The thing is, we don't even know how to begin to approach this problem. Paul Erdos has already commented on this problem.I know lol. This is one of the biggest mind gobbling problems to pure mathematicians apparently; WHY?
Only if I accept the axiom of choice. : PPPPPPPIf you have a small ball in 3 dimensional space, it is possible to decompose it as a union of a finite number of sets, which can be moved by rotations and translations such that the pieces never overlap and such that the final object constructed is an arbitrarily large ball.
Colloquially, one can cut a pea into a finite number of pieces and reassemble it into something the size of the sun.
Even if you don't accept the axiom of choice (which is a bit limiting, but some minority of mathematicians don't), you would not be able to prove that such a reassembling of the pea into the sun is impossible. (Because the axiom of choice is consistent with the other axioms of set theory.)Only if I accept the axiom of choice. : PPPPPPP