Re: counter example in analysis

On 11/8/2006 9:06 PM, Virgil wrote:


In Cantor's understanding all rational and even all real numbers exist.

And Eckard sets himself up as knowing better, while clearly ignorant of
the mathematical basis for Cantor's work.

I read Cantor's original papers. His basis is just intuitive guesswork.


This is however shallow reasoning because there is no limit (except such
phantasm like omega) to the rationals.

All numbers are equally phantasms.

This deliberately misleading claim does not differentiate between
countable genuine numbers and uncountable fictions.


Any singleton set is countable and any finite set is countable in the
mathematical meaning of the word.
Eckard's use of "countable" and "uncountable" is unaccountable.

An element of a set can itself be a set. If you will accept this idea
then you ought to also accept that single "numbers" like pi may be
uncountable ones.

When one speaks of a set being countable, on is talking about how many
members it has, not the nature of those members.

I spoke of countable and uncountable numbers, not sets. If pi is a
number at all, then it is an uncountable one. Neither does it fit into
the system of rational numbers, nor can it be expressed numerically with
any finite number of steps.


A set with one member is a countable set with one member no matter how
large that member may be as a set.

This is correct but distracting.


Eckard should get his head straight about such simple matters before
pontificating.

Pontificating on what? I always try to explain what I mean.


.

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