"He attributed it not to the geometry of 3-D space, but to the algebraic properties of the symmetries inherent to the sphere."

...which are clearly inconsistent with each other given the result.

in other words, the system that group theorists use to describe this sphere is not actually describing this sphere at all, but describing something else.

if it was actually describing this sphere, it couldn't come up with such a ridiculous result.

rejection of the system at such a deep level is far too traumatic for the average mathematician to even contemplate. it is what is.

i think that the evidence regarding the axiom of choice is that it is sometimes true, and that the focus should be on determining a different axiom that allows it to be true under certain circumstances. the axioms should be chosen in such a way that it is demonstrable that the axiom of choice cannot be used to derive this paradox because the condition in which it is true would fail.

however, i think this relies on a deeper geometric question that i cannot formulate and the requires empirical evidence to demonstrate. my faith in our geometric concept of space is very weak.

http://phys.org/news303637618.html