```Date: Jan 1, 2013 6:56 PM
Author: Virgil
Subject: Re: Uncountable Diagonal Problem

In article <8733dfe2-163a-4e34-b402-2f018fcac874@i2g2000pbi.googlegroups.com>, Graham Cooper <grahamcooper7@gmail.com> 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--
```