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.