Thursday, March 28, 2013

The principle of indifference and paradoxical decomposition

There are many paradoxes for the Principle of Indifference. Here's yet another. The Hausdorff Paradox tells us that (given the Axiom of Choice) we can break up (the surface of) a sphere into four disjoint subsets A, B, C and D, such that (a) D is countable, and (b) each of A, B, C and BC is rotationally equivalent. This of course leads to yet another paradox for the Principle of Indifference. Suppose our only information is that some point lies on the surface of a sphere. By classical probability, we should assign probability one to ABC (and even if that's disputed, because of worries about measure zero stuff, the argument only needs that we should assign a positive probability). By Indifference, we should assign equal probability to rotationally equivalent sets. Therefore, since P(ABC)=1, we must have P(A)=P(B)=P(C)=1/3. But by another application of Indifference, P(BC)=P(A). So, P(B)=P(C)=P(BC)=1/3, which is absurd given that B and C are disjoint.

Does this add anything to what we could learn from the other paradoxes for Indifference? I doubt it.

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