Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Uncountable Diagonal Problem
Posted:
Dec 30, 2012 6:27 PM
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In article
<4036660e-9527-479d-9c47-a1adf9d34d19@px4g2000pbc.googlegroups.com>,
"Ross A. Finlayson" <ross.finlayson@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.
--
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