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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