- In working with the Euclid proof primes

I found that any reasonable number past about 30 primes in the

product almost never stops in the factoring step.

43 primes in the product was the limit given in the A000945 definition as

being a computing limit.

It is reasonable to assume that an infinite prime might exist ,

but could never be computed.

One definition "operationally" of "Infinity" is the non-stopping Turing

machine.

It is one accepted in computer science at least.

Does a number that can't be computed actually exist?

Absolute randomness is also defined as a non-stopping Turing machine

in some books. Like the concept of perfect order

it is a Platonic ideal: the object that projects the shadows on the cave

wall.

There are two definite schools of philosophy here:

1) everything must have an operational definition

2) abstract ( uncomputable/ nonoperational) concepts can exist (Platonic

ideals)

The second of these is also used to justify concepts like "God"

and "faith"? Infinity can not be computed in finite time,

so it is a non-operational concept ( one ,unproven, accepted by faith or

axiom or definition).

It is a definite branching in mathematical theory

which some taught only mathematics

but no modern philosophy or science

seem to refuse to learn.

One has to be able to distinguish what is "real and computable"

and what isn't.

Theology causes wars,

but science

gives facts and answers questions.

Jose Ramón Brox wrote:

>If we assume that you can effectively keep running your machine for infinite time, then you will have counted aleph_0 natural numbers and aleph_0 prime numbers, and you could put them in a biyection: no natural number will be out of it.

--

>

>And you can not do the "difference of two infinities" because the operation is not defined in any way. You must do your definition first and get the conclusions after that. Anyway, when we talk about cardinals of sets, we say that the intersection of two infinite aleph_0 sets can be finite or infinite (so you cant define the difference as the cardinal of the intersection to get the result you want). For example: the intersection of natural and even numbers are the even numbers, an infinite set. The intersection of even and odd numbers gives the empty set, with cardinal zero. The intersection of even numbers and prime numbers gives the set {2}, with cardinal 1.

>

>I can't understand your proposal yet.

>

>Jose Brox

>

> ----- Original Message -----

> From: chasag@...

> To: primenumbers@yahoogroups.com

> Sent: Sunday, November 02, 2003 7:08 PM

> Subject: Re: [PrimeNumbers] Digest Number 1129

>

>

> Imagine a counting machine counting up and running for an infinite time

> ,After an eternity, a infinite large number n is reached and a very large number of

> primes have been logged. The counting machine also counts the number (n -

> number of primes). Thus the number of primes is n -(n - number of primes).This is

> a difference of two infinities leaving a finite result. The number of primes

> is not infinite but on the verge of infinity. This is where a prime gap might

> exist. The prime gap is incalculably high. The verge of infinity is a suborder

> of infinity.

> I know the logic of this is fuzzy, but somebody much more clever than me

> could develop a new kind of arithmetic.

>

>

> [Non-text portions of this message have been removed]

>

>

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Respectfully, Roger L. Bagula

tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :

URL : http://home.earthlink.net/~tftn

URL : http://victorian.fortunecity.com/carmelita/435/ - Roger Bagula wrote:

[Philosophical meandering deleted.]

> I don't know if any of this answers you list of questions or not.

No, it does not. Please post again, including my text and at

the appropriate point intersperse your answer to each specific

question that I asked.

> I really don't like to get in such discussions,

That is becoming ever more clear as time goes by.

> since math people seem to ignore any philosophical issues by

Close, but no cigar. Axioms and definitions are the foundations

> defining them away. Axioms and definitions are an answer

> to all their thinking problems?

of mathematical thinking. Rigorously correct logical arguments

are the building blocks placed on the foundations. If you wish to

be considered to be doing mathematics, please use clear and precise

logic.

> As a physical scientist ( chemist, physical scientist)

Ah, so you're not a mathematician and you are not interested in

> I'm not bound by those rules in my thinking.

participating in mathematics. Why, then, are you making so much

noise in an indubitably mathematical forum?

FWIW, my background is in the physical sciences. I have a BA in

chemistry from Oxford and my DPhil was for research in molecular

spectroscopy. I personally don't regard that as an obstacle to

contributing in a small way to a mathematical subject. I'm not

bound by the rules of mathematics any more than you are, but I

choose to follow them when communicating with mathematicians. If

you wish to converse with practitioners of other fields of study,

please do so but, please, do it in a relevant forum elsewhere and

use their rules to do so. Again, FWIW, I'm quite happy to talk

about quantum field theory or geometrodynamics, but not here.

> In other words is science more fundamental

A very good question and one well worth discussing, but not here.

> philosophically than mathematics at a metamathematical level?

It is not (IMO, the moderators may disagree) relevant to the

advertised aims of the forum.

Paul