```Date: Jan 4, 2013 1:18 AM
Author: Zaljohar@gmail.com
Subject: Re: The Distinguishability argument of the Reals.

On Jan 4, 3:59 am, fom <fomJ...@nyms.net> wrote:> 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.Dear fom I'm not against Uncountability, I'm not against Cantor'sargument. I'm saying that Cantor's argument is CORRECT. All what I'msaying is that it is COUNTER-INTUITIVE as it violates theDistinguishability argument which is an argument that comes fromintuition excerised in the FINITE world. That's all.Zuhair
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