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From: Archimedes Plutonium on 7 Jul 2010 02:33
Funny how infinity needs a discussion for both large infinity and a
separate discussion
for infinity on the small scale. Here is another example of Quantum
Duality in mathematics
in that infinity, which we usually think of as beyond large, that it
must be reconciled
with microscale and infinity there.

In July 2010, this month: I gave this as proof that large scale
infinity requires a boundary
such as 10^500 which says that finite is any integer smaller than
10^500, and 10^500 itself
is an infinite number as well as any number greater than 10^500.

I wrote in July 2010:


Theorem: In old-math, geometry had well-defined finite-line versus
infinite-line but
Algebra or Number theory was ill-defined with its finite-number
versus
infinite-number
and that is why mathematics could never prove Twin Primes, Perfect
Numbers,
Goldbach C. , Fermat's Last Theorem, Riemann Hypothesis and
thousands
of
number theory conjectures. In this theorem, we show there never can
be
constructed a infinite-line in geometry since the other half of
mathematics, the old-math never well defined
infinite-number versus finite-number.
Proof: Since old math does not recognize infinite-numbers, that no
matter how many finite number of line segments we put together, they
still will never
summon into an infinite-line-ray. However, if a precision definition
is given in
mathematics for geometry or algebra saying that finite-number means
all numbers less than 10^500 and 10^500 and beyond are infinite-
numbers.
Well, with that definition we can build an infinite-line-ray in
geometry by
adding together 10^500 units of line-segments of finite line
segments
building
an infinite-line-ray. QED

I referred to another proof of mine in 2009 where I said that
betweeness axiom
no longer held. So I need to refurbish this betweenness proof to prove
that in
small scale that you also need a boundary between finite and infinite
number,
and here it is 10^-500

--- quoting old post in part ---
Newsgroups: sci.logic, sci.math, sci.physics
From: Archimedes Plutonium <plutonium.archime...(a)gmail.com>
Date: Thu, 18 Jun 2009 21:51:09 -0700 (PDT)
Local: Thurs, Jun 18 2009 11:51 pm
Subject: proving the Betweenness Axiom contradicts the Parallel Axiom

Betweenness Axiom:
If A and B are any two points, then (1) there is a point C such
that A-B-C, and (2) there is a point D such that A-D-B.

Parallel Axiom:
Given a line and a point not on the line there exists one and only
one line parallel from that point to the given line.

Now I need to prove that those two Euclidean Geometry axioms
are contradictory. I did this earlier in this book by setting up a
triangle that becomes smaller and smaller and which would thence
have two 90 degree angles.

But let me try it using the same scheme a bit differently.

here I have a given line with a parallel line from that point not on
the
given line called A:

--------------------------A----------------------------

--------------------------------------------------------

Now what I do is form a right triangle using A
and two points on the given line called B and C
like this:

--------------------------A-----------------------------

--------------------------B------------------------C---

Now here is the interesting feature of the Old
Euclidean geometry axioms in that they are
contradictory. As I do the infinite-downward
-regression of Betweenness the C point
approaches infinitely close to the B point
such as this picture

--------------------------A-----------------------------

--------------------------BC---------------------------

Since the axiom of Betweenness never ends
means that C becomes B and the triangle
is merely a line segment AB and no longer
a triangle of ABC and before it becomes
a mere line segment it becomes a triangle
with two 90 degree angles.

Now if I went the other way of B approaching
C as such:

--------------------------A-----------------------------

--------------------------------------------------BC---

In this direction what ends up is a line segment
AC and where the B becomes C.

In the first case I have a triangle which has two
90 degree angles, and in the second
case I have a triangle whose angle sum is equal to
zero since side AB and BC vanished into becoming AC.

--- end quoting old post in part ---

Some minor adjustments to the above, that if infinity is without end
and
no boundary between finite and infinite number, then what happens in
the Microworld or small scale is that you end up with either a
triangle
that has angle sum greater than 180 degrees and two right-angles, or
you have a very slender scalene triangle whose tiny side, like in
Calculus,
making the side as tiny as we want to make it, so that finally, we end
up with
a triangle that has two sides of a scalene triangle as parallel since
they never
meet.

SUMMARY: in mathematics when we do not admit that there must be a
boundary
between finite number versus infinite number such as 10^500 large
scale and 10^-500 small
scale, that we end up not able to build a infinite line ray in
geometry simply because
infinite-number is not available. And on the small scale, there also
needs a boundary
between finite number versus infinite number, or else we have a
triangle sum greater
than 180 degrees and a triangle with two parallel sides.


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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