Date: Dec 30, 2012 4:51 PM
Subject: Re: Uncountable Diagonal Problem
On Dec 30, 1:33 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> "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. Yes, many perfectly reasonable theorems
are derived from them, both axiomatically, and matching intuition.
I'm not the first to suggest Cauchy insufficient, where of course I
will agree that Cauchy (or Cauchy/Weierstrass as it were) is a
perfectly reasonable framework for analytical results, just not
closure of all results.
A well ordering of the reals doesn't have uncountably many points in
their natural order.