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
- for every y if yRx, there is a z such that zRy, and
- for all u, v and z such that uRy, vRy, zRu and zRv, we have u=v,
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.