In article <email@example.com>, Graham Cooper <firstname.lastname@example.org> wrote:
> On Dec 31 2012, 9:27 am, 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. > > > > > LETS TRY! > > LIST > R1 0.11111111... > R2 0.22222222... > R3 0.01010101... > R4 0.99999999... > ... > > > DIAGONAL = 0.1209.... > > WHAT ARE ALL THE MISSING REALS VIRGIL? > > > HINT: you should be able to calculate 9*9*9*9 of them?
Way more than that!
As long as the digit replacement rule does not replace any digit with either a 0 or a 9, one can have as many as 8*7 = 56 different rules for any digit position, giving 4*56^8 nonmembers of your list.
And if your 4 listed elements are as periodic as they appear to be, there are uncountably many non-periodic others, though one cannot, of course, list them all.. > > JUST FROM THAT LIST! > > WOW! THERE REALLY ARE A LOT OF UNCOUNTABLE REALS!! > > Herc --