & -(y e y) (again taking x = y)
But with the axiom schema of separation, things look different.
Although Zermelo's axiom schema of separation gets supplanted by Fraenkel's axiom schema of replacement, the important thing to note here is that the first step toward axiomatization developed directly in response to early paradoxes. So although, as Anthony points out, the assertion of axioms seems like some kind of unfounded, "faith"-based step, in fact in this case it is the non-axiomatic description (Cantor's use of an unformalized version of the axiom of abstraction) that credulously asserts the existence of sets that turn out to be contradictory, and the positing of an axiom is instead a step back towards the demonstrable, towards presentation.
