Re: Cantor Confusion

On 12/7/2006 5:26 AM, Virgil wrote:
In article <4576FE21.8040606@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 12/6/2006 2:05 AM, Virgil wrote:
In article <1165345832.735910.255620@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:

Bob Kolker wrote:
There is no such thing as a countable integer, countable rational or
countable real.

In the sense you're trying to get across to the other poster, I
understand your point. But, just for the record, in a technical sense
in set theory, as integers, rational numbers, and real numbers are
themselves sets, it does make sense to say whether one of them is
countable or not. For example, where integers are defined as
equivalence classes of natural numbers, each integer is itself a
denumerable set. I am not necessarily endorsing anything the other
poster has said; I'm just adding the technical note that in a strict
set theoretic sense, even numbers are sets and thus it is meaningful to
talk about the cardinality of a number.

MoeBlee

In that particular sense, each real number as a Dedekind cut, being a
partition of the rationals into two sets, always has cardinality 2,
while each member of a Dedekind cut is countably infinite.

But each real number as a set of equivalent Cauchy sequences is
uncountable.

Conclusion: Dedekind cut's are self-delusive.

Why is any set with two clearly defined members self-delusive?

Virgil,

While I understood you refer to subsets, I would like to explain the
whole delusion first.

The "number" pi is definitely a merely fictitious element of continuum.
It is clearly defined by a geometrical problem which cannot be solved
numerically by means of a realistic, i.e. finite number of steps. There
is no possibility to reasonably quantify the amount of such fictitious
elements. The continuum of such "elements" is uncountable, no more and
no less than anything which is considered perfectly infinite. Notice:
Actual infinity means to abstractly include _all_ of indefinitely many
naturals, integers, rationals, irrationals, or reals. When I wrote
"abstractly", I meant it is impossible to reach infinity with counting.
Archimedes quasi defined natural numbers like someting that can
indefinitely be enlarged by just adding one more unit. Likewise
fractional numbers can be indefinitely reduced. So rational numbers
represent the Archimedean and Aristotelean notion of the potentially
indefinitely large and also the indefinitely small. Because the term
Archimedean has been given a deviating definition, I call such numbers
genuine numbers.
You may argue: The expression rational numbers is sufficient. Well, you
are correct. I intend to stress that only rational numbers including
intergers and naturals are genuine. Moreover, rational numbers loose
their property of being countable if they are embedded into the
continuum. It would not be wrong to interprete this loss of the property
to be countable as loss of existence. At least there is no possiblity to
decide inside the genuine continuum whether a fictitious "element"
belongs to the rationals or to the irrationals except via the defining
problem in each case. The primary continuum is strictly speaking amorph.
There is no structure available inside this continuum. Alleged
homomorphy is valid for rational quasi-reals. Ascribing the behavior of
genuine numbers to the reals is tempting but not justified. Already
Cauchy did not care about the categorical distinction between rationals
and reals. E. Heine "Die Elemente der Funktionenlehre", Crelles Journal,
Bd. 74 further encouraged to do so. I guess, there is indeed no
compelling reason to strictly obey the correct categories in practice.

What illusions I refer to?

1) Dedekind dreamed of making the rationals complete by addition of
numbers in between two rationals. This is neither possible nor necessary
because already systems of rational numbers are everywhere dense. It is
impossible to make reals rational, to make infinity a finite quantum,
and to resolve the continuum into countable points.

2) Dedekind imagined a line composed of single points. He argued: These
points are continuously ordered form left(small) to right (large). He
ignored that these points are just fictitious ones even if they
correspond to the solution of a geometrical problem. He was still
correct when he wrote that every rational number corresponds to only one
single point. Was he still correct in that there are indefinitely many
points which do not correspond to a rational number? Seemingly yes.
However, his idea that there are more reals than rationals tacitly
presumes: The entities of all rationals and all reals within a common
interval can be quantified and ergo can be compared with each other.

3) Dedekind as well as the majority of mathematicians believed to be
entitled to decide this question intuitively. It seems to be quite clear
to them that there are much more rational numbers than real ones because
the rational numbers are included within the reals. Consequently the
number or reals must be larger than indefinitely large.

4) Dedekind wrote: "Zerfallen alle Punkte der Geraden in zwei Klassen
von der Art, dass jeder Punkt der ersten Klasse links von jedem Punkt
der zweiten Klasse liegt, so existiert ein und nur ein Punkt, welcher
diese Einteilung aller Punkte ... hervorbringt". In brief: D. assumed
the line to consist of "all" points, and these points have to be located
either left or right with respect to just one selected point. He
admitted to be unable to prove this. Indeed this idea was wrong if we
allow for indefinitely many points. In order to select a point, we have
to have all points first. This is impossible.

5) Dedekind claimed to be in position to create real numbers by means of
his cuts, obviously with no avail. In order to know whether or not a
number is irreal, one has to define it first.

6) Admittedly up to now, I myself I was taken in by Dedekind's elusive
intuition. As did Stifel and Weyl, I correctly imagined the entity of
all real numbers continuous like a fog or a sauce while I imagined the
rationals as ordered single points. Wrong was just the expression "the"
rationals.
Any set of rational numbers corresponds to insulated points being
different from each other. "The" means all. However, all rationals are a
fiction, the same foglike fiction as are the reals. So the difference
between rational and real numbers is actually merely a categorical one.
In other words, it depends on the point of view. Take the position of
counting: Genuine numbers are considered countable even if they are as
dense as a fog. Take the opposite position: The genuine continuum is
considered to consist of uncountable reals while approximated by dots is
sufficient in practice.

7) I checked whether or not the difference between rationals and
irrationals is indeed merely a categorical one: If irrationality has
been proven by showing that a common divisor is missing, then this is
bound to quantities of finite size.
Example 2/2=1 but 2000000000...000000001/2000000000...000000000 =/= 1
In other words: I cannot confirm the difference between rationals and
reals, closed and open intervals, countable and uncountable, digital and
analog, etc. to persist where the realms of genuine numbers and the
genuine continuum are thought to meet each other.


Dr.-Ing. Eckard Blumschein
Electrical engineer, Uni of Magdeburg




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