Once when I was talking about Banach-Traski theorem (paradox) I said:
OK! This is Banach-Tarski's theorem which is against our intuition but provable from our intuitive axioms! It says you can decompose a small diamond to finite parts and then attach them together and build a diamond as large as a mountain!
Suddenly one of the students happily said:
Amazing! Let's do it!
And I said:
Ah! Unfortunately you never can do it! Go, think and try on it! I offer a full mark to the best research.
At the end of the course Students' reasons were very confusing and strange! Some of their arguments were based on physics, some others related to philosophy, some inspired by religious beliefs, etc. I have my own reasons about impossibility of paradoxical decompositions in the physical world but I don't know if they are the best possible reasons or not. Here I want to ask about any possible argument which I can use to convince my curious students about impossibility of paradoxical decompositions in actual world.
Question. What do you say to students who want to apply Banach-Tarski theorem in practice?!
Remark. Note that if we use an argument which destroys the bridge between mathematical and physical worlds in the case of Banach-Tarski theorem, we should be able to answer a question in the following form:
We know many parts of mathematics which use Axiom of Choice in their main theorems (e.g. Analysis, Linear Algebra, Differential Equations, etc.) have amazingly correct applications in the actual world. Why Banach-Tarski theorem is an exception? Is it really an exception?!