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 B∪C 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 A∪B∪C (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(A∪B∪C)=1, we must have P(A)=P(B)=P(C)=1/3. But by another application of Indifference, P(B∪C)=P(A). So, P(B)=P(C)=P(B∪C)=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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