Re: .999... ?= 1

From: Eckard Blumschein (blumschein_at_et.uni-magdeburg.de)
Date: 06/07/04 

Date: Mon, 07 Jun 2004 10:47:34 +0100

Lothar Brendel wrote:

>> I see zero having three aspects of equal value but different sign:
>> minus 0.0 over bar
>> just zero
>> plus 0.0 over bar.
>
>
> -0 = 0 = +0, no problem.
>
>
>> Both the negative and the positive description, from the left and from
>> the right, respectively are identical with just zero.
>
>
> Exactly.

Aren't you a bit careless? I wrote three aspects. You wrote -0 = 0 = +0.
You further wrote:

> The middle one is the same number as the other two. You could as well
> suggest to dispose "just zero" and use 1-1 everywhere.

While I do not expect people like Chapman who are specialized in theory
of numbers fighting for what they were thought by all means. I hope you
will be more independent, in principle. Do not consider it an attack on
numbers like 0 and 1 if I try to add some subtleties. Since language of
  mathematics has failed for centuries to satisfactory resolve the
issue, it might be worth pondering about alternatives. What does +0
mean? How does it differ from -0? We agree about the identity of their
value zero, no matter whether it is seen from the left or from the right.

While +0 > 0 > -0 would be a contradiction to the meaning of infinity, 0
is still located between +0 and -0.

Within IR+ you cannot see zero from the left. So one aspect is missing.
However, there is no compelling reason to exclude zero itself from Q+.
Likewise, +0 is missing within Q- but this does not imply that zero
itself must be excluded. Common decimal notation incorporates zero into
Q+ as well as into Q- (+0.000... and -0.000... as shown in Mathematica
definition of frac in contrast to Graham definition) because it
symmetrically counts from zero to the left and to the right. Perhaps
your teacher preferred to except zero from both parts as to avoid a
contradiction to the Trichotomy "law" that positive and negative values
exclude each other. I object that this "law" is among those which fail
at infinity and its reciprocal.

>> (within Q or IR and its two parts!) or can be used on demand while the
>> outer aspects are missing, of course, in case without a continuum.
>
>
> Gibberish...

No no. This is correct, and it makes perhaps at least as much sense as
surreal numbers.

>> To my understanding, a continuum is infinitely fine grained in the
>> sense, it cannot be resolved into a finite number of elementary numbers.
>
>
> _Your_ understanding, _your_ imagination of a continuum has nothing to
> do with the continuum defined in math.

Indeed, I didn't find yet much corresponding mathematical definition of
a continuum. Aleph-1 is an undecidable proposition within the tenets of
Cantor's set theory. I see a key problem hidden within the notion of a
set: x€A. While set theory arbitrarily assumes the existence of an empty
set, I wonder why it does not mention its reciprocal. There are genuine
infinitesimals:

http://mathworld.wolfram.com/NonstandardAnalysis.html

Wolfram gives continuity as follows:
"A general mathematical property obeyed by mathematical objects in which
all elements are within a neighborhood of nearby points."
A continuous function is described as follows:
"There are several commonly used methods of defining the slippery, but
extremely important, concept of a continuous function."
A function f(x) in a single variable x is said to be continuous at point
x0 if lim (x to x0) y(x) = y(x0).
|x-x0| < delta, |y-y(x0)| < epsilon are measures of closeness.

>> In other words, nobody should hesitate cutting straight through any
>> real number without harming it since two of the three aspects still
>> remain undamaged.
>
>
> Once again, "cutting numbers" is no mathematical concept, but a picture
> made up by you. How do you _define_ the mathematical operation of
> "cutting a number"?

I wrote cutting through a "real number", not through a number.
While Wolfram gave such exotic knowledge like centillion, I didn't find
much about the very properties of a number in general. A number is
merely said to be a member of a set. If I infer correctly, infinity
cannot be such a general number while I would tend to ascribe it to the
continuum c alias real numbers. So you can execute the usual cut across
c between plus infinity and the adjacent minus infinity without any
protest by number fetishists. I know: Set theory distinguishes between
open and closed borders. However, this is obviously not reasonable in
case of c.

>> IR+ has its zero as has IR- too.
>
>
> That's simply wrong. By virtue of their definiton, neither R^+ nor R^-
> contains the number zero.

Which definition do you refer to?

Eckard

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