Date: Dec 30, 2012 9:01 PM
Author: Virgil
Subject: Re: Uncountable Diagonal Problem
In article

<b933563c-4654-4759-b964-c3cd27e0a048@lb9g2000pbb.googlegroups.com>,

"Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:

> 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?

I see no reason why either need occur even in an explicit well-ordering

of the reals. Why do you?

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