Modality and the Axiom of Choice

Suppose that the set theory of our world is a Solovay model, where we don’t have the Axiom of Choice (AC), and where every subset of the reals is Lebesgue measurable. Now imagine that God picks out a line in space, and defines the Vitali equivalence relation for points on that line (where two points are equivalent if and only if the distance between them is a rational number). It is then surely within God’s power to create a particle of some unexemplified type T at exactly one point in every equivalence class. There is nothing incoherent about that! But if God did that, then there would surely be a set of the points containing a particle of type T. And that set would be a nonmeasurable Vitali set.

So what?

Well, prima facie, there are three possibilities about the existence of nonmeasurable sets:

1. Necessarily, there are no nonmeasurable sets.

2. Necessarily, there are nonmeasurable sets.

3. It is contingent whether there are nonmeasurable sets.

My argument strongly suggests that if there are no nonmeasurable sets, it is nonetheless possible that there are nonmeasurable sets. Hence, (1) is ruled out.

So we have an argument for the disjunction of (2) and (3).

Now, I think a lot of people have the intuition that mathematical facts are necessary. If so, then (3) is ruled out. They will see this as an argument for (2).

I don’t see it that way myself: I am quite open to contingent mathematical truths.

More generally, the argument shows that:

4. For any set of disjoint nonempty subsets of the reals, it is possible that there is a choice function.

Again, if the existence of pure sets is not a contingent matter, we conclude AC is true for all subsets of the reals.