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From: Archimedes Plutonium on 30 Jun 2010 04:48


Archimedes Plutonium wrote:

(snipped)

Now here is this proof after it is corrected with the Unique Prime
Factorization
theorem:

>
> 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
>

I want to say something further that I noticed and is probably a Lemma
disease
of reducto ad absurdum lemmas.

Notice that Euclid's translated proof appears to have a lemma of
contradiction,
and that Ore seems to have retained that lemma of contradiction.

But I said that the Euclid IP direct method needs no lemma of
contradiction at all
if you plugg in the Unique Prime Factorization Theorem UPFAT.

So what I am wondering is if the world of math has a proliferation or
reproduction of
lemmas of contradiction by all those who forget that there is some
theorem they should
be applying and not be applying a argument of contradiction. So that
if Ore had
realized he was using UPFAT, only he did not use UPFAT, and instead
argued there is
a prime factor with a lemma contradiction.

So that if Ore had realized or recognized he was using UPFAT, just
needed to state that
there exists a prime factor, not because P then P+1 has 1 divisible,
not because of that,
but because UPFAT was invoked and that Ore had not realized he was
using UPFAT. Not
realizing that UPFAT was used, then Ore launched a lemma of
contradiction.

So I am wondering whether a huge number of lemmas by contradictions in
other proofs
are used because the author invoked another theorem but did not
realize it and then launched
a needless lemma. I know alot of math proofs that seem to have strings
and strings of
lemmas. And for every lemma by contradiction, I would hazard to guess
the author invoked an
already established theorem, and did not realize he was using the
theorem and thus created excess baggage of a lemma.

So lemmas by contradiction are needless and heedless contraptions for
which the author should have listed the theorem invoked and kept the
proof as streamlined direct method.

Now if memory serves me, there are some mathematicians who when faced
with a lemma by
contradiction, will stop at that point in the proof and search around
and if they do not find
an existing theorem, will pause in the proof and actually state a
theorem and prove it there,
then picking up the original proof to continue. They do this because
they abhor most proofs, even lemmas
by contradiction.

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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