Indifference is dead, again

I've noted that given the Axiom of Choice, the Hausdorff Paradox kills the principle of indifference. But we don't need the Axiom of Choice to kill off Indifference in this way! Hausdorff's proof of his paradox[note 1] also showed, without at this point in the proof using the Axiom of Choice, that:

• There are disjoint countable subsets A, B and C of the (surface of the) sphere and a subgroup G of rotations about the center such that: (a) U=A∪B∪C is invariant under G, and (b) A, B, C and B∪C are all equivalent under rotations from G.

Suppose the only thing we know about some point x is that it lies in U. Now A, B and C are rotationally equivalent under rotations that U is invariant under, so our information about whether x lies in A or in B or in C is exactly on par. By Indifference, hypotheses that are equivalently related to our information get equal probability. Thus P(x∈A)=P(x∈B)=P(x∈C)=1/3 (that they equal 1/3 is due to the fact that the three hypotheses are mutually exclusive and jointly exhaustive). But A and B∪C are also rotationally equivalent under rotations that U is invariant under, so again by Indifference P(x∈A)=P(x∈B∪C)=1/2. And so 1/2=1/3, which is absurd.