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• ... This gets back to another misconception that I thought you had (I wrote about it last night) (infinite vs. unbounded). What does last asymptotic element
Message 1 of 16 , Nov 4, 2003
At 12:34 PM 11/4/2003, Roger Bagula wrote:
>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?

This gets back to another misconception that I thought you had (I wrote
about it last night) (infinite vs. unbounded). What does "last asymptotic
element" mean? There is no last prime - Euclid proved that. But just
because the size of prime numbers is not bounded that doesn't mean that the
size is infinite.

It seems that you want to take n=infinity in n*log(n) and you can't do
that. The primes are indexed by the positive integers 1, 2, 3,
... There is no prime whose index is infinity. Infinity is not in the set
of integers.
• ... asymptotic ... that the ... the set ... I think part of the confusion here is that we re used to the cardinality of a set of consecutive integers,
Message 2 of 16 , Nov 4, 2003
--- In primenumbers@yahoogroups.com, Jud McCranie <j.mccranie@a...> wrote:
> At 12:34 PM 11/4/2003, Roger Bagula wrote:
> >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?
>
> This gets back to another misconception that I thought you had (I wrote
> about it last night) (infinite vs. unbounded). What does "last
asymptotic
> element" mean? There is no last prime - Euclid proved that. But just
> because the size of prime numbers is not bounded that doesn't mean
that the
> size is infinite.
>
> It seems that you want to take n=infinity in n*log(n) and you can't do
> that. The primes are indexed by the positive integers 1, 2, 3,
> ... There is no prime whose index is infinity. Infinity is not in
the set
> of integers.

I think part of the confusion here is that we're used to the
cardinality of a set of consecutive integers, beginning at 1, being
the highest integer in that set. This works fine for counting coins,
for example. However, it only makes sense for finite sets.

A road with 10 mile markers ends at mile marker 10. A road with 500
markers ends at mile marker 500. A road that is infinitely long, and
has infinitely many mile markers, does not end, and needs no mile
marker numbered "infinity". And this is drifting off topic.

Nathan
• ... A logical problem brought about only by the way you choose to present it. If you have a set of primes which is infinite in number, then all it means is
Message 3 of 16 , Nov 4, 2003
--- In primenumbers@yahoogroups.com, Roger Bagula <tftn@e...> wrote:
> 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?
> 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 primes are infinite sets are either wrong
> or faulted if there is no infinite last element.
> It is a logical problem:

A logical problem brought about only by the way you choose to present
it. If you have a set of primes which is infinite in number, then all
it means is that for any number N, there exists an element p of the
set such that p>=N. (In fact there exist infinitely many such p, but
that doesn't matter too much.) It doesn't have to mean that there
exists a "last" element which is infinite.

Andy
• ... For one thing, the asymptotic limit of a set does not have to be a member of the set! You are assuming that it is, but that isn t true. In particular,
Message 4 of 16 , Nov 4, 2003
>At 12:34 PM 11/4/2003, Roger Bagula wrote:
> >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?

For one thing, the asymptotic limit of a set does not have to be a member
of the set! You are assuming that it is, but that isn't true. In
particular, the asymptotic limit of the set of primes isn't even an
integer, much less prime.
• ... I don t think it is off-topic, it clearly illustrates what I had been saying.
Message 5 of 16 , Nov 4, 2003
At 01:33 PM 11/4/2003, pakaran42 wrote:

>A road with 10 mile markers ends at mile marker 10. A road with 500
>markers ends at mile marker 500. A road that is infinitely long, and
>has infinitely many mile markers, does not end, and needs no mile
>marker numbered "infinity". And this is drifting off topic.

I don't think it is off-topic, it clearly illustrates what I had been saying.
• 1) Infinitely many primes. This is easy (read: euclidean proof...). 2) An infinite prime. I ll make up a definition, just for the fun of it: Let
Message 6 of 16 , Nov 4, 2003
1) Infinitely many primes. This is easy (read:
euclidean proof...).

2) An 'infinite' prime. I'll make up a definition,
just for the fun of it:

"Let p(x,n)=sum(i=0, n, ai*x^i)=a0+a1*x+a2*x^2+... be
a polynomial in the polynomial ring over the integers
such that for all n, p(x,n) is irreducible in this
ring. Then we say that p(x)=sum(i=0, infinity,
ai*x^i) is irreducible."

So this could be a potential definition of an
'infinite prime', although it doesn't seem to have any
immediate use...

Joseph.

--- Roger Bagula <tftn@...> wrote:
> A semantic distinction needs to be made:
>
> Infinitely many primes is distinct from an Infinite
> Prime.
>
> Although one seems to imply the other,
> they really involve two separate cases.

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• ... Well I wouldn t go that far. Here s a definition: If K is a field, then a prime of K is an equivalence class of valuations of K. An infinite prime is a
Message 7 of 16 , Nov 4, 2003
"Andrew Swallow" <umistphd2003@...> wrote:
>Please be kind enough to explain what you mean by an infinite prime.
>To the vast majority of mathematicians, it is absolutely meaningless.
Well I wouldn't go that far.
Here's a definition:
If K is a field, then a prime of K is an equivalence class of
valuations of K. An infinite prime is a prime of K that consists of
archimedean valuations.
(see http://planetmath.org/encyclopedia/InfinitePrime.html for more
details, for example)
Now, I'm not claiming that it's anything to do with what Roger is on
about, but it is a well-known definition of "infinite prime".
:-)
Chris
[ apologies if this appears twice - I tried to post this over 12 hours
ago, but it didn't appear ]
• ... If I was feeling immensely cheeky, I d point out that they re also known as infinite places , the substition of primes for places being called by some
Message 8 of 16 , Nov 5, 2003
> Here's a definition:
> If K is a field, then a prime of K is an equivalence class of
> valuations of K. An infinite prime is a prime of K that consists of
> archimedean valuations.
> (see http://planetmath.org/encyclopedia/InfinitePrime.html for more
> details, for example)

If I was feeling immensely cheeky, I'd point out that they're also known
as "infinite places", the substition of "primes" for "places" being called
by some an abuse of notation (from Swinnerton-Dyer's Brief guide to
algebraic number theory, p33)

But sorry, that's just nit picking. Perfectly true, and probably shows
up the gaps in my knowledge, the concept of infinite prime does have a
place in some parts of mathematics. But *not* as the limiting point of
the rational primes, which is what is being suggested in this thread.

But it's all good fun, :-)

Andy
• From: Roger Bagula [mailto:tftn@earthlink.net] ... Ah, now that one I can answer. You make the assumption that there is a last element. You have not proved
Message 9 of 16 , Nov 5, 2003
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
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