Re: Relative Cardinality

mueckenh@xxxxxxxxxxxxxxxxx wrote:
> Proginoskes wrote:
> > Your set of numbers does not allow the concept of "arbitrarily many"
> > objects; you can only count up to H, where H is the largest Muecken
> > number. "Arbitrarily many" means that for any number N, there are more
> > than N objects. So you can only say that there are more than H 1's, so
> > you can't tell the difference between a string of H+1 of them or an
> > "infinite" number of them.
>
> Wrong. There is no largest number. If your H is defined by 50 % of all
> memory available then I abbreviate it by a symbol using only small
> memory and go on.

Proginoskes is not wrong, though he doesn't understand your twisted
reasoning. Let me try to summarize what WM is trying to do:

There only exist finitely many Muecken numbers at any single time. At
a later time you can have other Muecken number. Actually by the above
paragraph you can realize any number as a Muecken number given enough
iterations. I just store some large number in my "memory" then
abbreviate as "H", then I can do things such as "H+1" which again fit
in my memory, and I can abbreviate "H+1" as "H" thus starting over.

Anyway, so Muecken is correct when he says there is no CURRENTLY
EXISTING number that is the largest because he can manufacture a larger
one which is suddenly exists, of coures one cannot fix a single time
since then a single largest one WOULD exist. Now Proginoskes is
correct since he's talking about all numbers that are potentially
Muecken numbers. While apparently that set (collection if WM doesn't
want to use the term "set") has numbers which are arbitrarily large,
that is, there is no largest member, (thus being infinite by all
mathematical standards), WM deems it finite by definition. Now I am
not able to understand this twist of logic other then perhaps
redefining "finite" to mean say "countable".

> By the way, if there was a largest encodable number then it would also
> apply to mainstream mathematics, because you have no other means to
> express large numbers than I. Do you believe that there is a largest
> number?

Once you define an encoding, then yes, there is a largest number that
can be encoded in such a way. For example if you assume there are only
finitely many particles in the universe, only finitely many positions
each particle could be in, and only finitely many energy states each
particle could be in (not that our universe is neccessairly like this,
but let's assume so), then there are only finitely many configurations
of the universe, each one could correspond to a number, and there would
be a largest such number.

I say "such number." Since it doesn't say that there is a largest
"natural number" or such nonsense. All the above says is that if there
are finitely many states the universe could be in, then the number of
those states is some natural number n. period.

BTW, even if there are only 10^100 particles, the configurations of the
universe using those 10^100 particles would allow to store arbitrary
numbers VASTLY greater then numbers with 10^100 digits.

> > > Cardinality was introduced as a measure of the number of elements. And
> > > when introducing it, Cantor mentioned something like clear thought. That
> > > is also the foundation of my definition.
> >
> > But your "clear thought" definition has different properties than
> > Cantor's definition, and should not be treated the same way, unless you
> > know it results in the same concept.
>
> It compares the number of rationals and the number of irrationals. That
> is all which is necessary. No other aspects are involved.

So if I come up with a new way to calculate pi like many cranks do, and
then find out that pi = 3.125 which is a popular result. Say I do this
by a new method of integration. Then if I compare and find out that by
integration using the Riemann definition and I get a different value,
should I then conclude that integration in any form is wrong? That is
analogous to what you do here.

See if I come up with a different definition for something and then I
find that it doesn't agree with the original one, all I know is that I
have something DIFFERENT, and not that the original one is "wrong" or
whatever.

Your WMcardinality (relative cardinality or whatnot) can reasonably be
defined rigorously if one wants (you have not done so, but it is
possible). Then you only have one WMcardinality for infinite sets
since all infinite sets have the same WMcardinality. Perhaps that is
more correct in your intuition to use as a measure of the "size" of the
set. That's perfectly fine. Cantor's cardinality is a DIFFERENT
definition, and perhaps it is not the correct way in your intuition to
use as a measure of the "size" of the set. You have however NOT shown
that Cantor's cardinality is inconsistent, you have ONLY shown that
WMcardinality is DIFFERENT. Hardly a memorable event to be recorded in
the anals of mathematics.

Actually we already have such a measure as WMcardinality defined in a
much easier manner. It's just that we say that an infinite set is
"infinite". No need for orderings and other such things.

Jiri

.

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