On Dec 30, 3:27 pm, Virgil <vir...@ligriv.com> wrote: > In article > <4036660e-9527-479d-9c47-a1adf9d34...@px4g2000pbc.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...@gmail.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. > --
So, the mega-sequences of the nested interval endpoints would end with side-by-side endpoints? Or, does any ordinally-indexed sequence of all of a segment of reals necessarily contain duplicates?