Re: Very basic mistakes

Thank you Balthasar. I am rather disappointed that nobody seriously took
issue so far. Victor did not understand anything what I am trying to point
to.

While my main interest relates to IR+, I also found out that - at least as
far as I can judge - the notions of number, real number, continuity, and
infinity were distorted in order to unnecessarily justify reasonable
pragmatism in an allegedly rigorous manner.

Let me try and summarize it in brief:

I agree with the mathematicians that "0.999... does not differ from 1.000...
iff ..." means that we assume all of indefinitely much nines or zeroes,
respectively, are given. However, I do not require an arbitrary definition
for that. According to my reasoning, infinity, continuum, irrational
numbers, real numbers, and non-linearity belong to an ideal realm that is
quite different from the ideally discrete realm of points and rational
numbers.

Those who are critical towards set theory are usually put into the drawer of
constructivism. While I found myself sharing some critical arguments with
e.g. Brouwer, I do not intend correcting set theory as did he.
Constructivists do perhaps not consequently apply Brouwer's insight that the
TND is only valid for finite quantities. They even substitute the law of
trichotomy (r<s or r=s or r>s) by "if r, u, n elements of IR with u<n, then
r>u or r> n".

Do not get me wrong. I appreciate the tacit substitution of genuine reals by
approximating rationals. I merely demand that one must not pretend actually
using a body of reals. If we decide that a number relates to a unity of e.g.
length of a line, then (strictly speaking) I see this precluding to
interpret it as a point.

Back to (as I claim correct) basics means:

Euclid: A point is what does not have parts.
eg Peirce: Each part of a (genuine) continuum has parts.
Stiefel: "dust of irrational numbers"
Weyl: "sauce of real numbers"
(no trichotomy, impossible to practice a well-ordering)
Aristoteles: Actual infinity cannot be reached.
Spinoza: Absolute infinity cannot be enlarged or exhausted.

and of course
Salviati:
.... in ultima conclusione, gli attributi di eguale
maggiore e minore non aver luogo ne gl'infiniti,
ma solo nelle quantità terminate.
IR>|>IR+=|=IR,
not smaller, not larger, not of equal size,
just no quantum but a property.

"Balthasar" <nomail@invalid> schrieb im Newsbeitrag
news:1qla5l1vjegb8.1gdy9v2jdtgnl.dlg@xxxxxxxxxxxxx
Am Thu, 11 Sep 2008 01:51:45 -0700 (PDT) schrieb
victor_meldrew_666@xxxxxxxxxxx:


My main topic is IR+. However you ignored what I am claiming.

You have neither defined IR+ (that's not a standard notation)

It is. But you may read it as R+.

IR+ := {x e IR : x > 0}.

See:
http://mathworld.wolfram.com/R-Plus.html


Mathematicians understand IR+ like IR but restricted to positive values.

Right.


Again, what is IR? That is not standard notation.

It is. "IR" (meant as a single symbol) denotes the set of real numbers:

See:
http://mathworld.wolfram.com/RealNumber.html


B.


.

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