corrected Infinitude of Primes proof; both direct and indirect methods

From: Archimedes Plutonium (a_plutonium_at_iw.net)
Date: 03/29/05 

Date: Tue, 29 Mar 2005 07:45:45 -0600

---- quoting a bit of the old File 106 of www.iw.net/~a_plutonium
regarding Infinitude of Primes proof ---

       PART TWO: Euclid's Infinitude of Primes Proof, my correction
                        thereof
Note: All of these IP proof attempts by many authors have been posted to
Usenet
starting 1993-1994. This first one of 2SEP94 header is kept but all the
others are deleted to save space.

From: Ludwig.Plutonium@dartmouth.edu (Ludwig Plutonium)
Newsgroups: sci.math,alt.sci.physics.plutonium
Subject: G.H.HARDY, FAILS TO GIVE A VALID PROOF OF EUCLID'S IP
Date: 2 Sep 1994 01:19:25 GMT
Organization: Dartmouth College, Hanover, NH
Lines: 69
Message-ID: (345uit$68n@dartvax.dartmouth.edu>

---- end quoting my old work ---

This is my latest up to date work:

------------------------------------------------------------------
Subject:
              Re: Correcting the invalid proof attempts of Euclid's
Infinitude of Primes, both direct and indirect methods
         Date:
              Mon, 28 Mar 2005 12:10:56 -0600
        From:
              Archimedes Plutonium (a_plutonium@iw.net>
     Reply-To:
              NOiwEMAIL
 Organization:
              www.iw.net/~a_plutonium
  Newsgroups:
              sci.logic, sci.math
    References:
              1

>
> ####
> ** posted 1993-1994 proofs of Infinitude of Primes by Archimedes
> Plutonium **
> Valid proofs of Euclid's Infinitude of Primes
> Here is the valid Direct proof of IP
> Infinitude of Primes Proof, *DIRECT Method*
>
> (1) Statement: Given any finite collection of primes 2,3,...,pn
> possessing a
> cardinality n Reason: given
>
> (2) Statement: we find another prime by considering W+1 = (2x3x...xpn)

> +1
> Reason: can always operate on given numbers
>
> (3) Statement: Either W+1 itself is a prime
> Reason: numbers are either unit, composite or prime
>
> (4) Statement: or else it has a prime factor not equal 2,3,...,pn
> Reason: numbers are either unit, composite or prime
>
> (5) Statement: If p is not prime, we find that prime factor
> Reason: Unique Prime Factorization theorem
>
> (6) Statement: Thus the cardinality of every finite set can be
increased
>
> Reason: from steps (3) through (5)
>
> (7) Statement: Since all/any finite cardinality set can be increased
by
> 1 more therefore the set of primes is an infinite set.
> Reason: going from the existential logical quantifier to the
> universal quantification
>
> ###
>
> Here is the valid Indirect proof of IP. (Using Hardy's terminology)
> Infinitude of Primes Proof, *INDIRECT Method*
>

Anyway here is a streamlined proof of IP indirect method:

Here is the valid Indirect proof of IP. (Using Hardy's terminology)
 Infinitude of Primes Proof, *INDIRECT Method*

(1) The prime numbers are the numbers 2,3,...,pn,... of set S
Reason: definition of primes

(2) Suppose finite, then 2,3,...,pn is the complete series set
Reason: supposition step

(3) Set S are the only primes that exist
Reason: from step (2)

 Note: This step follows immediately from the "Suppose primes are
finite".
And it is a very important step and because it is so much like the
statement
"Suppose the set S of primes is finite", for it is this statement
"The set S is all the primes that exist".
And it is this step that disallows any other prime factor search once
W+1
is
formed. W+1 is your only allowable prime candidate in the indirect
method,
otherwise you are self-contradicting your
own logic. The math logic is supposed to get you the contradiction, not
you, in your own illogic.)

(4) Form W+1 = (2x3x,...,xpn) + 1
Reason: can always operate and form a new number

(5) Is W+1 prime? yes
 Reason: Unique Prime Factorization theorem combined with the
 definition of what prime is.

(6) Contradiction
Reason: pn was supposed the largest prime yet we constructed a new
prime larger than pn

(7) Reverse supposition step and primes are infinite
Reason: steps (1) through (6)

Also, one thing nice about this streamlined version is that it is the
same
number of lines as the Direct proof method of 7 lines in all.

I would like for someone to render IP, both direct and indirect into
purely
Symbolic Logic and see if both methods require the same number of lines
of
proof.

Now there is nothing in mathematics to say that a valid direct method
compared to a valid indirect method should end up with the same number
of
lines.

However, in Physics, a subject I am working on serendipitiously today is

that of Principle of Least Action. And in an AtomTotality Physics has
mathematics as a tiny subset. Physics is King and mathematics is a
servant
in the house of physics.

So a principle of Least Action may state that every proof in mathematics

such as Infinitude of Primes when written out in Symbolic Logic must
have
the same number of lines for the Direct Method as well as the Indirect
Method.

Do I believe it true? Honestly I do not. I believe the Principle of
Least
Action would make the Direct and Indirect come close together in number
of
lines but not force them to be equal. A good test experiment is to get a

proof that is widely disparate between direct and indirect written in
Symbolic Logic.

I am going to change the proof of IP indirect method on my website of
www.iw.net/~a_plutonium to this streamlined version.

In the indirect IP the focus of attention is the fact of Pn being the
largest prime and so the instant P!+1 is formed it is also prime but
larger
than Pn which yields the contradiction. I could talk about P!+1 as
composite and prime simultaneously yielding the contradiction, but the
focus at the beginning of the proof was Pn was the largest prime and
yields
the contradiction faster. So in a Symbolic Logic write up of Euclid's
Infinitude of Primes indirect method I believe the number of lines is
shortened with the Contradiction yielded from the new prime larger than
the
old prime Pn.

But I suspect that Euclid's IP is not easy, perhaps even impossible to
write up in terms of Symbolic Logic. I suspect this to be the case
because
then that would explain why MathWorld and over half of college and
University professors of mathematics fail in writing a logically sound
IP
proof. I suspect something is not transferrable to Symbolic Logic. And
that
a Symbolic Logic write up of IP would not be a short 7 lines but more
like
several hundreds of lines. Maybe it is writeable in Symbolic Logic but
all
mathematicians to date were too lazy to perform the task.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies

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