From: Roger Bagula [mailto:

tftn@...]

> If you have a set of primes that is infinite in number, then how can

> the last asymptotic element at n*log(n) not be infinite, if n is

> infinite?

Ah, now that one I can answer. You make the assumption that there is

a last element. You have not proved this statement, you just assume

it. I assert that there is no such last element. The proof goes as

follows:

You now seem to accept that the set of primes is infinite, that is it

has cardinality aleph_0. In that case, each and every element of the

set can be labelled uniquely with an integer; this is the definition

of a set which has cardinality aleph_0. So, for any given N, no

matter how large, we can find the prime labelled by N. But, given

such a N, we can find another integer M which is larger than N. An

example of such an M is N+1. Therefore given a prime, no matter how

large, we can find its label AND we can find a larger prime which has

a larger label. Consequently, the sequence of primes is never ending;

we can always find larger and larger primes and we never reach the

last one. The last prime does not exist.

> And since it is an element of the set of primes, it is an infinite

> prime. So the proofs which are at least four in number that say the

And as it doesn't exist in the first place, all your subsequent

reasoning is meaningless.

Paul