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From: Archimedes Plutonium on 3 Jul 2010 02:55
Let me get back to the more important math, rather than chasing down
mistakes of logic
in Euclid's IP proof.

So we have the geometry side of mathematics with a precision
definition of finite-line
versus infinite-lines. Those definitions involve the fact that a
finite-line is a line segment
with two endpoints. The infinite-line involves two types called a
infinite-line-ray which has
a endpoint and a arrow to infinity at the other end. The infinite-line
has no endpoint, but
only two arrows.

So can we actually use the Geometry definition, since it is a
precision definitions, and use
them to formulate what the finite-number versus infinite-number
definitions should be? There
is nothing to say that numbers must be similar to lines. But there
must be some sort of
consistency or coherence between the definitions.

In a prior post I remarked that the AP-adics and the Hensel p-adics
could not be the infinite-number definition.

So what I must do is analyze how the Reals behave once we inject the
notion that
the boundary between finite number and infinite number is 10^500 due
to Physics.

Now is there any hint from geometry alone that there is a boundary
between finite-line and
infinite-line? Not that I can discern. Perhaps the idea that all
finite-lines have two endpoints, and these two endpoints somehow force
a boundary between finite lines and infinite lines.
But that does not look to be true. Then, the only other thing I can
inspect is the idea that
Geometry can never build a infinite-line from finite-lines unless
there exists an infinite-number.
That was the theorem I proved a few posts back.

So here we have a glimpse that we cannot use geometry line definitions
as a template
to defining finite number versus infinite-number since the infinite-
number is essential to
geometry, and geometry dependent on having infinite-numbers.

So what do the Reals and the Cartesian coordinate system look like
once a axiom
is injected into math, into the Peano axioms stating that there is a
boundary between
finite-number versus infinite-number and it is exactly 10^500.

Well, the Reals will no longer have continuity as a concept because
anything smaller
than 10^-500 leaves holes. And the graphs no longer need anything
beyond 10^500.

So starting at zero as a endpoint and then placing an arrow to this
line at 10^500
is that an infinite-line-ray? And having an arrow at (-)10^500 and
another arrow at
(+)10^500 is that an infinite-line?

There are vexing questions such as whether a infinite line ray is
formed by having
0 as end point and employing all the micronumbers between 0 and 1 such
as
10^-500, so do we have micronumbers as infinite? In modern math we say
call this
the limit for calculus but we do not call it a micro infinity. I am of
the opinion that
infinity means merely beyond the ability to do physics measurements so
that a
micro infinity makes just as much sense as a large scale infinity.

Like anything new, there are vexing new issues and questions. But look
at the benefit
of this new program. We no longer have unsolved problems anywhere in
mathematics,
nor do we ever have to search for a new proving techniques, because
there is now one
standard way of proving all Algebra or Number theory problems, run
through 10^500 of
those numbers.

And, unless I am mistaken, this new program vastly improves Calculus,
since in calculus,
the experts of that field spent most of their time on issues of
continuity. With this program,
continuity was never a mathematical concept, just as fire breathing
dragons are not a part
of biology.


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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