On Wed, 12 Jul 2006 21:37:30 -0600, Virgil <email@example.com> wrote:
>In article <firstname.lastname@example.org>, > email@example.com wrote: > >> >What about the set of all points on a perfect circle (ie all solutions to >> >x^2 + y^2 = 1). >> >> Well technically of course all solutions you point out define a >> perfect sphere not a perfect circle. > >With only two variables, x and y, there is no need to presume three >dimensions, but if one does, one gets a right circular cylinder, not a >sphere.
And as long as one gets to assume whatever one wants one gets whatever one wants to assume. There is no unambiguous right angle dimension to any other when fewer than three dimensions are involved. If you're talking about the set of all points equidistant from any other the figure is a sphere.
>If one doesn't assume three (or more) dimensions, one does get a circle.
Well the basic problem here is that it's a little difficult to discern what the author is complaining about in set theory. Sets of properties are perfectly useful. However typical set theory definitions which run along the lines of a "set of all points which . . ." do turn out to be a joke because they invariably rely on various geometric assumptions regarding figures such as planes, lines, etc. ~v~~