Re: counter example in analysis

In article <455204B5.9060607@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

Let's look if there is something to respond to.

On 11/7/2006 10:19 PM, Virgil wrote:


I first defined the interval of interest in terms of real numbers.
Reading just the beginning of my next sentence, you did not get aware
that I already referred to rational numbers when I correctly stated that
this interval contains arbitrarily many rational numbers:

There are c many numbers between
0 and 1, which means we can't have "arbitrarily many".

Since I referred to rationals, arbitrarily many is correct.

The "number" of rationals in any interval of positive length is not
arbitrary, it is fixed.

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.

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

All numbers are equally phantasms. If Eckard cannot deal honestly with
some of them it is because he does not understand any of them.

.......


such as the range of a convergent sequence. In fact,
the Cantor set is uncountable and not only fails to
be dense, it fails to be dense in every interval.

You are referring to so called Cantor's dust alias Cantor's
discontinuum, each element of which is just a single point, therefore
necessarily countable

Except that it is provably not capable of being counted, i.e., there
cannot be any mapping from N onto the Cantor set.

So I most probably did not correctly recall its tricky structure.

See this a confirmation of my thesis that each single element is either
countable or uncountable.

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.

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

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

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