I wrote earlier today:
> > It also seems you want me to provide a proof... Ok, here it goes.
> >
> > We say a natural number is prime iff it is greater than 1 and doesn't
> > have any divisor other than 1 and itself.
> > By the fundamental theorem of arithmetic, every number greater than 1
> > has a prime divisor.
> >
> > Let us suppose that the subset of all prime numbers is finite.
> > Then the product of all primes plus one, since it doesn't belong to
> > the subset of all prime numbers (it is greater than any of them) is
> > not a prime, but it has a prime divisor, since it is greater than 1.
> > But in fact, it doesn't have any divisor from the list of all primes.
> > Contradiction.
> >
Malcolm, I now realize what you were trying to do above. You were
trying to
do a Direct Euclid IP. It should have looked like this, with number
examples
for steps.
Earlier today I gave the Indirect methof of Euclid IP and here it is:
>
> Here is one for my proof of Euclid IP.
>
> (1) Definition of Prime number
> (2) Suppose the set of all primes is Finite
> (2*) Suppose 3 and 5 were all the primes that exist
> (3) Multiply the lot and add 1
> (3*) (3x5) +1 = 16
> (4) This new number is necessarily a new prime since we revert to our
> definition in (1)
> and all the primes divided into this new number leave a remainder of 1
> (4*) 3 and 5 are all the primes that exist and when divided into 16
> leave a remainder
> of 1 and by the definition of prime from (1), that 16 is indeed a new
> prime
> (5) Contradiction
> (6) Set of all primes is infinite
>
Malcolm, here is the direct method that you were attempting to do:
Euclid IP (direct method)
(1) Definition of Prime number
(2) Set of primes is this set {2,3,5,7,11,13,....} and we are out to
prove whether it is finite or infinite
(3) Given any subset of primes, that subset has a cardinality
(3*) a subset of primes such as {3,5} had cardinality of 2 since it
has two members
(4) The proof involves the increase in set cardinality of given any
subset of primes
and since we can increase the cardinality by one more prime, means the
set of
all primes is infinite
(4*) the crux of the proof is that like numbers, given any set of
numbers, add one to
the largest and you automatically increase the set cardinality and
thus an infinite set
(5) Every particular subset of primes, we multiply the lot and add 1.
(5*) The subset {3,5} gives (3x5)+1 = 16
(6) This new number "multiply the lot and add 1" can either be prime
itself or have
a prime factor because of (1) definition of prime
(6*) (3x5)+1 = 16 so this subset of cardinality 2, we have either 16
is prime or has
a prime factor. Obviously 16 is not prime and the prime factor is 2.
(7) Thus, given any finite set of primes we can augment that set with
a new prime
of either "multiply the lot add 1" or a prime factor of "multiply the
lot add 1"
(7*) 16 is not prime but 2 is prime and not in the subset {3,5}
(8) Since any finite subset of primes has the ability of augmentation
of a new prime
not in the list, means the set of all primes is an infinite set
There Malcolm, you were striving for the Direct Method, but you made
several mistakes. You
included a Suppose reductio ad absurdum, when you should have never
done so. There is no
"suppose" and a contradiction involved in the Direct method.
What is the difference between Direct and Indirect? Both rely on the
definition as first step.
Both require the use of "multiply the lot and add 1". The major
difference is that in the Indirect,
the new number formed by "multiply the lot and add 1" is necessarily
prime, and it can not have
a prime factor. Most math professors get it wrong and deliver an
invalid proof because in the back of
their minds they remember that 30031 has factors of 59x509 for the
subset where 13 is the
largest prime, but in the indirect method the number 30031 is a new
prime since
the set of all primes is {2,3,5,7,11,13}. The definition coerces you
to conclude 30031 is a new prime
not on the list of all primes and this new prime number 30031 is the
contradiction and discharge of the
proof. This is the mistake Niven, Zuckerman, Montgomery made in their
textbook, and the mistake that
Courant and Conway and Wikipedia and 30 other math professors make.
The mistake is a lack of Logical Continuity, just as if we were to
electrically wire a new house and
if the lines are not connected correctly there will be no flow of
electricity.
So the moment you do a Indirect Euclid IP and mention a search for a
"prime factor", you lost the proof
and have fallen off the mountain.
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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