On 3/16/2013 4:49 AM, WM wrote: > 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.
Where we come to the question of how you refer to points without an implicit use of infinity.
This, of course, comes back to how you refer singularly without an implicit use of infinity.