On Tue, 26 Feb 2013 17:54:19 GMT, Wayne Throop <firstname.lastname@example.org> wrote in <news:email@example.com> in rec.arts.sf.written,sci.math:
>: A story such as "The 4-D Doodler," that speculates about >: a fourth spacial dimension, one dimension beyond our >: known world of three dimensions, seemed promising to me, >: the dilettante, at first. Perhaps 4-dimensional space >: uses an exotic infinity. Alas, as Wolfram notes, it's >: the same old infinity. An infinity that's independent >: of spacial dimensions.
> It's c, or aleph-1 maybe, or whatnot.
It's c, or 2^\omega.
> It is not the infinity of natural numbers; it's bigger. > So there's at least two infinities, not the "same old > infinity". Similarly, the cardinality of the set of paths > in n>=2 d space (if I'm remembering a close enough > description) is bigger still.
No, it's also 2^\omega, since by definition a path is continuous.