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From: Archimedes Plutonium on 30 Jun 2010 21:41

Let me recap what this adventure is plunging into. We notice the old
math system never
defined with precision the difference between a finite number and an
infinite number.
So that 3 is a Peano axiom number as well as 3333....33333. So the
next question was, since
the old math is so poor in definitions, cannot even distinguish
between a finite number and
an infinite number, that how in the world such a messy system still
allows for a proof of
Infinitude of Primes but never any proof of Twin Primes? In other
words, the old math is
messy and ill-defined, so should that have prevented even a Euclid
Infinitude of Primes proof?
The answer is obviously not, in that even though it was messy, it
still allowed for a IP
proof. It even allows for a proof that the counting numbers are
infinite, since you add one more
to the highest number of any finite set.

But the boundary line for the old math of having a ill defined finite-
number versus infinite-number is that of Infinitude of Primes and Twin
Primes. If we define finite number as all
numbers less than 10^500, we have instantly a proof of Twin Primes,
since we find a pair
of twin primes larger than 10^500, and (10^500)+1 and (10^500) +3 are
the first two
candidates.

Now I said the reason that Twin Primes, FLT, Goldbach, Riemann
Hypothesis were different from Infinitude of Primes is that those were
two dimensional asking more than just whether the
primes were infinite. RH asks alot of complex questions of the state
of infinity. Goldbach and
FLT and Twin Primes deal with infinity but also deal with operations
of infinity such as adding in infinity. Whereas Euclid's Infinitude of
Primes is a one dimensional conjecture. And I said that this
complexity deals with the place-value of infinite-numbers.

But I maybe able to give a geometry reason for why Twin primes is
unprovable when finite versus infinite numbers are never defined
properly. In geometry we have finite lines as line segments. Now can
finite lines form to make a infinite-line? It seems that noone has
ever
asked that before. Since all finite lines are line segments, only an
infinite-number of finite
line segments can make an infinite line or infinite line ray. But old
math never defined "infinite
number". So in geometry we can never build a infinite line from that
of finite line segments
unless of course we define infinite-number. If we define the finite
number as less than 10^500
then if we had 10^500 one unit line segments we can put them together
to form an infinite line
or infinite line ray. Likewise, we can demonstrate that the addition
of all line segments of 1 + 2
+ 3 + 4 + . . + 10^500 forms a infinite line ray. We can also prove
that there is an infinitude of
prime segments since we add a prime that is larger than 10^500. We
also can prove an infinitude of Twin Prime segments by just adding a
twin prime larger than 10^500.

So how does Geometry in fact escape the dilemma mess of old math
Number theory that
never well defined finite-number versus infinite-number? How did
Geometry sneek past
Number theory with a well defined finite-line versus infinite-line?
Well most human minds
sense right and wrong with geometry far easier than they do with
algebra and quantities.
We know a line is finite since it has two endpoints. We know a line is
infinite if it has at least
one arrow rather than two endpoints. And no matter how many finite
line segments we add together, we never can turn those line segments
into a infinite line ray, unless we know what
an infinite-number is. If we define infinite number as greater than
10^500, then we can build
a infinite line ray out of finite line segments.

Sorry, this post is too long already, and I am not able to explain why
Twin primes has no proof
yet regular primes has a proof when the old system of math never
defined finite-number from
infinite-number. The geometry talk does not explain the cutoff from
Regular primes to Twin Primes unprovability. I am searching for a
geometry reason why Twin Primes is never provable, rather than the
Place-Value explanation. It may come down that I cannot explain it
without Place Value.


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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