Hausdorff Paradox

Theorem
There exists a disjoint decomposition of the sphere $\mathbb S^2$ into four sets $A, B, C, D$ such that $A, B, C, B \cup C$ are all congruent and $D$ is countable.

Proof 2
Whether you view this result as a veridical paradox or an antinomy depends your acceptance or otherwise of the Axiom of Choice.