I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five pieces. My question is :
"Is it possible to construct (mathematically) these five sets or is the proof more of an existence result and not a construction?" If it is possible to construct, I would really appreciate if someone can show the construction here.
Best Answer
No, it is not possible to explicitly construct the pieces. The pieces are, by necessity, non-measurable. This means that they cannot be Borel sets, which covers essentially every "explicit construction" technique you might think of.
Moreover, there are models of ZF set theory (without the axiom of choice) in which the Banach-Tarski paradox fails. So the construction must, necessarily, make use of some form of the axiom of choice. This means that an even wider range of construction techniques - those that can be carried out in ZF - are insufficient to form the decomposition.
Edit:
After clarification, it seems that one part of the question is to find an explicit proof that (using the axiom of choice) it is possible to get a decomposition using exactly 5 pieces. A proof of this is given by Francis Su's thesis on the paradox (PDF). Theorem 20 gives the proof of the five-piece decomposition, and by tracing back the previous results you can work out exactly what the pieces are. That thesis is, by the way, a wonderful reference for many other aspects of the paradox as well.