Wednesday, February 6, 2013

An argument for a version of the Axiom of Choice

This is an argument for the Axiom of Choice where the sets we're choosing from are all subsets of the real numbers. The argument needs the notion of really independent random processes. Real independence is not just probabilistic independence (if you're not convinced, read this). I don't know how to characterize real independence, but here is a necessary condition for it. If S is a collection of really independent processes producing outcomes, and for each s in S, Us is a non-empty subset of the range of s (where the range of a process here is all the outcomes that it can generate) then it is metaphysically possible that each member s of S produces an outcomes in Us. (This need not hold for merely probabilistically independent processes.)

Now, let U be a set of disjoint non-empty subsets of the real numbers R. Let N be the cardinality of U. It is surely metaphysically possible to have N really independent random processes each of which has range R. For instance, one might have a multiverse with N universes, in each of which there is a random process that produces a particle at a normally distributed point from the emitter, and the outcome of the process can be taken to be the x-coordinate of where the particle is produced.

Now, there is a one-to-one correspondence between the members of U and the random processes. If r is one of the random processes, let Ur be the member of U that corresponds to it (after fixing one such correspondence). By real independence, it is metaphysically possible that for all r, the outcome of r is in Ur. Take a world w where this is the case. In that world, the set of outcomes of our processes will contain exactly one member from each member of U, and hence will be a choice set. But what sets of real numbers there are surely does not differ between worlds (I can imagine questioning this, though). So if in w there is a choice set, there actually is a choice set.

Granted, this only gives us the Axiom of Choice for subsets of the reals. But that's enough to generate the Banach-Tarski, Hausdorff and Vitali non-measurable sets. It's paradoxical enough.


Anonymous said...

Don't you need AC to get a correspondence between U and the random processes?

Alexander R Pruss said...

I'm stipulating that the set of random processes has the same cardinality as U, and that just means that there is a correspondence. :-)