Was reading through this question and the answer given by @triple_sec lists some mindboggling results that are implied by the axiom of choice.
[Geometry] Banach–Tarski paradox. (The axiom of choice makes it possible to cut an object into a finite number of pieces in such a weird way that you can reassemble two copies of the same object of the same size!)
[Measure theory] Existence of sets that are not Lebesgue measurable.
How do we make sense of the Banach-Tarski problem? The reason I pointed out the existence of non-measurable sets made me think perhaps the paradox from Banach-Tarski theorem arises because we are "incorrectly/inconsistently" [I am not entirely sure how to formalize this notion] measuring these pieces of the ball.
We can't take a ball of chocolate and cut it up and re-assemble it into two balls of chocolate, both as large as the ball of chocolate was to start with, can we? Of course not!
Question: Is AC necessarely the culprit? Is the paradox invariant w.r.t the definition of measure (area/volume?) of a set?