Date: Nov 13, 2012 5:11 PM
Subject: Re: Cantor's first proof in DETAILS
In article <firstname.lastname@example.org>,
"LudovicoVan" <email@example.com> wrote:
> "Uirgil" <firstname.lastname@example.org> wrote in message
> > No values which are bounded below by a strictly increasing sequence and
> > bounded above by a strictly decreasing sequence are members of either
> > seequence.
> > Thus proving that, given any sequence of values in R, there must be
> > values in R not appearing in that sequence.
> I'll have a look at Zuhair's follow-up as soon as I manage, but let me for
> now just point out that the above argument is obviously bogus: the rationals
> too are dense (have the IVP as Zuhair has called it) and, by the very same
> argument, we have proved that the rationals too are not countable... see?
The difference being that a monotone but finitely bounded sequence of
rationals need not have a limit among the rationals but MUST have a
limit among the reals, a LUB or GLB.
Density is not enough distinguish between Q and R, but the GLB/ LUB
property is enough.
Any densely ordered interval of positive length having the GLB/LUB
property is uncountable.