Lwalk or Mr. Transfer P becoming the third human to do the only valid correct Indirect Euclid IP #4.10 Correcting Math [Logic]

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From: Tonico on 23 Jul 2010 18:46

On Jul 24, 12:52 am, "sttscitr...(a)tesco.net" <sttscitr...(a)tesco.net>
wrote:
> On 23 July, 21:00, Archimedes Plutonium
>
> <plutonium.archime...(a)gmail.com> wrote:
> > Archimedes Plutonium wrote:
> > > Transfer Principle wrote:
>
> (Much mindless gushing deleted)
>
> > So LWalk, I invite you to become the third human to first do a valid
> > Indirect proof.
>
> Only the first human to do so can be first to do a valid indirect
> proof.
>
> The third man to climb Everest cannot be the first
> to do so.
>
> Obviously, this is the precise logic
> that leads AP into to thinking w+1 is "necessarily"
> prime.
>
> Have you decided yet whether 1 is or is not a prime ?
>
> How can your proof be valid if you define 1 as a prime number but do
> not include 1 in your ascending list
> of assumed finite set of primes ?

Hello,hello! Do not question Archie outstanding, staggering and
bewildering mathematics and logic: if he says "..the third person to
first do..." then it is the third person to first do whatever!

And if he says W + 1 is a prime then it is a prime, and if it is
divisible by a non-unit integer then tough luck for that integer!

Tonio, aka the third human to first do something Archie said.
(Again, please: who was the second one?)

From: Androcles on 23 Jul 2010 18:54

"Tonico" <Tonicopm(a)yahoo.com> wrote in message
news:eef4be64-8e56-470c-9ee1-3097801e18d9(a)q12g2000yqj.googlegroups.com...
On Jul 24, 12:52 am, "sttscitr...(a)tesco.net" <sttscitr...(a)tesco.net>
wrote:
> On 23 July, 21:00, Archimedes Plutonium
>
> <plutonium.archime...(a)gmail.com> wrote:
> > Archimedes Plutonium wrote:
> > > Transfer Principle wrote:
>
> (Much mindless gushing deleted)
>
> > So LWalk, I invite you to become the third human to first do a valid
> > Indirect proof.
>
> Only the first human to do so can be first to do a valid indirect
> proof.
>
> The third man to climb Everest cannot be the first
> to do so.
>
> Obviously, this is the precise logic
> that leads AP into to thinking w+1 is "necessarily"
> prime.
>
> Have you decided yet whether 1 is or is not a prime ?
>
> How can your proof be valid if you define 1 as a prime number but do
> not include 1 in your ascending list
> of assumed finite set of primes ?

Hello,hello! Do not question Archie outstanding, staggering and
bewildering mathematics and logic: if he says "..the third person to
first do..." then it is the third person to first do whatever!

And if he says W + 1 is a prime then it is a prime, and if it is
divisible by a non-unit integer then tough luck for that integer!

Tonio, aka the third human to first do something Archie said.
(Again, please: who was the second one?)
=============================================
This clearly proves three comes after five.
http://www.youtube.com/watch?v=xOrgLj9lOwk

From: sttscitrans on 23 Jul 2010 20:11

On 23 July, 23:54, "Androcles" <Headmas...(a)Hogwarts.physics_z> wrote:
> "Tonico" <Tonic...(a)yahoo.com> wrote in message
>
> news:eef4be64-8e56-470c-9ee1-3097801e18d9(a)q12g2000yqj.googlegroups.com...
> On Jul 24, 12:52 am, "sttscitr...(a)tesco.net" <sttscitr...(a)tesco.net>
> wrote:
>
>
>
>
>
> > On 23 July, 21:00, Archimedes Plutonium
>
> > <plutonium.archime...(a)gmail.com> wrote:
> > > Archimedes Plutonium wrote:
> > > > Transfer Principle wrote:
>
> > (Much mindless gushing deleted)
>
> > > So LWalk, I invite you to become the third human to first do a valid
> > > Indirect proof.
>
> > Only the first human to do so can be first to do a valid indirect
> > proof.
>
> > The third man to climb Everest cannot be the first
> > to do so.
>
> > Obviously, this is the precise logic
> > that leads AP into to thinking w+1 is "necessarily"
> > prime.
>
> > Have you decided yet whether 1 is or is not a prime ?
>
> > How can your proof be valid if you define 1 as a prime number but do
> > not include 1 in your ascending list
> > of assumed finite set of primes ?
>
> Hello,hello! Do not question Archie outstanding, staggering and
> bewildering mathematics and logic: if he says "..the third person to
> first do..." then it is the third person to first do whatever!
>
> And if he says W + 1 is a prime then it is a prime, and if it is
> divisible by a non-unit integer then tough luck for that integer!
>
> Tonio, aka the third human to first do something Archie said.
> (Again, please: who was the second one?)

Archie Poo believes that the only human ever to agree with him is, I
think, Karl Heuer, a mysterious
and enigmatic character. Probably, one of Archie
Poo's multiple personalities actually intelligent
enough to read and write. So Archie's body is the substrate for the
first and second personalities
to first do a vaild indirect proof, even though he is unaware of this.

From: Androcles on 23 Jul 2010 20:24

From: Transfer Principle on 23 Jul 2010 23:41

On Jul 23, 1:00 pm, Archimedes Plutonium
<plutonium.archime...(a)gmail.com> wrote:
> Archimedes Plutonium wrote:
> > Not modify; and let me say, to render the valid proof indirect. All
> > other attempts of Indirect on Euclid Numbers were invalid proof
> > arguments because only when P-1 and P+1 are necessarily new prime
> > numbers is there a valid Euclid IP Indirect proof. So I am not
> > modifying anything, I am rendering the first valid Euclid IP
> > Indirect. And why would any intelligent mathematician, knowing that
> > Regular Primes infinitude is a more general theory than just the
> > subset of Twin Primes, why would any mathematician with his/her
> > thinking cap on, think that the Euclid method cannot yield Twin
> > Primes when it yields Regular Primes.
> (1) definition of prime number
> (2) hypothetical assumption, assume the primes are finite and that
> the sequence list is 2,3, 5, 7, 11, . . , p_k
> (3) multiply the lot and add 1, calling it W+1
> (4) W+1 is necessarily a new prime because of definition in (1)
> joining with the fact that
> division of W+1 by all the primes that exist in (2) leave a remainder
> (5) contradiction to (2) that p_k is the largest and last prime, for W
> +1 is now the largest prime
> (6) reverse supposition step (2) and primes are infinite
> LWalk, all you have to do to become the third human to have done a
> valid Euclid IP Indirect and the only humans to do a valid Indirect,
> since all Indirect Euclid IP has to have W+1 as necessarily prime.
> All you have to do LWalk is agree that the above is a valid Euclid
> Indirect.
> Just say yes, and then I will count you as the third human being to be
> able to do a valid proof indirect.

So AP is asking me to settle the dispute between himself and the
anonymous poster (on Google, he appears only via the email address
sttscitrans(a)tesco.net) regarding Euclid's infinitude of primes proof.

In particular, according to AP, if we assume that there are only
finitely many primes and W is their product, then W+1 must necessarily
be a prime number. But according to sttscitrans, W+1 must necessarily
_not_ be prime.

And so which side do I believe is right? Answer: _both_ are right!

In classical logic -- and I assume that we are using classical logic
here since we're discussing reductio ad absurdum here -- it is said
that falsity implies anything. Thus if P is false, then P->Q is said
to be true no matter what Q is. Thus, we have:

"If W is the product of all the primes, then W+1 is prime" is true --
it can't be composite, since no prime p_n can _divide_ W+1.

_and_

"If W is the product of all the primes, then W+1 is not prime" is
true -- it can't be prime, since no prime p_n can _equal_ W+1.

So W+1 is both prime and not prime, a contradiction. Therefore, there
exist infinitely many primes. Both AP and sttscitrans are right. QED

One note of contention between AP and sttscitrans regards the notion
of a unit. The ring Z has two units, 1 and -1, but AP, according to
sttscitrans, considers 1 to be prime, not a unit. (Notice that
according to the ancient Greeks, including Euclid, 1 wasn't a number
but was instead a unit, and we still consider 1 to be a unit to this
day, being neither a prime nor a composite number.)

The notion of a unit is critical to the infinitude of the primes, and
this is obvious when we consider the mathematician Furstenberg's
topological proof. In this proof each set of all multiples of a prime
p is a closed (in fact clopen) set, and so the complement in Z of
each such set -- i.e., the set of non-multiples of p -- must be an
open set. Notice that the intersection of all these open sets must be
the set of all integers not divisible by any prime -- in other words,
the set of all _units_. And if there were only finitely many primes,
then this would be a _finite_ union. Now the finite union of open
sets must itself be open, and any nonempty open set must be infinite,
but there are only finitely many units, namely 1 and -1. This is a
contradiction, and so there exist infinitely many primes. QED

As we can see, the key to the proof is that if there are finitely
many units, then there must be infinitely many primes. If there are
infinitely many units, then there could be finitely many primes. In
fact, this is exactly why we can't apply Furstenberg's proof to the
p-adics (Hensel p-adics, not AP-adics). In the p-adics, there are
infinitely many units, but only finitely many primes -- to be exact,
there is only _one_ prime, p itself. What Furstenberg's proof tells
us is that the set of units and the set of primes can't both be
finite (in any ring where we can define open and closed sets the way
that Furstenberg does).

Regardless of whether sttscitrans is correct that AP considers 1 to
be prime, we note that AP does implicitly use the fact that 1 and -1
are the only units anyway. For if p_1, ... p_k are taken to be
_positive_ primes, then W must be greater than or equal to 1 (with
equality in case that k=0, the empty product being 1). Thus W+1 must
be at least 2, hence not a unit since 1 is the largest unit. Since
W+1 isn't a unit, and since it can't be composite as no p_1, ... p_k
can divide it, it must be prime. And combined with sttscitrans's
proof that it can't be prime, we derive the contradiction that we set
out to derive in this indirect proof.

So I consider AP to be correct, but I won't receive credit for being
the third person to give a valid proof since I consider sttscitrans
to be correct as well -- and besides, Tonio has already agreed with
AP (though knowing Tonio, he was probably being sarcastic).

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