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From: Archimedes Plutonium on 8 Jan 2010 05:29
One thing about the coldness of winter is that I can get alot of
science posts sent off.
And yesterday was the first day in my life that I experienced
frostbite. It happened real
quickly with a blizzard wind. It happened on the cheeks of my face,
around the nose
and jaw. It feels like a sunburn with reddened skin. Looks like I need
to buy a face-mask
for just those occasions.

But anyway, the definition of ellipsis that I am veering towards is
that of two-dots signifies
a finite string of digits where finite is defined as 10^500 or below.
And for the three-dot
ellipsis signifies infinite string of digits and where infinite is
defined as far beyond 10^500.

Now using the ellipsis let us discuss the definition of Finite that
was used throughout the
history of mathematics. That definition was never given or stated in
Peano axioms and a
pity shame it was not stated because it made mathematics inconsistent.
That assumed definition of finite was that a number was finite if it
ends to infinity in a string of zeroes
leftwards. An example instantly illustrates this definition. The
number 765 is finite because
it is 0000...000765. So here we use the ellipsis to indicate an
infinity of zeroes and all the digits to the left of "7" are zeroes.

And the three-dot ellipsis can be used several times in a number for
example, 12...233...344
indicates an infinity of 2s and an infinity of 3s for that infinite
number.

And the definition of ellipsis includes the idea that we have an
ellipsis for a shorthand. We
obviously cannot write out a infinite number of all its digits so we
use a shorthand of
three-dots. And for finite numbers that are long we also use this
shorthand only we use
two-dots.

Now no mathematician ever gave a well-defined concept of Finite until
Physics defines
it as the largest number useable by Physics-- the Planck Units.

Now let me examine why the Peano Axioms of the Natural Numbers are
flawed and inconsistent and contradictory.

Peano never defined Finite but assumed it meant that the number ended
in a leftward string
of all zeroes, as noted above for 765 example.

Peano and later mathematicians all assumed the Natural Numbers as a
set was an infinite set
and they assumed it was infinite because you can endlessly add 1. So
the endless process planted into the minds of mathematicians that the
Natural Numbers formed an infinite set.
But, also, Peano assumed and all later mathematicians train-locked-
assumed with him that
every Natural Number of the Peano Axioms was a finite number. Even
though Peano never
defined Finite.

So, here is the contradiction, the inconsistency of the Peano Axioms.
If every Natural Number
was a Finite number then the set of all these Finite numbers cannot be
an infinite set. So Peano and all the followers focused only on the
idea that since the adding of 1 is endless that
the total set of all the Natural Numbers would be endless and thus
infinite. However, keep in mind, there was the other assumption that
each and every Natural Number had to be a Finite
Number.

So to Peano and his followers this set {0, 1, 2, 3, 4, 5, ...} was an
infinite set and where every
number in that set had to be a finite number. Can you see the problem,
can you see the contradiction and can you see the inconsistency?
Probably not. The contradiction is simply the fact that if every
element or member of a set is a Finite member, then the overall set
can
never be infinite. In order for a set to be infinite, it must contain
at least one member or element that is itself infinite.

Now most people would still not grasp that idea. So I will explain it
in another way.

We can add piecewise two elements of a set of numbers to form a more
compact set.
Using the Natural Numbers {0, 1, 2, 3, 4, 5, ...} I form the newer
compact set by adding
together adjacent numbers {0+1=1, 2+3=5, 4+5=9, ...} So is this newer
compact set still
infinite? According to Peano and his followers it is still infinite.
So I continue to compactify
the prior set {6, 14, ...} and according to Peano and followers that
set is also infinite. But
I compactify infinitely many times and what do I end up with? Well,
since each and every
Peano Natural Number is a finite number, what I end up with is a set
that has one number
and that number is a finite number.

So Peano Axioms destroys the set theory arithmetic of the union of
elements.

So how can one correct the Peano axioms so they are consistent? It is
easy. It is to
define Finite with a well-defined definition such as saying all
numbers less than 10^500
are finite and beyond is incognitum and far beyond is infinite. There
you have a consistent
Peano axioms. And the 10^500 comes out of Physics of the largest
number in physics
that can still carry on experimentation. And I have always thought of
mathematics as
a paper and pencil experiment. None of us can count to 10^500 and
still be alive.

And the most funny part of this story, is how long it took for anyone
to notice the huge
flaw of the Peano Axioms. Those axioms are about 150 years old, and
one would think
that no inconsistent and contradictory set of axioms could pass
undetected for 150 years.
One would think that as fast as I could get frostbitten in 1/2 hour
time that the Peano
axioms as flawed would have been realized by some astute student
learning them for the
first time. But I guess noone put these two together (i) every Peano
Natural was finite
(ii) finite was not well-defined.

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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