Significance of Banach Paradox

Question

What makes Banach paradox important?

Could we not just take the set of points defining a sphere and break them into two sets, one of rational and one of irrational points, so won't we again have two spheres with the same area?

Answer 1

It shows that we cannot define a measure (volume function) in $\Bbb R^3$ for all subsets that assigns $1$ to the unit ball, is finitely additive, and such that translated and rotated sets keep the same measure (because if we could, we could apply the sets from the paradox and derive $1=2$). It shows that $3$-dimensional space is more complicated than the plane (where such a measure does exist, assuming AC too).

Others will say that "of course" there must be such a volume (from a physic based intuition of $3$-space) and so AC (the axiom used to make the sets in the paradox) is the "culprit". I personally prefer the first interpretation. Stan Wagon has written a nice book on this "paradox" and its implications. I quite liked reading it.

Answer 2

The area of the set of rational points is zero, so we have not increased the total area with your construction. The point of the paradox is that we make a bijection between the points in one sphere and the points in a pair of spheres. If the pieces had computable areas, we would have doubled the area by moving the pieces.