On Mon, 15 Mar 2010, Mark J. Reed wrote:
> > Anything that can be made into a list is discrete.
> Not quite, since you can create lists whose members belong to
> continuous sets, e.g. real numbers. Anything that naturally forms a
> list, maybe.
A discrete non-finite set is isomorphic to the set of integers.
This implies that you can pick an ordering, although perhaps not one that is
computational useful. E.g. rational numbers can be mapped onto integers by
laying them out in a 2D array, much like Gauß's complex integers, and then
walking diagonally in one quadrant. If you have more than one quadrant, take
them in turns. Extends to 3D and higher in a logic fashion.
But totally useless as a "greater than/equal to/less than" comparison test.