Re: counter example in analysis

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.
BTW, I do not like the expression "c many", because I consider the
continuum uncountable. I know, Cantor declared the property of being
uncountable the second degree of a questionable quantity called
Maechtigkeit alias cardinality.

Admittedly, I run into trouble when trying to communicate to you that I
meant rational numbers already in the first part of the sentence. I was
in a hurry, and you did of course not know that I am regarding only
rational numbers genuine numbers:

these numbers can be different from each other and may
be called rational numbers.

At this point you're going to run into trouble when
communicating to other people. The term "rational
numbers" already means something specific. You should
choose a different term.

No no. I meant rational numbers in the same sense as did Cantor when he
designed his diagonal arguments. In particular, I am pointing to the
question what make every single rational number different from every
single real one. I do not know any mathematician who has a convincing
answer to this question.

Also, you never specified what your "rational numbers" are,

See above.

unless you're intending
this phrase to mean all the numbers between 0 and 1,
which seems like a pointless thing to do, pardon the pun.

I do not see a pun because I agree: Your expression "all numbers between
0 and 1" is poinless unless we may agree on the meaning of number.


There is no largest amount of them.

Do you mean there is no largest among them?

No, I mean what I wrote.

What you wrote doesn't make sense. If you're talking about a
set, it has at most one cardinal number (also at least
one if we're in ZFC), but since you used the plural
"them", I'm not sure what you mean.

Let me make it quite clear that I do not think in terms of what I am
respectlessly calling the illusion by Dedekind and Cantor. I fully
understand what you mean with cardinality. 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. You correctly concluded that "largest
amount of them" referred to the single elements.


In this case, I may and I have to distinguish between
open and closed intervals.

You can, but they might be the same, depending on what
set of numbers you're talking about.

Didn't I clearly specify that the case I was talking about was rational
numbers?

If you're talking
about the usual rational numbers, and b is an irrational
number between 0 and 1,

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. We are still not yet within IR.

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?


Between two numbers, there is always a potentially
infinite sauce-like continuum of locations.

I guess, except the use of the word "potentially" seems
misleading to me, besides being unnecessary.

While the distinction between potential and actual infinity is essential
for Cantor's set theory, and you may find it explained in detail e.g. in
the Book by Fraenkel 2nd ed. 1923 on p. 6, generations of students did
not learn about it. It is a pity but hopefully not an intentional
dulling of their minds.

Sauce-like reminds of Weyl. Stifel compared the irrationals with a fog.
This property of continuum is by no meand unnecessary to mention.


Nonetheless, according to Dedekind's wording, any
countable amount of rational numbers is overall dense.

No, and he never said this either. Perhaps you could
give a precise quote and reference?

I prefer Fraenkel 1923 because this book has a register: The original
wording was on p. 106 "ueberall dicht". Admittedly, my wording "any
countable amount of rational numbers" was incorrect. I meant: The
unlimited, potentially infinite amount of rational numbers it is thought
to be overall dense, and not even this corresponds to the original. Why
did I not follow to Cantors notion of "lineare Punktmengen" (linear sets
of points)? The given Defintion was: "A set of points is considered
overall dense if there is always one more point between two selected
points." I looked at how the points were described: "Eine genügend
starke Besetzung der Linie mit Punkten" [A sufficiently strong trimmings
of the line with points] as well as the metaphor of an amplifying
microscope clearly indicated potential infinity. The genuine continuum
cannot be resolved into points no matter how many times one zooms. So
the definition does not really refer to actual infinity, at least not
immediately.
My "any countable amount" should read "a potentially infinite amount".



Any finite set is countable and not dense.

Of course. On the other hand, uncountable implies 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?

If I consider the reals one by one, then they are not dense but countable.
If I consider them like a fictitious entitiy of all of them, then I
cannot have bijections, and they are to be considered dense as well as
uncountable.

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 and also necessarily it fails to be dense.
See this a confirmation of my thesis that each single element is either
countable or uncountable.


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

.

Relevant Pages

  • Re: (sketch of a) Proof that the set of Real Numbers doesnt exist
    ... >> an infinite set and its powerset ... > both rationals and reals are numbers between rational numbers. ... assigned to rational numbers and pairs of open sets assigned to irrational ...
    (sci.math)
  • Re: Cantor Confusion
    ... mathematicians are used to thinking on much higher ... the corresponding distinction between rationals and reals is ... between potentially infinite and perfectly infinite. ...
    (sci.math)
  • 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: Non-zero gaps between real numbers
    ... Maybe if you thought that axiomatizing an infinite set as regular/well- ... rationals, and moreso that between each pair of integers there are ... consideration of the integers, and rationals, and reals, then there ... the uncountability of the powerset of integers. ...
    (sci.math)
  • Re: Non-zero gaps between real numbers
    ... Maybe if you thought that axiomatizing an infinite set as regular/well- ... rationals, and moreso that between each pair of integers there are ... consideration of the integers, and rationals, and reals, then there ... the uncountability of the powerset of integers. ...
    (sci.math)