Re: An uncountable countable set

In article <1156843043.617422.7230@xxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
*** T. Winter schrieb:
In article <1156689126.335154.133150@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> muecke=
nh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter schrieb:
> > > > The first part of his first proof shows that a complete ordered=
field
> > > > has cardinality larger than the natural numbers. In his proof =
he did
> > > > not rely an any properties of the reals other than that they fo=
rm a
> > > > complete ordered field (he uses reals to exemplify).
...
> > Note what I wrote: "in his proof he did not rely on any properties of
> > the reals other than that they form a complete ordered field". What
> > other properties of the reals did he use?
>
> He had no field and he used no filed. What is a field without the
> axioms of the field? Nothing. Cantor disliked axioms if he did not hate
> them. His only concern were numbers, numbers, numbers and their
> *reality*. No fields and no axioms.

Perhaps. I thought we were discussing current set theory. And not what
in some time long ago was et theory. He may not have liked them, but in
current terminology "the first proof shows that a complete ordered field
has cardinality larger than the natural numbers".

In the terminology of the next century it may show even other things,
more general perhaps, with even more insight. I was discussing what
Cantor did. You were accusing me that I had not understood or misquoted
him.

Please reread the history of this discussion. I did not accuse you of
anything in this instance, only on misreading what I wrote. At some stage
I responded to somebody else (I think it was Virgil) that in the first proof
Cantor showed that complete ordered fields were not countable. You accused
me of misinterpretation.

The only case where I accused you of misquoting was when you left out
the first sentence of a paragraph that would be revealing. But, see
below.

> Aus dem in =3DA7 2 Bewiesenen folgt n=3DE4mlich ohne weiteres, da=3DDF

Quoting again only in part. I will quote the translation I gave, with
annotation of the complete paragraph, not only the part you like to quote:
In the article, titled: "Über eine Eigenschaft des Ombegriffs aller
reellen algebraischen Zahlen (Journ. Math. Bd. 77, S. 258) [hier
S. 115], can for the first time be found a proof of the theorem
that there are infinite sets that are not in bijection with the set of
natural numbers 1, 2, 3, ..., v, ..., or, as I say in general, do not
have the same cardinality as the natural numbers 1, 2, 3, ..., v, ... .
So apparently he is thinking that his first article proves the theorem
that there are sets that are not in bijection with the natural numbers.
Which is true.

Of course: The set of the reals and the set of the transcendental
numbers are such sets. So he has the right to speak generally about
"sets". Which mathematician would not like to generalize his theorems?

But note what he now claims the theorem actually *is*.

From what has been proven in section 2 follows that e.g. the
set of real numbers in an arbitrary interval can not be put in
a sequence w_1, w_2, w_3, ..., w_v, ... .
So he is now thinking that that proof was just an example for such a set.
Which it is. The "beispielsweise" is telling.
It is however possible to construct a much simpler proof for
that theorem that is independent from the observation of irrational
numbers.

Independent of the *necessary use* of irrational numbers.

Where in the German do you find "necessary use"? "ein viel einfacherer
Beweis liefern, der unäbhangig fon der Betrachtung der Irrationalzahlen
ist."

And the "that theorem" (in German "jenem Satze") can, in my opinion only
refer to the theorem mentioned in the first sentence in this paragraph.

That theorem the proof of which depends on irrational numbers.

Nope. That theorem is the theorem he stated in extenso in the first
sentence.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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