Re: counter example in analysis

In article <455051E9.7040408@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 11/6/2006 9:15 PM, Dave L. Renfro wrote:
Eckard Blumschein wrote:

My arguments are quite simple and hopefully compelling:
Let's consider like an example the interval of real
"numbers" between 0 and 1.
As long it contains just arbitrarily many numbers,

This is not true.

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.

BTW, I do not like the expression "c many"

Your likes and dislikes are irrelevant.

I am just not alwas sure how
to interprete the term "infinite set" since Cantor himself used it
sometimes for the countable single elements, sometimes for the belonging
fictitious and uncountable entity.

How about the Dedekind definition of infinite set, which fits both cases.
or the definitins which say not bijectable with any of the natural
ordinals?


You presumably will not understand why I consider rational numbers
different not just from the irrationals but from the real ones, too.
However even Dedekind's point of view excludes to mingle irrationals
among the rationals.
Not entirely. For each rational there is a corresponding Dedekind cut,
which performs that very mingling of rationals among irrationals.

then the open interval (b^2, b)
of rationals is equal to the closed interval [b^2, b]
of rationals.

This sounds interesting to me. Is this accepted for b=>0, too?

Only for such b in (0,1) for which b^2 is irrational.


Any finite set is countable and not dense.

Of course. On the other hand, uncountable implies dense.

Not necessarily, The Cantor "middle thirds" set is uncountable and
nowhere dense.

What about countably infinite, I see it a matter of the point of view.

It is easy
to get countably infinite sets that are not dense,

I would like to question the word "are". What about "are considered
like" instead?

The set {1/n: n in N} is countably infinite but not dense, having only
one accumulation point.

If I consider the reals one by one, then they are not dense but countable.
If I consider them like a fictitious entitiy
Fictitious entities in any mathematical sense are things which have no
existence within mathematics. So that Echard's continuum must be one of
them and he outens geometry.

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.


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.


I just envison two possibilities to cover the whole
line with points. The first one is motion. Remember
the fluxus of indivisibles for instance tought by
Pythagoreans, Cavalieri, Descartes, Torricelli and Newton.

I also remember Zeus, Aphrodite, Apollo, etc., but I wouldn't
use them to support a modern biblical argument for something.

You are disqualifying yourself.

Eckard Blumschein has many times disqualified himself from being able to
speak meaningfully about any of these matters.
.

Relevant Pages

  • Re: Rational numbers, irrational numbers: each dense in real numbers
    ... Among the notions of why there are "more" irrationals than rationals ... reals, ... Then, particular rationals of smaller numerators, after one and zero, ... as well in these infinite expansions have various ...
    (sci.math)
  • Re: Dense vs. Continuous
    ... mentioned that the rationals were "dense". ... function on the reals is a continuous function. ... If I were to graph the same function and said the domain ...
    (sci.math)
  • Re: Cantor Confusion
    ... but not if you apply the generally accepted definition. ... extended to rationals in order to allow division and include fractions, ... the irrationals cannot be located numerically. ... the reals vanish completely within the sauce of irreals. ...
    (sci.math)
  • Re: Cantor Confusion
    ... though hard to unveil intentional mistake by Dedekind? ... extended to rationals in order to allow division and include fractions, ... the irrationals cannot be located numerically. ... the reals vanish completely within the sauce of irreals. ...
    (sci.math)
  • Re: Rational/Irrational Numbers
    ... The consideration of rationals and irrationals, ... other sets that are NCD2 in the reals from "On Well-Orderingof Sets ... number line, or algebraics and transcendentals, or rationals, ...
    (sci.math)