Date: Dec 30, 2012 7:24 PM
Author: ross.finlayson@gmail.com
Subject: Re: Uncountable Diagonal Problem
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:

>

>

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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.

>

>

>

> > 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?