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Topic: Uncountable Diagonal Problem
Replies: 52   Last Post: Jan 6, 2013 2:43 PM

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 ross.finlayson@gmail.com Posts: 2,720 Registered: 2/15/09
Re: Uncountable Diagonal Problem
Posted: Dec 30, 2012 7:24 PM

On Dec 30, 3:27 pm, Virgil <vir...@ligriv.com> wrote:
> In article
>  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
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> > 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.
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> > 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?