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From: Archimedes Plutonium on 5 Jul 2010 18:02
I need to get back to my physics book, where I am in the middle of it
with "missing mass".

I was interrupted from that physics book by this:

2009 Mathematical Intelligencer magazine article:
> [0] Michael *Hardy* and Catherine Woodgold,
> "*Prime* *Simplicity*",  *Mathematical
> Intelligencer<https://mail.google.com/wiki/
Mathematical_Intelligencer>

Which when compared to my postings on this subject to sci.math from
1993
to 2009 was a "lifting of my postings" without attribute by the
Mathematical
Intelligencer (MI) article. One poster said that magazine editors are
"afraid" of
referencing the electronic sci. newsgroups. Well, they have to get
used to
it for the newsgroups are going to be a larger body of referencing
than most
individual books or magazines or periodicals.

In that article by Hardy/Woodgold, they do get across the true message
that
Euclid's proof was direct/constructive and not indirect/contradiction.
But the
article fails to show where most mistakes are made on the indirect
method,
and the article even suggests that Hardy/Woodgold and editors of MI
could
not do a valid proper Infinitude of Primes proof indirect method based
on their
inability to recognize that P+1 is necessarily prime in the
contradiction method.

It is a wonder that whenever supposedly logic persons are doing a
discussion
over the logic of Infinitude of Primes proof and then fail to give
both methods a
showing, side by side one another, and then lambast others for
committing errors.
Seems to me, if you are going to talk about Euclid's IP proof of
direct versus
indirect, the most logical article would show the two methods, but
here in
MI , Hardy and Woodgold and editors could only muster a showing of the
direct
method and then lambasting hundreds of mathematicians in that they did
a
indirect method.

So on the Internet of the science newsgroups, of sci.math and
sci.logic, I have
come up with a challenge that whenever anyone does Euclid's Infinitude
of Primes
Proof, that they do two proofs, one of the direct method and the other
of the
indirect method. I guess Hardy and Woodgold did not want to do that
ultra logical
exposition because, perhaps, maybe they felt they would be stealing
too much
of my sci.math postings without proper attribute in that Mathematical
Intelligencer
issue.

Without further delay, here are the two methods of proof of Euclid's
Infinitude of Primes.
And the major stumbling block is in the indirect that P+1 or I used W
+1 for Euclid's
number is ** necessarily prime **. Most authors, especially
mathematicians in books
make that mistake of thinking that P+1 in contradiction method is not
necessarily prime
for they cite some silly irrelevant example of 1+(2x3x5x7x11x13) =
59x509. That example
is actually part of the direct method proof where you have the list of
finite primes as
2,3,5,7,11,13 and where the constructive proof ends up fetching the 59
and 509 increasing
the set cardinality. But in the Indirect Method, we have to fetch a
new prime not on the list
of the supposed hypothetical list of all the primes in existence. In
the Indirect, we cannot go
scrambling around looking for a prime factor in P+1. Our only chance
of a new prime is P+1
itself. And it is the structure, the logical structure of the indirect
method (reductio ad absurdum) that the structure of logic allows you
to refer to step 1 where you defined a prime
number as divisible only by itself and 1, it is this definition in the
Indirect that permits you
to boldly claim that P+1 is necessarily prime. So when mathematics
professors writing books
on the Euclid Infinitude of primes and not recognizing that P+1 in
contradiction method is
necessarily prime, have failed to deliver a valid proof. For those
that cannot understand their
silly example of 59 x 509 is no example of the indirect method, well,
here is an example
to show them they are wrong. Start with definition of primes. Suppose
the set of all primes is finite and that 3 and 5 are the only primes
in existence. Thus we have 5 as the last and largest
prime. Form P+1 which is (3 x 5) +1 = 16. Now, 16 is necessarily prime
in this hypothetical
supposition space. Yet we all know that outside this supposition space
that 16 is not prime. But that makes no difference because Logic is
structure, and in this proof method, it is all about logical
structure. So that 16 is necessarily prime given the definition of
prime and the
supposition hypothesis and the contradiction follows from the fact
that 16 as prime is larger
than the largest supposed prime of 5. So it is no wonder that
hundreds, thousands of mathematicians themselves messed this up and
mixed the two methods. They forgot that it
is the logical structure that renders the proof and not irrelevant
examples.

And Euclid's IP, Direct or constructive in short-form goes like this:
 1) Definition of prime
 2) Given any finite set of primes
 3) Multiply the lot and add 1 (Euclid's number) which I call W+1
 4) Either W+1 is prime or we conduct a prime factor search
 5) this new prime increases the set cardinality by one more prime
 6) since this operation of increasing set cardinality occurs for
any
 given finite set we start with, means the primes are infinite set.


So in words, the Euclid Infinitude of Primes proof, Indirect in
short-
 form goes like this:


1) Definition of prime
 2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
 finite with P_k the last and final prime
 3) Multiply the lot and add 1 (Euclid's number) which I call W+1
 4) W+1 is necessarily prime
 5) contradiction to P_k as the last and largest prime
 6) set of primes is infinite.


DIRECT Method (constructive method), long-form; Infinitude of Primes
Proof


(1) Definition of prime as a positive integer divisible
 only by itself and 1.


(2) Statement: Given any finite collection of primes
2,3,5,7,11, ..,p_n possessing a cardinality n Reason: given


(3) Statement: we find another prime by considering W+1 =(2x3x...xpn)
 +1 Reason: can always operate on given numbers


(4) Statement: Either W+1 itself is a prime Reason: Unique Prime
Factorization theorem


(5) Statement: Or else it has a prime factor not equal to any of the
 2,3,...,pn
Reason: Unique Prime Factorization theorem


(6) Statement: If W+1 is not prime, we find that prime factor Reason:
We take the square root of W+1 and we do a prime search through all
the primes from 2 to
square-root of W+1 until we find that prime factor which
evenly divides W+1


(7) Statement: Thus the cardinality of every finite set can be
increased. Reason: from steps (3) through (6)


(8) Statement: Since all/any finite cardinality set can be increased
by one more prime, therefore the set of primes is an infinite set.
Reason: going from the existential logical quantifier to the
universal
quantification


INDIRECT (contradiction) Method, Long-form; Infinitude of Primes
Proof
and
the numbering is different to show the reductio ad absurdum
structure
as
given by Thomason and Fitch in Symbolic Logic book.


(1) Definition of prime as a positive integer divisible
 only by itself and 1.


(2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
 Reason: definition of primes


(3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
 with p_n the largest prime Reason: this is the supposition step


(3.1) Set S are the only primes that exist Reason: from step (3.0)


(3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
 form a new number


(3.3) Divide W+1 successively by each prime of
 2,3,5,7,11,..pn and they all leave a remainder of 1.
 Reason: unique prime factorization theorem


(3.4) W+1 is necessarily prime. Reason: definition of prime, step
(1).


(3.5) Contradiction Reason: pn was supposed the largest prime yet we
 constructed a new prime, W+1, larger than pn


(3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)


(4) Set of primes are infinite Reason: steps (1) through (3.6)


Archimedes Plutonium
http://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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