Re: Relative Cardinality



Proginoskes wrote:
> mueckenh@xxxxxxxxxxxxxxxxx wrote:
> > Randy Poe wrote:
> > > mueck...@xxxxxxxxxxxxxxxxx wrote:
> > > > As this would lead to strange results like Card(N) =
> > > > Card({Primes}),
> > >
> > > Of course, Card(N) does equal Card(Primes).
> > >
> > > Does WM think there is a natural number n such that the
> > > n-th prime does not exist?
> >
> > Yes, it is so. I am not sure, whether sequences like 111...111 with n
> > 1's or like 10^2n - 10^n + 1 do ever cease to supply primes now and
> > then.
>
> That is an irrelevant comment, because there are prime numbers which
> are not of that form (like 2).

It is no irrelevant but you have not yet understood my arguing. I
mentioned these numbers because numbers of that form (111...111)
definitely do exist. Pot. infiniteley many! If there are always some
prime numbers among them, then pot. infinitely many prime numbers can
be raised into existence too. Nevertheless we can never count up to
10^10^100, and, therefore, we cannot determine the 10^10^100-th prime
number - irrespective of how many can be found.

> Er ... Euclid proved that there are an _infinite_ number of primes.

Euclid did not talk of infinity. There are more than any given number,
he said.

> What do you find wrong with that proof?

Do you really think it necessary to demonstrate such things here?
Recently we had a proof that sqrt(2) is not rational. We should
concentrate on more general problems: It is impossible to label more
than 10^100 entities by all particles the univese supplies. Therefore
there cannot exist a set with more than 10^100 elements and we cannot
count up to 10^10^100.

Regards, WM

Regards, WM

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