Re: Well Ordering the Reals

In article <MPG.1ddbfeccc16c95dd98a692@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > In article <MPG.1ddac2788eafc47e98a685@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> >
> > > Daryl McCullough said:
> > > > Tony Orlow says...
> > > >
> > > > >From what I have read on Cauchy sequences, one needs to show that for
> > > > >any
> > > > >real
> > > > >epsilon, no matter how small, there is a maximum finite number of
> > > > >iterations it
> > > > >takes to get within epsilon of any other real value.
> > > >
> > > > That's what it means to have a Cauchy sequence converging to r.
> > > > That doesn't prove that r is in your set. For example, if Q is
> > > > the set of all rational numbers, then for every real number r,
> > > > and for any positive real number epsilon, you can find a number
> > > > q in Q such that |r-q| < epsilon. That doesn't prove that r is
> > > > in Q.
> > > >
> > > > --
> > > > Daryl McCullough
> > > > Ithaca, NY
> > > >
> > > >
> > > So, then, what exactly must one do to prove that a given set includes all
> > > the
> > > reals as opposed to just being dense in the reals? Do I need to use
> > > Dedekind
> > > cuts or what? I am looking for an example, but they seem rather hard to
> > > find.
> >
> > There are essentially two methods:
> > (1) Prove that for ever Cauchy sequence of values in your set, there is
> > a limit value in your set.
> > (2) prove that for every subset your set which is bounded above in your
> > set, there is a least upper bound in your set.
> >
> Okay, Virgil. I'll try to figure out how to do one of these. I am the world's
> foremost authority on TOmatics, but you are the expert in the standard world.
> Thanks.

You may have problems with (1) until you have a valid definition of what
it means for a sequence of your numbers to converge to zero. As that may
be tricky, I would suggest considering (2) first.

Also, for (2) you can restrict your attentions to only the positive
reals initially, and then do the negatives separately as a mirror image
with greatest lower bounds.
.

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