Peano Naturals are the +AP-adics once you define "finite" properly #472 new book 2nd edition: New True Mathematics
plutonium....@gmail.com
MoeBlee wrote:
> On Apr 27, 4:14 pm, lwal...@lausd.net wrote:
> > On Apr 27, 3:39 pm, Virgil <Vir...@gmale.com> wrote:
>
> Every object except the "zero" object.
>
> > as AP puts it. Would that be sufficient to
> > conclude that M is in fact the _standard_ model of PA?
>
> The standard model of PA is <w 0 S + *> where
> 'w' stands for omega
> 'S' for the succesor operation on w
> '0' for the empty set
> '+' for the addition operation on w
> '*' for the multiplication operation on w
>
> For a model to be an isomorphism of the standard model of PA requires
> having
> <X e f> to start with
> where
> e in X
> and f is a 1-1 function from X onto X\{e}
> and the induction principle holds.
>
> Then extend to the two "addition" and "multiplication" operations by
> recursive definition.
>
> MoeBlee
Not accurate. If one model is self-contradictory, then it is hard to
try to
measure it with a different model for being isomorphic.
If I tried comparing the taste of a orange with a plum to see which is
better flavor, but used a rotten orange to compare with a perfect
plum,
it is easy to see that you cannot compare at all.
Why would I want the AP-adics to be isomorphic with a rotten set of
Peano Naturals.
If MoeBlee, if Virgil were to define finite from infinite, then they
should be
able to recognize that 9999....9999 was a Peano Natural Number. That
Peano Natural Numbers are actually 1 to 9999...9999 including numbers
like 5000....777777 and 88888....4444.
Because Peano nor MoeBlee, nor Virgil are able to see that they have
no definition of "finite versus infinite" and that their understanding
of the
Peano Axioms is a hidden assumption that finite is whatever they deem
it to be, that the Peano Natural Numbers are a flawed gaggled set.
Once you define finite from infinite, then the Peano Natural Numbers
are
clearly seen as 1 to 9999...9999 and so, the Peano Natural Numbers are
another name for the +AP-adics. The two are one and the same set.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
plutonium....@gmail.com
I see that Google has dispersed this thread of over 1,000 posts, not
started
by me. Now maybe it is dispersed because it exceeded a 1,000 posts, or
I am
hoping that it is dispersed because Google is fixing sci.math of its
spam-ads
of obnoxious clothing and gym shoes. So I am hopeful that the
dispersion
is due to Google fixing sci.math.
But anyway, Tim Little remarked that my earlier posting of the axioms
for
AP-Reals and AP-adics was a 1st Draft, and I concur it needs a whale
of
improvements. When you do new math, it is expectant that there is a
turmoil
of constant improvements.
And I am rather disappointed with the comments so far as petty--
ellipsis. And petty as far as Peano Naturals isomorphic with +AP-
adics.
I was hoping someone would have seen the great new avenue--
that instead of extending the Peano Naturals to obtain the Rationals
and thence extending the Rationals to obtain the Irrationals and
thence
combining these three to form the Old Reals.
I was hoping to see a comment by someone who realized that I had
take the Reals into a brand new level. Instead of starting with
Naturals,
I started with the entire set of Reals as AP-Reals. Of course I had a
weapon that the Old Reals never had-- FrontView with BackView
which allowed me to start with all the Reals as the starter-set.
So I have all the Reals in AP-Reals.
Now I expand them or extend them into the AP-Reals-Rationals
by simply doing what I would call a Divisor Matrix. I take every
AP-Reals and make All Possible Divisor Arrangements such as
0d000...0001/0d000...0002. Now I do not have to worry what that
number is broken down. I just simply say-- there it is, it exists and
is
somewhere halfway in between 0 and 0d000....0001
Now the beauty of this is that in the Old Reals, the Rationals were a
subset of the Old Reals. In AP-Reals, the AP-Reals-Rationals become
a larger, more expanded set.
And I was disappointed that noone remarked on it.
And the further beauty is the AP-Reals-Rationals-Irrationals. This
expansion
or extension makes them a connected, dense, Total Ordered, Archimedean
Field Algebra.
And I define Irrational as per Continued Fractions. Only I change the
Fraction
part into Successive Roots.
So the square-root of 2d000...000 is as we know 1d414.....
but written as a AP-Real-Rational-Irrational it is written as
1d414...sqrt1.189... sqrt1.090.....
I have not yet worked out the details but the point is that Successive
Roots
converges to a limit of 1.
So that is the ultimate meaning of Rational versus Irrational, in that
a Rational
is merely a division of two AP-Reals. An Irrational can only be
written as a
Continued-Decimal.
So I was disappointed that only comments of a petty nature were given
and
that noone recognized how a simple adjustment--- the starter set for
Rationals
then Irrationals. Instead of the starter set as the Naturals, why not
use all the
Reals as the starter set.
plutonium....@gmail.com
a
present day technology, but when a new technology enters the scene
such as electrical power or transistor or computer, then society zooms
along. For math, the new technology is FrontView with BackView which
is opening up new vistas, new horizons in mathematics.
Here is the 2nd draft which I hope answers some complaints of the 1st
draft. And I added the third Number System-- (-)AP-adics because
there are three geometries, each wanting their native number systems.
________
AP-Reals
________
Using Wikipedia skeleton of the Peano axioms we simply change
a few items and end up with a axiomatization of the AP-Reals:
The first four axioms describe the equality relation.
1. For every AP-Real x, x = x. That is, equality is reflexive.
2. For all AP-Reals x and y, if x = y, then y = x. That is,
equality is symmetric.
3. For all AP-Reals x, y and z, if x = y and y = z, then x = z.
That is, equality is transitive.
4. For all a and b, if a is a AP-Reals and a = b, then b is also a
AP-Real. That is, the AP-Reals are closed under equality.
The remaining axioms define the properties of the AP-Reals. The
constant 0d0000....0000 is assumed to be a AP-Real, and the
AP-Reals are assumed to be closed under a "successor"
function S.
5. 0d000...000 and 0d000....0001 are AP-Reals.
6. For every AP-Real n, S(n) is a AP-Real. Meaning that every
AP-Real is obtained through successive addition of 0d000...0001
(7.) There is an additional axiom to provide for the Negative
AP-Reals and in which a negative sign is provided and in which
a Coordinate System which provides for a negative AP-Real
situated at the same spot as a positive AP-Real
(perhaps the concept that a Point in Geometry has
a negative hemisphere to the positive hemisphere) and where
< and > inequality ordering is provided by a absolute-value-first.
This is the Euclidean Geometry Axiom for the AP-Reals as
native numbers in that a positive next to a negative creates
a flat-plane-Euclidean Geometry.
Definitions for AP-Reals
___________________
(8) Define finite versus infinite as that of 10^500 or more as
infinite
for the macro-scale and 10^-500 or less for the micro-scale. We
use the best numbers out of physics to where measurement is no
longer possible such as the Planck units.
(9) Defining AP-Reals-Rationals: Now set AP-Reals-Rationals is formed
by division of all members by all other members in a division-matrix.
An example is 0d000...0002/0d000....0003
And unlike Old-Reals the AP-Reals-Rationals forms a larger set than
the AP-Reals by themselves.
(10) Defining AP-Reals-Rationals-Irrationals: using the same schemata
of Continued-Fractions we have something called Continued Decimals
so that the square-root of 2 is a succession of square roots:
1d414...sqrt1.189... sqrt1.090..... and it has to be worked out where
addition signs placed so that we have in the end 1 + 1 = 2. For
sqrt3 we have in the end 1 + 1 + 1 = 3 much like a Continued Fraction
setup.
(11) The AP-Reals by themselves are a Archimedean Well-Ordered Field
but when we reach the AP-Reals-Rationals-Irrationals they become a
Archimedean, Total Ordered, dense, connected Algebraic Field.
_________
+AP-adics, note these are the Positive AP-adics
_________
The first four axioms describe the equality relation.
1. For every +AP-adic x, x = x. That is, equality is reflexive.
2. For all +AP-adic x and y, if x = y, then y = x. That is, equality
is symmetric.
3. For all +AP-adic x, y and z, if x = y and y = z, then x = z. That
is, equality is transitive.
4. For all a and b, if a is a +AP-adic and a = b, then b is also a
+AP-adic. That is, the +AP-adics are closed under equality.
The remaining axioms define the properties of the +AP-adics.
The constant 0000....000 is imaginary since it is not positive
and assumed to be a +AP-adic, and the +AP-adics are
assumed to be closed under a "successor" function S using
0000....00001.
5. 000...000 and 000....0001 are +AP-adics.
6. For every +AP-adic n, S(n) is a +AP-adic. Meaning that every
+AP-adic is obtained through successive addition of 000...0001
(7.) There is an additional axiom that gives a Coordinate System for
the +AP-adics of its positive numbers only. The Coordinate System is
Elliptic geometry and the +AP-adics as the native numbers thereof.
Definitions for +AP-adics
___________________
(8) Define finite versus infinite as that of 10^500 or more as
infinite
for the macro-scale and 10^-500 or less for the micro-scale. We
use the best numbers out of physics to where measurement is no
longer possible such as the Planck units.
(9) Defining +AP-adics-Rationals: Now set AP-adics-Rationals is
formed
by division of all members by all other members in a division-matrix.
An example is 2000....0001/9999....9988
(10) Defining +AP-adics-Rationals-Irrationals: using the same schemata
of Continued-Fractions we have something called Continued Decimals
so that the square-root of 2H000...0000 which is the North Pole
after one complete rotation around a sphere is a succession of square
roots:
1H414...sqrt1H189... sqrt1H090..... and it has to be worked out where
addition signs placed so that we have in the end 1H + 1H = 2H. For
sqrt3H we have in the end 1H + 1H + 1H = 3H much like a Continued
Fraction
setup.
(11) The +AP-adics by themselves are a Archimedean Well-Ordered Field
but when we reach the +AP-adics-Rationals-Irrationals they become a
Archimedean, Total Ordered, dense, connected Algebraic Field.
_________
(-)AP-adics, note these are the Negative AP-adics
_________
The first four axioms describe the equality relation.
1. For every (-)AP-adic x, x = x. That is, equality is reflexive.
2. For all (-)AP-adic x and y, if x = y, then y = x. That is, equality
is symmetric.
3. For all (-)AP-adic x, y and z, if x = y and y = z, then x = z. That
is, equality is transitive.
4. For all a and b, if a is a (-)AP-adic and a = b, then b is also a
(-)AP-adic. That is, the (-)AP-adics are closed under equality.
The remaining axioms define the properties of the (-)AP-adics.
The constant 0000....000 is imaginary since it is not a negative
and assumed to be a (-)AP-adic, and the (-)AP-adics are
assumed to be closed under a "successor" function S using
-0000....00001.
5. 000...000 and -000....0001 are (-)AP-adics.
6. For every (-)AP-adic n, S(n) is a (-)AP-adic. Meaning that every
(-)AP-adic is obtained through successive addition of -000...0001
(7.) There is an additional axiom that gives a Coordinate System for
the (-)AP-adics of its negative numbers only. The Coordinate System is
Hyperbolic geometry and the (-)AP-adics as the native numbers thereof.
Definitions for (-)AP-adics
___________________
(8) It must be defined and noted that the carrying out of any
operation
in (-)AP-adics, the final number must be a negative signed number.
For example -8 x -6 = -48, or, for example the square-root of (-1)
is equal to -1. So the "i" number is no longer imaginary in (-)AP-
adics
but right at home.
(9) Define finite versus infinite as that of (-)10^500 or more as
infinite
for the macro-scale and (-)10^-500 or less for the micro-scale. We
use the best numbers out of physics to where measurement is no
longer possible such as the Planck units.
(10) Defining (-)AP-adics-Rationals: Now set (-)AP-adics-Rationals is
formed
by division of all members by all other members in a division-matrix.
An example is (-)2000....0001/9999....9988
(10) Defining (-)AP-adics-Rationals-Irrationals: using the same
schemata
of Continued-Fractions we have something called Continued Decimals
so that the square-root of (-)2H000...0000 which is the North Pole
after one complete rotation around a sphere is a succession of square
roots:
(-)1H414...sqrt(-)1H189... sqrt(-)1H090..... and it has to be worked
out where
addition signs placed so that we have in the end -1H + -1H = -2H. For
sqrt3H we have in the end -1H + -1H + -1H = -3H much like a Continued
Fraction
setup.
(11) The (-)AP-adics by themselves are a Archimedean Well-Ordered
Field
but when we reach the (-)AP-adics-Rationals-Irrationals they become a
Archimedean, Total Ordered, dense, connected Algebraic Field.