In article <firstname.lastname@example.org>, "Ross A. Finlayson" <email@example.com> wrote:
> Formulate Cantor's nested intervals with "mega-sequences" (or > transfinite sequence or ordinal-indexed sequence) instead of sequences > of endpoints. Well-order the reals and apply, that the sequences > converge yet have not emptiness between them else there would be two > contiguous points, in the linear continuum.
Not possible with the standard reals without violating such properties of the reals as the LUB and GLB properties: Every non-empty set of reals bounded above has a real number LUB. Every non-empty set of reals bounded below has a real number GLB. --