#95 Chapter 5, true Calculus weighs in on FrontView-BackView; new book 2nd edition: New True Mathematics
plutonium....@gmail.com
breakdown
with numbers like 1/3. But I am able to build the Calculus as which is
vastly more
clear than the old-calculus. The Old Calculus could not even explain
to anyone
how the derivative is inverse to integral. The New Calculus easily
shows how riding
up one-- the derivative creates the integral or how riding up the
integral creates the
derivative.
But the reason I want to talk about the New Calculus tonight is that
it may also
support the concept of FrontView and BackView. In the New Calculus we
can
have the smallest interval in which we want to differentiate and
integrate. In the
Old Calculus, the intervals chosen to find the derivative or perform
integration
were never as detailed as what the New Calculus can do.
So let me focus on the closed interval of 0 to 1, [0,1] on New Reals
and let my use the Identity function y = x so it delivers a right-
triangle
in the first quadrant as such:
|
|
| /
| /
|/____________
Now in New Reals the number closest to 0 is 0.000...00001 and the
next number is 0.000...0002 but remember that between these numbers
is a Real-fragment of 0.000...0000, then 0.000...000L, then
0.000...0001,
then 0.000...001L, then 0.000...0002, then 0.000...00002L
So we take the derivative of the function y = x at the point
0.000...0001
and we take the integral of that function over the interval 0 to
0.000....00001
Geometrically and pictorially what we have for the derivative is this
/ |
/__|
where the base of that right-triangle is 0.000...0001 metric distance
and the altitude of that right-triangle is 0.000...0001
So what is the derivative? Well of course 0.000...0001/0.000..0001 is
equal to 1
The derivative of y = x is 1
Now for the integral we know the area of a right-triangle is 1/2 b*a
and in this case we have 1/2x^2
which matches the antiderivative for y = x
So the integral is 1/2(0.000...0001 x 0.000...0001) = 0.000...0001L
In other words, all of Calculus is the derivative or integral of these
picket-fence
structures. Let us examine the picket fence at the point 0.999...99998
Pictorially it looks like this:
/|
/ |
---
| |
| |
| |
| |
__
It actually looks like a picket fence whereas in the first picture of
the point 0.000...0001 it was all a right-triangle. Here at the point
0.999...998
we have a slender rectangle and sitting atop the rectangle is the tiny
right-triangle.
Now remember I said that wedged between every two consecutive Reals
such as 0.999...9997, then 0.999...9998, then 0.9999...9999, then
1.000....0000 wedged between every one of those Reals is a Real-
fragment
with a second-decimal point designated by L so there is a
0.999...9997L , then 0.999...9998, then 0.999...99998L
So the Picket fence right-triangle has the hypotenuse that runs from
0.999...9997L to 0.999...99998L where the Real 0.999...9998 is between
the two Real-fragments.
Now the base of the rectangle is of course 0.000...0001 metric
distance
and the height of the rectangle is 0.999...99997L so the area of the
rectangle is 0.999...9997L x 0.000...0001 = 0.999...9997L
and the area of the right-triangle that sits atop the rectangle is
1/2(0.000...0001 x 0.000...0001) = 0.000...0001L
The derivative is again 0.000...0001/0.000...0001 = 1
and the integral is 0.999...9997L + 0.000...0001L = 0.999...9998L
You see Gilbert Strang in his 1991 textbook CALCULUS could never
do derivatives or integrals on Consecutive Reals, for he never knew
that Reals were consecutive.
So, if Calculus can do the integral and derivative to a Real point
in the graph of a function, then does that demand and necessitate
a BackView on Reals as well as a FrontView?
Well of course, said the reasoned man. We cannot perform a
integral or derivative of a point such as 0.999..... unless
we understand that 0.999.... is actually 0.999....9999
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
Sorry about that, that is a gross mistake
0.9 x 0.1 = 0.09
0.09 x 0.01 = 0.0009
so then 0.999...9997L x 0.000...0001 = 0.000...000L
> and the area of the right-triangle that sits atop the rectangle is
> 1/2(0.000...0001 x 0.000...0001) = 0.000...0001L
>
> The derivative is again 0.000...0001/0.000...0001 = 1
> and the integral is 0.999...9997L + 0.000...0001L = 0.999...9998L
>
So the integral is 0.000...000L + 0.000...0001L = 0.000...0001L
And that is what is expected of 999...9999 picket fences
of about 0.000...00001L summed would be 1/2 area
plutonium....@gmail.com
this book where
I do Algebra on the AP-adics which is Elliptic geometry and the
surface of a sphere
I thought I would respond to the below post of Dik Winter and tell him
why the
sphere cannot be a Algebraic Field.
--- old post of 2007 ---
Date: Thu, 25 Oct 2007 11:35:00 -0700
Subject: #204 degrees on circle, does it form a Ring and Field?; new
textbook: "Mathematical-Physics
Dik T. Winter wrote:
> In article <1193291445.348437.105...@v23g2000prn.googlegroups.com> a_plutonium <a_pluton...@hotmail.com> writes:
> > Dik T. Winter wrote:
> ...
> > > > (pi) X (pi) = (pi)^2 or if not satisfied
> > > But I assume that (pi) is an adic. If not, your adic arithmetic is
> > > not complete, while earlier you asked me to tell you what it was
> > > because you said you had "defined arithmetic" on the adics. Sorry,
> > > with that kind of arithmetic the adics do not even form a group with
> > > respect to addition.
> You do not answer to this?
> > > > It is 180 degrees radians X 180 degrees radians
> > > Makes no sense at all. What are "degrees radians"?
The AP-adics numbers are each equal to a degree measure. All of the
AP-
adics
that have their point-at-infinity as a 0 digit are 9percent or less of
180degrees
So that a number such as this 09999....999999 is 9 percent of 180
degrees.
Those with point at infinity of "1" digit such as .....1111111 is
between 19 percent
to 10 percent of 180 degrees. So, depending on what digit is in the
point-at-infinity
spot determines the angle and thus tells us immediately where the
number is
located on the Sphere.
> Neither do you answer this.
> > Question: you have a circle and for each point you give it an angle
> > and 0 is North Pole and
> > a complete circuit is 360 degrees.
> > One example of multiplication is 180 degrees X 180 degrees = 360
> > degrees.
> I would say that it is 180 degrees squared.
> > and, where 180 degrees + 180 degrees = 360 degrees
> > Do those degrees comprise a algebraic Ring or Field?
> But even when you correct it, you have not yet completely defined either
> multiplication nor addition. So, I cannot tell.
No, I am saying that given a sphere and then a line of longitude. Then
the North Pole
is both 0 and 2(pi) and the South Pole is (pi) and the numbers 1 to
999....9999 lie
on this longitude stretching from North Pole to South Pole and the
Antipode Semisphere
is imaginary of (pi), (pi) +1, (pi) +2,......(pi) + ..9999, 2(pi)
So, now every point on that line of longitude is also an angle,
starting at the North Pole
we are standing on imaginary 0 and one step forward is ....000001,
next ....000002, until
we reach 99999.....99999 which is one unit short of the South Pole.
Add 1 to 9999.....9999
and we are standing on the South Pole which is imaginary and is (pi)
and the next step
forward is (pi) +1 and so on.
So, give me that multiplication of say (pi) X (pi) is asking for 180 X
180 which is 32400 degrees
which is 90 degrees which is 50000......00000000 (unless it is the 90
degrees in the antipodal
hemisphere and then it would be (pi) + 5000....00000
Now you may ask why have 9999.....99999 one unit short of the South
Pole, and the answer is
that by doing so, I can have Hyperbolic Geometry included within the
very same Sphere, so that
this one sphere has included Elliptic Geometry and Hyperbolic Geometry
which is a very
valuable asset for it would be like Euclidean geometry with both
negative and positive Reals all
included in Descartes coordinate system.
> > Angles seem far easier to work with than arclength.
> And that was *precisely* what *I* did when I defined arithmetic on those
> points (when you measure angles in radians, arclength and angle are the
> same). Again, a point on the unit circle can be written in cartesian
> coordinates as (cos(x), sin(x)) for any x. Addition and multiplication
> you do on the x's, and you do both mod 2.pi.
> But apparently you do not even understand such simple math. It has
> been a long time since you were mathematical teacher in Australia.
I may not understand your abbreviated notation.
> And with my definition you *get* a ring, but the ring is quite strange.
> A true ring you get if you write each point as (cos(x.2pi), sin(x.2pi)),
> where 0 <= x < 1. Now again define the arithmetic as if you are working
> with the x's, but now you work mod 1. This way you get a field.
--- end old post of 2007 ---
The way I have the AP-adics set up to be the numbers that are native
and
intrinsic to Elliptic Geometry and thus the sphere surface. Is that
the AP-adics
must be a Algebraic Field because the Reals are an Algebraic Field and
the
only difference between Reals and AP-adics is that one is leftward
string
whereas the other is rightward string. So they are symmetrical, which
implies
that if Reals are an Algebraic Field the AP-adics must follow suit and
be also
an Algebraic Field.
Now the reason that Dik Winter's above will not work is because he
uses the entire
sphere surface. I use only a hemisphere and use the other hemisphere
only to catch
addition and subtraction spillovers, much like the sine or cosine
functions repeat.
So the full sphere cannot be a Algebraic Field for it is ruined by
associativity where
you cannot tell in which direction of the sphere to go. But when you
restrict the sphere
to a hemisphere then all the Algebraic Field properties work just
fine. So what
the AP-adics teaches us is that Elliptic Geometry and Hyperbolic
Geometry are
1/2 spheres or 1/2 of trumpets.
plutonium....@gmail.com
feel as though
I am not writing a rough-edition. This is the second edition but so
many bugs and
changes. That maybe by the 4th edition of this textbook to be smooth.
The chapter that probably most are looking forward to is this chapter
since so
many mathematicians of the past century were what one can call Algebra
fanatics.
That they sacrificed the more important subjects of mathematics such
as
geometry for that of algebra.
It is Geometry and Numbers which dominates mathematics and algebra is
mostly
a helper to Numbers, a cleaner-up of what the Numbers are.
It strikes me also as I write this chapter that the most important
operation is
probably multiplication rather than addition. For in Physics, energy,
momentum
and the dualities are in multiplication. And even in Calculus where
the antiderivative
of the identity function y = x that the antiderivative is 1/2 x^2 and
in physics the
kinetic energy is 1/2 mv^2.
In the 1st edition I was dwelling all on AP-adics, but in this edition
I had to change
even the Reals and call them the New Reals, so the algebra on New
Reals is
different so I need to include the New Reals with brand new
definitions of their
operations.
Already I made several mistakes in the below post yesterday with
multiplication.
I have no excuse for my mistakes other than to say that "habits die
hard". Recently
I was discussing that 0.000...0001 x 0.000...00001 was equal to
0.000...0001 since
it had exhausted all the place values of 10^(-)999...9998 place-value
and since I had
not yet discovered or learned of a Real-Fragment with second decimal
point so
that 0.000...0001 x 0.000...0001 = 0.0000...0000L
So I made many mistakes yesterday, and trying to make my old habit of
just
a few months ago die away.
Let me correct that since it is a derivative and integral of the point
0.000...0001
The neighbors of the point 0.000...0001 are 0.000...000L to the left
of 0.000...001
and 0.000...0001L to the right of 0.000...0001 and it is these two
Real Fragments
that the hypotenuse endpoints will be and where 0.000...0001 will be
embedded
in that hypotenuse line segment.
> >
> > So what is the derivative? Well of course 0.000...0001/0.000..0001 is
> > equal to 1
> >
That is still correct. Where the derivative is the dy/dx of the picket-
fence triangle
> > The derivative of y = x is 1
> >
> > Now for the integral we know the area of a right-triangle is 1/2 b*a
> > and in this case we have 1/2x^2
> >
> > which matches the antiderivative for y = x
> >
> > So the integral is 1/2(0.000...0001 x 0.000...0001) = 0.000...0001L
> >
No, it appears to me that the multiplication above should be
the smaller Real Fragment of 0.000...0000L
The above is multiplication on Reals, where it is a sort of Sequence
Convergence.
In the 1st edition I called it a Cauchy Sequence Convergence, but in
this edition
I am just going to call it a Sequence Convergence. Now I only showed
two
elements of the sequence above where I could have shown more:
What is 0.999...9997L x 0.000...0001 ??
0.9 x 0.1 = 0.09
0.09 x 0.01 = 0.0009
0.009 x 0.001 = 0.000009
0.0009 x 0.0001 = 0.00000009
.
.
.
etc etc
So the multiplication in Reals of an infinite string by another
infinite string
is a Sequence Convergence and what is 0.999...9997L x 0.000...0001 ??
It is according to that sequence convergence 0.000...000L. In other
words
it is not even the amount of 0.000...0001 but a Real Fragment of that
amount
and which that Fragment is greater than 0.000...0000 but less than
0.000...0001
Now let us flipp over to the AP-adics and define multiplication there
which
is very similar to the Reals between 0 and 1 and using the same sort
of
example except that 000...001 would be uninteresting compared to say
1111...1111
So what is 9999...9997 x 11111...1111 ??
Sequence Convergence has it:
97 x 11 = 1067
997 x 111 = 110667
9997 x 1111 =
99997 x 11111 =
.
.
etc etc
Unfortunately my calculator gives out very quickly but have enough
to know the answer is 1111....667 So in AP-adics
9999....99997 x 1111....1111 = 1111.....667
Now in Reals 0.999...9997 x 0.1111....11111 would equal
0.1111....6667 which is the mirror reflection symmetry of the
AP-adics multiplication where we can flipp over the Reals to
be the AP-adics and vice versa.
However, there is a difference because in Reals there are Real
Fragments
of second decimal point denoted by L between consecutive-Reals but
not in AP-adics where there are fractional-AP-adics denoted by the
radix
point. This is apparent in Reals as such
0.0000...00001 x 0.0000...00001 = 0.0000...0000L
and not apparent in AP-adics since the multiplication of AP-adic
Integers never leaves a fractional answer. Division in AP-adic
Integers can leave a fractional answer and that is taken care of
with the radix R point
>
>
>
> > and the area of the right-triangle that sits atop the rectangle is
> > 1/2(0.000...0001 x 0.000...0001) = 0.000...0001L
> >
> > The derivative is again 0.000...0001/0.000...0001 = 1
> > and the integral is 0.999...9997L + 0.000...0001L = 0.999...9998L
> >
>
> So the integral is 0.000...000L + 0.000...0001L = 0.000...0001L
>
> And that is what is expected of 999...9999 picket fences
> of about 0.000...00001L summed would be 1/2 area
>
>
> > You see Gilbert Strang in his 1991 textbook CALCULUS could never
> > do derivatives or integrals on Consecutive Reals, for he never knew
> > that Reals were consecutive.
> >
> > So, if Calculus can do the integral and derivative to a Real point
> > in the graph of a function, then does that demand and necessitate
> > a BackView on Reals as well as a FrontView?
> >
> > Well of course, said the reasoned man. We cannot perform a
> > integral or derivative of a point such as 0.999..... unless
> > we understand that 0.999.... is actually 0.999....9999
> >
Now I am defining the operations on Reals and AP-adics so that
their algebra, in the end are completed-Fields.
plutonium....@gmail.com
the
original with a (sic)
plutonium.archime...@gmail.com wrote:
(snipped)
>
> What is 0.999...9997L x 0.000...0001 ??
>
> 0.9 x 0.1 = 0.09
> 0.09 x 0.01 = 0.0009
> 0.009 x 0.001 = 0.000009
> 0.0009 x 0.0001 = 0.00000009
> .
> .
> .
> etc etc
I should have set that up better for it should have looked like this:
0.7 x 0.1 = 0.07
0.97 x 0.01 = 0.0009
0.997 x 0.001 = 0.000009
0.9997 x 0.0001 = 0.00000009
.
.
etc etc
Multiplication in AP-adics is clearcut and nothing troublesome where
the large integers after multiplying become smaller and we can see it
as in percentage multiplication that if you take say 98% x 99% your
answer is smaller. And the AP-adics can be considered a percentage
of the distance from North Pole on a sphere to South Pole so that the
number 555...5555 is 55% of the distance from NorthPole to SouthPole
on a longitude or the number 1000....0000 is 10% the distance.
Multiplication on New Reals has a new item in that there is a second
decimal
point and called a Real Fragment such as 1/3 is 0.3333....3333L
meaning
this is a Real Fragment and lies between the two consecutive Reals of
0.333...333 and 0.3333....33334 So that if we have this
multiplication
0.0000....00002 x 0.0000....0000999 the answer is going to be
2 x 10^(-)9999...9998 x 999 x 10^(-)999...9998 and since that is the
last
place value possible means the answer is 0.0000...0000L which is a
Real Fragment that lies between 0 and 0.000...00001
Now I must say something about multiplication in Positive AP-adics
versus
Negative AP-adics. The positive is clearcut normal but in the Negative
AP-adics, the answer of any operation comes out as a negative number.
So that all the numbers in Negative AP-adics which are the natural
intrinsic
numbers of Hyperbolic geometry, no matter what operation the answer is
a negative number.
So I think I covered multiplication, and what I can do if enough time
is show
a myriad examples that would be instructive.
The key mechanism in the operations is what I called a Sequence
Convergence
that applies to both the New Reals and AP-adics. And we have to get
used to
the idea of two decimal points in New Reals and in AP-adics. For the
New Reals
the normal well known decimal point and the L which signifies a Real
Fragment
lying between every two consecutive Reals. In AP-adics we have the
normal Radix
point that Hensel's p-adics had, and in additon we have a Hemispheric
decimal point
and I forgotten what I called it before so let me call it just H and
to be used only when
the AP-adic answer is in the other hemisphere such as 999...9999 +
8888...8888
is H88888....8887 signifying that the answer is 888...8887 in the
second hemisphere.
Likewise for addition in Negative AP-adics for example
(-)99999....9999 + (-)77777....77777 = (-)H7777....7776 which is (-)
777...7776
in the second hemisphere of a hyperbolic figure. In Positive AP-adics
it is
a Elliptic geometry figure.
Addition in Reals is the same as old Reals keeping in mind that 10^(-)
999...9998
is the last place value. Here there is no trouble with addition on New
Reals.
Now as to the Algebra of the above. I suspect both Reals and AP-adics
are
Completed, Well-ordered, Closed Fields. As my earlier post indicated
that
in Old Reals, it is impossible to make the points on a sphere as a
Algebraic
Field, but how I escaped that impossibility is by using only a
hemisphere of
a sphere surface, with the use of the second hemisphere as only a
addition
or subtraction spillover, much like sine and cosine periodic functions
repeat
once beyond 360 degrees, well, in fact they repeat once beyond 180
degrees
so I am following the periodicity of the trigonometric functions by
using only
one hemisphere. And this makes sense in that we have two geometries of
Elliptic and Hyperbolic conjoined or unioned to make Euclidean, so if
one hemisphere
is considered Elliptic then the other is considered Hyperbolic and
vice versa.
That leaves only subtraction and division and then if I feel energetic
to do
exponentiation and perhaps factorial.
In the chapter on Algebra I should go down a checklist of things like
commutative, associative, closed under addition, multiplicative
identity, the
usual list. But in that chapter I should also draw attention to the
idea that
rational versus irrational or that prime versus composite or
transcendental
versus algebraic are concepts that some of which no longer exist in
New
Reals or in any Number system of mathematics.
David R Tribble
> So I built the Reals anew, and shown where the Old-Reals are fake and
> breakdown with numbers like 1/3.
So you're saying that
1/3 + 1/3 + 1/3 = 1
doesn't work any more?
plutonium....@gmail.com
That is still true. The only complaint with 1/3 or 2/3 is when you
represent it as a decimal. It is not 0.333....
What I am saying is that 1/3 is not the same as 0.333..... or which is
0.333...3333 because neither accounts for that 1 remainder carryover.
On the other hand 1/2 is truly equal to 0.5000.....
In New-Reals, 1/3 is the same as 0.333....3333L called a Real Fragment
with its second decimal-point L which indicates there is "more to the
number"
and which is located between 0.333....3333 and 0.3333...33334
And where 0.333...333L x 3 = 1
or where 0.333...333L + 0.3333....3333L + 0.333...333L = 1
The Old Reals simply ignored precision that when you divide 1 by 3,
there is
a remainder 1 carryover at the point of infinity, the 10^(-)999...9998
place-value
and so 1/3 cannot be 0.333... for that does not account for the
remainder
carryover.
In the Old Reals they gave up on precision where in one instance
numbers
that evenly divide such as 1/4 or 1/2 had a true decimal
representative, but
numbers that did not divide evenly, were sweep under the rug and where
the three dots that indicate infinity kept everyone quiet and silenced
and
loss of precision in mathematics became entrenched.
So by insisting that there is a second decimal point denoted by L, we
recognize the remainder carryover in division. These are Real Number
Fragments and do not have the reality that a Real Number has because
the Reals are All Possible Digit Arrangements and so there is no
other Real between 0.3333....333 and 0.3333...33334. There is no
Real between those two consecutive Reals, so to recognize the division
and remainder 1 carryover of 1/3 we call it a Real Number Fragment and
denote it as 0.3333...3333L and know that it is situated between those
two consecutive Reals.
If we do not do this, we lose precision because we fully know there is
a
remainder carryover at the point of infinity. And we lose the Algebra
of a
completed closed Field on the Reals.
I call it a Real Number Fragment and it sort of acts like the
imaginary number
i as square root of (-1) is to the operation of square roots, for it
completes
the Reals to square-root.
The Real Fragment completes the Reals to division. So the L is a small
price
to pay to have the Reals completed as an Algebraic Field.
Mathematicians of the past all knew that there is a carryover
remainder of 1
when dividing 1 by 3. They were just lazy and sloppy by thinking that
the
three dots that indicates infinity would make the problem disappear.
And another great benefit of the Real Fragment is that it opens up the
Calculus
as never before in that one can actually now visualize the picket
fences as 2
dimensional picket fences and see how the derivative is truly inverse
of
integral. The L Real Fragment allows the Calculus to be performed on
individual points along the curve.
plutonium....@gmail.com
plutonium.archime...@gmail.com wrote:
>
> So by insisting that there is a second decimal point denoted by L, we
> recognize the remainder carryover in division. These are Real Number
> Fragments and do not have the reality that a Real Number has because
> the Reals are All Possible Digit Arrangements and so there is no
> other Real between 0.3333....333 and 0.3333...33334. There is no
> Real between those two consecutive Reals, so to recognize the division
> and remainder 1 carryover of 1/3 we call it a Real Number Fragment and
> denote it as 0.3333...3333L and know that it is situated between those
> two consecutive Reals.
>
Physics has a similar situation involving what is the fundamental and
lowest
particle of Nature. Not too long ago for millenium it was believed
that the
atom was the "last cut." But in recent history at the beginning of the
20th century
was found other particles called electron, proton and neutron. And
they
were then called the elementary particles of a cut lower than the atom
itself.
But let us ask a question. Are electrons, protons, neutrons, photons,
neutrinos etc etc are they independently existing particles or are
they
more like what the finger or toe or hair of a human is and that the
human
as one entity is the fundamental and lowest cut of a human as a
species.
We dare not call a fingernail as a "human". So in terms of biology, we
begin
to see that in Physics, the last cut is truly the atom itself, and
that stuff
like the proton, electron, neutron, photon are vestigial remnants of
the atom.
So that when I write out the complete matrix of all the Reals as All
Possible
Digit Arrangements and come to the Reals in that matrix of these three
consecutive Reals:
0.3333....33332 then 0.3333...3333 then 0.3333...33334 there are no
other Reals between those three consecutive Reals. Each is like an
atom
of independent existence.
But now we can understand that there can be a Real Fragment such as
0.333...3333L which is 1/3 that is situated between 0.333...333 and
0.333...33334
Perhaps we can draw the metaphor analogy of isotopes of atoms where
we have an isotope of hydrogen stuck between the hydrogen atom and
the helium atom.
So these analogies to Physics are helpful.
And perhaps in the future some may want to expand on the concept of
the second decimal point L by giving a finite string of digits, if
that is possible
where we could have 0.333...3333L1 and 0.333...33333L2 and
0.333...3333L3
and so on, where the L1 means a remainder 1 carryover and the L2 means
a
remainder 2 carryover. Whether that helps is some future applications
such as
perhaps the Calculus being refined even further.
As for me, I am satisfied with simply calling it L, signifying a
remainder carryover
between two consecutive Reals.
P.S. I named it L for limit because it throws the Limit into the
trashcan of the
Old Reals. It replaces the limit concept as excess and superfluous
baggage.
spudnik
for his "decimals," because the ambiguity
that's involved with fractions taht are prime
to the base, was covered by Stevin;
a huge chunk of classical numbertheory comes out
of this simple realization,
which Simon Stevin codified in "The Decimals" --
imagine that!
> That's still true. The only complaint with 1/3 or 2/3 is when you
> represent it as a decimal. It is not 0.333....
> What I am saying is that 1/3 is not the same as 0.333.....
> or which is 0.333...3333 because neither accounts
> for that 1 remainder carryover.
> On the other hand 1/2 is truly equal to 0.5000.....
thus:
That is the question.
> One thing you are overlooking is that the host is constrained.
> If you picked the car, he can pick which door to open, but
> if you did NOT pick the car, his pick of door
> to open is forced, since he can't open your door.
thus:
don't blame me for what Schroedinger's cat did
to your cosmic litter box, dood.
actually, what you infer is apparently
waht Galileo also didnt believe,
which was his actual heresy (atomism),
not his nonbelief in Ptolemy's hoax (because,
all semi-educated people knew
about the precession of the equinoxes).
you could effectively say that "because"
of the Uncertainty principle,
there is *only* relative vacuum, although
this has been reified into "zero-point energy"
et al ad vomitorium -- "aim if
for the cat's box!"
thus:
the only complicity that I have seen,
is the Scty. of Transort Mineta's testimony,
that Cheeny stopped the standard Air Force interceptions;
the plane-bombs were probably adequate incendiaries.
thus:
any congruence surd that repeats across the decimal point,
is equal to zero, such as ...9999.9999...; so,
why do you say that 0.349999... is not equal
to 0.350000...? because, the whole import of Hensel,
was to show that integers have the same properties
as decimals, properly oriented -- except that
he didn't use the commonsense ordering of the digits!
thus:
now, isn't that what the key method
of "remote viewers?" well, it depends;
the purveyors of it obviously have to resort
to some sort of hidden special knowledge,
as with magicians, to show some "results...."
anyway, that's the impression I got
from Ghost to Ghost BC radio;
there's a reason, they're called, Spooks!
thus:
no, no, no;
Fermat's unstated proof was not qualified
as to whether or not it applied to n=4, or
he thought at first that it did; clearly,
he wouldn't have needed to prove that case
with his infinite descent contradiction, if
he had already "had" the unstated proof. but,
isn't it clear that n=4 is very special,
after a while of consideration?
thus:
shouldn't it be clear that
photons are an artifact of the idea
of point-particles?... of course, iff
they exist, then they would have to be *the* particles
that actually were zero-dimensional points, but,
since they are waves, as shown by Young, Huyghens et al,
there is really no need for them,
except in the Pauli matrix formalism
of statistical bosons; eh?
Schroedinger's cat is dead --
long-live Schoredinger's cat!
--only 24 hours to impeach Trickier Dick from the N.Admin,
metaphorically typing, or Cheeny & Zbiggy, fo'mo' years;
Good Morning, Afghanistan!
... Good Afternoon, Sudan!
http://tarpley.net/bush12.htm
http://wlym.com/campaigner/8011.pdf -- Brits hate Shakes, Why?
http://www.wlym.com/~seattle/dynamis/
http://www.21stcenturysciencetech.com/current.html
http://www.rwgrayprojects.com/synergetics/plates/plates.html
http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=3163
http://wlym.com/campaigner/8011.pdf -- English, not! (see Psalms 46)
plutonium....@gmail.com
basically
the mechanism of the operations. What I do is take the old way of the
operations
on Reals and simply just make a Sequence Convergence to cover such
infinite
strings. Finite strings in the Old Reals were well defined, so I use
that, and
enhance and embellish the old way of operations, by simply making a
convergence
sequence. So let us see how it works for division.
Now division in AP-adics is going to be far easier than it is in New
Reals because
of the radix point in AP-adics takes care of any remainders. But
surprizingly,
division in Reals is messy and which the old-timers never really
noticed how
messy they were and never realized they had a major problem on their
hands.
Division in Positive AP-adics for example 99...7654321/555...555
Now the number 99...7654321 is about 99% of the distance from
NorthPole to SouthPole and the number 555...55 is about 55%
Remember AP-adics have a maximum place value of 10^999...99998
So the answer to the division in Positive AP-adics of 99...7654321/
555...555
is done like this:
654321/555555 = 1.17777
7654321/5555555 = 1.377777
87654321/55555555 = 1.5777777
987654321/555555555 = 1.77777777
.
.
.
So the convergence of that sequence is 000...0001r7
Remember the radix is a finite portion so we can expand finitely
that rightward string to the place value we want if r7 is not enough
for us in a application, but it is finite, not infinite and the radix
is what saves the AP-adics.
Let us try a different example of 9999...9999888/2222...22233
So the Convergence Sequence looks like this:
9888/2233 = 4.428
99888/22233 = 4.4927
999888/222233 = 4.49927
.
.
.
And the answer is 000...0004r49
In Negative AP-adics there is one glitch in that all divisions have
the negative sign. So that all numbers in Negative AP-adics, no
matter what the operation is, the answers always have a negative
sign. The negative sign merely is an indicator that the numbers
are all in Hyperbolic geometry. And for the Positive AP-adics there
can never be a negative signed Positive AP-adics. I have not reached
subtraction in AP-adics but when you subtract a larger number from
a smaller, it does not become a negative AP-adic in Positive AP-adics
but merely is a positive AP-adics that spans across the hemisphere and
located inside the other hemisphere. But I shall get to subtraction
soon.
So the AP-adics division is all straightforward and nothing unusual
going
on.
Now let us get to the Reals or New Reals and re analyze division.
The big mistake of the old timers of mathematics was that they swept
the remainder under the rug of imprecision. A division in Reals such
as
1/3 or 2/3. The old timers thought those answers were 0.3333...
and 0.666.... respectively. However the truth is far different because
they chose to ignore or not understand the remainder carryover in
both cases.
In New Reals, 1/3 is 0.3333....3333L and 2/3 is 0.6666....6666L
where the L simply is a second decimal point signifying there
is a tiny remainder at infinity that has to be accounted for. In Reals
we do not have the luxury of a radix in AP-adics to catch the
remainder carryover, so we impose a second-decimal-point
call it L for limit for it eliminates (sorry for the pun) the need
of the concept of limit in the old math. And I believe a simple L
is all that is needed even if there are different types of remainders
such as 1 in the case of 1/3 and 2 in the case of 2/3. I say L is
all that is needed because of the symmetry in AP-adics of the
second-decimal-point H which is needed in add subtract to
indicate the number is in the second hemisphere.
And, the need of a second-decimal-point L for Reals due to
division allows the Reals and Euclidean geometry to understand
the Calculus as Calculus was never able to be understood in the
old math with their Old Reals.
And the very most funny part of the above story is that mathematics
history went overboard on Algebra in the last century. They were
zealous
fanatics about Algebra, and is it so ironic that they failed to even
spot
that their fanaticism over algebra, that they could not even see the
huge
gaping hole of their Reals was flawed Algebraically. It is like an
army
on the battlefield ready to engage the enemy but instead commits mass
suicide by their own hands. So where the old timers of mathematics
went fanatic over Algebra, they failed to do a proper algebra at the
start.
The Old Reals were so flawed and blemished that they hardly obeyed a
single
rule for what amounts to being a Galois Field. All because of the huge
flaw
of division. That 1/3 is not 0.3333..... but is 0.3333....3333L
plutonium....@gmail.com
Cut mechanism
in Old Reals. Remember how the Old Reals were formed? Well you spend/
waste
a good year of your life at a University that treads you through a
avalanche of
definitions starting with Peano Axioms on to Natural Numbers, on to
Rationals
and then a Dedekind Cut Mechanism that allegedly creates Reals between
any
two given Reals so in the end you have what can be called *Absolute
Continuity*.
I am the first person to define and talk about Absolute Continuity
because it
really is nonexistant, only the old timers of math and the present day
brainwashed
math personnel do not yet realize that "absolute continuity" is
nonexistant.
The reason I bring up Absolute Continuity and Dedekind Cut is that it
is similar
to what I created as the mechanism for the Operations on Reals. I
created what
I call Convergence Sequence and it is very similar to Dedekind Cut but
with
a very obvious difference. The Dedekind Cut creates phony or bogus
entities
and calls them -- existing numbers such as a number between
0.3333....333
and 0.3333...33334. So the Dedekind Cut is a division that says there
are
an infinity of Numbers between those two shown. But the mechanism I
use
for division or multiplication as the Sequence Convergence or
Convergence
Sequence is a search or "looking for" the final answer which is
perhaps the
number 0.3333....3333 or perhaps ending with the number 0.3333...33334
or perhaps ending with the number such as 1/3 as 0.3333...33333L which
is
the only number between 0.333...3333 and 0.333...33334. So the
Dedekind
Cut method in the Old Mathematics of the Old Reals assumes there is an
infinity of Reals between 0.333...3333 and 0.333...33334 and then
proclaims
that the Dedekind Cut is an infinity of unkown Reals between them.
Whereas
the New Reals lists all the Reals from the start. I can zoom in on any
particular
Real Number and do a Calculus or geometry with that number and know
its every
neighbors. And then when I do an operation of say division like 1/3
then I fetch
or retrieve the existing number in the New Reals by the Convergence
Sequence.
Unlike the Old Reals in Old Math where the Dedekind Cut pretends and
wishy washy
assumes an infinity of Reals between any two given Reals.
In New Reals there is no such thing as Absolute Continuity that the
Dedekind Cut
pretends and assumes. In New Reals Continuity is defined as an
infinite supply of
numbers between two given numbers. For example, the interval 0.1 to
0.2 has an
infinity of Reals and an infinite supply of Reals between those two
numbers and hence
that is a Continuous Interval, however if we focus on two consecutive-
Reals in that
interval say 0.199999...99997 and 0.19999...99998 then there is an
obvious hole or
gap between those two Reals. So in that interval it is not continuous.
So in New
Reals we can have both continuity and discreteness or discontinuity
all bundled up
together. In New Reals we no longer have "absolute continuity" and
that was a bogus
concept all along, not only in mathematics but in physics there never
was
absolute continuity. And as there never was Absolute Space and Time in
physics
there never was that in mathematics.
Also, let me consolidate the second decimal point in Numbers. No
reason to have
more definitions than need be. And I myself often fail to remember
what I last called
the second-decimal-point in Reals and AP-adics. The last I can
remember I called
the second-decimal point in Reals as L and as H in AP-adics. So let me
consolidate
those two, for I would even like to drop the Radix r in AP-adics and
call it as a d
for decimal point so that I have only two symbols d and L, but there
is some confusion
with that and so I retain the radix r for AP-adics. Here are some
examples of
Real and AP-adics with their two decimal points.
0d3333....3333L which is commonly known as 1/3 where the L signifies a
remainder carryover and that number is called a Real Fragment for it
is not
a number as All Possible Digit Arrangements and it lies between two
Reals of 0.333...333 and 0.333...3334
0d333...3333 this number was thought to be 1/3 in Old Reals. In New
Reals
it is one example of All Possible Digit Arrangements
999999....99999r7 which is very close to the SouthPole and over 99%
but not quite
100% of the distance
-0d6666...6666L which is commonly known as -2/3
L9999....9999r7 which is very close to the NorthPole only it is in the
second and unused
hemisphere of Elliptic Geometry. The L indicates it is not in the used
hemisphere
but due to addition or subtraction has spilled over into the second
hemisphere.
Without further ado, let me define subtraction on AP-adics and Reals.
This is straightforward because the Sequence Convergence is not really
needed
as well as in addition the act of performing the operation does not
require a
convergence sequence. But there is one glitch with AP-adics in that a
subtraction
can end up in the second hemisphere such as this:
44444....44444 - 9999...99999
Now in Reals we would think the answer is a negative number. But in AP-
adics
the numbers are points on a sphere or hyperboloid figure and in
Positive AP-adics
such as the above example the final answer will end up being another
positive
AP-adic and be a number in the unused second hemisphere
4444...4444 - 9999...9999 = L55555...5555
where L indicates that the number is in the second hemisphere of the
Elliptic
geometry of Positive AP-adics
Now let us do the same subtraction in Hyperbolic geometry of Negative
AP-adics
(-)4444...4444 - (-)9999...9999 = L(-)555..5555
The only change is that we are dealing with a hyperbolic surface such
as
a pseudosphere and that the final answer is on the second hemisphere
of
the pseudosphere at the point (-)5555...5555
Now in New Reals subtraction is the same as in old Reals however there
is
one new item to contend with and that is the L second-decimal point.
So what is say this:
0d000...0000L - 0d0000...0001L
Those are two Real Fragments being subtracted. One is a fragment
between
0.000...0000 and 0.000...00001 and the second fragment is between
0.000...0001 and 0.000...0002
So what happens when we subtract Real Fragments?
I believe the answer is, for the above case, (-)0d000...0001
Since the metric between them is 0.000...0001 and that the L
drops out in the subtraction process.
Keep in mind that I am defining these operations so that in the end
they
form a Completed and Closed Galois Field in Algebra.
So that leaves three more operations I want to define as square-root,
exponentiation,
and factorial. I do not need to define these three but feel it fun in
doing so, especially
the factorial with the idea that 9999....99999 in AP-adics leads to
the North Pole
since the zeroes transform the answer into 0000....0000 and that is
imaginary in
AP-adics as the North Pole.
In New Reals the factorial is always a larger number and that is
because the leftward
string is always finite. In AP-adics, the factorial leads to the
NorthPole in Elliptic
geometry.
plutonium....@gmail.com
future editions I will make this
the first chapter so that the pragmatic use trumps all other concerns.
There are three major applications of this New Mathematics:
(a) corrects all the old math and their misguided conjectures and
their
mountains of mistakes and phony proofs
(b) the numbers of AP-adics and Reals shuffle back and forth between
Elliptic, Hyperbolic and Euclidean Geometry. So that a triangle formed
in Elliptic geometry unioned with the triangle associated in
Hyperbolic
geometry is the Euclidean triangle and were we can do this all with
the Numbers.
(c) sharpens the Calculus as it never was clear in the Old Reals.
And example of the application of (b).
We can have a triangle on the sphere surface with its AP-adic numbers
as the vertices. And because those AP-adics have a Hyperbolic
association
there are three Negative AP-adics numbers of a hyperbolic triangle
where we have Elliptic Triangle union Hyperbolic triangle = Euclidean
triangle
and we have Positive AP-adics + Negative AP-adics = (Reals) x
transpose factor
Let me try to clarify. We start with a triangle in Euclidean or
Elliptic geometry.
They have 3 vertices. Those vertices are three Reals or three +AP-
adics.
Given those three numbers as vertices we can find the associated AP-
adics
numbers as vertices. So the area of the triangle in Euclidean geometry
is the same as the area of the Elliptic subtract Hyperbolic geometry
and where
we know all of that from simply the uniqueness of the Numbers in AP-
adics
and Reals.
plutonium....@gmail.com
AP-adics is
dependent on exponent. Where in AP-adics, All Possible Digit
Arrangements
yields the largest AP-adic Integer of 999...9999 and so the exponent
cannot
exceed the largest integer thus the 10^999...9998 is the largest place-
value
and so the "1" digit in Reals 0d000...0001 is in the 10^(-)999...9998
place
value and ditto for the "7" in 0d999...9997.
Now I like to use "d" to signify decimal-point rather than the
commonly used
dot, because, it is often difficult to see the dot as decimal point
and many people
are sloppy in writing and so one often mistakes a dot as a decimal
point or
just cannot see the dot whereas a "d" is safer to write.
So the Numbers themselves determine the largest place-value and thus
the
exponentiation.
Exponentiation is a Convergence Sequence and for example in Reals:
0d6666...6666L^3 which is 2/3 to the exponent 3 would be:
d6^3 = d216
d66^3 = d287496
d666^3 = d295408296
d6666^3 = d296207416
d66666^3 = d296287407
.
.
.
And so the answer of what is 0d666...6666L^3, or (2/3)^3, is
0d296...6L
and where the L is retained
Now an example in AP-adics of L5555...5555r0 which signifies a
positive AP-adic in the second hemisphere with value of 555...5555 or
55% away from the South Pole and only 45% more to go to reach the
North Pole.
So we have this Convergence Sequence to find the answer
5^3 = 125
55^3 = 166375
555^3 = 170953875
5555^3 = 171416328875
55555^3 = 1.71462 x 10^14
.
.
.
So apparently the answer is going to be L171.....5 in the second
hemisphere.
I see no troubles with the exponent, but one must remember that in AP-
adics
the exponent can only be another AP-adics and we cannot be mixing up
Reals
with AP-adics. That was the trouble and problem of understanding the
famous
equation in mathematics that e^(i x 2pi) = 1. In that famous equation
which is
easily solved since both e and pi are 0 valued, but in the Old Math
with the old-timers
were never aware that they were mixing together Numbers of Reals and
number that
belong only to AP-adics. It would be similar as an analogy of mixing
oranges and apples
and calling them all citrus fruit.
plutonium....@gmail.com
plutonium.archime...@gmail.com wrote:
(snipped)
> And example of the application of (b).
>
> We can have a triangle on the sphere surface with its AP-adic numbers
> as the vertices. And because those AP-adics have a Hyperbolic
> association
> there are three Negative AP-adics numbers of a hyperbolic triangle
>
> where we have Elliptic Triangle union Hyperbolic triangle = Euclidean
> triangle
>
> and we have Positive AP-adics + Negative AP-adics = (Reals) x
> transpose factor
>
> Let me try to clarify. We start with a triangle in Euclidean or
> Elliptic geometry.
> They have 3 vertices. Those vertices are three Reals or three +AP-
> adics.
> Given those three numbers as vertices we can find the associated AP-
> adics
> numbers as vertices. So the area of the triangle in Euclidean geometry
> is the same as the area of the Elliptic subtract Hyperbolic geometry
> and where
> we know all of that from simply the uniqueness of the Numbers in AP-
> adics
> and Reals.
Now let me talk about the AP-adics as curved numbers whereas Reals are
straightline numbers. The actual axis of Reals as the Cartesian
Coordinate
Axis is a straightline axis but in AP-adics the axis is on a half
cylinder the
y-axis is a 180 degree arc.
And in AP-adics, the number 9999....9999 is so sharply bent that it is
a arc of 180 degrees or a semicircle arc. So in AP-adics we can
replace
numbers such as 5000...00000 becomes a pi/2 arc.
This gives added meaning and applications to the AP-adics. So that
when we
form a triangle in Euclidean Geometry say of a right-triangle whose
sides are
3,4, and 5, but if we form the same triangle on a cylinder surface in
Elliptic
Geometry with 000...00003, 0000....00004, 00000....0005 the triangle
is so
tiny and small that we have a hard time of seeing the curved or bent
sides of
the triangle.
But if we were to focus on a triangle whose sides are these in AP-
adics
3333....3333, 44444....44444, and 55555....55555 then the triangle has
such
enormously bent curved sides that we have no doubt of its being a
triangle in
Elliptic geometry and the associated Hyperbolic triangle with sides
(-)333....3333, (-)4444....4444, and (-)5555...55555 so to speak
straighten
out the concave outwards of the associated Elliptic triangle and as an
end result yielding a triangle in Euclidean Geometry.
So with AP-adics, these numbers can be given a radian measure as well
as
a numeric measure. So a number in AP-adics of 6666.....55555 has an
associated
pi radian measure and in that example of about 66% of pi.
Now one use of the idea that the AP-adics can be converted to radians,
unlike the
Hensel P-adics which cannot, is that the AP-adics would then be easy
to use
to figure out the associated Hyperbolic triangle given any Elliptic
triangle and
vice versa.
lwa...@lausd.net
> David R Tribble wrote:
> > Archimedes Plutonium wrote:
> > > So I built the Reals anew, and shown where the Old-Reals are fake and
> > > breakdown with numbers like 1/3.
> > So you're saying that
> > 1/3 + 1/3 + 1/3 = 1
> > doesn't work any more?
> That is still true. The only complaint with 1/3 or 2/3 is when you
> represent it as a decimal. It is not 0.333....
AP's comments here that 1/3 does not equal 0.333... is very
reminiscent of the writings of another so-called
"crank," MH Knowles (whom I have already mentioned in
several other threads). He referred to this concept as the
"Vanishing Remainder Paradox," and wrote:
"The rational 1/3 is transformed into the real (infinite
decimal expansion) 0.333... by successively multiplying 1
by 10, dividing that 10 by 3 getting a quotient of 3 with a
remainder of 1 which becomes the 10 for the next decimal
place, and so on... But that remainder of 1 vanishes from
theoretical view when we reach the Cantorianly completed
countable infinity of decimal places. What happens to it?
Does it somehow become an absolute zero after an infinite
number of divisions? A variant asks: what happens to the
remainder of 1 when it is divided by a natural number n as
n “goes to infinity”? Does it somehow become an absolute
zero when “divided by infinity”? These are some of the
paradigmatic questions for the “Vanishing Remainders
Paradoxes”. If the remainder does not become an absolute
zero, then 1/3 cannot be represented by 0.333... (they
cannot be strictly equal) and therefore 1/3 can not truly
be a real number; and if it does, this violates an as yet
unrecognized implicit conservation-type law, opening the
door to inconsistency in addition to paradox. E.g., if the
remainder of 1 when 1 is “divided by infinity” becomes an
absolute zero, one can derive that the reals are countable.
There are many other as yet overlooked paradoxes in
mathematics, some of which offer a basis for their joint
resolution, and for the problem of renormalization in
physics."
So both MHK and AP reject the equality 1/3 = 0.333..., and
for almost exactly the same reasons. Both MHK and AP pay
attention to the remainder of 10/3 as 1, and both want to
find a way to keep track of it -- and AP does this via his
L-notation, as the remainder 1 in 0.333...333L1.
Who came up with the idea first, MHK or AP? Notice that
though MHK's paper was written in 2004, he first posted it
on his website in 1995 -- which is right around the time
that AP started posted his AP-adics on sci.math. But since
MHK doesn't post on sci.math or update his page anymore,
we're left with AP to describe the theory.
Although AP has insisted that his AP-reals are not like
Robinson's hyperreals, I mention them for the sake of
Tribble and those who don't accept AP-reals.
Consider the subset of the rationals whose decimal
expansions are terminating, call it Z[.1]. We then take
the *-transform as defined by Robinson, to give a certain
subset of the hyperrationals, Z[.1]*.
By the Transfer Principle, we can determine some of the
properties of Z[.1]. Notice that ~(1/3)eZ[.1]*, but there
exists a hyperrational r whose standard part is in fact
1/3, such that reZ[.1]. Indeed, for _any_ real there
exists a hyperrational in Z[.1] with that real as its
standard part!
Of course, the AP-reals aren't exactly like Z[.1]*, since
the former has a minimum positive element (0.000...0001)
while the latter doesn't. Still, the concept that there
can be a set containing elements that are infinitesimally
close to a real (such as 0.333...333 and 0.333...3334)
yet miss the real itself (1/3) is similar.
plutonium....@gmail.com
Interesting read Lwal.
A few comments and questions:
(a) MH Knowles is "right-on" with his complaint over 1/3 as not
equalling 0d3333...... But did Knowles suggest a way out, or a
relief to the problem? Did Knowles ever go beyond complaining?
So what I did was notice the problem and then solve it by
saying the Reals have to have a second-decimal-point to capture
remainder carryovers and I simply call it now the L suffix or
second decimal point. I leave L by itself as with no digits
so that 1/3 = 0d333...3333L and 2/3 = 0d666...6666L and
I call these as Real Fragments since they are not one of
All Possible Digit Arrangements as is 0d3333...3333 and
0d3333....33334 for which 0d3333....3333L lies between.
And this is in harmony with the AP-adics second-decimal
point of L also, as in
L5000....0000r3 which indicates a AP-adic in the second
hemisphere about a tiny distance r3 north of the equator.
So question Lwal, did Knowles actually try to resolve the problem?
Or did he just outline and complain about the problem?
(b) Question on the HyperReals, can they provide a more clear picture
of the Calculus derivative and integral around a point such as
1/3 in the graph of a function? Because the AP-Reals for the first
time
in the history of mathematics is able to tell how and why the
derivative
is inverse to integral and actually perform the derivative and
integral
at a point location on the graph. So the question is, have the
HyperReals
ever facilatated the Calculus? I would say no, very quickly and that
the
HyperReals have never facilatated anything. To me, HyperReals,
Surreals,
are nothing but childish games, not mathematics.
plutonium....@gmail.com
Interesting read Lwal.
spudnik
that was explicitly deemed to be a feature --
by Simon Stevin in _The Decimals_!
see _Number Theory and Its History_ by Oystein Ore.
> (a) MH Knowles is "right-on" with his complaint over 1/3 as not
> equalling 0d3333...... But did Knowles suggest a way out, or a
thus:
I have no idea how many mathematicians have read this book, but
it is probably quantifiable in the zillions/ths,
mod a certifiable Don or two at Harry Potter PS#2..
If monsieur Munk is alive, he is over a hundred;
this book was published, as I recall, in '76, although
I came across it in the late '90s. Now, he is perfectly qualified
to be an amateur a la Fermat, having been an engineer
at NACA, the predecessor to NASA, where he worked
on aerodynamics. His main claim to fame is an early analysis
of (2d) sections of airfoils.
He wrote the first introduction for laymen on aerodynamics; since
it was published in the Great Depression, he used a "vanity press,"
Vantage Books, which he also used for his alleged proof
of Fermat's "last" theorem.
The insight that I got into Fermat is not acutally stated in the book
-- it goes with monsieur Munk's *vitae* --
suffice to say that the undisclosed theorem was probably amongst
his first insights into Diophantine eequations. You could guess that
it was a key to his Method.
The book is quite elementary and enjoyably written,
occaisionally funny and quite a tour de force. Now, although
Fermat made no known errors -- unless possibly,
you question the veracity of Wiles' proof of FLT, tee-hee --
that is not to say that monsieur Munk made no mistakes.
Underwood Dudley includes Munk in his chapter, in _Mathematical
Cranks_
(from MAA.org [*:]),
of "fermatistes," which you will have seen if you have ever used any
of the sci.math newsgroups. As he admitted, when I wrote to him,
it was obvious that he did not read the whole, slim book, but
jumped to the pen-intimate section from the table of content,
where the nib of the matter would be written.
There *is* a problem, there, but it may just be a peculiarity
of "English as a second language," because everything else is
beautiful,
including the remaining chapters that explain the workings of
congruence surds,
which is Munk's coinage. "They are not p-adic numbers."
Numbertheory is part of the *quadrivium*, which is Latin for
*mathematica*,
which is the four subjects of classical Greek science; as an example
of this "modular arithmetic," what is the meaning
of Platform Nine & Three-quarters?
Get this book back into print!
* The MAA's *Mathematics Magazine* is fantastic, and accessible at any
level.
plutonium....@gmail.com
and so I let it
simmer over periods of time and then by circumstances something
intersects with the
problem and it becomes clear as to how to solve it. In the 1st edition
of this book
I ran into a mystifying problem. I was doing cuts or cross-sections of
the sphere
and then rebuilding the sphere from those cross-sections of the
latitudes. I was
picturing hoola hoops that were successively smaller and stacking them
to build
a sphere.Why was I doing such a thing? Well, I was doing it because I
wanted
to stack cross-sections to get the psuedosphere of Hyperbolic
geometry.
And the experiment went totally counterintuitive. When something goes
totally
counterintuitive tells me I have more than just a problem to solve but
that I have
a very important problem.
So here is the problem I faced in 2007 that is totally
counterintuitive and mystical.
Take a sphere and cut it into slices so that at the end you have
slices of the latitudes
of the sphere, (the hoola hoops). And of course we can restack them
back to being
a sphere. But the problem I wanted to solve was that we have this pile
of cross-sections
of a sphere's latitudes and our goal is to stack them so that they are
a Pseudosphere.
Counterintuitive that the stacking of the latitude cross sections
seems to be the same
stacking as the sphere.
So in 2007 when I ran into this problem I had the hankering notion
that the solution was
that I cut out a large number of the sphere latitudes, say every ten
sphere latitudes
I removed and then stacked every other tenth cross section in order to
build the
Pseudosphere.
So is that the answer to my counterintuitive problem? That the sphere
cross section
latitudes have to be culled and every tenth cross section saved to
build the Pseudosphere
and the unused cross sections stay unused?
So have I thus theoretically built a Eccentricity in the 3rd dimension
for Eucl, Elliptic
and Hyperbolic geometry? That we have a Elliptic geometry object such
as a sphere
and we take infinity of cuts of latitudes and then cull those
latitudes and use only every
tenth sphere latitude in order to build a pseudosphere in hyperbolic
geometry?
Now I do not know if every tenth latitude is to be saved or whether
every third latitude?
But it seems to me as though that would be a solution to my mystifying
counterintuitive
problem I discovered in the 1st edition of this book in 2007.
If my above is correct then there would be a specific amount of
culling to produce a
Parabolic object in 3rd dimension from a sphere cuts of latitudes. And
then a further
culling for the Hyperbolic object in 3rd dimension. So I may have
discovered
eccentricity in geometry for 3rd dimension as latitude cross sections.
And of course
Euclidean geometry objects are not Conic Sections, are they, so there
is no
eccentricity concept for Euclidean objects.
plutonium....@gmail.com
Now we can take each sphere latitude cross section (each hoola hoop)
and give it a numeric value from 0000...00001 on up to 9999...9999
where the 000...00001 and 9999...99999 hoola hoop (latitude) are the
tiniest and are equal in diameter. And where the 5000....0000 hoola
hoop is the largest since it is the equator and the next largest
is the 4999....9999 and 5000....00001 hoola hoops which are equal in
diameter. So we have this infinite matrix of hoola hoops to build the
sphere
and we ask, which of them are culled to build a Paraboloid object?
Do we cull a constant number of the sphere hoola hoops say every
3rd hoola hoop we use to build the Paraboloid?
And how do we build the Pseudosphere? Is it a logarithmic culling?
Does
it involve Euler's number?
I do believe this stacking problem is the concept of eccentricity for
Conic
Sections in the 3rd dimension.
plutonium....@gmail.com
attack it on all sides.
What I am up to is to make an infinity of cross-section cuts of the
latitudes of a sphere
and then number each one of them as AP-adics Integers with 000...00001
to that
of 999...9999 where the largest latitude is the 5000...000 which is
the Equator.
Once I have cut the sphere into all these latitude cross-sections, all
of them
circles, then I want to restack them to build a pseudosphere. Here is
where the
counterintuitive comes in, in that to restack them to form the
pseudosphere from
the sphere, that it seems I have to do something such as select only
certain
latitude cross sections to use them and not use others.
So I wonder how to build an ellipsoid from these sphere latitudes. An
ellipsoid
is merely the topological deformation of the sphere, so it seems
likely that
I can build the ellipsoid from the sphere latitudes by using a
"spacer" between
each sphere latitude. Depending on how big the spacer is, the more
eccentric is
the ellipsoid.
So is that the answer? Is it a spacer? And that I use all the sphere
latitude
crosssections in building the pseudosphere or ellipsoid, and it is the
spacer
distance that then forms the desired object?
Not sure whether I can build a parabolic 3D object with sphere
latitudes? Maybe there
is some object that is a Parabolic-Pseudosphere ?? Anyone know?
Anyway, maybe I found a solution to this nagging counterintuitive
problem. The problem
of cutting a sphere into latitudes and then reassembly into other 3D
objects. Perhaps
it is the "spacing" of the cross-sections and this spacing related to
eccentricity in
2D.
lwa...@lausd.net
> Interesting read Lwal.
> A few comments and questions:
> (a) MH Knowles is "right-on" with his complaint over 1/3 as not
> equalling 0d3333...... But did Knowles suggest a way out, or a
> relief to the problem? Did Knowles ever go beyond complaining?
Not really. Knowles believed that the root of the problem
was set theory, ZFC, and the fact that there exist
bijections between sets and proper subsets of themselves,
or aleph_0+1 = aleph_0. So he suggested that one come up
with a new set theory to solve the problem -- but never
shows exactly how to do it.
Thus, I must read AP for attempts to solve the problem.
> And this is in harmony with the AP-adics second-decimal
> point of L also, as in
> L5000....0000r3 which indicates a AP-adic in the second
> hemisphere about a tiny distance r3 north of the equator.
Equator? Apparently AP has returned to the usual spheres
and pseudospheres as models for the elliptic and
hyperbolic geometries that he wants for his AP-adics.
> (b) Question on the HyperReals, can they provide a more clear picture
> of the Calculus derivative and integral around a point such as
> 1/3 in the graph of a function?
Possibly. The hyperreals can eliminate the use of deltas
and epsilons in proofs of calculus.
Of course, I already know that AP rejects the use of
hyperreals and surreals. I only mention them to help
people like Tribble understand why 1/3 isn't part of
the AP-reals -- because it isn't an element of the
set Z[.1]* either.
plutonium....@gmail.com
lwal...@lausd.net wrote:
> On Jan 5, 11:41�am, plutonium.archime...@gmail.com wrote:
> > Interesting read Lwal.
> > A few comments and questions:
> > (a) MH Knowles is "right-on" with his complaint over 1/3 as not
> > equalling 0d3333...... But did Knowles suggest a way out, or a
> > relief to the problem? Did Knowles ever go beyond complaining?
>
> Not really. Knowles believed that the root of the problem
> was set theory, ZFC, and the fact that there exist
> bijections between sets and proper subsets of themselves,
> or aleph_0+1 = aleph_0. So he suggested that one come up
> with a new set theory to solve the problem -- but never
> shows exactly how to do it.
>
The problem with 1/3 or 2/3 or 1/9 etc etc is not because of set
theory
but because of the construction of Reals and then Algebra. Old Reals
are a faulty construction and it only takes Algebra to display the
faulty
construction.
It is funny how Knowles and Tribble do not focus on Algebra rather
than
set theory for the ailment of 1/3.
If any mathematician at this very moment were to reflect on the
Dedekind
Cut method of creating the Reals, would instantly recognize it is a
construction
that seeks to attain "Absolute Continuity" so that the phrase "between
any two Reals is another Real" comes to life. But then does this
Absolute Continuity agree with "All Possible Digit Arrangements" or do
the two concepts contradict one another?
Is Dedekind Cut contradictory to All Possible Digit Arrangements or
are they
compatible?
Dedekind Cut creates the Reals by a Convergent Sequence.
All Possible Digit Arrangements simply says "here they are" now use
convergent-sequences on operations, not on creating the Reals.
This is where Algebra steps into the picture and decides whether
Dedekind
Cut creation of Reals is correct or bogus.
We take 1/3 which is 1 divided by 3 ad infinitum. Does 1/3 exist in
the Dedekind
Cut creation of Reals? No, for it is not 0.3333..... and that is the
only candidate
in the Dedekind Reals.
Is 1/3 existing in All Possible Digit Arrangements? Well, yes, with
the inclusion
of a second-decimal-point L where 1/3 is 0d333....3333L
And in the Dedekind Reals, they do not even have Reals such as
0.333...3334
because the entire Dedekind process is phony.
> Thus, I must read AP for attempts to solve the problem.
>
The problem with Dedekind Cut Reals is that it is not "All Possible
Digit Arrangements" And because it is not All Possible Digit
Arrangements
then the division operation on Dedekind Reals is not closed to
division
where remainder carryovers are ignored.
> > And this is in harmony with the AP-adics second-decimal
> > point of L also, as in
> > L5000....0000r3 which indicates a AP-adic in the second
> > hemisphere about a tiny distance r3 north of the equator.
>
> Equator? Apparently AP has returned to the usual spheres
> and pseudospheres as models for the elliptic and
> hyperbolic geometries that he wants for his AP-adics.
>
Yes, making some decent progress on stacking cross sections.
> > (b) Question on the HyperReals, can they provide a more clear picture
> > of the Calculus derivative and integral around a point such as
> > 1/3 in the graph of a function?
>
> Possibly. The hyperreals can eliminate the use of deltas
> and epsilons in proofs of calculus.
>
I do not know much if anything about hyperreals or surreals, but I do
know
this, that since they have been around they have not corrected the
Calculus.
Since they did not correct the Calculus by showing why the deriviative
is inverse to integral and by being able to differentiate or integrate
at a
particular point of the graph of a function, indicates to me that the
hyperreals and surreals are not mathematics but rather are imagination
without
truth value.
Since hyperreals and surreals do not, and can not correct Calculus in
any way
shape or form, is strong indication they are not mathematics but some
amusement game or trick, much like the ponzi games on wall-street.
> Of course, I already know that AP rejects the use of
> hyperreals and surreals. I only mention them to help
> people like Tribble understand why 1/3 isn't part of
> the AP-reals -- because it isn't an element of the
> set Z[.1]* either.
I have not injected Set theory into this book. There is not much need
to inject
Set theory because All Possible Digit Arrangements is the Set theory.
So if hyperreals
and surreals are just some tinkering with set-theory and based largely
on
Cantor-follies, well, hyperreals and surreals are just imagination and
not mathematics.
Summary: Old Reals created by Dedekind Cuts was seeking for "absolute
continuity"
but in its seeking, lost sight of the fact that the Old Reals were not
even
"All Possible Digit Arrangements" and that the Old Reals as Dedekind
Cuts was
not even Algebraically Closed to Division with 1/3 as a glaring
example.
plutonium....@gmail.com
plutonium.archime...@gmail.com wrote:
>
> > Of course, I already know that AP rejects the use of
> > hyperreals and surreals. I only mention them to help
> > people like Tribble understand why 1/3 isn't part of
> > the AP-reals -- because it isn't an element of the
> > set Z[.1]* either.
>
I need to correct Lwal on his above paragraph. In the Old-Reals, the
Dedekind Cut Reals that every math student is being taught at present
moment, the number 1/3 is not an element of their mathematics.
Because 1/3 to them is 0.3333...... So they ignored and forgot about
the 1 remainder carryover. So 1/3 is not an element in present day
mathematics.
But 1/3 is an element in the AP-Reals, because I do not forget nor
ignore the 1 remainder carryover and 1/3 is equal to 0.333...3333L
Math education today teaches a falsehood in every school and college
across the world when it says 1/3 = 0.3333.... because they ignore
and forget about the remainder. Not I, in the AP-Reals. I do not
ignore
nor forget there was a 1 remainder carryover and denote it as the
L suffix second-decimal-point.
This problem surfaces with another problem that has messed up the
minds
of nearly every student who studies mathematics with the famous riddle
of 0.9999...... In the old-math taught by old-fogeys they want to
insist that 0.999.... is the same as 1.000..... They are grotesquely
wrong.
In All Possible Digit Arrangements there is a Real Number 0.999...9999
and a different Real Number 1.000....0000 These two different Reals
differ by 0.0000....0001.
The Old Reals were not closed to division and did not form a Galois
Field Algebra because of their division problems.
>
> I have not injected Set theory into this book. There is not much need
> to inject
> Set theory because All Possible Digit Arrangements is the Set theory.
> So if hyperreals
> and surreals are just some tinkering with set-theory and based largely
> on
> Cantor-follies, well, hyperreals and surreals are just imagination and
> not mathematics.
>
> Summary: Old Reals created by Dedekind Cuts was seeking for "absolute
> continuity"
> but in its seeking, lost sight of the fact that the Old Reals were not
> even
> "All Possible Digit Arrangements" and that the Old Reals as Dedekind
> Cuts was
> not even Algebraically Closed to Division with 1/3 as a glaring
> example.
It occurred to me from the above that Set theory is not really a major
subject
of either Physics or Math. And this book has constantly emphasized
that
Mathematics is a tiny subset of Physics. But let me entertain the idea
that
Set theory is a tiny subset of Probability theory.
When I opened this book I defined the Numbers as All Possible Digit
Arrrangements
and I reached for Probability theory to make that concept come to life
as a Probability
Sampling Space for All Possible Digit Arrangements. I did not need nor
use
Set theory. So apparently, Probability Theory ranks higher in
importance to mathematics
than does Set theory and that Set theory appears to be a subset out of
Probability
theory because of its "Universal Space" concept.
Probability theory extends further than Set theory.
And about the only concept of Set theory that I used which does not
come from
Probability theory is the "union" concept as in
Elliptic geometry union Hyperbolic geometry yields Euclidean geometry
So, if I can get the union, intersection over into Probability theory,
then there is no
need to have a Set theory in Mathematics whatsoever, in that all of it
is within
Probability theory. I do not think that is too difficult of a chore to
see that union
and intersection are details of Probability theory.
plutonium....@gmail.com
this figure
overall is really a Hyperbolic hourglass shape if the sides of the
cones were
concave inwards instead of straightlines. So the Conic Sections is
really
Hyperbolic Geometry.
I am fascinated by the idea that a sphere can be cut or sectioned into
an
infinite latitude circles. So that if we had a collection of washers
or wire rings
that were progressively smaller and we stack them, we end up with a
sphere.
But now we disassemble those rings and asked to stack them so that
they
be a Conic Section or Hourglass shaped figure.
So what do we have to do to make the rings build not a sphere but
build
a Conic Section or an Hourglass, or a Pseudosphere?
What do we have to do for those same rings to build a Ellipsoid
instead of
the Sphere?
So is there some culling of the rings, that are used in building the
sphere
but not used in building the Ellipsoid or Pseudosphere? And is there
some
spacing between the rings to build an Ellipsoid or Pseudosphere?
The nice aspect of these thoughts is that they lend themselves to
actual
experimentation. That we can actually build a sphere from pieces of
paper
which circles were cut out and the circles progressively smaller in
diameter.
So how is it that the progressively smaller circles that compose the
latitudes
of the sphere, how are they thus stacked to build the Conic Sections
and the
Pseudosphere?
So we can actually do Experiments in mathematics to resolve this
problem.
plutonium....@gmail.com
plutonium.archime...@gmail.com wrote:
(snipped)
> than does Set theory and that Set theory appears to be a subset out of
> Probability
> theory because of its "Universal Space" concept.
>
The main concept of Set theory is "membership" but let us examine
Probability theory, in its main concept of the "universal probable
space".
When we roll a dice, its Probability Space is these six outcomes:
1,2,3,4,5,6
So is it apparent that Probability Space is the very same concept as
Set Membership? So if Probability has the same concepts as Set theory,
then we have a duplication of subjects where Set theory is nothing but
a
subset of Probability theory.
> Probability theory extends further than Set theory.
>
> And about the only concept of Set theory that I used which does not
> come from
> Probability theory is the "union" concept as in
>
> Elliptic geometry union Hyperbolic geometry yields Euclidean geometry
>
> So, if I can get the union, intersection over into Probability theory,
That should be easy to do in that union is considered by many as a
morphed
addition. And where set-intersection is considered by many as a form
of
subtraction of those without the identity function.
> then there is no
> need to have a Set theory in Mathematics whatsoever, in that all of it
> is within
> Probability theory. I do not think that is too difficult of a chore to
> see that union
> and intersection are details of Probability theory.
>
I believe Set theory was a minor subset of Probability theory all
along. But
when Cantor published his folly of transfinite infinities, that Set
theory
had to enlarge its role as a receptacle of bogus mathematics such as
power sets and hierarchies of bogus infinities.
Imagine for a moment a rational sensible person doing a Power-set on
All Possible Digit Arrangements. As the laughter reaches a crescendo.
plutonium....@gmail.com
lwal...@lausd.net wrote:
>
> Of course, I already know that AP rejects the use of
> hyperreals and surreals. I only mention them to help
> people like Tribble understand why 1/3 isn't part of
> the AP-reals -- because it isn't an element of the
> set Z[.1]* either.
I appreciate Lwal in taking the time to teach Tribble, but in the case
of the number
1/3, I must point out where the AP-Reals stand out as different from
the Reals that
both Tribble and Lwal and every book written on mathematics stands.
Lwal implies from his paragraph that the Reals of Dedekind Cut have a
number in the
Reals which represents 1/3 in decimal form. This Real to Tribble and
Lwal is
0.3333......
I argue against that. I say that the Dedekind Reals have no Real
decimal that
represents 1/3. That the Reals of Tribble and Lwal say that 0.333....
is 1/3
I say it is not, because 0.3333...... has no symbol to indicate where
the leftover
remainder of 1 went. Those same Reals do take care of 1/2 in that it
is truly
0.50000..... So these Old Reals, these Reals that Tribble and Lwal
were schooled
in are capricious in division that numbers like 1/2 are truly
0.5000.... but that
numbers like 1/3 are not 0.333..... since it dismisses the 1 remainder
carryover.
So the paragraph above by Lwal makes it look as though his Reals, and
Tribble
Reals of Dedekind Reals are all fine and dandy and that 1/3 =
0.3333....
and makes it look as though the AP-Reals are other than normal.
And where Lwal then implies in that paragraph that the AP-Reals do not
have
a number 1/3. When in fact the AP-Reals do have a number 1/3 but that
the
Reals of Tribble and Lwal do not have a number 1/3 in decimal form.
So the AP-Reals shine as correct but that the Reals of Tribble and
Lwal
are lacking and defective of a decimal form for 1/3.
Dedekind Reals want people to believe that 1/3 is 0.333..... but that
is a
fantasy since there was a 1 remainder that is never accounted for and
where
1/2 is 0.5000.... is accounted for. So the Reals that Tribble and Lwal
were
schooled in and had learned mathematics from, those Reals are
defective
because they do not have a number for 1/3 as decimal and that means
the
Reals that Tribble and Lwal were schooled in, are Algebraically
defective
to division in that they do not form a Closed Field since there is no
Real decimal form for 1/3 or 2/3 or 1/9 etc etc
The AP-Reals do account for the 1 remainder in 1/3 as 0d3333....33333L
where the L indicates a remainder carryover. Do account for 2/3 as
equal to 0d666...6666L and do account for 1/9 as 0d111...111L
And the AP-Reals are a Galois Algebraically Closed Field.
But the Reals that Tribble and Lwal were schooled in are not a Field
since
they are defective in division.
plutonium....@gmail.com
experiment of
stacking cross-sections. I actually am going to see, and stack cross-
sections
mentally with a pictorial of a sphere side by side with a tractrix-
pseudosphere.
A 2dimensional view of the sphere and pseudosphere in the first
quadrant.
Now in the sphere as circle and pseudosphere as tractrix curve, I need
only the
first quadrant and I am going to flip-around the circle as
x = sqrt(1- y^2) flip it around so the largest y point is not when x =
0
but when x = 1
I searched for a website that would give the formula of a tractrix, or
pseudosphere
http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node69.html
Tractrix curve given as x = (a)Ln(a + sqrt(a^2 - y^2)/y) - sqrt(a^2 -
y^2)
Here again I am going to flip that Tractrix curve around so the the
largest y point
is not when x = 0 but when x = 1 and also make the largest y point
equal to the
largest y point of the circle as y =1 at the point x =1.
Sounds confusing but my aim is simple to measure the height of the y
point
of the circle and tractrix as x goes from 0 to 1.
In other words, if the y point for the circle when x = 0.5 is that of
0.86 then
for the tractrix at x = 0.5 should be considerably less than 0.86
and how much less tells me immediately what the radius of the latitude
circle in the stacking to make the sphere and to make the
pseudosphere.
So, if the reader understands what I am saying above, will understand
that
instead of physically cutting ever smaller diameter circles to stack
on top
of one another to build a sphere and then using some of those circles
to
build a pseudosphere, instead of physically cutting out those circles,
what
I have done above is to mentally picture each point in the sphere and
pseudosphere
of its latitude-circle and compared them as per their radius.
Now the above is important but not as important as this next step
involved.
Since I know the New Reals or AP-Reals have frontview and backview and
the
same goes for the AP-adics. I want to trace back from x = 1 and y = 1
I want
to trace back as to what the radius of the associated circle to
pseudosphere
is at these points:
1d0000....000000
0d99999....99999
0d9999.....99998
0d9999.....999997
and then further down I want to compare the radius of circle to
tractrix
at these consecutive Real points:
0d500000....0001
0d50000....0000
0d49999....9999
I compare sphere to pseudosphere stacking of latitude circles
And I also compare in AP-adics so I compare this
99999....99999
99999....999998
9999....99997
9999....99996
and further down I compare
500....00001
5000....0000
49999....999
49999....99998
I should be able to plug those numbers into the formulas and be able
to give a radius-height of the curves associated with a radius of a
circle
cross-section.
So instead of physically doing the experiment on cut out paper of ever
decreasing radius circles and stacking them to form a sphere or
pseudosphere,
here I compare them mentally in a pictorial of their formulas.
And now my counterintuitiveness of the original problem disappears and
is
mollified. Mollified because I can easily see that the set of rings or
set of
circle cutouts of the sphere cannot be the same set of rings or circle
cutouts
for the pseudosphere. But that some of the sphere rings are used in
the
pseudosphere rings, or have duplicates in both sets. For example, when
x = 1 in the sphere y = 1 also and the same is true for the
pseudosphere
at x = 1 and y = 1. But then the next lower point for the sphere is
0d999...9999 in Reals and 9999...99999 in AP-adics and at that point
the y for the sphere is very much different than the y for the
pseudosphere.
And at the next lower point which is 0d999...9998 for Reals and
999...9998 for AP-adics, the y is again very different between sphere
and pseudosphere.
And this exercise allows me to see how the operations of add, subtract
multiply and divide works in New Reals and AP-adics as per the
formulas
of circle and tractrix.
plutonium....@gmail.com
Tonight I have some beautiful results, beautiful progress in math and
science.
Imagine going to a Math Store or Math Shop were we buy a sphere
or pseudosphere and in this shop we can only buy the latitude rings
for which when we bring them home we assemble them into a sphere.
The latitude rings are the world's thinnest sheet of metal possible
and are
circles and we stack them starting with the Equator ring. Now we
only build a semisphere because the Equator serves as a base and if
we build the whole sphere it would not stand on our table but roll
and topple. So we went to the shop and bought a sphere that consists
of nothing but latitude sheets of circles for which we stack from
the longest radius which is the Equator circle and then we stack
the next largest circle and stack all the others so we end up with
the semisphere or hemisphere.
Now in the sphere noted in the prior post was a radius of 1 and the
formula is x^2 + y^2 = 1
And I wanted that sphere only the first quadrant and where the
largest latitude ring or y = 1 be transposed to x = 1 and the smallest
latitude ring to be where x =0 and y = 0. I want the transposed sphere
so to accomodate the tractrix where the tractrix y = 1 and x = 1.
Now I did some calculations and good practice of my definitions
of add, subtract, multiply and divide and exponent and square
root on New Reals (AP-Reals) and these can be duplicated on
AP-adics.
So what is the smallest Reals of 0.000...00001 then 0.0000...00002
etc etc yield for the formula of the Circle and ultimately yield for
the
radius of the rings of latitude that build the sphere?
0.01 x 0.01 = 0.0001
0.001 x 0.001 = 0.000001
0.0001 x 0.0001 = 0.00000001
this convergent sequence yields that 0.000...00001 x 0.000...0001
is going to equal 0.0000...0000L and round off is 0.0000...0001
The convergent sequence further yields:
1 - 0.000000001 = 0.999999999 sqrt is 0.99999999
Likewise for the next smallest Real of 0.0000...00002, then
0.000...0003
then 0.000...00004
0.02 x 0.02 = 0.0004
0.002 x 0.002 = 0.000004
0.0002 x 0.0002 = 0.00000004
1 - 0.00000004 = 0.99999996 sqrt is 0.99999998
0.03 x 0.03 = 0.0009
0.003 x 0.003 = 0.000009
0.0003 x 0.0003 = 0.00000009
1 - 0.00000009 = 0.99999991 sqrt is 0.99999995
0.04 x 0.04 = 0.0016
0.004 x 0.004 = 0.000016
0.0004 x 0.0004 = 0.00000016
1 - 0.00000016 = 0.999999984 sqrt is 0.99999992
So now what have we arrived at? Or what have we got from the Math Shop
on building a sphere? We have the five largest Latitude rings or
circles
We have the Equator ring of radius 1, then the next largest ring of
radius 0.9999....99999, then the next largest ring of 0.9999....99998,
then the next largest ring of 0.9999...99995, then the next largest
ring
of 0.9999...99992
Now some readers are instantly thinking or complaining that these are
only
practical spheres or machine shop spheres, but that is not true. The
above are
the pure and clean theoretical spheres. The above are the purest and
most theoretical
of any and all spheres.
So the Sphere of mathematics, the sphere we have in the purest of
imagination
is a sphere whose Equator is a circle of radius 1, and whose next
circle is of
radius 0.9999...99999, then 0.9999...9998 then 0.9999...99995, and
then
0.9999...99992
And that the sphere does not have a latitude ring or circle of
0.9999...99997 nor
0.9999...99996 but misses those two rings.
Now the question is, which one of those rings to build the sphere, is
the next
ring other than the radius 1 ring to build the pseudosphere? The
tractrix ring of
radius 1 is the same ring used to build the sphere, so the question is
at point
0.0000....00001 then 0.0000....00002 then 0.0000...00003 what rings
are produced
by the pseudosphere-tractrix formula? Well, the ring radius of the
next ring after
the 1 ring is considerably smaller in radius than the sphere ring that
follows radius
1 which you remember is the 0.9999...9999 ring. Since the tractrix-
pseudosphere
has a natural-logarithm involved, it is considerably less than a ring
of radius
0.9999....99999. So I wonder what the first five radius rings of the
pseudosphere
are in association with this given sphere of radius 1? And how many of
those
sphere rings are used in the pseudosphere building process?
So one can now see that mathematics can actually tell you point by
point what the
latitudes are of a sphere, or pseudosphere or even a ellipsoid.
Never before have we been able to make a inventory of every latitude
on a sphere or
pseudosphere or ellipsoid.
I do not know the etymology of the word "calculus" was. But what I
have done above
reminds me of Calculus, only not that of finding area or rate of
change, but of
finding a stacking volume. So the above is a form of Calculus, and it
is possible
because I know every Real Number in any given interval. The Old Reals
could never
boast like this, and they could never perform like this.
plutonium....@gmail.com
Sorry, I have not yet figured out the radius circles of latitude for
the Pseudosphere that
shows us what the radius of the latitudes are for these specific x
points:
0.00000....0001
0.00000....0002
0.00000....0003
0.00000....0004
For the sphere the latitude radius of the unit sphere in 1st quadrant,
I figured out are these:
x = 0.00000....0001 then y = 0.9999...9999
x = 0.00000....0002 then y = 0.9999....9998
x = 0.00000....0003 then y = 0.9999....99995
x = 0.00000....0004 then y = 0.9999....99992
Now if we had done the same thing for a rectangular box whose cross
section
was a square in Reals Cartesian Coordinate System we would then have
these
astounding results:
Every cross section is a square and the building of the rectangular
box in
Euclidean geometry would involve every Real as such:
x = 0.00000....0001 then y = 1.0000....0000
x = 0.00000....0002 then y = 1.0000....0000
x = 0.00000....0003 then y = 1.0000....0000
x = 0.00000....0004 then y = 1.0000.....0000
Does the reader understand the significance? Probably not.
The significance is that Reals are only native to Euclidean geometry
and in that
geometry every Real Number builds a Euclidean cross section. But when
we
put a Elliptic or Hyperbolic object in Euclidean Cartesian Coordinate
System, then
in the stacking to build the sphere or pseudosphere have cross-
sections missing.
Missing in order to build the sphere or the pseudosphere. When we
build a rectangular
box in Euclidean geomerty using the Reals then no cross-sections are
missing
for every one is needed.
But when we build the sphere by cross sections in Euclidean Cartesian
Coordinate
System using Reals, as you can see above that we are missing these
radius latitudes:
0.9999....9997
0.9999....9996
0.999....99994
0.9999...99993
Those are missing in order for Euclidean geometry to accomodate a
sphere which
is alien to the geometry since it has no lines that are "curved lines"
Now if we go to AP-adics as a Coordinate System using AP-adics as
numbers
such as the sphere surface where the equator is a x-axis and the
Greenwich longitude
the y-axis then each AP-adic Integer starting with 0000...00001 then
000...0002
all the way up to 9999...99999 has a cross section that is a circle
latitude and there
are no cross sections missing.
So what do we learn from this? We learn that in order to have a
Elliptic or Hyperbolic
geometry object in a Euclidean Coordinate System that some cross
sections must
be missing in order to create the object in the first place.
plutonium....@gmail.com
I need to spend some time on this topic, for not quite sure I have a
handle
on it myself. The topic is totally new to math and being new often can
be
very misleading. So in a difficult situation like this, it is best to
retrace myself and walk through it again and again until satisfied it
is what I think
it is. For I maybe confusing myself with function or something else
and that of
intrinsic geometrical properties.
It is reasonable to think that a 3D object is the stacking of 2D
objects. And this
is supported by the very Calculus subject itself, for it is the
stacking of picket-fences
for 2D area. So the Calculus supports what I am doing. And further,
what I am
doing supports and sheds a new light on Calculus in being able to see
the integral
and derivative at each point along the curve and to show how they are
inverses of
one another. So I am supported by the Calculus.
So what I am doing is saying that the stacking of 2D objects makes a
3D object.
And the 2D objects I work with are the smallest objects possible in
all of mathematics
because they are consecutive Reals and consecutive AP-adics. This has
never
been done before in mathematics, because in Old-Reals they never knew
the Reals
were consecutive.
I am okay so far.
Now to build a rectangular box whose cross section is a square in 3D,
that the
stacking for every consecutive Real Number from 0.000...0001 to
0.000...0002
on up to 0.9999...99999 and then to 1.0000...0000 and then to
1.0000...00001
and so on and so forth, that all of these x-axis points have a cross-
section and
where the squares are 1 sq unit so the y for every x point is going to
be 1.
So the stacking is an infinity of squares whose side is 1. So every x
point has
a y point of 1.
Okay so far.
But now we graph a circle for the intention of it giving us latitude
cross-sections
which we will build a sphere with those cross-sections.
And as pointed out in earlier posts that the largest radius cross
section latitudes
are these:
1.0000....0000
0.9999.....9999
0.9999.....99998
0.9999....9995
0.9999....99992
So the question is. Have I made a mistake in thinking, not
appreciating function
or have I found a new way of viewing mathematics? That to build a
rectangular
box in Reals has a cross-section for every Real, but when building a
sphere
in Euclidean Cartesian Coordinate System with the Reals, that the
cross-sections
must have "missing spaces" such as the 0.9999...9997 and 0.999...9996.
That when
I walk into the Math Shop to buy a latitude ring kit to build a
sphere, there will by
no latitude of 0.9999....9997 radius. And if I bought a Pseudosphere
kit, the situation
becomes even larger with missing latitude rings.
So I need to take a pause for the moment, slap myself. Ask if I have
made some
silly mistake. Or, have I been correct? That I am looking at both
numbers and geometry
on their most primitive level of existence. That I am dissecting
Numbers to be
consective Reals and dissecting geometry to the smallest geometrical
objects in
existence? So which is it, am I on the path of truth or the path of
misconception?
When discovering new things, it takes some time to sink in whether it
was folly or
truth.
I need to find out what the latitude radius rings are for the
pseudosphere between
0 and 1 that is associated with x^2 + y^2 = 1. Maybe when I do that
the air will
clear more. If the gaps in pseudosphere matches the gaps in sphere,
then I probably
erred and committed folly.
plutonium....@gmail.com
plutonium.archime...@gmail.com wrote:
> plutonium.archime...@gmail.com wrote:
> > Sorry, I have not yet figured out the radius circles of latitude for
> > the Pseudosphere that
> > shows us what the radius of the latitudes are for these specific x
> > points:
> >
> > 0.00000....0001
> > 0.00000....0002
> > 0.00000....0003
> > 0.00000....0004
> >
> > For the sphere the latitude radius of the unit sphere in 1st quadrant,
> > I figured out are these:
> >
> > x = 0.00000....0001 then y = 0.9999...9999
> > x = 0.00000....0002 then y = 0.9999....9998
> > x = 0.00000....0003 then y = 0.9999....99995
> > x = 0.00000....0004 then y = 0.9999....99992
(snipped)
>
> I need to find out what the latitude radius rings are for the
> pseudosphere between
> 0 and 1 that is associated with x^2 + y^2 = 1. Maybe when I do that
> the air will
> clear more. If the gaps in pseudosphere matches the gaps in sphere,
> then I probably
> erred and committed folly.
I think I found a shortcut, that bypasses the equation for
pseudosphere
and all the transpositions to get it in the first quadrant such that
the highest
radius is 1 located at x = 1.
A shortcut in that the sine and secant function can serve as the
sphere circle
and pseudosphere-tractrix
So for some x points:
sine 0.9999 = 0.8414 secant 0.9999 = 1.8505
sine 0.99999 = 0.84146 secant 0.99999 = 1.85078
sine 0.999999 = 0.841470 secant 0.999999 = 1.850812
sine 0.9998 = 0.8413 secant 0.9998 = 1.8502
sine 0.99998 = 0.84146 secant 0.99998 = 1.85075
sine 0.999998 = 0.841469 secant 0.999998 = 1.850809
sine 0.0001 = 0.0000999999998 secant 0.0001 = 1.
sine 0.0002 = 0.00019 secant 0.0002 = 1.00000002
sine 0.0003 = 0.00029 secant 0.0003 = 1.00000005
sine 0.0004 = 0.00039 secant 0.0004 = 1.00000008
sine 0.0005 = 0.00049 secant 0.0005 = 1.00000013
So I substituted the trigonometric function of sine for a sphere
and the secant function for the pseudosphere and anyone
can sort of see that the sine restricted between 0 and 1 x-axis
is sort of like a circle, more like an ellipse, and that the secant
is very much like a hyperbolic curve of concave inward instead
of concave outward as the sine. And with the secant I merely
subtract 1 to give me the *Latitude circle radius*
So what does the above tell me or indicate to me as to what
latitude rings are missing in order to build a sphere and
pseudosphere?
The above indicates to me that very many more latitude circles
are missing in order to build the Pseudosphere because the
secant above has missing 1.00000003 and 1.00000004 and in the
space of what would be Six Consecutive Reals from 0.000...000
to 0.0000...0005 that the Secant has missing nine latitude rings
whereas the Sine has none missing.
The above is good news and supportive of the idea that I am on the
correct
path and that what I am doing here is the working of the smallest
intervals
of Real Numbers and the smallest geometrical objects. A metaphor to
physics
would be that I am doing to mathematics what quantum mechanics did to
physics
come 1901 when physics had no idea of the small scale physics and then
Quantum
Mechanics opened a window into the smallest of physics. And with the
AP-Reals
and AP-adics, I am opening a window into the smallest of Number
intervals and
the smallest of geometry objects.
plutonium....@gmail.com
plutonium.archime...@gmail.com wrote:
(snipped)
>
> A shortcut in that the sine and secant function can serve as the
> sphere circle
> and pseudosphere-tractrix
>
> So for some x points:
>
> sine 0.9999 = 0.8414 secant 0.9999 = 1.8505
> sine 0.99999 = 0.84146 secant 0.99999 = 1.85078
> sine 0.999999 = 0.841470 secant 0.999999 = 1.850812
>
> sine 0.9998 = 0.8413 secant 0.9998 = 1.8502
> sine 0.99998 = 0.84146 secant 0.99998 = 1.85075
> sine 0.999998 = 0.841469 secant 0.999998 = 1.850809
>
>
> sine 0.0001 = 0.0000999999998 secant 0.0001 = 1.
> sine 0.0002 = 0.00019 secant 0.0002 = 1.00000002
> sine 0.0003 = 0.00029 secant 0.0003 = 1.00000005
> sine 0.0004 = 0.00039 secant 0.0004 = 1.00000008
> sine 0.0005 = 0.00049 secant 0.0005 = 1.00000013
>
I am satisfied that what I am doing is correct and not a
misunderstanding.
So let me talk about what the above means for mathematics.
It all starts with ALL Possible Digit Arrangements where Euclidean
geometry Reals are All Possible Digit Arrangements of Reals
and where Elliptic and Hyperbolic Geometry are All Possible Digit
Arrangements of AP-adics.
All Possible Digit Arrangements forces the Reals to be Consecutive
Reals, meaning they are countable for every Real has a successor
and predecessor. Which means every two consecutive Reals
has a gap between them and is filled by a L suffix remainder
for example 0.000...00001 then 0.0000....00002 and between them
is 0.0000...00001L
Now imagine a Mathematics Store which you can buy a 3D sphere
or rectangular box in a kit form where it is cross sections that you
put together to build the sphere or pseudosphere or rectangular box.
So you go to the store and this store is a Euclidean Store for 3D
kits and you buy a sphere kit and a pseudosphere kit and a
rectangular box kit. Each kit has an infinity of cross sections
but each kit is not equinumerous. Example the Counting Numbers
infinity is the same as the infinity of Even Numbers but the
Counting Numbers are more numerous. So the Counting Numbers
are not equinumerous with the Even Numbers.
So when I get home to put together the Rectangular box kit cross
sections there will be 999....99999 cross sections for the open-
interval
of (0,1) but as for the sphere kit there will be I am guessing only
74999....9999 cross sections and for the Pseudosphere kit there
will by much less cross sections and guessing 6666....66666 cross
sections.
Why is that? How can that be so? It is because in Euclidean geometry
straightlines are truly what we think of as straightlines and so a 3D
object
that has straightlines will have a 999...9999 number of crosssections,
but
that a curved lined object such as a sphere or hyperboloid in
Euclidean
geometry to be able to have it as a 3D object means that some of the
cross sections of a total possible kit of 9999....99999 have to be
removed
or missing in order for the kit to create and build that curved object
in
Euclidean 3D space.
Likewise, if I went to the Math Shop or Math Store in Elliptic
Geometry and
purchased a kit to build 3D sphere and 3D rectangular box, then the
sphere
kit in 3D Elliptic would have 9999....99999 cross sections but the
Rectangular
box kit would have (I am guessing) only 6666....66666 cross sections.
So the overall picture is that in order to have a 3D object in
Euclidean geometry
that is containing curves, then the Euclidean Geometry must delete
cross sections
in order for the NonEuclidean object to exist in that space.
Let me also comment on the nature of the human mind as per this topic.
We can
visualize 4D by thinking of time as the 4th dimension. But none of us
is able to
picture 3D NonEuclidean geometry, let alone 4D NonEuclidean goemetry.
So
what I have done for 3D Elliptic and 3D Hyperbolic geometry is a tiny
step forward
into understanding of 3D NonEuclidean geometry. That there are cross-
sections
missing in order to have a sphere or pseudosphere in Euclidean 3D
geometry.
plutonium....@gmail.com
FrontView/
BackView
Chapter 2: Defining finite versus infinite, and continuous versus
discontinuous
Chapter 3 : Defining Reals & AP-adics
Chapter 4 : Euclidean Geometry = union of Elliptic & Hyperbolic
Geometries
Chapter 5 : Defining the operations add, subtract, multiply, divide,
exponentiation,
roots, factorial
Chapter 6 : Revising the Calculus where Limit is thrown out along
with
"absolute continuity"
Chapter 7 : Algebraic structure of Reals and AP-adics
Chapter 8 : 3D Elliptic and Hyperbolic geometry as cross-section
stacking
This book has become just one huge pile, having lost track of which
chapter I
am writing for. So I stopped and took the time to compile the
chapters.
Since I am on the subject of sine, secant etc I thought it appropriate
to discuss
continuity and discontinuity as per Gilbert Strang "Calculus" 1991 on
page 86
where he discusses f(x) = sin (1/x) and shows a huge black column as
the
function gets close to 0. Strang assumes correctly that the reason for
the
discontinuity is because the (1/x) gets so large so fast.
I agree with Strang that it is discontinuous as it approaches 0. But I
disagree as
to the reasoning why it is discontinuous. And this is a problem that
will plague
every function of division. And it seems as though division is the
worrisome
operation in calculus.
Strang of course is grounded in the Old-Reals and being thus grounded,
he
would say sin(1/x) is discontinuous as it approaches 0 because of the
wild
fluctuations concurrent with the huge numbers. I would say it is
discontinuous
for the simple reason that Reals have to be a finite string leftwards.
For me, the closest Real Number to 0 is 0.0000.....000001 which is a 1
in the
10^(-)9999....999998 place-value (actually I am not sure whether the
exponent
should be 99999....9999 or 9999....99998 but let me leave that for
another post).
And the next Real number is 0.0000...000002. Now Strang believes it is
discontinuous because 1/0.000...0001 is 10000....00000 and
1/0.0000....0002
is 20000....000000 which is so large of numbers that Strang does not
even
perceive them as members of his Old-Reals but so large that Strang
just calls
them "points at infinity or infinitesimals based on the Limit concept"
and thus
the function is so wild that it cannot have a sin value close to 0.
But the reason I say sin(1/x) is discontinuous near 0 is because Reals
have
to have a finite portion leftwards and when you divide
1/0.0000...00001 that
number is a infinite string leftward so it is no longer a Real Number.
Now here is the important question to ask Gilbert Strang versus
Archimedes
Plutonium. The question is where does Strang's picture of the function
y = sin(1/x) become a normal function away from that blackened area
near
0? Is it the Old-Real 0.1 for Strang or is it 0.09 or is it 0.00005
for Strang?
So Strang with his Old-Reals has to answer that question for he
obviously
graphs the function.
As for me, with New-Reals the answer lies in how I define those three
dots
that indicate infinite string and how I define finite versus infinite.
For example: 0.00000....00001 would be discontinuous for sin(1/x)
but the New Real 0.000001....0000 would be continuous for it is
10^(-)6 place value of a finite portion when used in sin(1/x)
So in New-Reals the function sin(1/x) becomes continuous from
10^(-)196 forwards. Where the exponent 196 is a Planck number
in physics of the smallest distance.
plutonium....@gmail.com
> obviously
> graphs the function.
>
I have another old Calculus textbook handy. Written by Ellis & Gulick,
3rd ed., 1986, Calculus with Analytic Geometry.
I do not want to "appear" to be picking on just Strang. And funny how
it is that continuity is discussed on page 86 of Strang's book and
continuity of the function 1/x is discussed on page 85 of Ellis and
Gulick. As if all Calculus textbooks were written in sync with some
robotic rhythm. I guess when writing a Calculus textbook every writer
takes his old favorite and just rewrites the pages so the same
chapters
and same concepts only different authors.
Strang on page 86 discusses the discontinuity of y = 1/x^2
Ellis and Gulick on page 85 discusses the discontinuity of y = 1/x
They both go into the concept of Limit as handling the discontinuity
and obviously both use the Old-Reals and to them the discontinuity
is merely a point-wise discontinuity at 0.
But with New Reals we have new insights and new knowledge.
Strang shows us on page 86 that the function sin(1/x) is a band-wise
discontinuity stretching outwards from the 0 point on the x-axis to
about 10^(-)196 of a band of discontinuity.
You see the problem? Strang, Ellis and Gulick have the Old Reals in
mind where there is an infinity of Reals between any two given Reals,
or as the coined phrase I use is "absolute continuity".
Well, in New Reals the Reals are discrete when we microscope in on
them
and the Real after 0 is 0.0000....00001 and the next one is
0.0000....00002
so there are gaps and holes between consecutive Reals.
But the mistake that Strang, Ellis and Gulick make is a mistake that
they should
have been cognizant of even before I ever posted about the New Reals.
Strang, Ellis and Gulick are very much aware that Reals, their Old
Reals have
a finite string leftward of the decimal point and an infinite string
rightwards.
So what happens when you divide 1 by say the irrational number
0.1011001110001111..... and the answer is that you end up with another
infinite string, but since it is division that you end up with a
number that defies
your very definition of a Real Number as a finite string leftward.
So does the above not only sink the functions 1/x and 1/x^2 but nearly
every
function that has a division operation involved? Is every function
with division
discontinuous throughout their graphs?
Well it may appear gloomy for Strang, Ellis and Gulick and everyone in
Calculus
but there is a simple remedy. The remedy is what Physics has for the
concept
of duality. In physics matter is both particle and wave
simultaneously. And a number
in mathematics is both finite and infinite simultaneously. So a
duality of
finite to infinite and a duality of discrete with continuous
simultaneously solves the
troubles.
We simply say that enough of the graph of 1/x or 1/x^2 looks like what
Strang
and Ellis and Gulick have graphed that we pay no attention to the
myriad holes
or gaps of discontinuity because the Reals already have holes and gaps
between
every two consecutive Reals.
P.S. So I am not berating Strang or Ellis and Gulick since the general
understanding
of the Old Reals was "absolute continuity". But I am alarmed that
these gentlemen
should have had enough of a logical mind to realize that some of those
divisions
ended up with infinite strings to the leftward when their Old Reals
were defined as
"finite leftward strings". So like a fire alarm, the alarm should have
gone off on them.
> As for me, with New-Reals the answer lies in how I define those three
> dots
> that indicate infinite string and how I define finite versus infinite.
>
> For example: 0.00000....00001 would be discontinuous for sin(1/x)
> but the New Real 0.000001....0000 would be continuous for it is
> 10^(-)6 place value of a finite portion when used in sin(1/x)
> So in New-Reals the function sin(1/x) becomes continuous from
> 10^(-)196 forwards. Where the exponent 196 is a Planck number
> in physics of the smallest distance.
>
I suspect our concept of function is now very much eroded since the
discovery of New Reals. But what has not eroded and stands very
bright and shining and gives so much more meaning to mathematics
is the concept of All Possible Digit Arrangements. That is the most
beautiful concept for the last 2 centuries, not just the last 2
decades.
But that concept will never dimish or get rusty or corroded but
concepts
of the old math such as "limit" such as "absolute continuity, that
between
any two Reals is a third Real", those concepts were build on
quicksand.
plutonium....@gmail.com
It starts with the AP-adics as All Possible Digit Arrangements
which delivers the largest integer in the world as
9999....99999 which is an infinite integer and there are
no more integers beyond 9999....99999 so what is that
number expressed in decimal form of exponentiation?
I have put this idea off until now since I really do not have
a final answer as of yet and said it was 10^999...99998
power for the digit 7 in the number 7999....9999.
So here is my problem. All Possible Digit Arrangements
forces a number such as 9999...9999 to exist and be the
last and largest integer. So how do I denote that last and
largest integer of say its 7 digit in the number 799...999?
Is that digit 7 in the 10^999...9998 place-value?
Well, whatever place-value the "7" digit is, it has to transfer
to the New Reals where the number 0.9999...9997 digit is
in the 10^(-) place value. So is it a exponent power of
9999...99998? It could not be 999...9999 exponent for that
would mean 10^9999...9999 had to be larger than 999...99
itself, would it not?
There is nothing in the finite realm that can serve as a
model for what the exponent must be. Let us suppose
1,000 was the South Pole and that 999 was the last and
largest number, then all we would need is three
place-values of -,-,- where we fill in each of those three with
all possible arrangements of the ten digits and so
10^2 covers the entire domain. So how does 2 relate to
999? Or how does 3 relate to 9999, or 4 relate to 99999?
Seems to me that an infinite number for the exponent is
required and that no finite number as exponent can cover
All Possible Digit Arrangements in infinity.
So can I say that 799...999 has a 7 in the 10^9999...99998
place value? And thus say that
0.999...9997 has a 7 in the 10^(-)9999...9998
place value?
Maybe an argument case can be made for the number
10000....0000 which is about 10% of 9999...9999 lacking
only one unit from exactly 10%. So can we argue that
7999...99999 is in the 10^1000...0000 place value
and that 7 in 0.9999...9997 is in the 10^(-)1000...000
place value?
Maybe what I argued recently about the function
sin(1/x) with its band of discontinuity near 0, not because
of wild fluctuations, but because of a breaking of the very
definition of a Real Number as only a "finite portion
leftwards of the decimal point". We cannot violate definitions
of Real Number. And so, maybe exponentiation
is tied to this problem of sin(1/x) as to where does the
band of discontinuity start? Is it a distance of 10^(-)300
from 0? Where 10^(-)300 is a Planck distance that is
the smallest distance of noteworthiness in physics?
So do I link finiteness in Physics with infinity in mathematics?
Seems reasonable to me.
plutonium....@gmail.com
but I did not
give any examples of "finite AP-adics" multiplied by infinite AP-adics
and I should spend
a considerable time on this aspect. And today is a good day because
there is a blizzard
outside for which the weather forecasters, forecasted 8 hours too
early.
I said the AP-adics are numbers on the hemisphere of a sphere where
000...0001 is
one unit away from the North Pole and 9999...9999 is one unit away
from the
South Pole.
And in multiplication I gave examples such as 999...9999 x 9999...9999
and I said multiplication is well behaved since the product ends up
being a
number between 1 and 999...9999. That is because I breakdown these
infinite strings as a percentage:
999...9999 is over 99% of the way to the South Pole
5000...0000 is exactly 50% of the way to the South Pole
1111...1111 is approx 11% away from the North Pole
1000....0000 is precisely 10% away from the North Pole
0000.....00088888 is so close to the North Pole that we consider it 0%
away
Now we can refer to the New Reals for multiplication so what is
0.9999...9999 x 0.000...0001 and then ask
what is 99999....9999 x 0000...00001?
Now most would think that 9999....9999 x 0000....00001 is going to
be 999...9999 because, well, they look upon 0000...00001 as 1 and as
the identity to multiplication.
But in Reals what is say 0.01 x 0.09 and it is not 0.09 is it. It is
0.0009
And if we take say 99% and multiply by 0% we do not end up with
99% do we.
So the answer is in 999...9999 x 0000....0001 is the convergence
sequence
01 x 99 = 099
001 x 999 = 00999
0001 x 9999 = 0009999
So the Sequence Convergence to 999...9999 x 0000...00001 =
0000.....999
or barely moved beyond the North Pole of about 0%
This is different than what I wrote in the first edition of this book
and this
also impinges heavily on the definition of a prime number in AP-adics
Now in Reals, the multiplication of any two Reals between 0 and 1 such
as
0.0000....0001 x 0.9999....99999 has no such trouble finding that the
answer
is 0.0000....000L in other words a number so much smaller than even
0.0000...00001 itself, for the answer is a Fragment of a Real between
that of
0.0000...0000 and 0.0000...00001 which I denote as 0.0000...0000L
Now here it becomes discernable to me that the multiplicative identity
in Reals
as 1 is called into question.
Good thing I save Algebra for near the end.
But here, the exponentiation and roots and factorial operations makes
it
discernable that 1 behaves like 0 and 0 behaves like 1 when we
translate from
Reals into AP-adics and vice versa.
In Reals successive Roots converge to the number 1. In AP-adics
successive
roots converges to 0. In Reals successive exponentiation becomes a
larger
number due to the finite string, but in AP-adics successive
exponentiation
approaches zero.
plutonium....@gmail.com
perform and simple
to describe. It is as easy as multiplication on Reals in the open
interval of 0 to 1.
So that for example given two Reals in that interval say 0.5555.... x
0.67777...
is that of 55% x 67% and the answer is 0.37.... or 37% Likewise in AP-
adics
of two infinite strings of 5555....55555 x 67777...7777 where one is
55% beyond
the North Pole heading for the South Pole and 67% beyond the North
Pole
and heading for the South Pole that the multiplication of these two AP-
adics
is that of
55 x 67 = 3685
555 x 677 = 375735
5555 x 6777 = 37646235
and the Convergent Sequence is that of 37.....35
But I wanted to spend time with multipliers of 1,2,3 and the smaller
AP-adics.
In the 1st edition of this book I often made the mistake of thinking
that
3333....3333 x 3 = 9999....9999 which was a huge blunder. For that was
really 3333...3333 x 0000...00003 and the zeros cancelled out most of
the
threes to infinity. In other words the real multiplication of
3333...333 x 000...003
was this:
33 x 03 = 099
333 x 003 = 00999
3333 x 0003 = 0009999
33333 x 00003 = 000099999
And so the convergent sequence is 0000...9999 which in terms of
percentage
is a mere tiny more than 0%
Likewise for Reals in interval 0 to 1 where we have 0.0000....0003 x
0.3333...3333
is not going to be 0.9999....9999 but is going to be a number that is
a tiny bit
more than 0.0000....000009 and I denote as a Real Fragment as
0.000...009L
and in terms of percentage is 0% x 33% which would be in practical
terms
0%.
Now this may shed light on the definition of what it means to be a
prime number
for the Counting Numbers as I said that 9999....9997 is the world's
largest prime.
But I have to reconsider what is 9999....9999/0000...00003
If I look at the Reals in the open interval 0 to 1 with
0.9999...999/0.0000...0003
Do I look at it as 99% divided by 0% And then instead if I had
0.9999...9999/0.3000...0000 would I look at that as 99% divided by
30% which would be 330%
And then the bigger picture of division by 000...0002 as even or odd
AP-adics,
for I would not want to lose the concept of even and odd number.
So I need to pause here to straighten this out.
plutonium....@gmail.com
the boys, or another
phrase-- the devil is in the details.
It is no secret that I am at a crossroads of heavy problems here. I
have a division that
is acting up and misbehaving in AP-adics and Reals. I have a number in
Reals
as 0.0000...0001 and in AP-adics as 0000....00001 that looks and acts
as though they
are 1 but cannot be 1. And what caused this latest mess is the
multiplication of
0000...00001 in AP-adics with 9999...9999 so is the answer 9999...9999
or is it
closer to zero?
After thinking about it all night long, I think I come to a reasonable
fixer.
In messy spots like this, I look and see what I want to save above all
else
and what I want to save above all else is that of All Possible Digit
Arrangements.
The Old Reals never had a number like 0.0000....0001 but the New Reals
are All Possible Digit Arrangements and must have such a number.
My fix for the problem involves geometry and area for multiplication
and then I
define division as the sides of the geometrical object for which the
area is begot.
For example in New Reals of the closed interval 0 to 1 is represented
as this
square:
_______________
| |
| |
| |
| |
| |
| |
|______________|
0 1
Now if we want the area we have 1 x 1 = entire square
But say we want 0.9999...999 x 0.99999....9999
which would be this:
_______________
--------------------------
| ||
| ||
| ||
| ||
| ||
| ||
|______________||
0 1
Where we have a sliver on the x-axis and a sliver on the y-axis that
is
1x1 -(9999...9999 x 9999....9999)
So what is this multiplication in New Reals of 0.0000...0001 x
0.9999...9999
and it is the area of a sliver of the above box such as this:
_
||
||
||
||
||
||
|-|
Now instead of Reals and Cartesian Coordinate System I can do the same
for AP-adics using the hemisphere of a Cylinder where the x-axis is a
Greenwich
longitude and where the y-axis is the semicircle North Pole.
So the question in AP-adics as Elliptic Geometry, what is the
multiplication of
0000....00001 x 9999...99999 ?
In the first edition of this book I thought that answer was
9999...9999 but I now
have realized that was a mistake and the answer is 0000.....9999 or 0%
distance
from North Pole to South Pole.
So my problem is how to reconcile 0000....00001 and is it 1 of Reals?
And is
there a multiplicative identity in AP-adics and what about
0.000...0001 in New
Reals. Tough problems, but I think they are manageably solved. And I
believe
the solution is to tie multiplication with a geometrical-
multiplication such as
the area of figures.
In the solution, I believe I can demonstrate why division by 0 is
undefined
and why exponent 0 delivers 1 and why multiplication by 0 delivers
zero.
Of utmost importance is to hold onto All Possible Digit Arrangements.
plutonium....@gmail.com
multiplication
using percentages as a the basis of the multiplication. I am not using
percentage
as a basis but as a guide. I am using geometry as a basis.
I shall refer to this square figure for Reals and for AP-adics I shall
refer to a
half-cylinder where the x-axis is a straightline and where the y-axis
is
the half-circle so these figures can be visualized for both and
interchanged.
______________
| |
| |
| |
| |
| |
|_____________ |
0 1
In New-Reals let us take for example 0.50000...001 x 0.6000...0001
would be this figure
______________
| |
|______ |
| | |
| | |
| | |
|______|______ |
0 1
Where the area is 0.5000...0001 x 0.600...0001 = 0.3000...00001
where the area of 50% x 60% is roughly 30% not counting that
tiny "1".
So in AP-adics where the area is on the surface of a half cylinder
we have the same results of 50000....0001 x 6000...00001
= 30000....00001 and interpret it as 50% along the cylinder
longitude from North Pole (circle) and 60% along North Pole
half-circle produces a cylinder area of roughly 30% of the half
cylinder surface.
So area and percentage are great aids in understanding but
the understanding I want to achieve are these two troublesome
numbers in AP-adics:
0000....00001
10000....0000
and this troublesome number in New Reals:
0.000....00001
They are troublesome because I need a multiplicative identity in AP-
adics
and cannot discern if 000...0001 is that number or whether I have to
make
the South Pole the imaginary 1. I have called the South Pole many
things
prior to this as pi and 2pi and e. But maybe the South Pole is
imaginary
1 and the North Pole is imaginary 0.
The trouble with the Reals 0.0000....0001 is that in multiplication of
such:
0.0000...00001 x 0.99999...99999 is about 0% in area
but in AP-adics 000...00001 x 9999....9999 is troublesome. Is it
9999...9999 or is it 0000....9999 which is near 0%.
When we look at Hensel P-adics, his multiplicative identity is
0000...00001 and where 9999....99999 x 0000....00001 is equal
to 9999....9999. So the AP-adics will differ in a major way from the
Hensel P-adics.
And in Hensel P-adics, they would have 000...0003 x 3333...3333
as equaling 9999...9999 or (-1) but in AP-adics
0000...00003 x 3333....333333
_______________
| |
| |
| |
| |
|| |
||_____________ |
Where the above picture depicts a half cylinder surface and the point
000...0003 just a tiny bit on the x-axis as the North Pole half circle
and extending 33% of the distance on the y-axis to the South Pole
is depicted as that tiny sliver of area.
So that in AP-adics 0000...00003 x 3333....33333 is not equal
to 9999...9999 but is equal to 0000....9999 as begot by this
Convergence Sequence:
03 x 33 = 099
003 x 333 = 00999
0003 x 3333 = 0009999
ad infinitum
So the area of the sliver is about 0% of the surface and the
multiplication
is that of 0% x 33% should equal not 99% but that of 0%.
So here is a vast difference in the Hensel definition of
multiplication in
P-adics versus multiplication in AP-adics.
So I have to hammer out the New Reals troublespot of 0.000...00001
and what is the multiplicative identity in AP-adics.
I have a gut feeling that since the multiplicative identity in Reals
is
the number 1 which is the successor of 0.99999....9999 that the
South Pole which is one unit distance away from 9999....9999
in AP-adics that the South Pole is an "imaginary 1 as multiplicative
identity"
Now the troubles with division can be worked out by defining division
as the sides of the "figures area" begot in multiplying. So that
when we multiply in New Reals 0.5000...0001 x 0.6000...0001
and get 0.3000...0001 that if we divided 0.3000....00001 by
0.6000...0001 we end up with 0.5000...0001.
So I am looking to define division as "obtaining the sides of the
figures-area"
plutonium....@gmail.com
because of the
troubles in defining multiplication and division and having a
multiplicative identity
in AP-adics.
A few questions. Since this book espouses that Euclidean Geometry is
the broken symmetry of two geometries of Elliptic and Hyperbolic. In
Physics
if you wait a few minutes with a neutron it breaks its symmetry apart
and becomes
a proton and electron with some energy. So in Physics we have
Neutron = proton + electron
In math, similarly we have: Euclidean Geometry = Elliptic union
Hyperbolic geometry
Now the New Reals allow for a Calculus and explains the Calculus so
much better
than the Old-Reals where they could not even explain why derivative is
inverse
to integral. But the Calculus is not transferable to the Elliptic
Geometry nor
Hyperbolic Geometry because of "lines" in NonEuclidean Geometry.
So I have a slew of questions about Galois Group theory of Field
Structure.
How closely connected is Galois Field theory with the Calculus?
Can we have a Calculus on numbers which are not a Field?
I do not know if anyone ever thought about that. Or if anyone ever
thought about
the links between Calculus and Field theory.
Because if Euclidean Geometry is the only geometry that can have a
Calculus
and since the Reals are a Field, then must the Elliptic Geometry and
Hyperbolic
Geometry as solo, not be a Field since they cannot have a Calculus?
I wondering about this as to my efforts in forcing Elliptic geometry
to be a Field.
So if Euclidean Geometry is the only geometry that allows a Calculus
and the
Euclidean Geometry is the Algebraic Field on Reals. And since Elliptic
and Hyperbolic
cannot have a Calculus, then does that preclude them from being a
Algebraic
Field?
If I had some sort of answer to that question, I would not spend so
much time in
forcing the Elliptic and Hyperbolic geometry AP-adics to be a Field.
I doubt that any mathematician before this post has ever even
considered that thought
of the relationship of Calculus to Field theory. That you have to have
a Field in
order to have a Calculus. But once you have a Field, it does not
suffice to form a
Calculus. And if you have a number system that cannot accomodate a
Calculus, then
is it necessarily not a Field.
plutonium....@gmail.com
AP-adics. I should
start it off with one big question. The question of what should ALL
Possible Digit Arrangements
yield as an Algebraic Structure. For that concept alone is bigger than
the subject of
Mathematical Algebra. Algebra is merely the defining of operations
upon numbers.
But you cannot have Algebra if you do not have numbers. So numbers
come first
and thus All Possible Digit Arrangements precedes and is bigger than
is algebra.
So in the last post, I was questioning the interrelationship of
Calculus with Algebra.
Wondering how important Calculus was in Algebra and vice versa. For it
seems as
though Calculus of finding tangents and slopes and area are
operations.
But the very best way to start the Chapter of Algebra on Reals and AP-
adics is
to ask the first and most important question of all. How does All
Possible
Digit Arrangements affect Algebra? Does that concept require there to
be
Groups, Rings, Fields etc etc? I think that a retracing of actual math
history
is very instructive in that the Group, Ring and Field were devised and
discovered
to solve the quintic problem. The Quintic problem can be considered
what? Is it
a problem of Euclidean Geometry? Or was the quintic a curve in space
and thus
a NonEuclidean problem of the past?
My next post is to ask what are these multiplications in New Reals.
1 x 1 = 1
1 x 0.0000....00001 = ?
0.0000...0001 x 0.0000....00001 = 0.0000...00001
Answers to those questions could upset the entire subject of Algebra.
The
Algebra of the past as assembled by Galois.
That the Algebra of the future is dependent not on some ancient riddle
of a
quintic problem, but an Algebra of the future that is based on All
Possible
Digit Arrangements as the definition of Reals in the first place.
plutonium....@gmail.com
different
formats will project the final image. As long as the reader can
understand where
the image is intended is enough. As they say, a picture speaks a
thousand
words. And as long as I can communicate the ideas in a picture means I
am
more than half way to the end goal of clear understanding.
This is a very important post to this book and to this adventure in
understanding
because if I do not get a Algebra for multiplication on New Reals and
AP-adics
then I have a long way to go to the clear understanding.
So in this post I examine these four multiplications in Reals:
0.000...0001 x 0.000...0001 = I expect 0.0000...0001
1 x 1 = 1
0.000...0001 x 1 = well, this is the big question
0 / 0 = well, I expect that to be 0, as well as 1/0 = 0
Now those four questions are also put to the AP-adics as these:
000...0001 x 0000....00001 = 0000...0001
South Pole x South Pole as to it being imaginary 1
0000....00001 x 99999...99999 = I expect 0000....9999
0000....00001 x South Pole = ?
10000....0000 x 0000....000001 = I expect the same as what
the Reals of 0.1000.... x 0.0000....0001 equals symmetrically
_____________
| |
| |
| |
| |
| |
|____________ |
0 1
So I have the square in the first quadrant of Euclidean Cartesian
Coordinate
System in the closed interval 0 to 1
And obviously
1 x 1 is a 1 square area so that multiplication is alright so far
0.999...9999 x 0.999...9999 would be the next largest square in
that above picture and its area would be 0.9999.....1
Now the smallest area in that square above would be 0.0000...0001
x 0.0000...00001 and be the tiniest square in that 1 x 1 square
and have an area of 0.0000....00001 square.
Now the next largest area in that above square would be a rectangle
of 0.000...0001 x 0.000...0002 and be a rectangle whose area
is 0.000....0002 square
The 0.000...0001 x 0.0000....0001 multiplication looks like this tiny
square
_
|_|
And the 0.000...0001 x 0.000...0002 area looks like this tiny
rectangle:
_
| |
|_|
Alright so far and everything is alright with the old math and the Old
Reals
except when we ponder about 0.0000....00001 x 1
So what I have endeavored to do is define multiplication as that of
Area
with Percentage of area in a given Space and the space involved here
is the 1 x 1 square.
So that 0.0000....0001 x 0.0000...00001 = 0.000...0001
and where 1 x 1 is the entire area of the Space and where
its next largest area is 99% x 99% which would be 0.9999...9999 x
0.9999....99999
But the huge question all hinges on what is
0.0000....0001 x 1 = ?
We can describe it as the area of a long slender rectangle as such:
___________
|| |
|| |
|| |
|| |
||___________|
Do you see that long slender rectangle to the very left of the image?
That long slender rectangle is representative of 0.0000....0001 x 1
It obviously has an area so it must have an answer as to what its
product is
Now an easier question is 0.1000...0000 x 1 which would be
10% x 100% or 0.10000....0000 and we can picture it as
the square of 1 x 1 divided into ten equal columns and that
one of those ten columns is 0.1000...000 x 1
So, what is 0.000....00001 x 1 = ?
It is a column and has an area but what area and what product?
If we go to AP-adics we have the similar question, the symmetric
question of 000...00001 x 9999....9999 and the answer there
would fool most every reader into thinking it was 9999...9999 when
in fact the zeroes cancel out most of the "nines" leaving
the answer as 00000....99999
So in the Reals, is the product 1 x 0.0000....00001 that of
0.0000....1000
As I said repeatedly above, the multiplication of 1 x 0.0000...0001
has an area and must be accounted for.
Ultimately the question will come back around asking whether Algebra
of Group and Ring and Field are adequate theoretical concepts?
In that the concept of All Possible Digit Arrangements is a superior
concept over the old and ancient quintic with Galois Group theory.
Why try to hammer down a vastly superior concept of All Possible
Digit Arrangements in order to satisfy some ancient math of the
quintic?
David R Tribble
> 0.999...9999 x 0.999...9999 would be the next largest square in
> that above picture and its area would be 0.9999.....1
So sqrt(0.999...1) = 0.999...999 ?
What is sqrt(0.999...999)?
> Now the smallest area in that square above would be 0.0000...0001
> x 0.0000...00001 and be the tiniest square in that 1 x 1 square
> and have an area of 0.0000....00001 square.
So sqrt(0.000...001) = 0.000...001 ?
That would give you an x such that sqrt(x) = x and x > 0.
I also wonder, what is sqrt(999...999)?
plutonium....@gmail.com
David R Tribble wrote:
> Archimedes Plutonium wrote:
> > 0.999...9999 x 0.999...9999 would be the next largest square in
> > that above picture and its area would be 0.9999.....1
>
> So sqrt(0.999...1) = 0.999...999 ?
>
> What is sqrt(0.999...999)?
>
Operations in either New Reals or AP-adics are always Sequence
Convergences:
sqrt0.9 = 0.94
sqrt0.99 = 0.994
sqrt0.999 = 0.9994
which implies sqrt 0.9999...9999 = 0.9999....9999
plutonium....@gmail.com
David R Tribble wrote:
> Archimedes Plutonium wrote:
> > 0.999...9999 x 0.999...9999 would be the next largest square in
> > that above picture and its area would be 0.9999.....1
>
> So sqrt(0.999...1) = 0.999...999 ?
One note about 0.999...1 for that is only a abbreviated answer
of 0.9999...9999 x 0.9999...999 because as you can see
in these:
0.99 x 0.99 = 0.9801
0.999 x 0.999 = 0.998001
that there are some zeroes digits before the "1" which I chose
to not write.
No, the sqrt of 0.9999....1 would be
a Convergence Sequence of such:
sqrt 0.91 = 0.95
sqrt 0.991 = 0.995
sqrt 0.9991 = 0.9995
So the final Convergence would be 0.9999....5
>
> What is sqrt(0.999...999)?
>
Operations in either New Reals or AP-adics are always Sequence
Convergences:
sqrt0.9 = 0.94
sqrt0.99 = 0.994
sqrt0.999 = 0.9994
which implies sqrt 0.9999...9999 = 0.9999....9999
Archimedes Plutonium
plutonium....@gmail.com
Fields,
how important is it for there to be a "multiplicative identity"
because the Reals
do not have such a specimen. Although the number "1" comes close to
acting
and behaving as such but from time to time with very tiny numbers
breaks apart.
Keep in mind that the Galois theory was invented to conquer the
quintic of
old and ancient mathematics, but was it really necessary to have a
multiplicative identity for a Field or Ring?
And that raises an interesting question about Algebra altogether with
not only
the Quintic but with Fermat's Last Theorem on Counting Numbers.
In All Possible Digit Arrangements, the Fermat's Last Theorem is full
of
counterexamples such as any of the Hensel P-adics. But what about the
quintic of old and ancient mathematics. What happens when we have the
full and true Reals to work with, not the half-deck of Reals that
Galois
worked on with Group theory? What happens when we have numbers such as
0.0000...0001 and 0.0000...0002 in the quintic of old?
So what is 1 x 0.0000...00001 = ?
Well, using AP-adics we get a idea of what it is in:
000...00001 x 9999...9999 = 00000....9999
Let me call those three infinity dots as infinculum, so there is an
infinity of zeroes then the infinculum and then a string of 9s that
is infinite also. And what I propose is that this number
0000....99999 or better yet this number 0000....10000 is the
smallest infinite string.
So that the multiplication of 1 x 0.0000....0001 = 0.0000.....10000
Now that is a historic discovery because it is the first time anyone
has shown that the Reals, whether the old Dedekind Reals or the
new AP-Reals, where the number "1" breaks down and does not
deliver a multiplicative identity.
Every college and university that teaches mathematics would say that
all multiplications involving 1 yield a identity but in the case above
I have shown where 1 times another number yields a third new number.
And the trouble with the old math and why they never saw this coming
was because they looked upon a Real number as only half of what it
truly is as a FrontView and no BackView and because the old math
never set its foundation on rock solid ground that Reals are All
Possible
Digit Arrangements.
So there is a Real Number that is equal to 0.0000....100000
The Old mathematics missed All Possible Digit Arrangements and in
doing so they missed FrontView with BackView and in doing so, they
missed the Calculus in how the derivative is inverse to integral.
And all sorts of falsehoods crept into mathematics such as Cantor
phony baloney.
Now below is a picture of 1 x 0.0000...0001 = 0.0000...10000
___________
|| |
|| |
|| |
|| |
||___________|
0 1
That slender rectangle is 1 x 0.0000...0001 It has a area and thus
it has a product. The area cannot be that of 0.0000....0001 since
that is the area of the smallest possible square in the entire
Universe
as 0.0000...00001 x 0.0000....0001 and pictured as a tiny square
in the lower left corner near 0.
_
||
-
Poorly pictured by me above.
So the slender rectangle cannot be area of 1 for that is the area of 1
x 1.
And since it cannot be area of 0.000...0001 it must be some other
Real Number between 0 and 1 and that Real Number which has never
before been discussed in mathematics is the number 0.0000.....10000
plutonium....@gmail.com
3rd edition of this
book is put the notation in the first chapter not the 5th. But it was
these operations
of multiplication etc that prompted the notations.
Use the example of of 0.9999...9999 x 0.9999...999 because as you can
see
in these Sequence Convergence that the product:
0.99 x 0.99 = 0.9801
0.999 x 0.999 = 0.998001
0.9999 x 0.9999 = 0.99980001
etc, etc
The product is going to end up being 0.99999.....0001
And the same goes for AP-adics of 9999...9999 x 9999....9999 =
9999....00001
But can I make the notation that can bring out as much of the number
as possible? Can I have a notation that brings out the "8" digit?
Or better yet, can I have a notation that brings out the "1" digit
in 0.0000.....1...0000 where 1 x 0.000...001 = 0.0000.....1...000
Define infinity as three or more dots as such 0.999... and
call those three or more dots as the "infinitculum"
Define finite as two dots such as 99..9 to indicate say 99999999
and call the two dots as "finitculum"
So, now can I display 0.999...9999 x 0.9999....9999 much better so
that it includes the "8"? Let me try.
0.9999...9999 x 0.9999...999 = 0.9999....99800...0001
There, I believe I displayed fully what the product of 0.999...999 x
0.9999...999 was.
Now I defined the Reals and AP-adics as All Possible Digit
Arrangements
and that concept is perhaps the very most important concept in this
entire
book, perhaps in all of mathematics other than geometry.
The most foul and dead thing in present day mathematics is the cobbled
together
nature of Numbers and the Reals going through a ridiculous creation
process
where they start with Peano axioms on Counting Numbers and proceed
through
thousands of definitions for rational and irrational and Dedekind Cut.
It is like building
a city from scraps at a dump yard in a adjacent city. Whereas I
contruct the
Numbers of mathematics by simply saying ALL Possible Digit
Arrangements. Like
building a city by calling up the adjacent city and ordering millions
of concrete block
and brick and steel and cement. I build an orderly process. The old
way by the old
fogeys of mathematics has thousands of stupid and silly definitions
and cobbles
them together. I build the Reals in 5 minutes in a High School
Classroom. The Old
Math builds the Reals in a College sophmore course taking an entire
year to go
from Peano Axioms through Dedekind Cut. And in the end, mine is
correct and the
old-fogey's end up with Reals that cannot even explain why Derivative
is inverse
to Integral.
All Possible Digit Arrangements forces there to be a Largest AP-adic
which
turns out to be 9999.....99999 and forces there to be a Smallest and
Largest
Real in the Open-interval 0 to 1 as that of 0.0000....00001 and
0.9999...9999
respectively.
Now I said that such a forcing of a largest AP-adics as 9999...99999
forces
a decimal place value notation that 9999....99997 is in the
10^999...9998
place value for the "7" digit and that "1" is in the 10^(-)999...9998
place-value for the Real 0.0000....0001
Does that make sense as a notation? Well, the place-value cannot
exceed the
largest AP-adic so that in a sense we can perceive that 9999...99999
is a larger
number itself from that of 10^9999...9998. Larger because this is no
longer finite
world but an infinite world which tends to defy our common intuition.
We have a
sense that All the Counting Numbers should be a larger set than just
the Even
Counting Numbers 1,2,3,4,5,6,7, .... should seem to be larger set than
2,4,6,.... but Cantor did get a few things correct in his huge pile of
mostly mistakes.
Cantor did understand that infinity is different from the concept of
measure.
So can I represent the Real Number of
0.12345678910111213141516....9999
Or can I represent the AP-adic of 9999....1716151413121110987654321
I seem to remember a name of Champerowne's number (excuse the
spelling)
So is there a sequence of digits in there that holds every infinite
string?
Is there a string in either of those numbers-- Real or AP-adic of
this string 5555....555555 inside of 9999....987654321 ?
Well of course there is and for its successor and predecessor of
555...5556 and 5555....555554
Seems as though our commonsense is defied that a Real Number between
0 and 1 has every one of the single numbers incorporated inside itself
as a number.
It defies our common sense and let me try to display that number using
the notation.
0.123456789101112....5555...55545555...5555..5555...5556.....9999999
Of course there was a passage in there for numbers such as
1111...11112 or
7777....66666 as infinite strings.
The point is that the infinity defies our common sense, and we can
only cling to
correct concepts such as All Possible Digit Arrangements because noone
can
doubt that those are one of All Possible Digit Arrangements. And the
concept that
infinity is nonstop or never ending.
So that we can define a notation place-value of the 0.0000....00001 as
the 10^(-)
9999....99998 place-value for the "1" digit and not be worried that
the exponent
seems to be a larger number than the number itself, for it is not,
since this is
the terrain of the infinite.
Now with this new notation, let me see if I can better write what this
number is:
1 x 0.0000....00001 = 0.0000.....1000...000
Yes, that is a accurate description of what I am going to call the
"smallest infinity"
Now I have to look up what the Planck length and other numbers that
mark the
out of bounds of measurement in physics. Are they exponents of 10^60.
And how
many atoms to fill the Universe? Are they 10^180 ? I believe I posted
what the
world's largest physical number is as the number of Coulomb
Interactions to hold
together a plutonium atom was of the order of 190! (that is a
factorial) and that
number is huge.
What I am thinking is that mathematics leads us to this number, purely
from
mathematics-- 0.0000.....1000...000 or in AP-adics
0000....10000....000
So can I somehow define that "1" digit with that of Physics last
measure?
That where physics stops and ends in measuring is where infinity of
mathematics
begins, and that "1" digit in either the Reals or AP-adics is
connected to Physics
end of measure.
David R Tribble
>> What is sqrt(0.999...999)?
>
Archimedes Plutonium wrote:
> Operations in either New Reals or AP-adics are always Sequence
> Convergences:
>
> sqrt0.9 = 0.94
> sqrt0.99 = 0.994
> sqrt0.999 = 0.9994
>
> which implies sqrt 0.9999...9999 = 0.9999....9999
What happened to the 4 digit?
Using Windows Calculator, I get, where s(x) is sqrt(x):
s(0.9) = 0.94868329805051379959966806332982
s(0.99) = 0.99498743710661995473447982100121
s(0.999) = 0.99949987493746091013572606111579
s(0.9999) = 0.99994999874993749609347654199058
s(0.99999) = 0.99999499998749993749960937226560
s(0.999999) = 0.99999949999987499993749996093747
etc.
Wouldn't that imply that s(0.999...999) should be something
like 0.99999...949...99874, (whatever that could possibly mean)?
spudnik
we can toss the rest of it, because it changes, BUT
the nines get longer, to which we apply Stevin's algebra.
you know, the part where he said that
it ~converges to 0.9999...1,
was teh first time that I saw that
he wasn't just BSing us. (and,
you know what the tilde means, hereinat,
in the gnu pedagogy .-)
> What happened to the 4 digit?...
> Using Windows Calculator, I get, where s(x) is sqrt(x):
> s(0.9) = 0.94868329805051379959966806332982
> s(0.99) = 0.99498743710661995473447982100121
> s(0.999) = 0.99949987493746091013572606111579
> s(0.9999) = 0.99994999874993749609347654199058
> s(0.99999) = 0.99999499998749993749960937226560
> s(0.999999) = 0.99999949999987499993749996093747
thus:
what does the tilde (~) mean?
> From the description of Stirling-numbers 2'nd kind (see *1)
> we know, that the equation with matrix/vectors of infinite size
> V(x)~ * U = V(e^x-1)~
--Marching too Darfuria, Darfuria --
with Trickier Dick Cheeny from the N.Admin.!!
plutonium....@gmail.com
Yes, I agree, thanks for carrying it out further and surprized to see
a persistent "4".
plutonium....@gmail.com
that you
have a multiplication with "1" where the product is not a
multiplicative identity and thus
the entire structure of Reals-- the Old Reals is called into question.
1.00000....00000 x 0.0000....000001 = 0.00000....00100....00000
I am going to give this number a name as the "AP's smallest infinity".
What I mean by smallest infinity is that the predecessor of this
number
is a "finite number" as that of 0.00000......099..999. I say smallest
infinity
because we attach the largest possible number in Physics measurement
and we end up with that finite predecessor number.
But the reason for this post is not to gloat over that number but to
ask
a serious question of the old math. When Galois discovered Group
theory
to solve the quintic, how essential is "multiplicative identity" to
Group theory?
And could the quintic be solved without a multiplicative identity.
And another question while on the subject. Has anyone applied Hensel's
P-adics to the quintic? Because I have a deep seeded suspicion that
the
quintic as we know is not over with and that the quintic reminds me
alot
about Fermat's Last Theorem when P-adics are taken as the Counting
Numbers
and that FLT has a boatload of counterexamples in P-adics. So have I
resurrected
the quintic of old math and that the quintic has new life and that it
has
a different answer in light of New Reals, since the New Reals do not
have
a "absolute multiplicative identity" but rather a quasi-multiplicative-
identity.
So is anyone expert on quintic and Galois theory to know how central
is
multiplicative identity, for the above multiplication clearly shows
that old
belief is no longer true.
Ken Quirici
If what you're trying to do is find an expression that uniquely
describes s(2) then since it's irrational you can't. I'm not
sure AP cares about this fact.
But it's fascinating that it seems to develop patterns the further
down
into the calculation you go.
For example based on the calcs you show one can START a
description of s(2) by:
.9...949...98749....93749...
The repeating patterns seem to emerge out of a random chaos of digits
as you
read from the left and continue the series of calculations. However it
would
require a REAL calculator to be more helpful and I don't have one.
David R Tribble
> In the past few days I have made a remarkable discovery in mathematics
> that you have a multiplication with "1" where the product is not a
> multiplicative identity and thus
> the entire structure of Reals-- the Old Reals is called into question.
How can anything you demonstrate with your AP-reals possibly
affect the standard reals? The two are utterly different.
You are applying the false logic of:
"Here, I'll define this new system A, which is entirely different
from standard arithmetic. In this A, we can see that <some
operation with some values> exhibits <certain behavior>.
So this proves that the standard reals have a problem."
This is also known as a "straw man fallacy" or simply a
"non sequitur".
David R Tribble
> a serious question of the old math. When Galois discovered Group
> theory to solve the quintic, how essential is "multiplicative identity" to
> Group theory?
The requirements for a set S and an operation # to qualify as a
group (S,#) are:
closure, associativity, identity, and invertibility.
See these:
http://en.wikipedia.org/wiki/Group_(mathematics)
http://en.wikipedia.org/wiki/Group_theory
http://en.wikipedia.org/wiki/Galois_theory
Do your AP-reals have all these requirements?
I certainly does not look like they do for multiplication.
David R Tribble
> And could the quintic be solved without a multiplicative identity.
Since Galois' proof uses the idea of symmetric polynomials,
which require permutations that include an identity permutation,
the answer would certainly be no.
You might want to read this fine book, which has a chapter
on group theory and symmetry:
Unknown Quantity: A Real and Imaginary History of Algebra
by John Derbyshire
http://www.amazon.com/exec/obidos/ASIN/0452288533
http://product.half.ebay.com/Unknown-Quantity_W0QQprZ57087447QQtgZinfo
plutonium....@gmail.com
David R Tribble wrote:
> Archimedes Plutonium wrote:
> > In the past few days I have made a remarkable discovery in mathematics
> > that you have a multiplication with "1" where the product is not a
> > multiplicative identity and thus
> > the entire structure of Reals-- the Old Reals is called into question.
>
> How can anything you demonstrate with your AP-reals possibly
> affect the standard reals? The two are utterly different.
>
There is only one Reals. What you call "standard Reals" is only a
half baked enterprise. Physics had a similar situation as far as
what is "light". For centuries it was argued particle and wave.
Finally Quantum Mechanics wedded the two into one explanation.
I am filling in the other half of what the Old Reals missed and
throwing
out the trash of what the Old Reals added which was never true.
> You are applying the false logic of:
Funny, from a man who has little logic.
> "Here, I'll define this new system A, which is entirely different
> from standard arithmetic. In this A, we can see that <some
> operation with some values> exhibits <certain behavior>.
> So this proves that the standard reals have a problem."
>
How one week Tribble says his Old Reals are All Possible Digit
Arrangements.
Then I show a possible digit arrangement of 0.00000....0100...0000
Do you see and understand that such a number is a possible digit
arrangement
of one particular Real Number? Do you see that it was formed from
1.00000.....00000 x 0.00000.....000001
And thus and therefor the Old Reals, the Standard Reals, the Peano and
Dedekind
Reals, the Reals taught in every college and university today, have a
Real Number
that is 0.0000....00001000.....00000 and for which that Real number
defies the
existence of the number 1 as a "absolute multiplicative identity"
In the multiplication of 1 x 0.0000....000002 it also destroys 1 as
the absolute
multiplicative identity for it equals 0.00000.....000200....000000
So one week Tribble says the Standard Reals is All Possible Digit
Arrangements
and the next week, when it looks bad for his arguments, he says the
Reals
are not All Possible Digit Arrangements.
With logic of that calibre, Tribble should not open his mouth.
> This is also known as a "straw man fallacy" or simply a
> "non sequitur".
You are not an intellect but a heckler. You do not know the first
thing about
logic.
plutonium....@gmail.com
Your standard Reals do not have those properties because numbers such
as these
0.0000....00001, 0.0000....00002, 0.00000....00003 do not obey
multiplicative
identity:
1 x 0.0000....00003 = 0.0000....00300...0000
Your Standard Reals have numbers that are neither rational or
irrational such as
this Real Number 0.10000....000005
Your Standard Reals has numbers that are neither algebraic nor
transcendental
such as 0.1000....00001000.....00001
Your Standard Reals has numbers that are neither even nor odd such as
0.
The message I am conveying to you, for you are rather a slow learning
bloke,
is that your Reals you learned in college which you call the Standard
Reals
were only half baked.
And when Galois devised his Group theory, the numbers he had known
were only a dabbling fraction of what the Reals truly are. And thus,
his
entire Group theory of rings and fields are shot full of holes.
Galois Group theory is as dilapidated and broken as is the Caloric
theory of
heat when Galois was alive, and which was devised in physics.
Sure, dullard students of math like Tribble will put flawed theories
on a pedestal
and claim them as truths forever, because blokes like Tribble never
had a
math intelligence in the first place.
plutonium....@gmail.com
Judging from your posts of the past, I have a difficult time of
believing that you
possessed the above ideas in your mind, and that rather, someone else
fed you the above idea via email.
Your above post is not the usual Tribble post.
My sense of the Galois theory for Reals such as the quintic problem,
is that it is
"local mathematics" but not universal mathematics. It is not the stuff
of theorems
of mathematics but of isolated bits and pieces.
The number 1 in Reals is most often the multiplicative identity. But
in All Possible
Digit Arrangements, there are cases in which 1 fails to be the
multiplicative identity.
What this tells me, is that the concept of All Possible Digit
Arrangements is far and
vastly superior than any algebra that wants to try to describe all
those numbers.
When I make the case that the Reals have no Universal Multiplicative
Identity.
And the Reals have numbers that are neither rational nor irrational,
and have
numbers that are neither algebraic nor transcendental.
Where the Reals of Old never even had a decimal representation of 1/3
that was
true.
Then all mathematicians of this age should start to wonder that the
Galois theory
was so blown out of importance.
Perhaps within the safe and quant confines of Half-baked Reals where
everyone
goes hush quiet when they write 1/3 = 0.3333..... and never say
anything once
they write the infinitculum "...." Where the mathematician becomes not
a learned
respected person but an ostrich with head in the sand.
Under those conditions do they think they have a Galois theory that is
significant
when in fact, Galois theory is a shallow restricted and localized
mathematics.
plutonium....@gmail.com
can avoid it all
for I have that sort of discipline but I chose not to avoid it because
most readers are really
not good at mathematics or ideas, but can follow a discussion for its
anger and emotion.
For me, emotion and anger is a waste of science, but I see its use for
readers.
So to get past the emotion, and let me dwell for a moment on the
Algebra of Galois theory.
This is probably the largest rathole of mathematics of the 20th
century. Physics had
a rathole also in General Relativity and Black Holes which were thus
spawned by GR.
But let me say what Algebra for the future will likely look or
resemble. I see where
Galois theory is mostly thrown out the window as remote and isolated
pockets of
mathematics. What I call localized math or algorithmic math.
I see in the future where Algebra is focused on geometry as a
foundation. That we
have multiplication based on area of Euclidean or NonEuclidean
figures. So that
multiplication of 1 x 0.5000....00000 is half of the area of 1 x 1 and
where
1 x 0.0000....00001 is a area of a tiny and tall slender sliver that
is
0.0000....00001000.....00000
So Algebra of the future is not built for a foundation of wrestling
with some ancient
outdated problem of quintic, but where Algebra is based with a solid
foundation in
Geometry of area for multiplication.
And the trouble with Galois theory is not only was it misguided as for
foundation on
some piddly paddly quintic problem, but that it never addressed truly
major problems
of mathematics of larger subjects such as Geometry.
It grieves me to no end in my own formal education when I decided to
major in
mathematics and was accosted by having to learn those old Greek ideas
of
using only a straightedge and compass. Remember those excercises? Do
we
do that in Physics, where we spend time on pretending the Earth is
stationary
and the Sun revolving and angels flying to keep it in orbit? And yet
still in the
21st century plenty of math time is spent on straightedge compass
exercises
and where we have Algebra based on a quintic that is Medievil
mathematics.
To say that mathematicians are conservative is an understatement. They
are
antedeluvian Ordovician conservative. If not for the other sciences,
mathematicians
would still be in the horse and buggy days.
The world needs an algebra that is not grounded in the ancient past of
quintic
and Galois theory, but needs to shuck that old stuff as bygone old and
fetid
math. The world needs Algebra based on a new and greater problem. The
problem
of Geometry and Numbers based on All Possible Digit Arrangements.
Already I have shown that All Possible Digit Arrangements makes clear
and
clarifies the Calculus, showing clearly how the derivative is inverse
to integral.
And the Algebra should play a leading role in that clarity of
Calculus. But unfortunately
the Algebra is so bogged down in its ancient past of quintic and
Galois theory
that Algebra is more of a modern day math nuisance than anything
useful and
helpful.
David R Tribble
>> How can anything you demonstrate with your AP-reals possibly
>> affect the standard reals? The two are utterly different.
>
Archimedes Plutonium wrote:
> There is only one Reals. What you call "standard Reals" is only a
> half baked enterprise. [...]
> How one week Tribble says his Old Reals are All Possible Digit
> Arrangements.
You didn't understand what I wrote the first time, so why do
you keep pretending that you understand it now?
> Then I show a possible digit arrangement of 0.00000....0100...0000
> Do you see and understand that such a number is a possible digit
> arrangement of one particular Real Number?
How could I? You still have not explained what all those dots
are supposed to mean, nor how an infinite sequence of
digits can have a last digit.
David R Tribble
>> This is also known as a "straw man fallacy" or simply a
>> "non sequitur".
>
Archimedes Plutonium wrote:
> You are not an intellect but a heckler. You do not know the first
> thing about logic.
>
> Sure, dullard students of math like Tribble will put flawed theories
> on a pedestal
> and claim them as truths forever, because blokes like Tribble never
> had a math intelligence in the first place.
You really made my day, because that's the funniest thing I've
heard all week. I'm tearing up now from laughing so hard.
Thanks for that.
plutonium....@gmail.com
that the Reals
and Galois Group theory collide into ruin. That Algebra, true algebra,
on Reals
is based on Geometry, not on some old and ancient problem of quintic.
Geometry Based Algebra:
Well, first of all, let us be clear as to the purpose and function of
algebra as a
subject of mathematics. It is to define precisely the operations of
the Numbers
of mathematics. So the realm of algebra is an accurate defining of
the
operations that are used in mathematics. But algebra as a subject is
smaller
than Geometry or Numbers or even that of Probability theory.
Probability theory
is more crucial in the creation of Numbers than is Algebra. So Algebra
ends up being
not only a subset of Geometry and a subset of Numbers but a subset of
Probability theory.
Now the two key operations on numbers is addition and multiplication,
for subtraction
and division are begot as inverses.
Now a proper development of Algebra using Geometry as the foundation
would
be to say that:
Addition equals the geometrical length
Multiplication equals the geometrical area or length x length
So we have the Reals as All Possible Digit Arrangements.
And we have the Reals between 0 and 1 as depicted by this
square box in the 1st quadrant of Cartesian Coordinate System:
_____________
| |
| |
| |
| |
| |
|____________ |
0 1
Now there is a number in the Reals for which when we add it to itself
continually
that it is able to produce all the Positive Reals except 0.
I am going to call it the Additive Closure Element and it is the
number
0.0000....00001
Starting with 0 and 0.0000....00001 and by continually adding, it
allows
anyone to traverse over every Positive Real Number.
And this element is the reason that Reals are closed to addition. And
it is
this element that the Reals are consecutive Reals, meaning that they
each
possess a successor and predecessor in the open-interval of 0 to 1.
So addition as Algebra is geometric length and multiplication is area
or
length x length.
So what is 0.0000...00001 x 0.0000....00001 = ?
It must be 0.0000.....00001
And what is 1 x 0.00000.....00001 = ?
In other words the length of 1 x length of 0.0000....0001.
It cannot be 0.000...0001 because that is the tiniest square box in
all of the
Positive Reals. So it must be that tiny sliver of a strip that is 1.
length distance
long and 0.0000...00001 length wide as depicted here to the leftmost
of the
unit box.
___________
|| |
|| |
|| |
|| |
||___________|
0 1
So the area of that sliver of a rectangle of 1 x 0.0000....0001 must
equal a Real Number and the only candidate is 0.0000....00100...000
That means the Reals do not have, nor can they have what I call
an Absolute-Multiplicative-Identity, but can have a Quasi-
Multiplicative
Identity.
As the Reals get larger than 0.0000....010.....00000 then multiplying
by 1 yields a multiplicative identity but any Real between 0 and
0.0000....010....0000 when multiplied by 1 is a third new number.
And the reason is obvious, Numbers can only have one absolute-identity
to either
addition or multiplication but not both. Reals have a absolute-
additive-identity
in the form of 0. Reals have a quasi-multiplicative-identity in the
form of 1, and
this multiplicative identity breaks down as you get closer to zero.
plutonium....@gmail.com
I guess obvious to me, but not most readers. Let me explain: if you
define addition geometrically as length and then go and define
area as length times length, you obviously have length entangled
into both addition and multiplication. And thus multiplication is a
dependent function of addition:
Addition = length
Multiplication = length x length
So when you have an operation that is a dependent function of another
operation, it stands to reason that a "identity" for one cannot be
independent
of the other. So that you can have a Unique and Absolute Identity in
addition but you cannot have such a identity for Multiplication since
multiplication
is dependent on addition. And we see this feature of Multiplicative
identity
breaking down as the numbers in Reals gets closer and closer to zero,
since
zero is the Absolute Additive Identity.
spudnik
for wasting an inordinate amount of my precious bodily fluids
-- that is to say, valuable life-span --
on your so-called new math -- truly, and
you may not even be reading this.
what possible value can we get, from arguing
about "real Galois theory," when
it was manifestly situated in the complex plane?
or was it a quaternionic bicycle!?... yeah;
all o'your **** is pablum, or vise-versa. that is,
inherently nonsequiter/misnomer/oxymoron-ish, because
there is no way that you will ever deign
to actually apply it to "the physics,"
that you insist is primary -- and it is!
thus quoth:
And the trouble with Galois theory is not only,
was it misguided as for a foundation on
some piddly paddly quintic problem, but
that it never addressed the problem
of Schroedinger's Headgasket.
plutonium....@gmail.com
Easy to see that you have Additive Closure on Reals with
0.0000....000001 serving as
1 and continual adding begets all the Reals. Sort of like a Peano
axiomitization of the
Reals, where we even have Math Induction over the Reals given that
true for
0.0000....00001 then 0.0000....00002 and suppose true for Real N then
show true
for Real N+ 0.00000....00001, conclude true for all the Reals.
But let me dwell upon multiplication more. Do the Reals have a
multiplicative
closure considering that 1 is only a quasi-multiplicative-identity and
that 0.0000....0001
serves as the identity to multiplication for numbers close to zero.
Let me define closure to multiplication as that of every Real Number
is the product
of two different Reals. Example: 1 x 0.1 = 0.1 and 1 x
0.1234567891011...
2222.....3333.....44444.....5555....6666....7777....88888....99999 =
0.1234567891011...2222.....3333.....44444.....5555....6666....7777
....88888....99999
But what about the number 0.0000....00001? Well it is begot only by
multiplication in the form of 0.000....001 x 0.000....0001 =
0.000...001
And so it seems to defy mine own definition of closure to
multiplication
for the only way to arrive at 0.0000...001 is to multiply it by
itself. For
multiplying it by 1 yields 0.0000...00100...0000
Another example is that of 0.000...002 and 0.000...0003 where in both
cases, multiplying them by 0.000...0001 yields their identity.
So between these two multipliers of either 1.000...0000 or
0.0000....00001
I arrive at the product which covers every Real Number other than
zero,
and zero also obeys.
But there is one problem in that 0.0000....0001 is only begot from
itself in
multiplication and not by 1.
So what is the meaning of that?
One thing super nice about Reals as All Possible Digit Arrangements is
that
we have them all there, altogether , all at once and can look at them
and
decipher their characteristics, their traits and features. Whereas in
Old Reals,
they never had a full view of all the Reals, and they tended to
understand the
Reals as to some human valued-feature. Humans value being "closed
under
addition or multiplication". And humans valued there being one unique
multiplicative
identity when it turns out that the Reals never had a absolute-
multiplicative identity, but
two of them in the form of quasi-multiplicative-identities as 1 and
0.000...0001
So, yes the Reals are closed under multiplication because between the
two
quasi multiplicative identites we cover every Real number.
So why does All Possible Digit Arrangements yield a single unique
Additive Identity
but two quasi multiplicative identities? I think the answer is because
addition
is length in geometry and multiplication is length x length or area
and thus dependent
on addition and being dependent requires that multiplication cannot be
solo unique
but duo quasi.
Now does this harken back to the Calculus also? Is the integral
somehow more
fundamental than the derivative since the integral is another form of
addition whereas
the derivative is multiplication? I am trying to search through the
Calculus to see if
the integral seems to be the starting point of Calculus rather than
the derivative. All
I can remember on this topic is that whenever anyone discusses the
various forms
of Calculus, it is always in terms of integral such as the Riemann
integral or Lebesgue
integral.
P.S. So the Closure concepts on Reals is supporting evidence that the
Reals
do not have a unique multiplicative identity in the form of 1 but
rather has
two quasi multiplicative identities where 0.0000....0001 is that
second entity.
Now I did highlight the fact that only the Real Number 0.0000....00001
is begot
by multiplying itself whereas we can get 0.000....0002 by multiplying
itself
with 0.000...0001, and the same goes for 0.000...0003.
So what is the significance of the idea that only 0.0000....0001
stands out like that?
Geometrically it is simply the understanding that 0.000....00001 is
the smallest
nonzero positive Real Number. Just as 1 was the smallest nonzero
positive Counting
Number and so to get 1 via multiplication we end up with 1 x 1 = 1 ,
and
likewise only 0.000...00001 x 0.000...0001 = 0.000...0001.
So do we have a special name for this characteristic of 0.000...0001?
plutonium....@gmail.com
Multiplicative Identities
on the Reals. Both should be called Quasi-Multiplicative-Identities
and these two
Reals are 1.0000....00000 and 0.00000....000001
The reason that Reals have two quasi-multiplicative-identities is
easily understood
since addition is geometrical length of the adding of the Additive
Inductor in Reals
of 0.0000....00001. So the length of any Real Number is composed of
the addition
of "so many" 0.0000....00001 units. And since multiplication is area
or
length x length means that multiplication is a dependent variable of
addition.
Because multiplication is a dependent variable of addition, means that
multiplication
cannot possibly have a unique independent multiplicative identity, but
at best have
a duo quasi multiplicative identities.
Now the Reals have Mathematical Induction because given any Real
Number we have
its successor and its predecessor and the Mathematical Inductor
Element is the
Real number 0.0000....00001. In my next post I will give some
Mathematical Induction
proofs over Reals.
But here I want to dwell on the idea that because Multiplication in
Reals Algebra
has no unique Multiplicative Identity but has two quasi identities
means that the
division in Real Algebra must have a hole or gap in it, where one
number behaves
very much out of place. And this number is zero in that division by
zero is undefined.
The reason it is undefined goes back to the fact that Reals are All
Possible Digit
Arrangements. And when you have All Possible Digit Arrangements that
the "best"
algebra over All Possible Digit Arrangements ends up having one
Additive Identity,
and two quasi Multiplicative Identities which causes there to be a
hole or gap in
division which is that 0 cannot be a divisor, but left undefined.
plutonium....@gmail.com
I have numerous math books lying around the house and funny how only
two of them
have any examples of Math Induction proof. I went and looked at what
Wikipedia
had to say over Math Induction and a pretty good page. Quoting a good
passage here.
--- quoting Wikipedia on Math Induction ---
This method works by first proving the statement is true for a
starting value, and then proving that the process used to go from one
value to the next is valid. If these are both proven, then any value
can be obtained by performing the process repeatedly. It may be
helpful to think of the domino effect; if one is presented with a long
row of dominoes standing on end, one can be sure that:
1. The first domino will fall
2. Whenever a domino falls, its next neighbor will also fall,
so it is concluded that all of the dominoes will fall, and that this
fact is inevitable.
--- end quoting Wikipedia ---
But Math Induction of the past was designed because it features the
"finite string
on Reals" and so it is going to be well behaved. But there is a
feature of the Old
Math about Math-Induction which is false. They believe their proofs
ended by them
saying "true infinitely many cases". Well, it is not true infinitely
many cases because
the finite portion on Reals, the leftward string was initially defined
as being *finite*
and finite only, so that you cannot thus break your definition by
saying that Math
Induction over finite Reals is going to give you an infinity of true-
cases.
What you can do is do Math Induction over the AP-adics which do go to
infinity.
So let me talk through a Mathematical Induction over finite-Reals, AP-
adics and the
for All the Reals. And make it a simple Math Induction Proof of this
statement:
a) Finite Reals starting with 1 alternate between odd and even
b) Positive AP-adics starting with 000....0001 alternate between odd
and even
c) Positive Reals starting at 0.0000.....00001 alternate between odd
and even
The proof of (a) is only a finite set and cannot be an infinite set
since the Reals
leftward string is defined finite, but Math Induction works on finite
sets and the proof
involves simply seeing the basis set as 1 is odd, 2 is even, 3 is odd,
and 4 is even.
The Induction step is that suppose when N is odd that N+1 is even and
the
proving step is then to show that (N+1) +1 and (N+1) +2 is forced to
be odd and
even respectively and this is easily shown through manipulation of the
Math Supposition
step. So we conclude it true over the Finite set of Finite-Reals
The proof of (b) is an infinite set and we do the same as in (a) only
it ends up
being true for an infinite set.
Now we come to (c) and this is probably the first written literature
on Mathematical
Induction on the total set of Real Numbers, not just the finite-Reals
taken as a
substitute of the Counting Numbers, but the total Reals. The first
ever written
math literature on Mathematical Induction on REals because it is the
first time
anyone has had a Inductor Element on REals and that number is
0.000...0001.
Given 0 and 0.0000....00001 then by addition of endlessly adding
0.0000...0001 I
thence build all the positive Real Numbers. So all the Reals have a
Mathematical
Induction that can be applied.
Proof that All the REals alternate between a Even and an Odd Number.
Now here I
am going to define zero as being even and of course the next Real
Number beyond
zero is 0.0000....00001 and it is odd (shown in prior posts). Now the
next Real Number
is 0.0000....0002 and it is even and the next is odd in 0.0000...0003
and the next
is even in 0.0000....0004. Those will form the basis step of
Mathematical Induction.
Now the Supposition step is to assume Real Number N is even then (N +
0.000...001)
is odd. Now to show that N + 0.0000...0001 + 0.000...0001 is even and
N + 0.0000...0001 + 0.000....0001 + 0.000...0001 is odd and we use the
same
manipulation as earlier and thus proving that over All the Reals,
which is an infinite
set that they alternate between being an Even REal and an Odd Real to
infinity.
It is interesting that even and odd is retained over All Reals but
that the characteristic
of Rational versus Irrational no longer exists in All Reals and that
Math Induction
on All-Reals does not provide a Math Induction Proof that Reals must
be either
rational or irrational.
plutonium....@gmail.com
wherein that system
is the construction of All Possible Arrangements of its Elements, is a
system that
is ordered and countable and must possess Mathematical Induction.
Try it with two digits, try it with three, with ten. Try it both
finite and infinite, no matter.
Every such system borne from All Possible Arrangements is a system
that is orderly
sequential and where every element has a predecessor and successor.
All Number Systems that are numbers of Geometry possess Mathematical
Induction.
So that the Reals are borne of All Possible Digit Arrangements means
the Reals
are borne of the continual addition of 0.0000....00001 to that of 0.
And the Reals
are Euclidean Geometry numbers, and that the NonEuclidean Geometry
Numbers
of AP-adics must also possess All Possible Digit Arrangements and thus
a Mathematical Induction.
plutonium....@gmail.com
Example: A Sum Formula
Theore. For any positive integer n, 1 + 2 + ... + n = n(n+1)/2.
Proof. (Proof by Mathematical Induction) Let's let P(n) be the
statement "1 + 2 + ... + n = (n (n+1)/2." (The idea is that P(n)
should be an assertion that for any n is verifiably either true or
false.) The proof will now proceed in two steps: the initial step and
the inductive step.
Initial Step. We must verify that P(1) is True. P(1) asserts "1 = 1(2)/
2", which is clearly true. So we are done with the initial step.
Inductive Step. Here we must prove the following assertion: "If there
is a k such that P(k) is true, then (for this same k) P(k+1) is true."
Thus, we assume there is a k such that 1 + 2 + ... + k = k (k+1)/2.
(We call this the inductive assumption.) We must prove, for this same
k, the formula 1 + 2 + ... + k + (k+1) = (k+1)(k+2)/2.
This is not too hard: 1 + 2 + ... + k + (k+1) = k(k+1)/2 + (k+1) = (k(k
+1) + 2 (k+1))/2 = (k+1)(k+2)/2. The first equality is a consequence
of the inductive assumption.
--- end quoting from http://zimmer.csufresno.edu/~larryc/proofs/proofs.mathinduction.html
Trouble with the above is that the Counting Numbers for which
Mathematical Induction
was created, trouble with that is that the Counting Numbers are all
"finite specimens"
and so the conclusion that it is true over an infinite set is simply
false.
Mathematical Induction was actually a method that was for Infinite
Sets and the conclusion
"for every" implied an infinite set conclusion.
So whereas the above example from CSU Fresno was meant for only the
finite
Reals of its finite leftward portion and not the entire set of Reals,
where the finite Reals
are synonymous with Counting Numbers was too narrowly formulated.
Here I show where Mathematical Induction is far more vast than the
narrow application
over finite specimens.
The sequence on Reals is this:
0.0000....00000, then 0.0000....00001, then 0.0000...00002, then
0.0000...0003
and the reader gets the meaning of that sequence.
Now we substitute 0.0000....00001 whereever there was a 1 in the above
and substitute 0.0000....0002 whereever there was a 2 in the above,
etc etc
Now look at the formula 1 + 2 + ... + n = n(n+1)/2 and substitute
gives us:
0.000...0001 + 0.000...002 + ... + n = n(n+0.000...001)/
0.0000....0002
Now I ask the question, does the above switch I made from just the
Finite-Reals
to that of All the Reals obey the same Mathematical Induction Proof?
Well, when n = 0.0000...0001, the same proof result as the CSU Fresno
result.
And whereas my result for All-Reals is an infinite set result, not
just a finite-set
result.
So, what I can do is whereever there is a Math Induction proof, just
simply switch
out the 1 and 2 etc etc for that of 0.000...0001 and 0.000...0002 etc
etc and achieve
a endresult of a proof over an infinite set not a finite set as the
old math induction
proofs.
plutonium....@gmail.com
Let me point out an error of mine. My Induction is a improvement over
old math with their loss of recognition that of Finite Portion Reals.
But my above is not a proof by Math Induction over All-Reals but only
a infinite-band-width of Reals. Specifically those Reals whose
FrontView
is a 0-band. And this applies to AP-adics also.
Define all those Reals between 0 and 1 whose frontview is
0 as the 0-band where it is approx 0% of [0,1]
1 as the 1-band where it is approx 10 to 20% of [0,1]
2 as the 2-band where it is approx 20 to 30% of [0,1]
3 as the 3-band
4 as the 4-band
5 as the 5-band
6 as the 6-band
7 as the 7-band
8 as the 8-band
9 as the 9-band
A number such as 0.0000....0001 belongs in the 0-band
as well as 0.01000...000. A number such as 0.50000...0000 belongs
in the 5-band.
The same scheme holds for the AP-adics.
Thus the above Math Induction proof of n(n+1)/2 holds true for only
the
0-band in Reals and the 0-band in AP-adics. It is an improvement over
the Old Reals and Counting Numbers since it does deliver an infinite
set as an answer, but it is not Universal over All REals nor All AP-
adics.
My earlier Math Induction proof that the Reals and AP-adics Integers
alternate
between being even and odd is a Universal Math Induction proof.
> So, what I can do is whereever there is a Math Induction proof, just
> simply switch
> out the 1 and 2 etc etc for that of 0.000...0001 and 0.000...0002 etc
> etc and achieve
> a endresult of a proof over an infinite set not a finite set as the
> old math induction
> proofs.
>
So far I have found only one Universal Math Induction Proof-- Reals
alternate
between even and odd. Are there more? I am guessing yes.
But Math Induction that pits addition against multiplication is not
going to yield
a Universal Mathematical Induction proof because, well, addition is
length and
multiplication is length x length or area and it is not a universal
math truth where
you have the summation of length the same as the area. And most math
induction
attempts are over issues of pitting addition to multiplication.
Math Induction proofs that pit the characteristics of the Reals or AP-
adics such as
alternating even to odd, those questions are Universal Reals or
Universal AP-adics.
Conjecture: prove that the Reals have gaps or holes between
consecutive Reals such as
0.9999...99999 to 1.0000....00000 to 1.0000...00001. Well it is
obviously true for
Universal Reals. So it is these types of propositions that can become
theorems of
mathematics using Mathematical Induction
Conjecture: Between any two given Reals, all those Reals between are
Algebraic
Reals, where none is a transcendental-number. This is another example
where
Math Induction proves the statement since we can use math induction
because all
the Reals were built from either All Possible Digit Arrangements which
translates
into piecewise building by continually adding 0.0000....00001
Similar conjectures proven by Math Induction for AP-adics.
The message is that although Math Induction now delivers an infinite-
answer-set
that does not contradict the definition of the finite portion on
Reals, nevertheless,
Math Induction is confined to bandwidths of proofs, except for those
cases that
talks of the Universal set of Reals or AP-adics.
David R Tribble
> Trouble with the above is that the Counting Numbers for which
> Mathematical Induction
> was created, trouble with that is that the Counting Numbers are all
> "finite specimens"
> and so the conclusion that it is true over an infinite set is simply
> false.
That's profound. I'm sure we all would love to see your
proof of that. You could start with something along the
lines of showing us which counting numbers get left out.
plutonium....@gmail.com
Here is where we need Lwal to be a referee of math history. Was I
the first person to write that a set is infinite only if it contains
an
infinite number of infinite specimens? I do recall I wrote that a set
must contain at least one member that is infinite specimen. My
dates of claims goes to 1993.
But I also claimed in the 1990s that the Counting Numbers never are
an infinite set because every one of its members was forced to be
finite. So if all members are finite, the set can never be infinite.
Only
if some members are infinite can a set bridge across and be infinite
set.
Now let me describe that situation so that dullards like Tribble can
understand.
Two kinds of ice cream, either the uncovered vanilla equalling finite
number or the chocolate coated vanilla equalling infinite number.
Correspondingly a set is finite if it is uncovered vanilla and
infinite
if it is chocolate covered.
So the Counting Numbers or the Finite leftward portion of Reals as
such 1,2,3,4,5, 6, .....
Each of its members is uncoated plain vanilla and the set overall has
no chocolate ever.
Now the set of Reals in the closed interval [0,2] has three members
that are considered finite as 0, 1, and 2 and the others are infinite
rightward strings and all chocolate covered.
So now, working with a set that is forced to have every member as a
finite
number 0,1,2,3,4,5,...... it is impossible for that set to be infinite
since all
its members are defined as finite and thus the set can never be
infinite set.
The proofs in Old Math such as Primes are infinite or Mathematical
Induction
on Natural Numbers, and even the Peano Axioms on Natural Numbers fail
to axiomatize finite from infinite. All those alleged proofs are
flawed and erred
because they cannot conclude an "infinite set answer" since none of
their
numbers are infinite specimens. If every specimen is plain old vanilla
then
the entire set is vanilla, but if some specimens are chocolate covered
then
the set overall has chocolate and infinite.
Lwal, I may have been, or not have been (Muschenbrook?? spelling) the
first to discover that a infinite set must contain
infinite members. Although I did not accurately state it as such for I
may
have said "at least one member must be infinite". So someone needs to
unravel
whether AP was the first to discover that a set has to have infinite
specimens for the
overall set to be infinite. My math history on the Internet starts in
1993. So if there
is someone in published print of similar idea that an infinite set
must contain infinite
members, then let it be known, for otherwise I claim it as my
discovery and which
is a rather important discovery because it shows the shoddiness or the
fraying of
the common math at its corners and really the heart of mathematics.
Whenever
math becomes less than precision, then math has failed.
But I am highly confident that I am the first to discover that all of
set theory
is a tiny subset of Probability theory. For the concept of All
Possible Digit
Arrangements as the defining concept of Numbers comes from the
Probability Space in Probability theory and so does the concept of
membership comes from Probability theory. So Probability theory is
vastly
more general and has Set theory as a subset thereof.
Jesse F. Hughes
> Here is where we need Lwal to be a referee of math history. Was I
> the first person to write that a set is infinite only if it contains
> an infinite number of infinite specimens?
Lwalker does not know math history or crank history. Why rely on him
as an authority?
--
"And yes, I will be darkening the doors of some of you, sooner than you
think, even if it is going to be a couple of years, and when you look
in my eyes on that last day of work at your school, then maybe you'll
understand mathematics." -- James S. Harris on Judgment Day
Ken Quirici
This folks is your typical LP 'proof' - it starts out on what
it takes to be a promising track, then flounders, totally
lost, and ends up lamely simply reasserting what it's trying
to prove. The 'proof' in other words, begging the question,
a circular argument, etc, etc.
Now for the benefit of dullards like LP here's a REAL proof:
Proposition: LP is a moron.
imagine we have a pile of shinola, and a pile of the other stuff.
We can see that the pile of the other stuff is a moron,
because it can't prove any mathematical theorem or
assertion whatsoever, ever. Nor can it do any sort of
calculation whatsoever, ever.
We know LP isn't a pile of shinola, so he must be a moron.
QED
David R Tribble
>> Trouble with the above is that the Counting Numbers for which
>> Mathematical Induction
>> was created, trouble with that is that the Counting Numbers are all
>> "finite specimens"
>> and so the conclusion that it is true over an infinite set is simply
>> false.
>
David R Tribble wrote:
>> That's profound. I'm sure we all would love to see your
>> proof of that. You could start with something along the
>> lines of showing us which counting numbers get left out.
>
Archimedes Plutonium wrote:
> Two kinds of ice cream, either the uncovered vanilla equalling finite
> number or the chocolate coated vanilla equalling infinite number.
> Correspondingly a set is finite if it is uncovered vanilla and
> infinite if it is chocolate covered.
>
> So the Counting Numbers or the Finite leftward portion of Reals as
> such 1,2,3,4,5, 6, .....
>
> Each of its members is uncoated plain vanilla and the set overall has
> no chocolate ever.
So you're saying that if you have a batch of vanilla scoops, the
batch itself is vanilla.
> Now the set of Reals in the closed interval [0,2] has three members
> that are considered finite as 0, 1, and 2 and the others are infinite
> rightward strings and all chocolate covered.
So is you have a batch of both vanilla and chocolate scoops, the
batch itself is chocolate. Got it.
> So now, working with a set that is forced to have every member as a
> finite number 0,1,2,3,4,5,...... it is impossible for that set to be infinite
> since all its members are defined as finite and thus the set can never be
> infinite set.
So since the set of all finite numbers can never be an infinite
set, it must be a finite set. And since all finite sets have a finite
number of elements, perhaps you could tell us what the finite
size of the set of all finite counting numbers actually is. Otherwise,
we are tempted to believe that it is not really a finite set after
all.
plutonium....@gmail.com
foundation of
Numbers is that it gives a largest and smallest nonzero positive
number in both
Reals and AP-adics. In Reals it is the numbers 0d0000...0001 and
0d9999...9999
in open-interval (0,1) and in AP-adics Integers it is 0000....0001 and
9999...99999
So the foundation concept forces us onto these smallest and largest
numbers.
Now does that force us to further conclusions? Does it force us into
determining
the exponent base 10 place-value? I am hoping it does and as of yet
still
unsure of what it does bring. For "finite numbers" the decimal place-
value
exponent is quite clear where the "one's place" then the "ten's place"
then
the "hundred's place" etc etc.
So far I simply have sluffed off this topic, for I still feel not sure
of a determination.
So far I have called the "7" digit in a number such as the Real
0d999...9997 or
in the AP-adic of 7999....9999 as the 10^(-)9999...9998 and
10^9999...99998
place-value respectively. Now I could not call the "7" as the
9999...9999
exponent, thinking that there is no difference between the number
9999...9999
and 10^9999...99999 itself. So that did not make commonsense and
defies
intuition, but we are here at infinity and there is little intuition
that is reliable
or dependable.
But maybe, just maybe, that the "7" digit in those examples is really
in the 10^(-)9999...99999 place value and 10^9999...99999 place value
respectively. Perhaps at infinity, the place value in terms of
exponentiation
converges with the so called "length of the number 9999...9999.
I was putting off this topic in hopes that some insight may strike me.
Recently
I did have some insight. Consider Champernowne's number (spelling?) of
the Reals 0d123456789101112131415...... As you can see it has every
counting number in a sequence. Now if we examine the AP-adic of
Champernowne's number 1234567891011121314151617.........999999
we can easily recognize that it must go through not only the so called
finite numbers such as 1 and 2 and 3 etc but through the infinite
numbers
such as 2222....22222 and 343434....343434 etc etc
So now, with infinite numbers containing other infinite numbers, would
it
be plausible or reasonable to say that the exponentiation can also be
the largest number of 9999....99999? So that a number such as
7999....99997 with its "7" digit is in the 10^9999...9999 place-value?
Because at infinity, size no longer has any meaning, does it? Where
the even numbers 2,4,6,8,.....999...9998 are just as infinite as all
the AP-adic
integers of 1,2,3,4,.....9999....9999.
So does the foundation of All Possible Digit Arrangements force the
place-value
exponentiation to be 10^9999....99999 or 10^(-)9999....9999 in the
case
of 7999...9999 and 0d9999....99997 for the "7" digit in those
respective
AP-adic and Real numbers? I am beginning to think so, but I need some
more confirming or supporting evidence.
plutonium....@gmail.com
David R Tribble wrote:
>
> So you're saying that if you have a batch of vanilla scoops, the
> batch itself is vanilla.
>
I am using a metaphor to help dense people understand easier. I do
not know what Muschenbrook (excuse the spelling) used if he used
any metaphors. But I am not the only person who sees that there is a
logical mess-up in the conventional math.
The Old Math with the Old Reals and with the Peano Natural Numbers
all define the Reals as a "finite portion leftwards" and define the
Natural
Numbers as *only finite*. So they all insist, demand that these are
finite and flunk out of school any student who protests otherwise.
And students should protest because the teachers demand that those
are finite numbers. But then when they do series sums or sequences
or proofs such as the Infinitude of Primes, they demand that the
students
accept that the set of Primes is an infinite set or that the Natural
Numbers
are an infinite set.
They demand every Natural Number is finite, yet that the set of all of
them is
infinite.
But is that logical? Is that commonsense? If every member is forced to
be
**finite** then can the set be infinite?
The key concept is *every* member. If Natural Numbers demands every
member be finite, then no matter what happens a set of Natural Numbers
is never going to be infinite.
Now I do not know how Muschenbrook came to this same conclusion, that
a set has to have infinite specimens in order for it to be a infinite
set candidate.
And my metaphor is pretty good. Consider plain vanilla as finite and
consider
if there are chocolate covering or chocolate chips in a bar then it is
infinite.
And for the entire set, if there is no chocolate present, the set can
only be
finite with finite members. But if there is chocolate present then the
set is
infinite.
>
> > Now the set of Reals in the closed interval [0,2] has three members
> > that are considered finite as 0, 1, and 2 and the others are infinite
> > rightward strings and all chocolate covered.
>
> So is you have a batch of both vanilla and chocolate scoops, the
> batch itself is chocolate. Got it.
>
Obtuse people when they do not want to learn, or find themselves
losing
the discussion, resort to mockery.
>
> > So now, working with a set that is forced to have every member as a
> > finite number 0,1,2,3,4,5,...... it is impossible for that set to be infinite
> > since all its members are defined as finite and thus the set can never be
> > infinite set.
>
> So since the set of all finite numbers can never be an infinite
> set, it must be a finite set. And since all finite sets have a finite
> number of elements, perhaps you could tell us what the finite
> size of the set of all finite counting numbers actually is.
Wherever you stop counting is the finite size.
>Otherwise,
> we are tempted to believe that it is not really a finite set after
> all.
You are frightened by infinity which is beyond the finite, and it is
the infinite
that is making your mind spin out of control.
P.S. The AP-adics Integers do go to infinity and it is here where we
have
a proof or disproof of things such as the Infinitude of Primes. We do
not
have a proof of the Infinitude of Primes with Peano Natural Numbers
because
all of those are axiomed as "finite" and we do not have Infinitude of
Primes
on the finite portion of Reals because they are defined as finite.
And as I said earlier, that straight lines in Euclidean geometry never
go to
infinity and parallel lines are not parallel out to infinity. The
parallel lines
in Euclidean geometry occur because there is an infinity of points
between 0 and 1 or 1 and 2 or 2 and 3. So the finiteness of 2 and 3
have
an infinity of points between them and the parallelism between two
lines
in Euclidean geometry is due to the same distance held between those
infinite points between 2 and 3 for both parallel lines. So lines in
Euclidean
Geometry never go to infinity, and that parallelism is upheld because
of a
constant distance of separation between two finite but parallel lines.
If Mathematics is the science of precision, then for much of the 20th
century
and for the first 9 years of the 21st century the math community has
been
horribly lax on precision and have done a pitiful job.
Ken Quirici
So if someone restarts the counting where you left off, what then?
All of a sudden the size of the Natural Numbers increases?
If you say the natural numbers have n elements, I say no, they have
at least n + 1. If you say they have n + 1, I say no, they have
at least n + 2.
Actually it's YOU who's afraid of the infinite. In particular,
the leap from the finite to the infinite. Unbounded infinity.
You can deal with the infinite if it's sitting there in front
of you DELIMITED somehow, so it's nice and safe.
Thus you understand there are an infinite number of reals between
1 and 2.
You understand that there are an infinite number of digits
in 2222......222.
In fact your brilliant notion of front-view and back-view is
because you have to limit infinity SOMEHOW so it's nice and
safe (did I say that already?)
But you cannot comprehend an infinity that is not somehow
'bounded'.
Like 1,2,3,4,.....
Like 1,3,5,7,9,.....
Like a straight line extending PAST any arbitrary end points.
Consider the set of AP-adics:
000..............001
000..............002
000..............003
etc.
You will accept that this set is infinite because in your notation
(and it's only notation so far, there's no substance to it yet, you
have done none of the due diligence you must do to make them
mathematical
objects) each number is infinite.
But you don't grasp this simple fact:
this set can be placed in a trivial 1-to-1 correspondence with
1,2,3,......
You don't grasp that a 1-to-1 correspondence between two sets
means they have the same size? If the 000.....0000n sequence
is infinite then the 1,2,3,....n set MUST ALSO be by that
simple 1-to-1 correspondence:
0000............01 1
0000............02 2
0000............03 3
etc.
plutonium....@gmail.com
Maybe I can get away with this idea that 9999...99999 =
10^9999....99999
and that 0d0000...00001 in Reals is where the "1" digit is in the 10^
(-)999...9999
place value?
I would like to be *forced into* those understanding rather than be
wondering if
or dabbling into that condition.
I mean forced, just as I was forced to see that there must be a
largest integer
of 9999...9999 and there must be a smallest nonzero positive Real as
0d0000...0001. Forced into those ideas because of All Possible Digit
Arrangements.
Likewise, I would like to be forced into whether 9999...99999 equals
10^99999....9999 or forced out of them being equal.
Are they equal because here in the domain of the infinite, the largest
integer has
to equal the largest possible place-value? In finite domain we have
the "one's place
value, then the ten's place value then the hundred's place value, etc
etc"
But we are no longer in the finite domain. So that in the infinite
domain, the decimal
place value, does it force upon me to say that the 59999....99999, the
"5" digit is
in the 10^9999....9999 place-value? Or is the "5" digit in a smaller
place value
such as 10^999...9998 or even much smaller say that of
10^0000....010...0000
where the number 0000....000100....00000 is the smallest infinity
digit arrangement
I am not sure, but I am sure that many more people will be pondering
over this.
lwa...@lausd.net
> > proof of that. You could start with something along the
> > lines of showing us which counting numbers get left out.
> Here is where we need Lwal to be a referee of math history. Was I
> the first person to write that a set is infinite only if it contains an
> infinite number of infinite specimens? I do recall I wrote that a set
> must contain at least one member that is infinite specimen. My
> dates of claims goes to 1993.
Most likely, AP was first. The oldest other "crank" of which
I'm aware wrote in 1995, meaning that AP has priority.
> Two kinds of ice cream, either the uncovered vanilla equalling finite
> number or the chocolate coated vanilla equalling infinite number.
> Correspondingly a set is finite if it is uncovered vanilla and
> infinite if it is chocolate covered.
Ironically, AP wasn't the first to come up with the ice
cream analogy. The ice cream analogy was jointly invented
by WM and Brian Chandler. To them:
A "vanilla" set is a finite set (just like AP's definition).
A "strawberry" set is a countably infinite set (unlike AP).
All other sets are "chocolate."
lwa...@lausd.net
> Archimedes Plutonium wrote:
> > Each of its members is uncoated plain vanilla and the set overall has
> > no chocolate ever.
> So you're saying that if you have a batch of vanilla scoops, the
> batch itself is vanilla.
Note that WM/Brian Chandler were actually the first to come up
with the ice cream analogy. To them:
A "vanilla" set is a finite set (just like AP's definition).
A "strawberry" set is a countably infinite set (unlike AP).
All other sets are "chocolate."
Notice that in the context of ZFC, a "chocolate" set (using
WM/Chandler's definition) is an uncountable set. In other
theories this need not be so (most notably ZF+~AC).
> > So now, working with a set that is forced to have every member as a
> > finite number 0,1,2,3,4,5,...... it is impossible for that set to be infinite
> > since all its members are defined as finite and thus the set can never be
> > infinite set.
> So since the set of all finite numbers can never be an infinite
> set, it must be a finite set. And since all finite sets have a finite
> number of elements, perhaps you could tell us what the finite
> size of the set of all finite counting numbers actually is. Otherwise,
> we are tempted to believe that it is not really a finite set after
> all.
Ed Nelson came up with a theory of potential Aristotelian infinity,
which is essentially PA-Induction. In this theory, there exists
numbers so great, such as 2^^2^^5, that one cannot effectively prove
them to be counting numbers. But one cannot name the largest counting
number in this theory, nor can one assign a counting number to be the
cardinality of the set of all counting numbers.
Also, Nelson has a theory ZFC+IST. In this theory, there exists
finite standard naturals and infinite nonstandard naturals. But one
cannot even form the set of all finite standard naturals (Illegal Set
Formation), much less find its cardinality to be a counting number.
The existence of the two Nelson theories implies that just because AP
can't give a finite size of the set of all finite counting numbers in
his theory, as Tribble is asking him to, it doesn't mean that the
set of all finite counting numbers must necessarily be infinite.
David R Tribble
>> So you're saying that if you have a batch of vanilla scoops, the
>> batch itself is vanilla.
>
Archimedes Plutonium wrote:
> I am using a metaphor to help dense people understand easier. I do
> not know what Muschenbrook (excuse the spelling) used if he used
> any metaphors. But I am not the only person who sees that there is a
> logical mess-up in the conventional math.
William Muckenheim (WM). And he's as wrong as you are, but for
different reasons. He rejects the notion of infinity altogether.
Are you sure you want to name him as an ally in the defense of your
beliefs?
> The Old Math with the Old Reals and with the Peano Natural Numbers
> all define the Reals as a "finite portion leftwards" and define the
> Natural
> Numbers as *only finite*. So they all insist, demand that these are
> finite and flunk out of school any student who protests otherwise.
If 1 is not a finite number, what is it?
Some definitions of "finite" stipulate that for finite x, x+1 > x.
Do you know of a finite natural where this is not the case?
> They demand every Natural Number is finite, yet that the set of all of
> them is infinite.
> But is that logical? Is that commonsense? If every member is forced to
> be **finite** then can the set be infinite?
> The key concept is *every* member. If Natural Numbers demands every
> member be finite, then no matter what happens a set of Natural Numbers
> is never going to be infinite.
The logic is pretty elementary.
1. Start with 0, the first natural, and which is obviously a finite
number.
2. For every natural n, there also exists the natural n+1.
This gives us all the naturals. Continuing,
Theorem. The set of all naturals (N) is not finite.
Proof.
1. Assume the contrary, that N is a finite set (as per AP).
2. Since all finite sets of naturals have a largest member
(which is pretty intuitive and easily proven), designate
the largest member of N to be the natural m.
3. Since m is the largest member of N, all other members
of N must be less than m. (Otherwise it wouldn't be the largest
member.)
4. But m+1 is also a natural (see above), so it is also a member
of N.
5. And m+1 > m, which contradicts (3).
6. Therefore assumption (1) must be false. So N cannot
be a finite set.
QED.
Now you can show us all how smart you are and find the flaw
in the logic above.
David R Tribble
> And my metaphor is pretty good. Consider plain vanilla as finite and
> consider if there are chocolate covering or chocolate chips in a bar then it is
> infinite. And for the entire set, if there is no chocolate present, the set can
> only be finite with finite members. But if there is chocolate present then the
> set is infinite.
So if the set contains only a single element, and that element is
an infinite number (chocolate), the entire set must be infinite?
I don't follow that.
> Obtuse people when they do not want to learn, or find themselves
> losing the discussion, resort to mockery.
Could you be refering to the way you insult people by calling
them dense, obtuse, and dullards? Just a guess on my part, but
it's possible that people might find you a tad more interesting if
you stopped doing that.
David R Tribble wrote:
>> So since the set of all finite numbers can never be an infinite
>> set, it must be a finite set. And since all finite sets have a finite
>> number of elements, perhaps you could tell us what the finite
>> size of the set of all finite counting numbers actually is.
>
Archimedes Plutonium wrote:
> Wherever you stop counting is the finite size.
Are you really saying that wherever you decide to stop counting,
that's the point where you've counted /all/ of the naturals?
Or are you trying to say that when you run out of naturals to
count, it's at that point you've reached the size of the set of all
of them? Which point is that?
Your answer is not very mathematically precise.
> You are frightened by infinity which is beyond the finite, and it is
> the infinite that is making your mind spin out of control.
You are the one saying that the set can't be infinite, not us.
Does that mean that you are frightened that it is an infinite set?
Stop throwing around insults.
David R Tribble
>> Here is where we need Lwal to be a referee of math history. Was I
>> the first person to write that a set is infinite only if it contains an
>> infinite number of infinite specimens? I do recall I wrote that a set
>> must contain at least one member that is infinite specimen. My
>> dates of claims goes to 1993.
>
LWalker wrote:
> Most likely, AP was first. The oldest other "crank" of which
> I'm aware wrote in 1995, meaning that AP has priority.
Only if you limit your research to Usenet and Internet postings.
You really can't conceive of anyone else coming up with this
idea at any time between the invention of decimal notation and
AP's first postings on the Internet?
> Ironically, AP wasn't the first to come up with the ice
> cream analogy. The ice cream analogy was jointly invented
> by WM and Brian Chandler. To them:
> A "vanilla" set is a finite set (just like AP's definition).
> A "strawberry" set is a countably infinite set (unlike AP).
> All other sets are "chocolate."
Yeah, and I invented "red" and "green" sets a couple of years
before that for precisely the same reason. Big deal.
David R Tribble
> Also, Nelson has a theory ZFC+IST. In this theory, there exists
> finite standard naturals and infinite nonstandard naturals. But one
> cannot even form the set of all finite standard naturals (Illegal Set
> Formation), much less find its cardinality to be a counting number.
>
> The existence of the two Nelson theories implies that just because AP
> can't give a finite size of the set of all finite counting numbers in
> his theory, as Tribble is asking him to, it doesn't mean that the
> set of all finite counting numbers must necessarily be infinite.
It certainly does in ZFC.
plutonium....@gmail.com
Well I can provide a "natural size" keyed in with physics. It hinges
on
this new number recently discovered in All Possible Digit
Arrangements.
I call it the "smallest infinity number". It sort of acts as the
bridge between
the finite world and the realm of the infinite. It is a number that
the mind
of a expert in mathematics and the mind of one who hates mathematics,
the two minds want such a number to hold onto.
The in AP-adics is 00000....00001000.....00000 and in Reals this
number between 0 and 1 is 0d0000....00001000.....00000
This number is demanded of, and forced to exist due to All Possible
Digit Arrangements and is the number required in the multiplication
in Reals of 0d0000....00001 x 1 = this number
and in +AP-adics this number is demanded of by a multiplication
of this 000....000001 x 10000....00000 = this number.
So both Reals and AP-adics demand for this number, and what it means
is that it bridges the finite realm with the infinite realm.
Now I need to look into physics as the largest meaningful physical
number to
exist. Because if noone or nothing is able to count beyond such a
number, whether
there is not enough time in the Universe or whatever physical reason,
then, that is
the end of "finiteness" and the beginning of infinity.
From what I have located in physics, that the number of Coulomb
interactions to hold
together a Plutonium Atom is of the order of between 192! and 231!
which are huge
numbers for which nothing in the Cosmos could ever count. So in effect
that is the end
of finiteness whether we are doing physics or doing mathematics and
the beginning
of infinity.
So what I propose is that this number 0000....00001000....00000 with
its "1" digit
is in the place value of that largest physics number
plutonium....@gmail.com
Funny, how metaphors, and ease of teaching or talking about finite and
infinite were duplicated in the imagery of vanilla and chocolate, just
so it
is easy to teach and comprehend but that the major idea that an
infinite set
must have infinite-members was not easily duplicated.
Mistaken paths are trodded by many and true paths have few who
trod down them.
I need to look up what physics numbers are the largest or smallest
numbers
possible that retain some physical meaning. Then I proceed to
flag this special Real Number of 0d000....00100...0000 and this
special
+AP-adic Integer of 0000....0000100....0000 for those are the smallest
infinity
numbers. Where 000....00000999...999 is the predecessor and it is a
Finite Integer
just as 0000....00001 is finite. But where 0000...0001000...00000 the
next number
is an infinite integer.
So what I propose is that the largest physics number of physical
meaning and peg
it as 0000....000099999....9999 where adding 1 more to that number
delivers
the smallest infinity number of 000...0001000....00000
I believe the largest number in physics is about 10^200 of the Planck
numbers, but
I have to check on that. I worked out in the 1990s that the number of
Coulomb
Interactions that keep a plutonium atom together is of the order of
192! to 231! which
are numbers larger than 10^200. I think 231! is about 10^500.
So what I propose is that since Physics is exhausted of meaning beyond
10^500,
that we peg 0000....0000999...9999 as 10^500 and thus adding 1 more to
that
delivers 0000....0001000....00000
This makes sense for in effect what I have done here is say that
Finiteness is
equivalent to being of Physics Meaning, and beyond that is the realm
of infinity
where we no longer have Physical meaning. Where we can no longer
count, and
so it makes no difference anyway since we can no longer count there.
And that is
what finite means in the first place-- it has a physics reality.
plutonium....@gmail.com
David R Tribble wrote:
> David R Tribble wrote:
> >> So you're saying that if you have a batch of vanilla scoops, the
> >> batch itself is vanilla.
> >
>
> Archimedes Plutonium wrote:
> > I am using a metaphor to help dense people understand easier. I do
> > not know what Muschenbrook (excuse the spelling) used if he used
> > any metaphors. But I am not the only person who sees that there is a
> > logical mess-up in the conventional math.
>
> William Muckenheim (WM). And he's as wrong as you are, but for
> different reasons. He rejects the notion of infinity altogether.
> Are you sure you want to name him as an ally in the defense of your
> beliefs?
>
You are the one wrong.
At least Muckenheim got a parcel of truth in this world when he spoke
of a
set can only be infinite if it contains infinite specimens. If a set
contains only
finite specimens, it cannot be infinite.
(snipped)
>
> The logic is pretty elementary.
> 1. Start with 0, the first natural, and which is obviously a finite
> number.
> 2. For every natural n, there also exists the natural n+1.
>
> This gives us all the naturals. Continuing,
Oh, okay, so you reference the Peano Axioms, eh. So tell me, which
Peano axiom says that 99999 is a natural number but where
99999....999999 is not a natural. So explain which Peano axiom
resolves that issue.
And when you finish with that, explain where the Peano axioms says
that
the n+1 stops at a number because the next number is 10000....000000
or the next number is 0999....99999 or the next number is
0999....99998
etc etc.
So, Tribble, explain how the Peano axioms tells you whether 1111 is
natural
but 1111....1111 is not natural. How it explains that 0000....0007 is
natural
but that 70000....0000 is not natural.
So point to the Peano axioms that provide you with that discerning
eyeball.
>
> Theorem. The set of all naturals (N) is not finite.
Hold on a minute. You need to show first, how the Peano Axioms
tell you that 7777 is a natural but that 77777.....77777 is not a
natural
> Proof.
> 1. Assume the contrary, that N is a finite set (as per AP).
> 2. Since all finite sets of naturals have a largest member
> (which is pretty intuitive and easily proven), designate
> the largest member of N to be the natural m.
> 3. Since m is the largest member of N, all other members
> of N must be less than m. (Otherwise it wouldn't be the largest
> member.)
> 4. But m+1 is also a natural (see above), so it is also a member
> of N.
> 5. And m+1 > m, which contradicts (3).
> 6. Therefore assumption (1) must be false. So N cannot
> be a finite set.
> QED.
>
> Now you can show us all how smart you are and find the flaw
> in the logic above.
And you want me to stop calling you dullard, while you turn around
and act childish about a foolish attempted proof.
No proof is needed in this conversation about finite or infinite
because it is
at the axiom level and the level of definition.
So answer the above question put to you-- how is it that you can tell
whether
88888 is a natural number and not 8888....8888. What Peano axiom are
you
using to decipher between those two? And when do you know in your n, n
+1
that all your n+1 numbers are not infinite-integers. How does your
beloved
Peano axioms tell you that your n+1 has always kept you in "finite
integers."
plutonium....@gmail.com
versus infinite.
With trying to make sense of where the so-called Peano finite Natural
Numbers steps
into becoming the Infinite Integers of the AP-adics. I have made some
remarkable
progress because I can safely say that this number is relevant:
0000....00100....00000 for it is the multiplication of 0000...0001 x
10000....00000
and that number is what can be termed the "smallest infinity". It is
the bridge to
where Finite stops and Infinity begins. And the Reals have a similar
number
in the form of 1 x 0d000...00001 = 0d000....00100...0000
Now the only problem remaining is to ask what decimal place-value is
that
"1" digit in 0000...0001000.....0000 for which the finite numbers ends
and
becomes infinite-integers from thereon out?
Well, I have an answer for that. So my focus and concentration for the
past
16 years pays off grandly.
I have often said that Physics is king and math is a subset of
physics. So does
Physics have a bridge for which physics beyond that bridge makes no
more sense
and is out of reach? Well, yes in the Planck units of measure. So the
idea is that
what is the largest number or smallest number in Physics that has
physical reality
such as experiment or measure. For if we can no longer experiment or
measure
then we can no longer have a Physics there. And since Physics is top
and has
Math as a subset, then if Physics stops at some large number, well, at
that number
mathematics is no longer finite but infinite.
So here I have discovered a new definition for the concept of finite
and infinite. That
finite is Physics realm of measureability and if physics cannot
measure in a realm
then it is an infinite realm.
Now I looked up Wikipedia for the largest numbers in physics (or
smallest) and
found this Wikipedia page:
--- quoting Wikipedia ---
http://en.wikipedia.org/wiki/Planck_units
Table 2: Base Planck units Name Dimension Expressions SI equivalent
with uncertainties[1] Other equivalent
Planck length Length (L) l_\text{P} = \sqrt{\frac{\hbar G}{c^3}} 1.616
252(81) × 10^−35 m
Planck mass Mass (M) m_\text{P} = \sqrt{\frac{\hbar c}{G}} 2.176 44
(11) × 10^−8 kg 1.220 862(61)× 10^19 GeV/c2
Planck time Time (T) t_\text{P} = \frac{l_\text{P}}{c} = \frac{\hbar}
{m_\text{P}c^2} = \sqrt{\frac{\hbar G}{c^5}} 5.391 24(27) × 10^−44 s
Planck charge Electric charge (Q) q_\text{P} = m_\text{P} 2 \pi \sqrt
{G \varepsilon_0} = \sqrt{\hbar c 4 \pi \varepsilon_0} 1.875 545 870
(47) × 10^−18 C 11.706 237 6398(40) e
Planck temperature Temperature (Θ) T_\text{P} = \frac{m_\text{P} c^2}
{k} = \sqrt{\frac{\hbar c^5}{G k^2}} 1.416 785(71) × 10^32 K
--- end quoting Wikipedia ---
Further on down Wikipedia gives derived Planck units with
Planck volume as
Planck volume Volume (L3) l_P^3 = \left( \frac{\hbar G}{c^3} \right)^
{\frac{3}{2}} 4.22419 × 10^-105 m3
And further down Wikipedia gives Planck pressure as
Planck pressure Pressure (LM-1T-2) p_P = \frac{F_P}{l_P^2} = \frac
{\hbar}{l_P^3 t_P} =\frac{c^7}{\hbar G^2} 4.63309 × 10^113 Pa
So we can sort of sense that numbers in Physics give out at about
10^44, or 10^113
And some in physics have estimated that the number of atoms to tile
and fill the volume
of the Cosmos is around 10^60 x 10^60 x 10^60 = 10^180
But I personally like the number of Coulomb Interactions that goes on
inside a
individual plutonium atom with its 231 elementary particles inside.
Those Coulomb
Interactions keeps the atom in order and my estimates of the
permutations of all
231 elementary particles is the number 231!/2
That is a huge number 232!/2 which is approx
10^500, where 170! is 10^306
So I am feeling confident and safe to say that the number 10^500 is
the limit
to where you have Physics going on and its inverse of 10^(-)500.
Since Physics is king and above mathematics where math is a subset of
physics.
And since physics runs out at 10^500 or 10^(-)500
Then I place the value of 0000....0001000.....0000 of its "1" digit
place value as
that of 10^500
and likewise for the Real Number 0d000...0001000....0000 of its "1"
digit as
being 10^(-)500.
So I found what I was looking for in the past 16 years of a bridge
number which
spans from the finite world into the infinite world, where finite
numbers meet the
infinite numbers.
So if you add 1 to that of 10^500 you have the Infinite Integer of
000....00001000...001
and if you subtract 1 from 10^500 you have the Finite Integer of
0000....00009999....99999 which is 9d99 x 10^499
So here I married Physics with Mathematics, where Physics is the king
and mathematics the reflection of what physics is. Where Physics gives
birth to geometry because the Universe is one big atom and atoms have
shape and size. And Physics gives birth to Numbers because atoms
are numerous. And Physics gives birth to there being a finite and
infinite
and the finite is what is measurable for physics. If Physics cannot
measure
then it is in the realm of the infinite.
The old definition of finite and infinite was that one comes to an end
whereas the
other is endless. The new definiton of finite versus infinite is that
where Physics
measurement and experiment can go is finite and where measurement is
beyond
reach for physics is infinite. And for the Counting Numbers it is
where
0000....00010000....0000 is the number 10^500
David R Tribble
> So what I propose is that the largest physics number of physical
> meaning and peg
> it as 0000....000099999....9999 where adding 1 more to that number
> delivers the smallest infinity number of 000...0001000....00000
Then there is that rule that for every finite natural k, k+1 is
also a finite natural. I guess that rule is broken now?
> So what I propose is that since Physics is exhausted of meaning beyond
> 10^500, that we peg 0000....0000999...9999 as 10^500 and thus adding 1 more to
> that delivers 0000....0001000....00000
You seem to have a limited imagination. What prevents the number
(10^500)^(10^500) from being a finite natural? Too many digits?
Too many divisors? Because we've run out of names for all the
numbers smaller than it?
> And that is what finite means in the first place-- it has a physics reality.
Where in physical reality does the number 7 exist? I don't recall
ever stubbing my toe on a 7.
David R Tribble
>> The logic is pretty elementary.
>> 1. Start with 0, the first natural, and which is obviously a finite
>> number.
>> 2. For every natural n, there also exists the natural n+1.
>> This gives us all the naturals. Continuing,
>
Archimedes Plutonium wrote:
> Oh, okay, so you reference the Peano Axioms, eh. So tell me, which
> Peano axiom says that 99999 is a natural number but where
> 99999....999999 is not a natural. So explain which Peano axiom
> resolves that issue.
I can't, for the simple reason that you won't tell us what "999...999"
is supposed to mean. What is the "..." in the middle of the digits
supposed to mean?
Until you answer that question, we don't have the faintest idea of
how "999...999" is supposed to be a number.
> And when you finish with that, explain where the Peano axioms says
> that the n+1 stops at a number because the next number is 10000....000000
> or the next number is 0999....99999 or the next number is
> 0999....99998 etc etc.
I can't answer that question, because those aren't numbers.
Tell us where the Peano Axioms say that "0999...999" is a number.
> So, Tribble, explain how the Peano axioms tells you whether 1111 is
> natural but 1111....1111 is not natural.
I can't, because you haven't told us how "111...111" is a natural.
Show us where the Peano Axioms tell you that "111...111" is a natural.
Show us where the Peano Axioms tell you what the "..." between the
digits is supposed to mean.
David R Tribble
>> Theorem. The set of all naturals (N) is not finite.
>> 1. Assume the contrary, that N is a finite set (as per AP).
>> 2. Since all finite sets of naturals have a largest member
>> (which is pretty intuitive and easily proven), designate
>> the largest member of N to be the natural m.
>> 3. Since m is the largest member of N, all other members
>> of N must be less than m. (Otherwise it wouldn't be the largest
>> member.)
>> 4. But m+1 is also a natural (see above), so it is also a member
>> of N.
>> 5. And m+1 > m, which contradicts (3).
>> 6. Therefore assumption (1) must be false. So N cannot
>> be a finite set.
>> QED.
>
Archimedes Plutonium wrote:
> No proof is needed in this conversation about finite or infinite
> because it is at the axiom level and the level of definition.
You seem to be saying that N cannot be infinite because of the
way N is defined "at the axiom level" and "at the definition level".
What does that mean? How would you define N?
Maybe you are saying that there is something about the members
m and m+1 in N that causes N to be finite "at the axiom level"
or "at the level of definition"? What would that be?
Or maybe you're saying that you're just going to ignore the
proof and you don't care whether it's right or wrong?
plutonium....@gmail.com
The old way of defining Finite versus Infinite was to think of finite
as an end,
where the number stops and Infinite where there is no end and the
number
never stops. Well, that idea served science since the Ancient Greek
mathematics, but today we need a revised and refurbished definition. A
definition
is pretty bad, pretty awful when it says that finite ends and infinite
never ends.
Where A is this and B is not this. Where definition A defines
definition B preceded
by a not. Those sort of definitions are the poorest of definitions.
Another example
is Algebraic and Transcendental as not-algebraic.
So I define Finite as its new modern day science definition as the
mathematical
point of 0000....000100....00000 in AP-adic Integers as equal to
10^500 and
equal to 10^(-)500 in the rightward radix string.
And in Reals the finite leftward portion is 10^500 and beyond that is
infinite.
And in Reals rightward string the Real 0d0000...0100....0000 is 10^(-)
500
and beyond that is the infinite portion of the string.
I define finite and infinite as to the superior science of physics,
since physics
creates all of mathematics in the first place, then physics would
determine
where and what is the cutoff between finite and infinite.
So the new definition of Finite in science is that largest and
smallest numbers
of physics significance. As best as I can determine from physics, that
beyond
10^500 is no longer able to be measured or counted or anything else of
physics.
So beyond 10^500 is the realm of the infinite. So the AP-adic of
0000.....00001000....0002 where the 1 is in the 10^500 place-value is
an infinite
integer and so is its predecessor of 000....0001000....00001
Likewise for the Reals in that the Natural Numbers of the Peano Axiom
System
only go up to 10^500. To go beyond is to use the +AP-adic Integers.
Infinite is thus defined as unable to Physics apply measure or
experiment.
This is rather handy in discussing astronomy and the universe, because
if
we know about it or gather facts and data, those are within and below
the number 10^500. So the question of whether the Universe is finite
or
infinite is a ridiculuous question because it is both simultaneously,
since
there are things we do not know and cannot know, meaning it is in the
infinite realm.
plutonium....@gmail.com
David R Tribble wrote:
> Archimedes Plutonium wrote:
> > So what I propose is that the largest physics number of physical
> > meaning and peg
> > it as 0000....000099999....9999 where adding 1 more to that number
> > delivers the smallest infinity number of 000...0001000....00000
>
> Then there is that rule that for every finite natural k, k+1 is
> also a finite natural. I guess that rule is broken now?
>
>
> > So what I propose is that since Physics is exhausted of meaning beyond
> > 10^500, that we peg 0000....0000999...9999 as 10^500 and thus adding 1 more to
> > that delivers 0000....0001000....00000
>
> You seem to have a limited imagination. What prevents the number
> (10^500)^(10^500) from being a finite natural? Too many digits?
> Too many divisors? Because we've run out of names for all the
> numbers smaller than it?
>
Your limited comprehension.
Physicists determine the limit of physics. Since physics is superior
to math,
that is the limit.
>
> > And that is what finite means in the first place-- it has a physics reality.
>
> Where in physical reality does the number 7 exist? I don't recall
> ever stubbing my toe on a 7.
No, you banged your head 7 times on the floor when you were a toddler
thus crippling your ability to comprehend science.
plutonium....@gmail.com
David R Tribble wrote:
> David R Tribble wrote:
> >> The logic is pretty elementary.
> >> 1. Start with 0, the first natural, and which is obviously a finite
> >> number.
> >> 2. For every natural n, there also exists the natural n+1.
> >> This gives us all the naturals. Continuing,
> >
>
> Archimedes Plutonium wrote:
> > Oh, okay, so you reference the Peano Axioms, eh. So tell me, which
> > Peano axiom says that 99999 is a natural number but where
> > 99999....999999 is not a natural. So explain which Peano axiom
> > resolves that issue.
>
> I can't, for the simple reason that you won't tell us what "999...999"
> is supposed to mean. What is the "..." in the middle of the digits
> supposed to mean?
It means the same as your 0000.....00000 which you called zero and
which you said was obviously finite. Then there is 00000.....0001
which
you called 1 and said was obviously finite.
So now you are being pitifully feeble and evasive. Pretending as if
you know that 0 is 0000....00000 or 1 is 0000....00001 but acting
stupid when 9999....99999 or 3333...3333 is given.
>
> Until you answer that question, we don't have the faintest idea of
> how "999...999" is supposed to be a number.
>
Some more "acting stupid". So you know 6 is 0000....0006
but when you are asked where the Peano Axioms forbids
60000.....00000 you go into a Tribble act stupid routine.
But this is not new for you have been acting stupid for the
past years you posted to my threads. So we do not expect
you to change.
The answer which you can never forthcome is that Peano Axioms
are defunct. They are shot full of holes. The Peano Axioms do not
discern whether an integer is finite or infinite.
These numbers are all Peano Axiom Numbers:
0000....00000
0000.....00001
77777....77773
99999....959595
The greatest flaw in the Peano Axioms is why Tribble acts stupid
when questions are applied. The Peano Axioms never distinguished
between whether a number is finite or infinite. Peano forgot to make
an axiom specifically to distinguish whether a number is "finite"
or "infinite" and modern day mathematicians constantly sweep that
mistake under the rug hoping everyone ignores it. And when they cannot
ignore it, they pull silly and stupid Tribble type rejoinders.
plutonium....@gmail.com
Stephen Horne wrote:
> On Mon, 19 Jan 2009 20:41:09 -0800 (PST),
> plutonium....@gmail.com wrote:
>
> >But I also claimed in the 1990s that the Counting Numbers never are
> >an infinite set because every one of its members was forced to be
> >finite. So if all members are finite, the set can never be infinite.
> >Only
> >if some members are infinite can a set bridge across and be infinite
> >set.
>
> ...
>
> >So the Counting Numbers or the Finite leftward portion of Reals as
> >such 1,2,3,4,5, 6, .....
>
> ...
>
> >So now, working with a set that is forced to have every member as a
> >finite
> >number 0,1,2,3,4,5,...... it is impossible for that set to be infinite
> >since all
> >its members are defined as finite and thus the set can never be
> >infinite set.
> >The proofs in Old Math such as Primes are infinite or Mathematical
> >Induction
> >on Natural Numbers, and even the Peano Axioms on Natural Numbers fail
> >to axiomatize finite from infinite. All those alleged proofs are
> >flawed and erred
> >because they cannot conclude an "infinite set answer" since none of
> >their
> >numbers are infinite specimens.
>
> You seem to be arguing that because each natural number that you can
> count up to must be finite, that there can be no infinite numbers in
> the set of natural numbers.
>
> That is, as an inductive proof...
>
No there is no proof involved at this point of the discussion. This is
at the
definition and axiom level, not at proving grounds level.
The Reals are defined as rightward infinite string with "finite"
portion leftwards.
That is the definition that every mathematician today uses.
The Peano Axioms never even defined nor gave an axiom as to what
"finite"
differs from infinite, yet the Reals were built from the Peano Axioms.
So what I am doing is defining Finite from Infinite and using Physics,
the
superior subject, the king of sciences. Mathematics is but a tiny
subset
of Physics.
So Physics says that at 10^500, there is no more mathematics to
measure.
There is no time of 10^500 seconds because the Planck time is
exhausted
at 10^44 seconds.
So, if you cannot do any physics at 10^500 or 10^(-)500 and since math
is
but a subset of Physics. Well, put 2 and 2 together.
Finite means it has Physics meaning, physics experiment, physics
measuring.
If there is no physics able to be involved, well, it is in the
infinite realm.
So, where the Peano Axioms apply and are true is from 1 to 10^500 in
integers.
Beyond 10^500 is where 0000....00010000....00001 begins and is
infinite.
Likewise for Reals between 0 and 1 in that the Real Number
0d000...00100...0000
is 10^(-)500. Anything smaller than that is not measurable by Physics
and thus
in the infinitesimal range.
Stephen Horne
plutonium....@gmail.com wrote:
>The greatest flaw in the Peano Axioms is why Tribble acts stupid
>when questions are applied. The Peano Axioms never distinguished
>between whether a number is finite or infinite. Peano forgot to make
>an axiom specifically to distinguish whether a number is "finite"
>or "infinite" and modern day mathematicians constantly sweep that
>mistake under the rug hoping everyone ignores it. And when they cannot
>ignore it, they pull silly and stupid Tribble type rejoinders.
The definition of infinite is well known and built into the name...
infinite => not finite
The set of finite natural numbers is trivial to define using
induction...
finite (1)
forall n > 1 : finite (n) if finite (n-1)
And of course Peanos axioms basically define the set of natural
numbers as follows...
natural (1)
forall n > 1 : natural (n) if natural (n-1)
Since the natural numbers are effectively defined using an induction
process, similar induction processes can be used to reason about the
set of natural numbers. There can be no barrier impassable to
induction, since the natural-numberness-induction would be equally
unable to pass any such barrier.
Clearly all natural numbers are finite. Since Peanos axioms define the
natural numbers, they have no need to define infiniteness - there can
be no infinite values in the set of natural numbers. To complain about
this is like complaining that Peano failed to define axioms for
imaginary numbers.
Anyway, the within-Peanos-axioms approach is to state that the size of
the set of natural numbers cannot be given as a natural number (or
equivalently, as a finite number). If you declare the size of the set
of natural numbers to be infinite, then you are using an additional
axiom - the widely understood definition of the word "infinite" -
which is perfectly valid. For the size of the set of natural numbers
to be infinite is no problem - Peano was only defining the *members*
of the set of natural numbers and their properties.
The only oddity is that I understood the set of natural numbers to
exclude zero, as in my description above, but apparently the current
definition includes zero. If they wanted to reference that set, I
don't know why they couldn't use the term "whole numbers" which, IIRC,
always used to refer to the set of non-negative integers back when I
was in school - and yes, that was in the 20th century (1980s).
Strange.
Not that it matters. A name is just a name. Switching names is fine,
so long as you don't confuse the concepts that the names reference.
plutonium....@gmail.com
David R Tribble wrote:
> Archimedes Plutonium wrote:
> > No proof is needed in this conversation about finite or infinite
> > because it is at the axiom level and the level of definition.
>
> You seem to be saying that N cannot be infinite because of the
> way N is defined "at the axiom level" and "at the definition level".
> What does that mean? How would you define N?
What I am saying is that the Peano Axioms do not define finite
from infinite and thus the Peano Axioms are flawed because the
Infinite Integers in AP-adics is the same set as the Peano Axioms
delivers. The N and N+1 or Successor cannot stop and say these
are only finite integers.
So when I ask you why you confirm that 000...0001 is a Finite Integer
but that 11111....11111 is not. It is only your opinion as to what you
want to say is a Peano Integer, because there is nothing in Peano
axioms that allows you to discard 1111....11111 or 8888...88888
yet keep 00000....00088 as a finite integer.
>
> Maybe you are saying that there is something about the members
> m and m+1 in N that causes N to be finite "at the axiom level"
> or "at the level of definition"? What would that be?
>
You still do not get it.
--- Quoting Wikipedia on Peano Axioms ---
The axioms
When Peano formulated his axioms, the language of mathematical logic
was in its infancy. The system of logical notation he created to
present the axioms did not prove to be popular, although it was the
genesis of the modern notation for set membership (∈ from Peano's ε)
and implication (⊃ from Peano's reversed 'C'). Peano maintained a
clear distinction between mathematical and logical symbols, which was
not yet common in mathematics; such a separation had first been
introduced in the Begriffsschrift by Gottlob Frege, published in 1879.
[4] Peano was unaware of Frege's work and independently recreated his
logical apparatus based on the work of Boole and Schröder.[5]
The Peano axioms define the properties of natural numbers, usually
represented as a set N or \mathbb{N}. The first four axioms describe
the equality relation.[6]
1. For every natural number x, x = x. That is, equality is
reflexive.
2. For all natural numbers x and y, if x = y, then y = x. That is,
equality is symmetric.
3. For all natural numbers 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 natural number and a = b, then b is
also a natural number. That is, the natural numbers are closed under
equality.
The remaining axioms define the properties of the natural numbers. The
constant 0 is assumed to be a natural number, and the naturals are
assumed to be closed under a "successor" function S.
5. 0 is a natural number.
6. For every natural number n, S(n) is a natural number.
Peano's original formulation of the axioms used 1 instead of 0 as the
"first" natural number. This choice is arbitrary, as axiom 5 does not
endow the constant 0 with any additional properties. However, because
0 is the additive identity in arithmetic, most modern formulations of
the Peano axioms start from 0. Axioms 5 and 6 define a unary
representation of the natural numbers: the number 1 is S(0), 2 is S(S
(0)) (= S(1)), and, in general, any natural number n is Sn(0). The
next two axioms define the properties of this representation.
7. For every natural number n, S(n) ≠ 0. That is, there is no
natural number whose successor is 0.
8. For all natural numbers m and n, if S(m) = S(n), then m = n.
That is, S is an injection.
These two axioms together imply that the set of natural numbers is
infinite, because it contains at least the infinite subset { 0, S(0), S
(S(0)), … }, each element of which differs from the rest. The final
axiom, sometimes called the axiom of induction, is a method of
reasoning about all natural numbers; it is the only second order
axiom.
9. If K is a set such that:
* 0 is in K, and
* for every natural number n, if n is in K, then S(n) is in
K,
then K contains every natural number.
--- end quoting Wikipedia on Peano Axioms ---
There is nothing in the Peano Axioms that says what is "finite" versus
"infinite"
And those nine axioms say that the Natural Numbers are
0000....000000 to 9999.....999999
What Peano missed, and the flaw of his axioms is that he needed a
tenth axiom
to separate Finite from Infinite.
That tenth axiom should look like this:
10. Define finite as from All Possible Digit Arrangements of Natural
Numbers, wherein
0000....0000 is the smallest and 9999....99999 the largest. Then that
gives
these two Natural Numbers 0000....00001 and 1000....00000. Now define
multiplication
and where 10000.....00000 x 0000....00001 = 00000....000100....000000
This number 00000....00001000...0000 is the template number for the
smallest
infinity in Natural Numbers.
Now we go to Physics, since physics is superior to mathematics and
where all of
math is a subset of physics. We have Physics define "finite" from
"infinite". What
is the largest or smallest number in Physics to where experiment,
observation, testing,
measurement cease to exist? That number from Planck units and Coulomb
Interactions
is the number of around 10^500.
If there is no physics after 10^500 in practical terms then there is
no mathematics.
Thus we define finite as the point in which physics stops to be
measured or tested
or experimented. So we can no longer do physics at 10^500 and since
physics
creates mathematics, we cannot do math beyond 10^500. That is not
saying that
there is no physics or math beyond 10^500 but that it is a barrier and
that barrier
is the point at which "finite math ends" and "infinite math begins".
So the 10th Peano Axiom so that the Natural Numbers are finite Natural
Numbers
the 10th axiom should look like this:
10. The previous nine Peano Axioms allow for all integers from
0000....0000
to 9999....9999, but we want to define Finite from Infinite and we
determine
that the Integer 0000....000100....00000 is the turning-point integer
to where
finite integers end and infinite-integers begin. And since the most
current
physics data puts the number of 10^500 as the limit to physics
measurement.
So all numbers smaller than 10^500 are Finite Integers and beyond
10^500
are Infinite Integers.
Now that also solves the problem on Reals with the definition of Reals
as
finite portion leftward, decimal point, infinite portion rightwards.
The finite portion leftwards can only be integers from 0 to 10^500
This also solves the problem with the radix point in AP-adics for that
finite
portion is only 10^(-)500
lwa...@lausd.net
> On Thu, 22 Jan 2009 12:45:48 -0800 (PST),
> plutonium.archime...@gmail.com wrote:
> >The greatest flaw in the Peano Axioms is why Tribble acts stupid
> >when questions are applied.
> exclude zero, as in my description above, but apparently the current
> definition includes zero. If they wanted to reference that set, I
> don't know why they couldn't use the term "whole numbers" which, IIRC,
> always used to refer to the set of non-negative integers back when I
> was in school - and yes, that was in the 20th century (1980s).
> Strange.
I agree with Horne wholeheartedly. I learned it that way too:
N = {1,2,3,4,...} (naturals)
W = {0,1,2,3,4,...} (wholes)
Of course, what happened is that in _set theory_, there is no
debate that:
omega = {0,1,2,3,4,...} (finite ordinals)
and if we define the naturals to be the finite ordinals, then
there's no debate that zero is a natural number.
Nowadays, set theorists seldom, if ever, use the term "whole
number," and often on sci.math, starting the naturals with
one is associated with the so-called "cranks."
Hey, I'm all for identifying the finite ordinals with the
"whole numbers" -- especially since the letter W for "whole"
looks a lot like an "omega" symbol anyway.