I spent the last week trying to get clear on the logical interconnections between a number of results about probabilities that are relevant to formal epistemology and that use a version of the Axiom of Choice in proof, such as:
- For every non-empty set Ω, there is an ordered field K and a K-valued probability function that assigns non-zero finitely additive probability to every non-empty subset of Ω.
- For every non-empty set Ω, there is a full finitely additive conditional probability on Ω (i.e., a Popper function with all non-empty subsets normal).
- The Banach-Tarski Paradox holds: one can decompose a three-dimensional ball into a finite number of pieces that can be moved around and made into two balls of the same size.
- There are Lebesgue non-measurable sets in the unit interval [0,1].
The proof from BPI to (1) is standard--just let K be an ultrapower of the reals with an appropriate ultrafilter. That from (1) to (2) is almost immediate: just define the conditional probabilities via the ratio formula and take the standard part. Pawlikowski's proof of Banach-Tarski easily adapts to use (2) (officially, he uses Hahn-Banach). Finally, Foreman and Wehrung show in ZF that every subset of Rn is Lebesgue measurable iff every subset of [0,1] is. But it follows from (2) that not every subset of R3 is Lebesgue measurable.
This has important consequences. Without the Axiom of Choice, one can prove that either (a) there are sets that have no regular probabilities no matter what ordered field is chosen for the values and no full conditional probabilities, or (b) the Banach-Tarski Paradox holds and hence there are no rigid-motion-invariant probabilities on regions of three-dimensional space big enough to hold a ball. And in either case, Bayesianism has a problem.