Thursday, September 8, 2011

The Axiom of Choice

For any relation R and world w we can ask the following question: Is it the case that for all x, y and z such that xRy and yRz, we also have xRz?  If the answer is affirmative, we say that R is transitive at w.

Likewise, for any relation R and world w we can ask the following question: Is it the case that for every object x such that
1. for every y if yRx, there is a z such that zRy, and
2. for all u, v and z such that uRy, vRy, zRu and zRv, we have u=v,
there exists an object x* such that for every y such that yRx there is a unique z such that zRx* and zRy?  If the answer is affirmative, we could say that R is choosy at w.

Now, it would be silly to ask: "Is transitivity true?"  Transitivity is not the sort of thing to be true.  Some relations are transitive at a world (and some are transitive at all) and some aren't.  Likewise, it would be silly to ask: "Is choosiness true?"  Choosiness is not the sort of thing to be true.  Some relations are choosy at a world (and some are choosy at all) and some aren't.

As it turns out (not by chance--I rigged it), the Axiom of Choice in a set theory is equivalent to claim that the membership relation in that set theory is choosy.  But just as it is nonsense to ask if transitivity or choosiness is true, I rather like the view that it's nonsense to ask if the Axiom of Choice is true.  We can ask if a particular relation satisfies the Axiom of Choice, i.e., is choosy at some world, or at all worlds, but why think there is a distinguished relation that we can call "the membership relation" and that we can ask about the choosiness of?

I am fairly naively inclined to take this quite far in mathematics, along the lines of ifthenism: Mathematicians simply prove necessary conditionals like that if a relation is Zermelo-Fraenkelish and choosy, then it's Zorny, or--for a much more difficult example--that if a relation is Peanish, it is finally-Fermatish.  This is a thesis about mathematical practice, not mathematical truth.  But I really don't know much philosophy of mathematics (I know a lot more mathematics than philosophy of mathematics) and this version of ifthenism may be untenable.