Das Kalenderblatt 090812

11/08/2009 - 21:54 von WM | Report spam
This does not, of course, prove anything inconsistent in Cantor’s
notion of an infinite
cardinal. But it does show that it cannot automatically be assumed
that any actual infinite
will automatically have a corresponding infinite number, an infinite
cardinality. That is, it
raises questions about the applicability of Cantorian transfinite
mathematics. In any case
where there is a requirement of a recursive connection between any
pair of the things
numbered, the Cantorian conception of the infinite will not be valid.
This is because the
set N of natural numbers ordered by the relation > (is ‘greater than’)
is recursively
connected if and only if every number is finite. If limit ordinals
(Cantor’s omega, omega^2 etc.) are included, recursive connectedness
fails.

[Richard Arthur: Leibniz and Cantor on the Actual Infinite (2001)]

Philosopher and Leibniz scholar Richard Arthur's critique of Cantor's
arguments for an actual infinity

Gruß, WM
 

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#1 Herbert Newman
12/08/2009 - 02:11 | Warnen spam
Am Tue, 11 Aug 2009 12:54:44 -0700 (PDT) schrieb WM:

Lustig, was Herr Professor Dr. Mückenheim so alles postet. :-)

[Richard Arthur:]

This does not, of course, prove anything inconsistent in Cantor's
notion of an infinite cardinal.



Of course not. :-)

Auch wenn Herr Mückenheim das bekanntlich anders sieht! :-)

Zitat:

"Set theory with the axiom of infinity is self contradictory."

(Prof. Dr. Wolfgang Mückenheim, Fachhochschule Augsburg, in sci.logic am
21. Juli 2009)

But it does show that it cannot automatically be assumed that any
actual infinite will automatically have a corresponding infinite
number, an infinite cardinality.



Ja; aber das tut ja auch niemand. :-)

That is, it raises questions about the applicability of Cantorian
transfinite mathematics.



Nun, aus philosophischer Sicht gibt es gut wie NICHTS, das keine Fragen
aufwirft. :-)

(Und es ist natürlich das gute Recht eines Philosophen, solche Fragen zu
stellen.)

Dazu fàllt mir gerade ein, dass Herr Professor Dr. Mückenheim
WEDER Philosoph, NOCH Mathematiker ist. :-)

In any case where there is a requirement of a recursive connection between any
pair of the things numbered, the Cantorian conception of the infinite will not
be valid.



Das kann gut sein. Nur gut, that in the context of the usual systems of set
theory, like ZFC for example, THERE IS NO SUCH REQUIREMENT. :-)

[Richard Arthur: Leibniz and Cantor on the Actual Infinite (2001)]



Ja, schön.

Aber Herr Professor Dr. Mückenheim kann schon noch Philosophie von
Mathematik unterscheiden, oder? (Oder nicht?)

Philosopher and Leibniz scholar Richard Arthur's critique of Cantor's
arguments for an actual infinity.



Was ist?! Diese Schwachsinnsbehauptung ist nun wohl wieder auf WMs Mist
gewachsen. Kriegen Sie eigentlich noch IRGENDWAS gebacken, Herr Professor
Dr. Mückenheim?

Hint: "...arising out of my work on Leibniz, I have been defending an
account of the actual infinite that is a rival to the Cantorian account,
but which eschews infinite sets." (Richard T. W. Arthur)


Herbert

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