Date: Jan 3, 2013 7:59 PM
Author: fom
Subject: Re: The Distinguishability argument of the Reals.

On 1/3/2013 10:30 AM, Zuhair wrote:

> By the way I might be wrong of course, I'll be glad to have anyone
> spot my error, my analogies might simply be misleading.


All right.

Why did Dedekind make his investigations?

Why did Bolzano feel compelled to prove the
intermediate value theorem?

Why was Cauchy careful to not say that the
fundamental sequences converged into the
space from which their elements had been
given?

I realize that you are not talking about
those subjects. But you are taking them
to the garbage heap -- along with every
plausible piece of mathematics that uses
the completeness axiom for the real numbers.

You cannot prove the fundamental theorem
of algebra without results from analysis.
It requires the existence of irrational
roots for polynomials and the intermediate
value theorem. So, you are tossing
algebra onto the same heap with analysis.


Now, there is a circularity in the topology
of real numbers. If you want to have

x=y

it must satisfy the axioms of a metric
space. But those axioms are too
strong.

Go get yourself a copy of "General Topology"
by Kelley and read about uniformities and
the metrization lemma for systems of relations.

What you will find is that the metric space
axioms (the important direction associated
with pseudometrics) depend on the least upper
bound principle.

One can simply view it as fundamental sequences
being grounded by cuts. It is not circular
in that sense. It simply makes Dedekind prior
to Cantor.

Before you continue with this mess, you should
take some time to learn what it means for two
real numbers to be equal to one another.

It is not the Euclidean algorithm.