On 15 Mrz., 20:17, Virgil <vir...@ligriv.com> wrote: > In article > <3e85e851-e926-4564-a6fe-68ba9270b...@g16g2000vbf.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 13 Mrz., 18:40, Frederick Williams <freddywilli...@btinternet.com> > > wrote: > > > Some mathematicians do reject the > > > axiom of choice, but I do not know if any have done so because of > > > Banach-Tarski. > > > Every mathematician does so! Notwithstanding any intermediate > > abracadabra the result is that v = 2V and that is wrong in > > mathematics. > > The 'number' of points in one sphere equals the 'number' of points in > sphere with twice the radius, so that v = 4v, even in ordinary geometry, > without any axiom of choice required.
We are not interested in these "numbers" of points, but in the "points" themselves. It is a difference whether you count by a a silly and inconstent mwthod or if you fix point by point.