Re: infinity ...


David R Tribble wrote:
> Albrecht S. Storz wrote:
> >> [...]
> >> Since there is no biggest number and since there is no infinite number,
> >> the size of the set of numbers in form of sets of #s is undefined as
> >> the biggest natural number is undefined.
> >>
> >> But the sequence of the sets of # fullfill the peano axiomes. So this
> >> set must be infinite.
> >>
> >> The cardinality of a set is not able to be infinite and "not defined"
> >> at the same time.
> >> This is the contradiction.
> >
>
> David R Tribble wrote:
> >> I don't see the contradiction. The size of the set is "not defined"
> >> to be the same as any natural number, and the set size is obviously
> >> infinite. This is no contradiction, since no natural number is
> >> infinite.
> >>
> >> The thing that is "not defined" is the largest natural, which obviously
> >> does not exist. But the set size is infinite, and is nicely defined
> >> by an infinite cardinal.
> >>
> >> You seem to be mixing the two concepts of "natural" and "cardinal"
> >> numbers to create a supposed contradiction, but that does not work.
> >
>
> Albrecht S. Storz wrote:
> > You are not able to understand that there is no difference between
> > numerals and sets.
>
> I have no problem seeing the correspondence between natural numbers
> and von Neumann sets. But neither of these are the same as
> cardinalities, which are not numbers, but measures (sizes) of sets.

Natural numbers are sets. Why be so delicate about this? It's not only
a correspondence between them. It's identity. Only a fool is unable to
see that fact if the numbers are shown in unitary (1-adic) system.

And now the cardinality. The definition of cardinality bases on sets.
The existence of infinite sets bases on definition. The meaning of the
cardinality of an infinite set is unknown.
Coincidently natural numbers and cardinalities are undistinguishable in
finity. Cardinality is just a artificial concept with no sens and
meaning.
Define the existence of unicorns and be happy.


Regards
AS


>
>
> > My sketches shows this exactly.
> > Cantor proofs his wrong conclusion with the same mix of potential
> > infinity and actual infinity. But there is no bijection between this
> > two concepts. The antidiagonal is an unicorn.
> > There is no stringend concept about infinity. And there is no aleph_1,
> > aleph_2, ... or any other infinity.
>
> For that to be true, there must be a bijection between an infinite
> set (any infinite set) and its powerset. Bitte, show us a bijection
> between N and P(N).

.

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