Prev: history of Series and why Peano did not use Series for Successor #325 ; Correcting Math
Next: the characteristics and features of Infinity #331 Correcting Math
From: Archimedes Plutonium on 23 Jan 2010 02:08

Up until the last several posts, I thought the Successor Axiom was
going to survive the
changes, but it looks as though it needs refurbishing also. It looks
as though Peano presumed the Natural Numbers as Counting Numbers
within the Successor function, whereas the Series
was the better means of defining succession. It is the Series that
makes no distinction as to
finite and infinite numbers and a Successor Function prejudices or
biases what the number
is. Hensel shows us that with the p-adic axiomatics using a Series
that the numbers are both
finite numbers and infinite numbers and that is how the Natural
Numbers should be also.

So here is a quick summary of the changes needed to make the Peano
Axioms a consistent
set.

(1) The creation of two numbers of both 0 and 1 and not just a solo 0.
This is needed since we
need a metric distance of 1 for the Series (successor axiom).

(2) Since the p-adics shows us that going to infinity is like that of
circling around a globe. Another way to think of this is that in
Physics, the Atom Totality does not go to infinity but
is in a elliptic geometry shape and that means if we go far enough we
end up coming to the
starting point. So that axiom of Peano that says 0 has no predecessor
is wrong. For that
the largest possible Natural Number of 9999....9999 when adding 1
becomes 0 again.
So that the successor of 9999....9999 is 0.

(3) Again, the Successor function needs to be replaced with Series so
that
1 + 1 + . . . . + 1 represents all the Natural Numbers beyond 0 of
finite stopping and that
this Series also included infinite-numbers.

(4) Define Finite-number as equal to or less than 10^500, anything
beyond is an infinite-number. Caveat: we can include a Incognitum as a
region between finite and infinite and
we can well-define the Incognitum as a territory in which Algebra is
still obeyed of its
operations but beyond the Incognitum, algebra is untrustworthy. We use
a 100-Model
where multiplication ceases at 11x9 since 12x9 is larger than the
largest number of 99.
So the Incognitum is about 10% of all the numbers of all-possible-
digit-arrangements
where 9999.....99999 is the last and largest Natural Number.

(5) Mathematical Induction is superfluous and can be derived as a
theorem from the
other axioms.

Now I need to talk about what the above does to my earlier book of AP-
adics where
I was trying to synthesis this geometry theory:

Eucl geom. = Elliptic geom unioned to Hyperbolic geom.

and where I said that was equivalent to the Number systems
of Reals, AP-adics, and Doubly Infinites.

However, I can see alot of changes to that theory, not the geometry
aspect
but to the Number Systems aspect. Because I have turned into a
*finitist*
and the Doubly Infinites or Reals or AP-adics no longer have much
meaning
since mathematics is confined to only 10^500 for meaning.

Now this helps that other book of mine because there is no longer any
need
for me to wrestle with Algebra because Algebra decays after it
reaches
10^500. It is meaningless to consider what is 5000....0000 x
6666....6666.

So Algebra is no longer a problem. But there is a further help in the
fact that
geometry is also confined to the finite region.

So in the equation above of Eucl = Ellipt unioned Hyperbolic we draw
a
small triangle on a sphere surface (which is elliptic triangle since
the sides
are concave outward) and on the underside of the sphere we draw the
corresponding hyperbolic triangle whose sides are concave inward and
so if we were to place
these two triangles together (unioned) they cancel their concavity and
yield a
Euclidean triangle. So this facet is made easier with the finite
restriction.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
 | 
Pages: 1
Prev: history of Series and why Peano did not use Series for Successor #325 ; Correcting Math
Next: the characteristics and features of Infinity #331 Correcting Math