Re: Cantor Confusion

In article <45793B7C.4050105@xxxxxxxxxxxxxxxxxxx> Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> writes:
On 12/8/2006 1:54 AM, Dik T. Winter wrote:
In article <45781DC1.1000804@xxxxxxxxxxxxxxxxxxx> Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> writes:
....
> According to my reasoning, any really real number is not unique but must
> rather be void because even the tiniest interval is thought to contain
> indefinitely not just many rational numbers but indefinitely much of
> real numbers.

But what you state here for real numbers is also valid for rational
numbers. So you make not clear why in one case they are "void"
(whatever that may mean) and in the other case they are not void.

I beg your pardon. Though the matter is clear to me, I fear it is
difficult to explain because mathematicians are not used to think in
terms of notions with higher abstraction.

It is difficult to explain because you provide no definitions. It has
nothing to do with abstraction.

Christian Betsch was bought
for 1,000,000.00 Mark as to spread confusion about the "as if" and the
notion "fictitious". So meanwhile every dull mathematician tells and is
even teaching that all numbers are fictitious.

I had never heard of Christian Betsch. Who paid him that sum of money,
and when? However, it appears that that discussion is mostly philosophical,
but not mathematical.

While the imaginary numbers are obviously different from the ordinary
numbers, the corresponding distinction between rationals and reals is
more subtle.

That is quite clear. The square of an imaginary number is negative, for
ordinary numbers it is non-negative.

Perhaps it is most helpful to declare the reals just
fictions, while the rationals, including naturals and integers, are
genuine numbers.

That is not helpful at all. This is like saying "they are different
because they are different". But I ask you (I did ask Wolfgang
Mueckenheim already several times, and each time he gave no answer)
to give a mathematical definition of "number".

The difference between rationals and reals corresponds to the difference
between potentially infinite and perfectly infinite.

As "potentially infinite" and "perfectly infinite" is no terminology in
mathematics, I fail to se the correspondence.

Genuine positive
numbers can be chosen as large or as small as you like. There is no
upper and no lower limit to their potential to have larger and smaller,
respectively, values. In brief: They are called infinite. The word
"they" is somewhat mistakable here, because it refers to the abstract
category of rational numbers. It does not refer to particular numbers
and also not to the fictitious entity of all rational numbers.

"They" makes in this case absolutely no sense.

Infinity is in some sense the opposite of being infinite.

Eh? Do you care to rephrase this? Infinity is in some sense finite?
In what sense?

Infinity has
been thought to include all of anticipated indefinitely many elements.

Eh? I think you mean oo. In current mathematics that one has been
delegated to topology (in the one-point compactification). In the
remainder it is only a left-over without implicit meaning.

Hilbert was quite correct when he wrote that infinity evades our
imagination. There seems to be a considerable portion of mathematicians
who blueeyedly claim to be able to imagine all of infinitely many
elements.

They do not claim that. What they claim (and I claim I can) is imagine
the *set* of infinitely many elements.

They are simply able to switch off what I call alert common
sense. Something which has only the property to be endless by definition
cannot actually be perfect. Taken as a quantitative measure, infinity is
selfcontradictory. Perfect and imperfect at a time exclude each other.
Cantor called infinity "aktuales Unendlich", and he naively suggested
the possiblity to reach and even exceed it: oo, oo+1, oo+2, ...

He quickly switched to omega, a switch you apparently have not yet made.
He did that switch because he thought there would be confusion because
the difference between what was known as oo and what he initially called
oo. And it is just that confusion that traps you.

Nonetheless, infinity is a reasonable concept.
The fictitious indefinitely large number oo can precisely describe a
fictitious quality: something that cannot be enlarged and not exhausted
either: oo + a = oo.
In IR+ the nil is the dual fictitious number to oo: 0=1/oo.

What is nil + nil? Anyhow, what you are describing is not a field, so
all theorems that depend on the field theorems have to be rewritten.

Likewise, irrational numbers like pi are mere fictions. They are
categorically different from rationals. Therefore they are strictly
speaking useless in practice, since nobody can handle indefinitely much
of decimals. Do not worry, pi stands for a precisely defined though
impossible to fulfill task, and there are enough substitute rational
numbers which come as close as you like.

But when I do calculations with (say) sqrt(2), I do *not* want close
rational numbers. Those close rational numbers would fail to show
what I might like to show, or do show what is false. Have a look at
the number field sieve for factoring large integers. One of the most
challenging tasks (mathematically seen) is the finding of the exact
square root of an algebraic number in terms of the number field. And
that is only for factoring integers.

It is moreover interesting that with older variants of the sine function
in computer libraries you could easily detect what rational approximation
to pi they were using. I have once done so for an implementation where
the expected results where not as they should be. But working with
rational approximations is the field of numerical mathematics, i.e.
mathematics applied to numerical calculations. Not all of mathematics
falls in this field, that is an extremely short-sighted view.

> Therefore, unreachable the very nil on top of the nested
> intervals has not any significance at all.

I can not parse this statement, so I have no idea what it means.

I meant: "Therefore, the unreachable very nil ..."
Unreachable means it has to have a perfectly infinite amount of decimals:
0.00000000000000000000000000000000000000000000000000000000000...
This quality cannot even be expressed numerically.
No significance at all is categorically different from effectively no
significance.

You are again focussing on representation. Mathematics is not concerned
about representation. It is numerical mathematics that is concerned
about that, but it is solidly founded in true mathematics.

> It cannot even be
> distinguished from numbers 0- and 0+ left and right from it,
> respectively, because the diffence is zero.

What "It"?

The very nil between 0+ and 0-.

I have not yet seen a proper definition of either 0+ and 0-, so I
fail to see what you are telling here. You use a lot of terms without
any definition of the terms used. And if I do not know what 0+ and
0- are, I have also no idea what is between them.

> So I agree with the
> Bourbakis perhaps for the first time: 0+ and 0- are indiscriminable in
> IR. However among the rationals, the nil is the first negative number
> according to my reasoning and my old encyclopedia.

Oh. Well in English nil is not a number, I think you mean zero. But
pray quote your old encyclopedia where it is stated that it is the first
negative number, and I think we can outline where you are wrong.

I do not have it at hands. Nonetheless I recall the compelling
reasoning: Naturals and positive reals start at the unity 1.
1, 1+1, 1+1+1, ...
1/1, 1/(1+1), 1/(1+1+1), ...
(1+1)/(1+1+1), ...

Yes, so what? You apparently think that non-positive is negative. That
is false, both in Bourbaki mathematics as in non-Bourbaki mathematics.
In Bourbaki mathematics 0 is both positive and negative, in non-Bourbaki
mathematics 0 is neither. So please get your hands on your old
encyclopedia and verify it.

Notice, I am sceptical towards v. Neumann because of his gay character.

And this takes the cake. Such remarks mark you as unqualified for
scientific discussions. The character of von Neumann has nothing
to do with his qualifications. Do you resent that Germany lost the
war because the gay Turing was instrumental in decoding Enigma?

Sheesh.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
.

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