Triviality

For almost three years, I’ve occasionally been thinking about a certain mathematical question about infinity and probability arising from my work in formal epistemology (more details below). I posted on mathoverflow, and got no answer. And then a couple of days ago, I saw that the answer is trivial, at least by the standards of research mathematics. :-)

It’s also not a very philosophically interesting answer. For a while, I’ve been collecting results that say that under certain conditions, there is no appropriate probability function. So I asked myself this: Is there a way of assigning a finitely additive probability function to all possible events (i.e., all subsets of the state space) defined by a countable infinity of independent fair coin tosses such that (a) facts about disjoint sets of coins are independent and (b) the probabilities are invariant under arbitrary heads-tails reversals? I suspected the answer was negative, which would have been rather philosophically interesting, suggesting a tension between the independence and symmmetry considerations in (a) and (b).

But it turns out that the answer is positive. This isn’t philosophically interesting. For the conditions (a) and (b) are too weak for the measure to match our intuitions about coin tosses. To really match these intuitions, we would also need a third condition, invariance under permutations of coins, and that we can’t have (that follows from the same method that is used to prove the Banach-Tarski paradox). It would, however, have been interesting if just (a) and (b) were too much.

Oh well.