Re: An uncountable countable set


*** T. Winter schrieb:

In article <1156842732.043353.163480@xxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter schrieb:
...
> > > Cantor states (analogously) that *all stairs exist*, that the width of
> > > all ot them is L, but that none of them has hight L. Width [0, 1] and
> > > height [0, 1).
> >
> > A quote please. But indeed, the width of all of them is L, but there is
> > none of them with width L. And the heighth of all of them is L, but there
> > is none of them with heighth L. So width [0,1] and heighth [0,1]. That
> > is, if you define both as the smallest box containing the completed stair.
> > If you define both as the largest numbers that can be obtained, you get
> > width [0,1) and heighth [0,1).

You are trying to make it difficult, using UTF-8. But this one takes the
cake.

What is UTF-8?
> I agree with you. But Cantor said (Werke, p. 409): "So stellt uns
> beispielsweise eine veränderliche Größe x, die nacheinander die
> verschiedenen endlichen ganzen Zahlwerte 1, 2, 3, ..., v, ...
> anzunehmen hat, ein potentiales Unendliches vor, wogegen die durch ein
> Gesetz begrifflich durchaus bestimmte Menge (=EF=81=AE) aller ganzen
If "=EF=81=AE" is UTF-8, the Unicode character is U+F06E, which is in the
private area of characters. I have no idea what that symbol stands for,
so I will modify it to N.
> endlichen Zahlen N das einfachste Beispiel eines aktual-unendlichen
> Quantums darbietet.

Again translated (why do you post so much German in an English speaking
newsgroup while you should know that most readers are not able to read
German?):

Because I have the German text available and because I do not want to
be blamed of mistranslating.

Cantor: So while a changing quantity x that successively takes the
various values of finite numbers 1, 2, 3, ..., v, ... , is a
potential infinite, on the other hand, a through the axioms completely
determined set (N) of all integral finite number is an example of an
actually finite quantity.

Not through the axioms, but through "a law" (ein Gesetz).

Nice that you found the quote I have alluded to, and that you did deny
of existing, but that I could not find back.

What Cantor is stating here (and I did already indicate that in an
earlier response), is, translated to current set theory:
The set N is potentially infinite,

No. The changing quantity is here a variable.

the size of N is actually infinite.
(In current terminology a set is not a quantity.)

Cantor's changing quantity is a variable. A set is never potentially
infinite according to Cantor.

Regards, WM

.

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