In article <firstname.lastname@example.org>, "Ross A. Finlayson" <email@example.com> wrote:
> On Dec 30, 1:33 pm, Virgil <vir...@ligriv.com> wrote: > > In article > > <2fc759b9-3c22-4f0b-83e0-bf9814a3f...@y5g2000pbi.googlegroups.com>, > > "Ross A. Finlayson" <ross.finlay...@gmail.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. > > -- > > > Those are definitions, not derived. Maybe they're "wrong", of the > true nature of the continuum.
if false for your "continuum" then that continuum is not the standard real number field.
> > A well ordering of the reals doesn't have uncountably many points in > their natural order.
But, if one could find an explicit well-ordering of the reals, it would have to contain all those uncountably many reals in SOME order. --