PLUTONIUM INTEGERS
Alexander Abian
In ZF classical set theory, the natural numbers 0, 1, 2, 3, ....
are (and are called) finite ordinals which are the same as finite
cardinals. The smallest transfinite ordinal (which is also the smallest
transfinite cardinal) is "omega"
Now, if you present a set-theoretician with a picture, say, like
(1) .....256341765439800654
where the sequence of the digits expands to the left with NEVER HAVING
A BEGINNING, and if you insist that the set-theoretician gives a meaning
to (1), most probably the set-theoretician will say that :
the meaning of .....256341765439800654
is
limit of 4, 54. 654, 0654, 00654, 800654, 98800654 .... = omega
or lim of 800654, 439800654, 65439800654, 41765439800654 ... = omega
and some other variations but always yielding "omega"
So, for the set-theoretician:
(2) .....256341765439800654 = omega
But then again if you ask the set-theoretician to give a meaning to the
beginningless sequence, say,
(3) .....777789055433211533
Again, most probably the set-theoretician will say that:
the meaning of .....777789055433211533
is
(4) limit of 3, 33, 533, 1533, 211533, 3211533,.... = omega
or limit of 1533, 211533, 5433211533, 55433211533 .... = omega
So for the set-theoretician:
(5) .....777789055433211533 = omega
Being well aware that it is not inconsistent to assign the same
limit to different looking sequences, nevertheless it is quite
justifiable, rational, logical, imaginative and insightful to
consider arithmetics of items such as the beginningless sequences
where (2) and (5) do not have the same meaning and do not have
identical arithmetical values.
Clearly beginningless sequences such as (2) and (5) are not
ordinal or cardinal numbers. They are not also integers in
the usual arithmetical sense. But since, as will be remarked later
that arithmetically the beginningless sequence such as
..........000000000000001999 will play the role of the integer 1999
and motivated by Archimedes Plutonium's earlier introduction of the
present idea we will refer to any beginningless sequence of digits as a
PLUTONIUM INTEGER
The major significance of Plutonium integers lie in the fact that in
Plutonium arithmetic (2) and (5) will not have the same value
and thus, for the first time there will be continuum many infinite
integers (none being an ordinal number)
PLUTONIUM ARITHMETIC
To develop the arithmetic of Plutonium integers the very first
step is to define the addition and multiplication of two
Plutonium integer.
For the sake of brevity I will let "ap" stand for "Archimedes Plutonium"
So in order to define addition of two ap-integers one has to define
the process of how to determine the precise digit at any decimal point
of the sum of the given two ap-integers based on their represent-
ations:
The following example will show the process
The sum
(6) .........6984410098 + ......7209876554
is defined as follows: add segmentwise based on the ordinary arithmetic
8 + 4 = 12
98 + 54 = 152
098 + 554 = 652
0098 + 6554 = 6652
10098 + 76554 = 86652 (7)
410098 + 876554 = 1286652
4410098 + 9876554 = 14286652
84410098 + 09876554 = 94286652
.........+ ........ = ........
Now, the sum of two ap-integers appearing in (6) is UNAMBIGUOUSLY and
UNIQUELY DEFINED AS an ap-number as follows:
start with the digit in the first decimal place of the first sum in
the column (7), i.e., 2, then to the left of 2 insert the 2-nd digit
of the second sum appearing in (7), i.e, 5 then to the left of 5 insert
the 3-rd digit of the third sum appearing in (7), i.e., 6, then to the
left of 6 insert the 4-th digit in the forth sum appearing in (7), and
follow the procedure (called DIAGONAL process) to obtain the Plutonium
integer
.....94286652
Thus (6) is answered as follows
(7) ........6984410098 + ......7209876554 = .....94286652
The above shows that addition of Plutonium integers is WELL defined
and the corresponding sum is a Plutonium integer.
The process beautifully works for the sum of "ordinary finite
integers"
Let us find the sum of, say, 67 + 34
First Plutoniumize 67 and 34 as:
....00000067 and ....00000034
Now apply the above procedure
7 + 4 = 11
67 + 34 = 101
067 + 034 = 0101
0067 + 0034 = 00101
........ + .... = ....
Applying the above diagonal process we obtain
(8) OOOOOO101
now,deplutoniumize (8), we obtain 101 and thus as expected in the
usual arithmetic
(9) 67 + 34 = 101
I am too tired and exhausted. Multiplication in Plutonium
arithmetic an be similarly defined based on ordinary multiplication
of the final finite segments.
Then Plutonium REAL Numbers can be defined as a picture with a
decimal point with two infinite sequences of digits one expanding
to the left and the other to the right of the decimal point.
In conclusion PLUTONIUM INTEGERS have well defined arithmetic and
may found some interesting applications. Their pure intellectual
significance is the fact that there are continuum many Plutonium integers
and as stated for example (2) and (5) represent two distinct
Plutonium integers neither being equal to "omega".
I am exhausted and tired and I leave to Archimedes Plutonium and others
to develop Plutonium integers and real numbers and their arithmetic and
analysis.
There could be some typos and minor oversights.
Alexander Abian
--
-------------------------------------------------------------------------
ABIAN TIME-MASS EQUIVALENCE FORMULA T = A m^2 in Abian units.
ALTER EARTH'S ORBIT AND TILT TO STOP GLOBAL DISASTERS AND EPIDEMICS.
JOLT THE MOON TO JOLT THE EARTH INTO A SANER ORBIT.ALTER THE SOLAR SYSTEM.
REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT TO CREATE A BORN AGAIN EARTH(1990)
THERE WAS A BIG SUCK AND DILUTION OF PRIMEVAL MASS INTO THE VOID OF SPACE
Archimedes Plutonium
ab...@iastate.edu (Alexander Abian) writes:
Glad to see you are back Dr. Abian, I missed you in all these years.
But I am afraid that I am not interested in (Adics = Natural Numbers)
or any version thereof, until about Autumn term of College when the new
school year starts. I feel it my duty to teach and warn students about
mathematics, especially Euclid's proof of the Infinitude of Primes. I
try to steer students away from math as a career and into a more
rewarding career of engineering or one of the sciences.
= In ZF classical set theory, the natural numbers 0, 1, 2, 3, ....
= are (and are called) finite ordinals which are the same as finite
= cardinals. The smallest transfinite ordinal (which is also the
smallest
= transfinite cardinal) is "omega"
=
= Now, if you present a set-theoretician with a picture, say, like
=
= (1) .....256341765439800654
=
= where the sequence of the digits expands to the left with NEVER
HAVING
= A BEGINNING, and if you insist that the set-theoretician gives a
meaning
= to (1), most probably the set-theoretician will say that :
= the meaning of .....256341765439800654
= is
=
= limit of 4, 54. 654, 0654, 00654, 800654, 98800654 .... =
omega
=
= or lim of 800654, 439800654, 65439800654, 41765439800654 ... =
omega
=
= and some other variations but always yielding "omega"
=
= So, for the set-theoretician:
=
= (2) .....256341765439800654 = omega
=
=
= But then again if you ask the set-theoretician to give a meaning to
the
= beginningless sequence, say,
=
= (3) .....777789055433211533
=
= Again, most probably the set-theoretician will say that:
=
= the meaning of .....777789055433211533
= is
=
= (4) limit of 3, 33, 533, 1533, 211533, 3211533,.... = omega
=
= or limit of 1533, 211533, 5433211533, 55433211533 .... = omega
=
= So for the set-theoretician:
=
=
= (5) .....777789055433211533 = omega
=
=
= Being well aware that it is not inconsistent to assign the same
= limit to different looking sequences, nevertheless it is quite
= justifiable, rational, logical, imaginative and insightful to
= consider arithmetics of items such as the beginningless sequences
= where (2) and (5) do not have the same meaning and do not have
= identical arithmetical values.
=
= Clearly beginningless sequences such as (2) and (5) are not
= ordinal or cardinal numbers. They are not also integers in
= the usual arithmetical sense. But since, as will be remarked later
= that arithmetically the beginningless sequence such as
=
= ..........000000000000001999 will play the role of the integer
1999
=
= and motivated by Archimedes Plutonium's earlier introduction of the
= present idea we will refer to any beginningless sequence of digits as
a
=
= PLUTONIUM INTEGER
=
= The major significance of Plutonium integers lie in the fact that in
= Plutonium arithmetic (2) and (5) will not have the same value
= and thus, for the first time there will be continuum many infinite
= integers (none being an ordinal number)
=
= PLUTONIUM ARITHMETIC
=
= To develop the arithmetic of Plutonium integers the very first
= step is to define the addition and multiplication of two
= Plutonium integer.
=
= For the sake of brevity I will let "ap" stand for "Archimedes
Plutonium"
=
= So in order to define addition of two ap-integers one has to define
= the process of how to determine the precise digit at any decimal
point
= of the sum of the given two ap-integers based on their represent-
= ations:
=
= The following example will show the process
=
= The sum
=
= (6) .........6984410098 + ......7209876554
=
= is defined as follows: add segmentwise based on the ordinary
arithmetic
=
= 8 + 4 = 12
= 98 + 54 = 152
= 098 + 554 = 652
= 0098 + 6554 = 6652
= 10098 + 76554 = 86652
(7)
= 410098 + 876554 = 1286652
= 4410098 + 9876554 = 14286652
= 84410098 + 09876554 = 94286652
= .........+ ........ = ........
=
I do not know how you come up with these schemes Abian but they are
neat and original. It begs the question of whether the P-adic Integers
as a system of numbers can be *improved upon*. Whether we are at the
development stage of the p-adics when human culture was at the
development stage of say having Rationals but not Irrationals, or at
the historical development stage when humanity had the Reals but did
not have imaginary numbers. I do not know if the P-adic Number System
is all there, as the Reals/Complex Number System is all there. And when
I see one of your latest constructions Abian, it reminds me of that
question of whether the P-adics are all there.
= Now, the sum of two ap-integers appearing in (6) is UNAMBIGUOUSLY
and
= UNIQUELY DEFINED AS an ap-number as follows:
=
= start with the digit in the first decimal place of the first sum
in
= the column (7), i.e., 2, then to the left of 2 insert the 2-nd
digit
= of the second sum appearing in (7), i.e, 5 then to the left of 5
insert
= the 3-rd digit of the third sum appearing in (7), i.e., 6, then to
the
= left of 6 insert the 4-th digit in the forth sum appearing in (7),
and
= follow the procedure (called DIAGONAL process) to obtain the
Plutonium
= integer
= .....94286652
=
= Thus (6) is answered as follows
=
= (7) ........6984410098 + ......7209876554 = .....94286652
=
= The above shows that addition of Plutonium integers is WELL defined
= and the corresponding sum is a Plutonium integer.
=
= The process beautifully works for the sum of "ordinary finite
= integers"
=
= Let us find the sum of, say, 67 + 34
=
= First Plutoniumize 67 and 34 as:
=
= ....00000067 and ....00000034
=
= Now apply the above procedure
=
= 7 + 4 = 11
= 67 + 34 = 101
= 067 + 034 = 0101
= 0067 + 0034 = 00101
= ........ + .... = ....
= Applying the above diagonal process we obtain
=
= (8) OOOOOO101
= now,deplutoniumize (8), we obtain 101 and thus as expected in the
= usual arithmetic
=
= (9) 67 + 34 = 101
=
= I am too tired and exhausted. Multiplication in Plutonium
= arithmetic an be similarly defined based on ordinary multiplication
= of the final finite segments.
=
= Then Plutonium REAL Numbers can be defined as a picture with a
= decimal point with two infinite sequences of digits one expanding
= to the left and the other to the right of the decimal point.
=
= In conclusion PLUTONIUM INTEGERS have well defined arithmetic and
= may found some interesting applications. Their pure intellectual
= significance is the fact that there are continuum many Plutonium
integers
= and as stated for example (2) and (5) represent two distinct
= Plutonium integers neither being equal to "omega".
=
= I am exhausted and tired and I leave to Archimedes Plutonium and
others
= to develop Plutonium integers and real numbers and their arithmetic
and
= analysis.
=
= There could be some typos and minor oversights.
=
= Alexander Abian
=
= --
=
------------------------------------------------------------------------
-
= ABIAN TIME-MASS EQUIVALENCE FORMULA T = A m^2 in Abian units.
= ALTER EARTH'S ORBIT AND TILT TO STOP GLOBAL DISASTERS AND
EPIDEMICS.
= JOLT THE MOON TO JOLT THE EARTH INTO A SANER ORBIT.ALTER THE SOLAR
SYSTEM.
= REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT TO CREATE A BORN AGAIN
EARTH(1990)
= THERE WAS A BIG SUCK AND DILUTION OF PRIMEVAL MASS INTO THE VOID OF
SPACE
Abian, I do not want to get involved with P-adics at this time. I am
working on Conic Section Systems and whether the Conic Sections are
*unique* to Euclidean Geometry. Question Abian: do you know of any
theorem or mathematics that implies that Riemannian Geometry and
Lobachevskian Geometry are incapable of having a Conic Section System?
I do not know if you are -not a geometer- Dr. Abian, but if you can
help me on that Conic Section question, I would be grateful. Thanks
James Kibo Parry
sci.physics.relativity, Alexander Abian (ab...@iastate.edu) wrote:
>
> Subject: PLUTONIUM INTEGERS
Ma! Now Archie's done gone posted under Abian's name!
No, wait, it really is Alexander Abian. Weird. Abian seems to have
endorsed Plutonium's wacky math -- perhaps this means Archimedes Plutonium
will adopt Dr. Abian's wacky astronomy. Naah, the distinguished
Mr. Plutonium would never adopt a wacky idea like blowing up the Moon.
> [...snipped: a finite amount of blather about infinite strings of digits...]
>
> But since, as will be remarked later
> that arithmetically the beginningless sequence such as
>
> ..........000000000000001999 will play the role of the integer 1999
With Martin Landau playing the role of Captain Kirk, and Barbara Bain
playing the role of Nurse Chapel! Only without the stunning acting
ability of Gene Roddenberry's wife!
> and motivated by Archimedes Plutonium's earlier introduction of the
> present idea we will refer to any beginningless sequence of digits as a
>
> PLUTONIUM INTEGER
And adding or subtracting or multiplying or dividing them will be
a PLUTONIUM OPERATION, or a PuOP for short.
PuOP! PuOP! PuOP!
I WILL PAY ALL THE LEGITIMATE SCIENTISTS IN THE WORLD A DOLLAR EACH IF
THEY UNANIMOUSLY VOTE TO PETITION THE QUEEN OF ENGLAND TO PUT "PuOP"
INTO THE ENCYCLOPEDIA BRITANNICA.
(Just don't tell her that Archie Plutonium has declared that her country
is a peninsula. The big question is, a peninsula of WHAT? Ireland?
Europe? McDonaldland?)
> The major significance of Plutonium integers lie in the fact that in
> Plutonium arithmetic (2) and (5) will not have the same value
> and thus, for the first time there will be continuum many infinite
> integers (none being an ordinal number)
>
> PLUTONIUM ARITHMETIC
>
> To develop the arithmetic of Plutonium integers the very first
> step is to define the addition and multiplication of two
> Plutonium integer.
PuOP! PuOP! PuOP!
> For the sake of brevity I will let "ap" stand for "Archimedes Plutonium"
No, in Plutonium Arithmentic you have to call him ".........apapapapapapapap".
Please fix your horrendous mistake, otherwise we will never be able
to perform PuOPs on Archimedes Plutonium.
> So in order to define addition of two ap-integers one has to define
> the process of how to determine the precise digit at any decimal point
> of the sum of the given two ap-integers based on their represent-
> ations:
Hey, how do the "ap-integers" correspond to those numbers made up five
years ago by LUDWIG plutonium? Would those be lp-integers, which would
be made obsolete by cd-integers? Would lp-integers not freeze-frame
as well as sp-integers and ep-integers in my VCR?
> The following example will show the process
>
> The sum
>
> (6) .........6984410098 + ......7209876554
>
> is defined as follows: add segmentwise based on the ordinary arithmetic
>
> 8 + 4 = 12
> 98 + 54 = 152
> 098 + 554 = 652
> 0098 + 6554 = 6652
> 10098 + 76554 = 86652 (7)
> 410098 + 876554 = 1286652
> 4410098 + 9876554 = 14286652
> 84410098 + 09876554 = 94286652
> .........+ ........ = ........
>
> Now, the sum of two ap-integers appearing in (6) is UNAMBIGUOUSLY and
> UNIQUELY DEFINED AS an ap-number as follows:
>
> start with the digit in the first decimal place of the first sum in
> the column (7), i.e., 2, then to the left of 2 insert the 2-nd digit
> of the second sum appearing in (7), i.e, 5 then to the left of 5 insert
> the 3-rd digit of the third sum appearing in (7), i.e., 6, then to the
> left of 6 insert the 4-th digit in the forth sum appearing in (7), and
> follow the procedure (called DIAGONAL process) to obtain the Plutonium
> integer
> .....94286652
Wow! You have just invented ORDINARY ADDITION!!!
Perhaps you should get out your yellow highlighter (you know, the one that
says "Crayola") and draw a circle around the diagonal numbers on your screen
so you can look at them more closely. Notice how the first seven numbers
in the diagonal are always exactly the same as the seven digits in the
seventh row?
Provided you lick that fatal flaw about ignoring the carried '1', your
method could one day become every bit as good as ordinary addition!
> Thus (6) is answered as follows
>
> (7) ........6984410098 + ......7209876554 = .....94286652
>
> The above shows that addition of Plutonium integers is WELL defined
> and the corresponding sum is a Plutonium integer.
No it doesn't. You added a number with EIGHT dots to a number with SIX
dots and got a number with FIVE dots. The number on the right should
have had THIRTEEN dots, you bozo!
> The process beautifully works for the sum of "ordinary finite
> integers"
>
> Let us find the sum of, say, 67 + 34
>
> First Plutoniumize 67 and 34 as:
>
> ....00000067 and ....00000034
>
> Now apply the above procedure
>
> 7 + 4 = 11
> 67 + 34 = 101
> 067 + 034 = 0101
> 0067 + 0034 = 00101
> ........ + .... = ....
> Applying the above diagonal process we obtain
>
> (8) OOOOOO101
> now,deplutoniumize (8), we obtain 101 and thus as expected in the
> usual arithmetic
>
> (9) 67 + 34 = 101
>
> I am too tired and exhausted.
Maybe you should buy a calculator. If you already have one, I recommend
you go over to the Big Kids department and invest in one that has a plus
key that really works.
Also, what's the PuOP sum of tired and exhausted? Let's calculate it correctly:
d + d = dd
ed + ed = eded
red + ted = redted
ired + sted = iredsted
tired + usted = tiredusted
(throw away the "exha" part by accident,
then to keep things symmetrical also
accidentally discard the same number
of letters, "tire", from the result)
tired + exhausted = dusted
"I don't want to sound biblical but it is THE MATTER TO DUST."
(Alexander Abian, February 1997)
> Multiplication in Plutonium arithmetic an be similarly defined based on
> ordinary multiplication of the final finite segments.
>
> Then Plutonium REAL Numbers can be defined
I can't wait for the Imaginary Plutonium Numbers.
> as a picture with a decimal point with two infinite sequences of digits
> one expanding to the left and the other to the right of the decimal point.
What about above and below? What about numbers with two decimal points?
What about numbers arranged to make pictures of kitties?
> In conclusion PLUTONIUM INTEGERS have well defined arithmetic and
> may found some interesting applications. Their pure intellectual
> significance is
less than their Pure Chewing Satisfaction!
> the fact that there are continuum many Plutonium integers
> and as stated for example (2) and (5) represent two distinct
> Plutonium integers neither being equal to "omega".
>
> I am exhausted and tired and I leave to Archimedes Plutonium and others
> to develop Plutonium integers and real numbers and their arithmetic and
> analysis.
HOORAY! HE SAID "OTHERS"! I CLAIM ALL THE EVEN PLUTONIUM NUMBERS!
ARCHIE CAN HAVE ALL THE ODD ONES! ESPECIALLY THE VERY ODD ONES!
> There could be some typos and minor oversights.
Yes, you SHOULD have oversight.
> Alexander Abian
>
> --
> -------------------------------------------------------------------------
> ABIAN TIME-MASS EQUIVALENCE FORMULA T = A m^2 in Abian units.
> ALTER EARTH'S ORBIT AND TILT TO STOP GLOBAL DISASTERS AND EPIDEMICS.
> JOLT THE MOON TO JOLT THE EARTH INTO A SANER ORBIT.ALTER THE SOLAR SYSTEM.
> REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT TO CREATE A BORN AGAIN EARTH(1990)
> THERE WAS A BIG SUCK AND DILUTION OF PRIMEVAL MASS INTO THE VOID OF SPACE
How can I trust your theories to be accurate when you can't even
make your signature symmetrical?
-- K.
Also, I think there might be something
vaguely silly about the term "Big Suck",
but I am not sure. Please diagram a
Big Suck so that I may ascertain whether
or not it is a silly term.
P.S. PuOP! PuOP! PuOP!
Alexander Abian
att: C. Blue
and a note to A. Plutonium
In article <djenny-0606...@ppp-207-214-211-182.sntc01.pacbell.net>,
China Blue <dje...@my-dejanews.com> wrote:
>[|>>> Subject: PLUTONIUM INTEGERS
>
>[|>Now you too? I believe that eight plus six is fourteen, not thirteen.
>[|>But I could be mistaken in what you are trying to do. The operation +'
>[|>defined by x+'y=x+y-1 is a conjugated version of addition (being (x-1)
Abian answers:
Plutonium arithmetic is a nontrivial proper extension of the ordinary
(Peano Arithmetic)
The significance of Plutonium arithmetic is that it has continuum
many integers (and not only countably many integers as in the
ordinary Arithmetic)
As a proper extension of ordinary Arithmetic, the latter is isomorphic
to a proper subset of the Plutonium Arithmetic. Indeed every ordinary
integer can be identified with a Plutonium integer (the process is called
Plutoniumization) but not conversely. . Thus, say
3 is plutoniumized as .........0000000003
Now your comment that "in Plutonium mathematiocs 6 + 8 = 13" IS NOT
CORRECT. You have to plutniumize both 6 and 8 and then apply the
diagonal procedure of Plutonium to obtain the sum, as follows
......00008 + .....000006
8 + 6 = 14
08 + 06 = 014
008 + 006 = 0014
..... + ..... = ....
and the diagonal of the last column in the above yields the
Plutonium integer
......0000000014
which when deplutoniumized yields 14 (and not 13, as you suggested)
Of course for addition of two ordinary integers you don't have to
go thru the above procedure of Plutoniumizing and then deplutoniumizing .
You just add as in ordinary Arithmetic, i.e., 6 + 8 = 14.
The point in the above explanation is just to prove that Plutonium
Arithmetic preserves the rules and deductions of the ordinary
Arithmetic
The interlocutor continues:
>
>[|>new. This is just a formal way of justifying a renaming of the
>[|>integers.
>
>No, it's a renaming of reals. Map the reals bijectively into (0,1) and
>then reverse their decimal expansion. Or something like that. It's not so
>much as wrong as pointless.
>
Abian answers
I already remarked that Plutonium Integers cannot be a renaming of
the ordinary integers since there are countably many ordinary integers
but continuum many Plutonium integers. Every ordinary integer can be
renamed as a Plutonium integers (i.e. plutoniumized) BUT NOT CONVERSELY !
and this is the novelty of Plutonium integers and their arithmetic
Similarly, the above suggestion of the "most probably by your
intended bijection map into (0, 1) etc " is also INCORRECT. Since
the two distinct Plutonium integers ..99999999990 and ..00000000001
would map to the same real.
I would like to reiterate that PLUTONIUM INTEGERS AND THEIR ARITHMETIC
HAVE SOLID MATHEMATICAL CONTENT AS A NONTRIVIAL PROPER EXTENSION
OF THE ORDINARY ARITHMETIC. THEY EXTEND THE ORDINARY ARITHMETIC TO A
SET OF CONTINUUM MANY DISTINCT PLUTONIUM INTEGERS.
Note to Archimedes Plutonium.
____________________________
Plutonium intrgers and their Arithmetic HAVE NO DIRECT
CONNECTION with p-adic numbers and their arithmetic.
Plutonium integers are nontrivial proper extension of ordinary
arithmetic to continuum many integers.Thus, roughly speaking they
provide a model of "ordinary-like" arithmetic which has the
cardinality of the Continuum.
This fact may have interesting and valuable mathematical content
Also, I called Plutonium integers and Plutonium Arithmetic
after some of your ideas posted 3 or 4 years ago (?) the basic
idea came from you (I don't remember whether or not they were
in connection with your p-adics) In any event Plutonium integers
and their arithmetic have no direct connection with p-adic numbers and
their arithmetic. For me it was proper and legitimate to call
what I posted recently as "Plutonium integers and their arithmetic".
I hope you have no objections in my naming so. If you have any
please advise me so.
Archimedes Plutonium
ab...@iastate.edu (Alexander Abian) writes:
= att: C. Blue
= and a note to A. Plutonium
=
= In article
<djenny-0606...@ppp-207-214-211-182.sntc01.pacbell.net>,
= China Blue <dje...@my-dejanews.com> wrote:
= =[|=== Subject: PLUTONIUM INTEGERS
= =
= =[|=Now you too? I believe that eight plus six is fourteen, not
thirteen.
= =[|=But I could be mistaken in what you are trying to do. The
operation +'
= =[|=defined by x+'y=x+y-1 is a conjugated version of addition (being
(x-1)
=
= Abian answers:
=
= Plutonium arithmetic is a nontrivial proper extension of the
ordinary
= (Peano Arithmetic)
= The significance of Plutonium arithmetic is that it has continuum
= many integers (and not only countably many integers as in the
= ordinary Arithmetic)
=
= As a proper extension of ordinary Arithmetic, the latter is
isomorphic
= to a proper subset of the Plutonium Arithmetic. Indeed every
ordinary
= integer can be identified with a Plutonium integer (the process is
called
= Plutoniumization) but not conversely. . Thus, say
=
= 3 is plutoniumized as .........0000000003
=
= Now your comment that "in Plutonium mathematiocs 6 + 8 = 13" IS
NOT
= CORRECT. You have to plutniumize both 6 and 8 and then apply the
= diagonal procedure of Plutonium to obtain the sum, as follows
=
= ......00008 + .....000006
=
= 8 + 6 = 14
= 08 + 06 = 014
= 008 + 006 = 0014
= ..... + ..... = ....
= and the diagonal of the last column in the above yields the
= Plutonium integer
= ......0000000014
=
= which when deplutoniumized yields 14 (and not 13, as you
suggested)
=
= Of course for addition of two ordinary integers you don't have to
= go thru the above procedure of Plutoniumizing and then
deplutoniumizing .
= You just add as in ordinary Arithmetic, i.e., 6 + 8 = 14.
=
= The point in the above explanation is just to prove that Plutonium
= Arithmetic preserves the rules and deductions of the ordinary
= Arithmetic
=
= The interlocutor continues:
=
= =
= =[|=new. This is just a formal way of justifying a renaming of the
= =[|=integers.
= =
= =No, it's a renaming of reals. Map the reals bijectively into (0,1)
and
= =then reverse their decimal expansion. Or something like that. It's
not so
= =much as wrong as pointless.
= =
=
= Abian answers
=
= I already remarked that Plutonium Integers cannot be a renaming
of
= the ordinary integers since there are countably many ordinary
integers
= but continuum many Plutonium integers. Every ordinary integer can be
= renamed as a Plutonium integers (i.e. plutoniumized) BUT NOT
CONVERSELY !
= and this is the novelty of Plutonium integers and their arithmetic
=
= Similarly, the above suggestion of the "most probably by your
= intended bijection map into (0, 1) etc " is also INCORRECT. Since
=
= the two distinct Plutonium integers ..99999999990 and
..00000000001
= would map to the same real.
=
=
= I would like to reiterate that PLUTONIUM INTEGERS AND THEIR
ARITHMETIC
= HAVE SOLID MATHEMATICAL CONTENT AS A NONTRIVIAL PROPER EXTENSION
= OF THE ORDINARY ARITHMETIC. THEY EXTEND THE ORDINARY ARITHMETIC TO A
= SET OF CONTINUUM MANY DISTINCT PLUTONIUM INTEGERS.
=
= Note to Archimedes Plutonium.
= ____________________________
=
= Plutonium intrgers and their Arithmetic HAVE NO DIRECT
= CONNECTION with p-adic numbers and their arithmetic.
But can they be linked to the p-adic numbers? Can they be made to be
isomorphic? Can these Plutonium Integers be so devised as to get rid of
the base dependency of p-adic integers? And that one can then visualize
all the p-adics at once, rather than disjoint union of 2-adics, 3-adics
etc.
Abain, is your construction scheme, a substitution of operators for
that of bases in p-adics? You have a triangle-segment definition of
addition and multiplication for base 10, and let us say it is
isomorphic to 10-adics. Then, is there a different scheme of addition
and multiplication that would yield the 2-adics and be isomorphic to
the 2-adics, and then on to the 3-adics?
If you can do that, the beauty of your construction would then be to
visualize all of the P-ADIC NUMBERS at once. The mind can picture Reals
at once and geometrically, and that is perhaps the greatest beauty of
the Reals, to be able to picture them geometrically, but with the
p-adics, no-one can picture them geometrically. Karl Heuer, about 5
years ago said the p-adics are more like a tree. But I find that
unsatisfactory. The best I can come up with so far as to picturing the
P-ADICS is that of nested balls, or balls inside of balls. Smaller
balls inside of bigger balls. But that is only a hunch and perhaps
wrong.
Can your construction scheme, Abian, be extended as to create the
2-adics, and the 3-adics and the 5-adics and the 6-adics etc. Perhaps
by changing operation schemes, you can replace the changing bases of
p-adics?
Let me give the name INFINITE INTEGERS to all of these construction
schemes. Thus, p-adics are Infinite Integers, and so are Plutonium
Integers.
Question, do the p-adics have the greatest mathematical content of
any Infinite Integers?
=
= Plutonium integers are nontrivial proper extension of ordinary
= arithmetic to continuum many integers.Thus, roughly speaking they
= provide a model of "ordinary-like" arithmetic which has the
= cardinality of the Continuum.
= This fact may have interesting and valuable mathematical content
=
= Also, I called Plutonium integers and Plutonium Arithmetic
= after some of your ideas posted 3 or 4 years ago (?) the basic
= idea came from you (I don't remember whether or not they were
= in connection with your p-adics) In any event Plutonium integers
= and their arithmetic have no direct connection with p-adic numbers
and
= their arithmetic. For me it was proper and legitimate to call
= what I posted recently as "Plutonium integers and their
arithmetic".
= I hope you have no objections in my naming so. If you have any
= please advise me so.
=
= Alexander Abian
Alexander, I need your help on a geometry problem. A reference will
do. I need to know whether the Conic-Section-System is unique to
Euclidean Geometry. Or whether a Conic-Section-System exists in
Riemannian Geometry also, (and likewise for Lobachevskian Geometry)? I
believe it is unique to Euclidean Geometry. I need to know whether any
mathematician has well-defined Conic-Section-System. And if there are
any theorems about Conic Section Systems.
Harvard's McIrvin
ki...@world.std.com (James "Kibo" Parry) writes:
Sender: mmci...@world.std.com (Matthew J McIrvin)
Approved: mmci...@world.std.com (sci.physics.research)
= Physics Harvard University
= Faculty of the Department of Physics
= Henry Ehrenreich, Clowes Professor of Science
= Gary J. Feldman, Frank B. Baird, Jr. Professor of Science
= Daniel S. Fisher, Professor of Physics and Professor of Applied
Physics
= Melissa Franklin, Professor of Physics
=
= Ma! Now they done gone posted under Ehrenreich's name!
=
= No, wait, it really is Feldman. Weird. Feldman seems to have
= endorsed Ehrenreich's wacky math -- perhaps this means Feldman
= will adopt Ehrenreich's wacky astronomy. Naah, the distinguished
= Mr. Feldman would never adopt a wacky idea like blowing up Harvard.
=
= With Fisher playing the role of Captain Kirk, and Franklin
= playing the role of Nurse Chapel! Only without the stunning acting
= ability of Gene Roddenberry's wife!
=
= And adding or subtracting or multiplying or dividing them will be
= a Harvard OPERATION, or a Har for short.
=
= Har! Har! Har!
=
= I WILL PAY ALL THE LEGITIMATE SCIENTISTS IN THE WORLD A DOLLAR EACH
IF
= THEY UNANIMOUSLY VOTE TO PETITION THE QUEEN OF ENGLAND TO PUT "Har"
= INTO THE ENCYCLOPEDIA BRITANNICA.
=
= (Just don't tell her that Fisher has declared that her country
= is a peninsula. The big question is, a peninsula of WHAT? Ireland?
= Europe? McDonaldland?)
=
= Har! Har! Har!
=
= No, in Harvard Arithmentic you have to call Franklin
= ".........apapapapapapapap".
= Please fix your horrendous mistake, otherwise we will never be able
= to perform Hars on Harvard.
=
= Hey, how do the "ap-integers" correspond to those numbers made up
five
= years ago by Fisher? Would those be Fe Fi Fo Fum-integers, which
would
= be made obsolete by cd-integers? Would Fi-integers not freeze-frame
= as well as sp-integers and ep-integers in my VCR?
=
=
= -- K.
=
= P.S. Har! Har! Har!
A problem created by the computer revolution of people communication
such as Internet and WWW is that of how we define "freedom of speech",
"ad hominem", and "stalking" with relation to this new way of people
communication. AOL, the nation's largest ISP has made it clear that ad
hominem is not allowed, and therefore the most extreme form of ad
hominem-- that of stalking is not allowed by AOL. Yet, some little
rinky dink ISP such as "std.com" can get away with stalking. They have
no functional abuse desk.
And whereas I have posted in the thread "HARVARD vs. MIT in Stalking"
that one post by Mr. Dupree regards to stalking is it all takes to
convince some people at MIT to halt the stalker of Decobert of a month
or two. However, for a rogue ISP of std.com, a similar post by Mr.
Whatcott in 1998
Date: 29 Mar 1998 00:00:00 GMT
Message-ID: <6flhie$j...@enews4.newsguy.com>
is ineffective with a Parry type stalker of 6 years duration. After 2
months, MIT says enough of a stalker. But after 6 years, Parry with his
Harvard connections is allowed to continue to stalk.
If ad hominem and stalking are forbidden by AOL, then these little
tin pot ISPs of std.com should have the same rules of etiquette.
Charles H. Giffen
[snip]
>
> To develop the arithmetic of Plutonium integers the very first
> step is to define the addition and multiplication of two
> Plutonium integer.
>
> For the sake of brevity I will let "ap" stand for "Archimedes Plutonium"
>
> arithmetic an be similarly defined based on ordinary multiplication
> of the final finite segments.
Perhaps the biggest problems with ap-integers for real applications
are these:
(1) The ap-integers have no good ordering: the
negative of ...32875092 is ...67124908. Since their
sum is ...00000000 (= ap-zero), if there were a good
ordering, one of these would be positive and one would be
negative -- but which is which?
(2) The product of non-ap-zero ap-integers can be
ap-zero, so one cannot enlarge the ap-integers to be a
field. Here are the tails of a pair of ap-integers
whose product is ap-zero:
...4375 ...3152
[ Exercise: show this is true ].
Indeed, ap-integers are nothing more, nor anything less, than
the inverse limit of the sequence of finite rings and surjective
homomorphisms:
... --> Z/10^{k+1} --> Z/10^k --> ... --> Z/100 --> Z/10 .
For any (ordinary) integer m > 1, let Z_m denote the
inverse limit of
... --> Z/m^{k+1} --> Z/m^k --> ... --> Z/m^2 --> Z/m .
Then ap-integers are just Z_10. There seems to be nothing
special about Z_10 (except for humankind's predilection
towards base 10 arithmetic). One might just as well have
considered Z_15 or Z_60 or Z_112. Only when m is
a power of a prime, m = p^s, has Z_m = Z_p ever been
regarded as having truly nice properties -- being the
classical p-adic integers, as introduced by Hensel. For
then Z_p is a complete local ring, and its field of
quotients Q_p (the p-adic numbers) is just the completion
of the rationals with respect to the p-adic valuation.
There is a rich theory of "p-adic analysis" which tends to
vaporize when one tries to extend its ideas to Z_m for
m not a prime power.
The ap-integers, while perhaps cute or amusing to some, are
really nothing special in the world of mathematica or of
physics.
--Chuck Giffen
Pertti Lounesto
> Then ap-integers are just Z_10. There seems to be nothing
> special about Z_10 (except for humankind's predilection
> towards base 10 arithmetic). One might just as well have
> considered Z_15 or Z_60 or Z_112. Only when m is
> a power of a prime, m = p^s, has Z_m = Z_p ever been
> regarded as having truly nice properties -- being the
> classical p-adic integers, as introduced by Hensel.
The Dozenal Society of Great Britain is devoted to promoting Z_12.
See http://www.shaunf.dircon.co.uk/shaun/metrology/tgm.htm.
No Z_p, p prime, has a society devoted to its promotion. On the
contrary, promotors of Z_12 say that it is indeed satis_factory_.
Archimedes Plutonium
Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:
> If you can do that, the beauty of your construction would then be to
> visualize all of the P-ADIC NUMBERS at once. The mind can picture Reals
> at once and geometrically, and that is perhaps the greatest beauty of
> the Reals, to be able to picture them geometrically, but with the
> p-adics, no-one can picture them geometrically. Karl Heuer, about 5
> years ago said the p-adics are more like a tree. But I find that
> unsatisfactory. The best I can come up with so far as to picturing the
> P-ADICS is that of nested balls, or balls inside of balls. Smaller
> balls inside of bigger balls. But that is only a hunch and perhaps
> wrong.
>
> Can your construction scheme, Abian, be extended as to create the
> 2-adics, and the 3-adics and the 5-adics and the 6-adics etc. Perhaps
> by changing operation schemes, you can replace the changing bases of
> p-adics?
>
> Let me give the name INFINITE INTEGERS to all of these construction
> schemes. Thus, p-adics are Infinite Integers, and so are Plutonium
> Integers.
>
> Question, do the p-adics have the greatest mathematical content of
> any Infinite Integers?
Irony and the occurrence of ironies is perhaps the very best
indication of a God, a superintelligence far above the intelligence of
humanity. I remember reading philosophy articles by philosophers and
logicians giving their arguments for the existence of a God, none of
which are convincing. However, the existence of ironies and
coincidences are too common and too frequent. A case in point is this
thread.
> Alexander, I need your help on a geometry problem. A reference will
> do. I need to know whether the Conic-Section-System is unique to
> Euclidean Geometry. Or whether a Conic-Section-System exists in
> Riemannian Geometry also, (and likewise for Lobachevskian Geometry)? I
> believe it is unique to Euclidean Geometry. I need to know whether any
> mathematician has well-defined Conic-Section-System. And if there are
> any theorems about Conic Section Systems.
Karl Heuer back in 1993-4 suggested that the p-adics are
geometrically described as a "tree". I never liked that description.
And although I have not proved it, I intuit that the p-adics are
geometrically nested sphere surfaces or ellipses. P-adics are
Riemannian geometry in other words.
Well, the irony occurred last night shortly after I typed in the
above reply to Abian. The irony I had after writing the conic section
system asking Abian for a reference, is that I wrote that p-adics are
nested balls.
Would it not be ironic, very ironic that when this entire subject or
subjects are settled in the future, say a couple centuries in the
future. That the p-adics when described geometrically are not nested
balls but rather are conic section systems!!!
In other words, God, 231Pu, brought Abian to post that, and for me to
reply as such in order for God to bring together the idea that the
p-adics are geometrically the conic sections
Alexander Abian
In article <375BF2FC...@virginia.edu>,
Charles H. Giffen <ch...@virginia.edu> wrote:
>Alexander Abian wrote:
>[snip]
>> PLUTONIUM ARITHMETIC
>>
>> To develop the arithmetic of Plutonium integers the very first
>> step is to define the addition and multiplication of two
>> Plutonium integer.
>>
>> For the sake of brevity I will let "ap" stand for "Archimedes Plutonium"
>
>Perhaps the biggest problems with ap-integers for real applications
>are these:
>
>(1) The ap-integers have no good ordering: the
>negative of ...32875092 is ...67124908. Since their
>sum is ...00000000 (= ap-zero), if there were a good
>ordering, one of these would be positive and one would be
>negative -- but which is which?
Abian answers:
Depends on what do you mean by "good order".
If A and B are two ap-integers (i.e., Plutonium integers)
then one can define
A < or = B iff for every final finite segment fA of A
there is a final finite segment fB of B such that
fA is < or = in the ordinary arithmetic.
The above order is transitive but need not be antisymmetric.
There is nothing wrong with that there are many examples of
such orders.
The fact that the sum of two nonzero ap-integers is the zero
ap-integers or even the fact that every ap-integer has a
additive inverse only makes them to have richer structure.
In Boolean rings every element is its own additive inverse.
Same goes for the existence of divisors of zero in Plutonium
Arithmetic.
As far as disadvantages of having 10 instead of a prime in
Z_k, is concerned, I do not necessarily share the usual
opinions. Nonprime numbers have many more desirable and useful
properties than the prime numbers have.
Dr. Giffen contiues:
>The ap-integers, while perhaps cute or amusing to some, are
>really nothing special in the world of mathematica or of
>physics.
>
>--Chuck Giffen
Abiam answers:
Analogous things were said about nonEuclidean Geometries.
Supposedly Gauss, afraid of being ridiculed did not
announce some examples of his non Euclidean Geometries !!
And later regretted it very, very much!
Dr. Giffen, overall your comments are appropriate and they
will be very useful to Archie Plutonium who suspected the
validity of your comments in connection with the relationship
of ap-integers to the p-adic ones.
As far as I am concerned I still think that ap-integers have
quite interesting and rich structure and hopefully its
arithmetic (or algebra) will be developed by some people.
I won't be able to further study the ap-integers and I am
glad that at least it seems to be an interesting and nontrivial
subject.
Dr. Giffen, Thanks for your comments.
Uncle Al
>
> In sci.psychology.theory, sci.math, sci.physics, posting under someone else's
> name, Archimedes...@dartmouth.edu wrote:
> >
> > In article <kibo-06069...@ppp0b002.std.com>
> > ki...@world.std.com (James "Kibo" Parry) writes:
> > Sender: mmci...@world.std.com (Matthew J McIrvin)
> > Approved: mmci...@world.std.com (sci.physics.research)
[snip]
> That's odd, it didn't say that when I posted it. But I don't mind because you
> were clever enough to leave in the "<kibo-06069...@ppp0b002.std.com>"
> Message-ID so that anyone can click on it to see what I actually said.
> (To save them the trouble, I'll just summarize it below:)
>
> +----------------------------------------------------------------------------+
> | Archimedes Plutonium is a pinhead who can't even misquote people properly. |
> +----------------------------------------------------------------------------+
> (CLIP & SAVE FOR FUTURE REFERENCE)
[snip]
Send complaints of libel to Dartmouth president:
James.E...@dartmouth.edu
There was Kibo long before there was much of anything else. That has
weight.
--
Uncle Al Schwartz
http://www.mazepath.com/uncleal/
http://www.ultra.net.au/~wisby/uncleal/
http://www.guyy.demon.co.uk/uncleal/
http://uncleal.within.net/
(Toxic URLs! Unsafe for children, Democrats, and most mammals)
"Quis custodiet ipsos custodes?" The Net!
James Kibo Parry
name, Archimedes...@dartmouth.edu wrote:
>
> In article <kibo-06069...@ppp0b002.std.com>
> ki...@world.std.com (James "Kibo" Parry) writes:
> Sender: mmci...@world.std.com (Matthew J McIrvin)
> Approved: mmci...@world.std.com (sci.physics.research)
>
> = Faculty of the Department of Physics
> = Henry Ehrenreich, Clowes Professor of Science
> = Gary J. Feldman, Frank B. Baird, Jr. Professor of Science
> = Daniel S. Fisher, Professor of Physics and Professor of Applied
> Physics
> = Melissa Franklin, Professor of Physics
> =
> = Ma! Now they done gone posted under Ehrenreich's name!
That's odd, it didn't say that when I posted it. But I don't mind because you
were clever enough to leave in the "<kibo-06069...@ppp0b002.std.com>"
Message-ID so that anyone can click on it to see what I actually said.
(To save them the trouble, I'll just summarize it below:)
+----------------------------------------------------------------------------+
| Archimedes Plutonium is a pinhead who can't even misquote people properly. |
+----------------------------------------------------------------------------+
(CLIP & SAVE FOR FUTURE REFERENCE)
And you seem to be very confused about which headers do what. Maybe you should
try fabricating headers only after someone explains to you that the "From:"
and "Sender:" are normally the same person?
> [...] After 2 months, MIT says enough of a stalker. But after 6 years,
> Parry with his Harvard connections is allowed to continue to stalk.
Who is this "Parry" guy at Harvard who's stalking you? And how can he be
stalking you from all the way over there in Boston, where you said Harvard was?
Tell you what, since I live in Boston, I'll run over to Harvard and give him
your message, the next time Harvard moves its campus from Cambridge to Boston.
> If ad hominem and stalking are forbidden [...]
WOOOOOOO!!!! I USED AN AD HOMINEM!!!!!! IN PUBLIC!!!!!!!!!!!!!!!!!!!!!!
NEXT, MAYBE I'LL USE SARCASM!!! OR MAYBE I WON'T!
BECAUSE I WOULD *NEVER* USE SARCASM!
'S'ok, I can prove in a court of law that you're a pinhead. Your honor,
I'd like to enter this balloon as evidence... now, Mr. Plutonium -- if
that IS your real name -- please touch the balloon to the top of your head...
-- K.
Of course, that demonstration could
go horribly wrong if his head
explodes and not the balloon.
James Kibo Parry
Archimedes Plutonium (Archimedes...@dartmouth.edu) replied to himself:
>
> Archimedes Plutonium (Archimedes...@dartmouth.edu) wrote:
> >
> > If you can do that, the beauty of your construction would then be to
> > visualize all of the P-ADIC NUMBERS at once. The mind can picture Reals
> > at once and geometrically, and that is perhaps the greatest beauty of
> > the Reals, to be able to picture them geometrically, but with the
> > p-adics, no-one can picture them geometrically. Karl Heuer, about 5
> > years ago said the p-adics are more like a tree. But I find that
> > unsatisfactory. The best I can come up with so far as to picturing the
> > P-ADICS is that of nested balls, or balls inside of balls. Smaller
> > balls inside of bigger balls. But that is only a hunch and perhaps
> > wrong.
Yeah, maybe you've got it backwards. Try putting the bigger balls on the
inside of the smaller balls. It should be easy if they let you have Nerf
balls in there, although I don't know, maybe even those are sharper than
you're allowed to play with.
> > [...]
> >
> > Let me give the name INFINITE INTEGERS to all of these construction
> > schemes. Thus, p-adics are Infinite Integers, and so are Plutonium
> > Integers.
> >
> > Question, do the p-adics have the greatest mathematical content of
> > any Infinite Integers?
Question, if the Infinite Integers include Plutonium Integers and all
other construction schemes, does that mean they include Lego Integers?
Boron Integers? Duct-Tape Integers? Zinc Integers? Incorrect Integers?
> Irony and the occurrence of ironies is perhaps the very best
> indication of a God, a superintelligence far above the intelligence of
> humanity.
Do you mean dramatic irony or comedic irony?
I think the existence of dramatic irony just proves that God skipped
head to the last page of the book to see who the murderer was.
The presence of comedic irony proves that God is A WACKY CHIMP!
> I remember reading philosophy articles by philosophers and
> logicians giving their arguments for the existence of a God, none of
> which are convincing. However, the existence of ironies and
> coincidences are too common and too frequent. A case in point is this
> thread.
WRITE YOUR CONGRESSMAN, COINCIDENTCES ARE TOO COMMON AND TOO FREQUENT!
BAN ALL COINCIDENCES FOREVER!
> Karl Heuer back in 1993-4 suggested that the p-adics are
> geometrically described as a "tree". I never liked that description.
> And although I have not proved it, I intuit that the p-adics are
> geometrically nested sphere surfaces or ellipses. P-adics are
> Riemannian geometry in other words.
Ah, so you also have the hierarchical filesystem of your Mac set up
as a series of ellipsoids rather than as a tree.
So, Arch, on your map of the world that shows England as a peninsula,
did you represent the countries as nested ellipsoids or as spirals?
> Well, the irony occurred last night shortly after I typed in the
> above reply to Abian. The irony I had after writing the conic section
> system
You misspelled "comic section system". Hope this helps. I know you didn't
mean "conic section system" because I think someone else already invented
parabolas. I think it was back in the seventies because they didn't have
parabolas when Chuck Jones was drawing those Road Runner cartoons.
> asking Abian for a reference, is that I wrote that p-adics are
> nested balls.
>
> Would it not be ironic, very ironic that when this entire subject or
> subjects are settled in the future, say a couple centuries in the
> future. That the p-adics when described geometrically are not nested
> balls but rather are conic section systems!!!
>
> In other words, God, 231Pu, brought Abian to post that, and for me to
> reply as such in order for God to bring together the idea that the
> p-adics are geometrically the conic sections
And God sent me here because you need a hug. *HUG* ON BEHALF OF GOD, I LUV U!
-- K.
Eww, now there's slime all over my arms.
Tell God to hug Archie himself next time.
Beable van
Uncle Al <uncl...@hate.spam.net> wrote:
>
> There was Kibo long before there was much of anything else. That has
> weight.
KIBO HAS INERTIA!! THE EQUIVALENCE OF TIME AND KIBO!
KIBO MUST BE REORBITZED TO FORM A BORN-AGAIN DURIAN!
SOFT LAND KIBO ON A CHERRY PEZ TO FORM A NEW UTOPIA!
KEEP IT OUT OF SCI.HCEM FOOLS!!!11!11!!1 -- UNCLE AL
DOIDY DOIDY DOIDY DOIDY!!! DOIDY DOIDY DOIDY DOIDY!!
cheers
beable van polasm
--
_____________Have_you_SMASHED_the_STATE_today?____________
S___M___A___S___H______T___H___E______S___T___A___T___E___!
__L___E___G___A___L___I___Z___E______C___R___I___M___E___!
http://beable.webjump.com/index.html
Sent via Deja.com http://www.deja.com/
Share what you know. Learn what you don't.
Archimedes Plutonium
Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:
> If you can do that, the beauty of your construction would then be to
> visualize all of the P-ADIC NUMBERS at once. The mind can picture Reals
> at once and geometrically, and that is perhaps the greatest beauty of
> the Reals, to be able to picture them geometrically, but with the
> p-adics, no-one can picture them geometrically. Karl Heuer, about 5
> years ago said the p-adics are more like a tree. But I find that
> unsatisfactory. The best I can come up with so far as to picturing the
> P-ADICS is that of nested balls, or balls inside of balls. Smaller
> balls inside of bigger balls. But that is only a hunch and perhaps
> wrong.
>
> Can your construction scheme, Abian, be extended as to create the
> 2-adics, and the 3-adics and the 5-adics and the 6-adics etc. Perhaps
> by changing operation schemes, you can replace the changing bases of
> p-adics?
>
> Let me give the name INFINITE INTEGERS to all of these construction
> schemes. Thus, p-adics are Infinite Integers, and so are Plutonium
> Integers.
>
> Question, do the p-adics have the greatest mathematical content of
> any Infinite Integers?
Irony and the occurrence of ironies is perhaps the very best
indication of a God, a superintelligence far above the intelligence of
humanity. I remember reading philosophy articles by philosophers and
logicians giving their arguments for the existence of a God, none of
which are convincing. However, the existence of ironies and
coincidences are too common and too frequent. A case in point is this
thread.
> Alexander, I need your help on a geometry problem. A reference will
> do. I need to know whether the Conic-Section-System is unique to
> Euclidean Geometry. Or whether a Conic-Section-System exists in
> Riemannian Geometry also, (and likewise for Lobachevskian Geometry)? I
> believe it is unique to Euclidean Geometry. I need to know whether any
> mathematician has well-defined Conic-Section-System. And if there are
> any theorems about Conic Section Systems.
Karl Heuer back in 1993-4 suggested that the p-adics are
geometrically described as a "tree". I never liked that description.
And although I have not proved it, I intuit that the p-adics are
geometrically nested sphere surfaces or ellipses. P-adics are
Riemannian geometry in other words.
Well, the irony occurred last night shortly after I typed in the
above reply to Abian. The irony I had after writing the conic section
system asking Abian for a reference, is that I wrote that p-adics are
Harvard's McIrvin
ki...@world.std.com (James "Kibo" Parry) writes:
Sender: mmci...@world.std.com (Matthew J McIrvin)
Approved: mmci...@world.std.com (sci.physics.research)
> Physics
> Harvard University
> Faculty of the Department of Physics
> Gerald Gabrielse, Professor of Physics
> > Yeah, maybe you've got it backwards. Try putting the bigger balls on
> > the inside of the smaller balls. It should be easy if they let you have
> > Nerf balls in there, although I don't know, maybe even those are sharper
> > than you're allowed to play with.
>
> Ah, you're kidding, right?
>
> I mean, what stuff did you take to cause you to beleive that?
>
> Besides, it's CLEARly obvious that the Universe is actually
> a large cheese omlette, and the stars are just pepper stuck
> in the cheese.
>
> (Can I get coffee with that, please?)
Looks like not a single medical doctor in all of Boston area can
treat the stalker Parry
Charles H. Giffen
Hi Pertti! -- I should have expected as much, I suppose. Thanks
for the amusing information on Z_12. There are ad(d)ic(t)s of
all kinds!
--Chuck Giffen
twi...@sound.net
Archimedes...@dartmouth.edu (Harvard's McIrvin) wrote:
>
> Looks like not a single medical doctor in all of Boston area can
> treat the stalker Parry
>
Well, we have to catch him first. That's the tricky part.
He's quite slippery, you know.
--Terri
--
Also, just when you think you have them all
down------Llama spit comes around in one form or
another.
-- Frederick Hassen 5/23/99
Dag Oidy
Beable "van" Polasm <bea...@my-deja.com> wrote:
> In article <375C7E48...@hate.spam.net>,
> Uncle Al <uncl...@hate.spam.net> wrote:
> >
> > There was Kibo long before there was much of anything else. That
> > has weight.
>
> KIBO HAS INERTIA!! THE EQUIVALENCE OF TIME AND KIBO!
> KIBO MUST BE REORBITZED TO FORM A BORN-AGAIN DURIAN!
> SOFT LAND KIBO ON A CHERRY PEZ TO FORM A NEW UTOPIA!
> KEEP IT OUT OF SCI.HCEM FOOLS!!!11!11!!1 -- UNCLE AL
> DOIDY DOIDY DOIDY DOIDY!!! DOIDY DOIDY DOIDY DOIDY!!
PL0T T0 DEFAME MY G00D NAME?
time for tubby bye-bye,
Dag Oidy
--
WHAT!!!!1!!1!!! H0W Am 1 eVeR g01nG to f1T A
HEFTY-kEWL ASCII SW0RD 1n f0uR LINZ?!?!?!?
China Blue
> Depends on what do you mean by "good order".
>
> If A and B are two ap-integers (i.e., Plutonium integers)
> then one can define
>
> A < or = B iff for every final finite segment fA of A
> there is a final finite segment fB of B such that
> fA is < or = in the ordinary arithmetic.
I respond:
Let X = 0...010...0 and Y = 0...020...0. All finite suffixes of X and Y
are equal, but X and Y appear to be inequal.
Suppose I define real fractions as a fraction point followed by an
arbitrary number of digits. How would I define +? It would require
propagating some carry from some infinitely least signficant digit. In
fact reals are defined as the limits of cauchy sequences of rational
fractions. Addition in reals is defined by the limit of the addition of
sequences of rationals. Addition of rationals of is a nice finite
operation, and the addition of any two sequence elements is a nice
finite step. Lots of itty bitty finite steps which then go through the
now well understood notion of a limit.
The only definition of your infinite integers I've seen is pile on
digits to the left of the fraction point. You're going from finite to
infinite in one big step with no notion of limits of sequences to guide
my intutions. You can answer my question by declaring X and Y are not
infinite integers but an ill-defined abuse of notation. But unless you
have some kind of definition like they're the limit of an anticauchy
sequence of integers, and that the limit is unique. At the moment the
only proof rule seems to be Proof by Referral to Abian.
Zeno's paradoxes and all the rest show the danger of casually going
finite to infinite.
> The fact that the sum of two nonzero ap-integers is the zero
> ap-integers or even the fact that every ap-integer has a
> additive inverse only makes them to have richer structure.
I respond:
Is it? For ...999 + ...001, you have all finite suffixes equal to 0, but
also a carry propagating to some infinitely remote leading digit? Is it
okay to eventually discard the carry? At which point? Is anything I'm
saying sensible?
--
There is no sanctuary.
David Pacheco
(James "Kibo" Parry) says...
> -- K.
>
> Of course, that demonstration could
> go horribly wrong if his head
> explodes and not the balloon.
>
This would be a demonstration going "horribly wrong"... how?
-dp.
Unless it were
*my* balloon
China Blue
[|>KIBO MUST BE REORBITZED TO FORM A BORN-AGAIN DURIAN!
[|>SOFT LAND KIBO ON A CHERRY PEZ TO FORM A NEW UTOPIA!
[|>KEEP IT OUT OF SCI.HCEM FOOLS!!!11!11!!1 -- UNCLE AL
And the colourred girls sing,
[|>DOIDY DOIDY DOIDY DOIDY!!! DOIDY DOIDY DOIDY DOIDY!!
Make a post on the wild side.
--
Smeagol, Smeagol, Smeagol. I thought we had an understanding.
I do you a favour, you do me a favour. Just one little ring.
-Don Sauroni, Godfather of the Rings.
CACS: http://homestead.dejanews.com/user.smjames/index.html
text: http://www.geocities.com/SoHo/Studios/5079/index.html
David DeLaney
>[|>KIBO HAS INERTIA!! THE EQUIVALENCE OF TIME AND KIBO!
>[|>KIBO MUST BE REORBITZED TO FORM A BORN-AGAIN DURIAN!
>[|>SOFT LAND KIBO ON A CHERRY PEZ TO FORM A NEW UTOPIA!
>[|>KEEP IT OUT OF SCI.HCEM FOOLS!!!11!11!!1 -- UNCLE AL
>And the colourred girls sing,
>[|>DOIDY DOIDY DOIDY DOIDY!!! DOIDY DOIDY DOIDY DOIDY!!
>Make a post on the wild side.
If Abian decides there was a BIG SUCK associated with Kibo, I'm not
sure I want small children reading sci.physics to be exposed to it.
Look at what happened to poor Hanna-Maria, after all!
Dave "criminal kibonian grandparents" DeLaney
--
\/David DeLaney d...@panacea.phys.utk.edu "It's not the pot that grows the flower
It's not the clock that slows the hour The definition's plain for anyone to see
Love is all it takes to make a family" - R&P. VISUALIZE HAPPYNET VRbeable<BLINK>
http://panacea.phys.utk.edu/~dbd/ - net.legends FAQ/ I WUV you in all CAPS! --K.
Noah A Christis
>If Abian decides there was a BIG SUCK associated with Kibo, I'm not
>sure I want small children reading sci.physics to be exposed to it.
>Look at what happened to poor Hanna-Maria, after all!
^^^^^^^^^^^
ym: Tuxedo Sam, the penguin from kitty paradise. His brothers, Tam
and Ham are his best friends. They speak penguinese. After
subduing Chip (a small seal), the trio forced him to be the
test subject for numerous physics experiments. Sam's home base
is in Antarctica, where he works for the UN. He is currently
studying French in Action with Nick Bensema.
Daniel Buettner
[ snip ]
> ym: Tuxedo Sam, the penguin from kitty paradise. His brothers, Tam
> and Ham are his best friends. They speak penguinese. After
> subduing Chip (a small seal), the trio forced him to be the
> test subject for numerous physics experiments. Sam's home base
> is in Antarctica, where he works for the UN. He is currently
> studying French in Action with Nick Bensema.
I just wanted to let you know that when I first read this I was
thinking of Tennessee Tuxedo. I knew in the back of my mind that
the image of him didn't match up with the name Tuxedo Sam, but
what could I do? All I could think of was that one episode when
Bullwinkle and the gang were abducted by the robot people who
put them on trial. It culminated with the robot jury shouting
"to the melting pot!!!!!1!!1!!" which was quite frightening.
I don't actually remember much else about the show, even though
I watched it before school every day for most of a year.
--
~
~
~
"Daniel Buettner" line 4 of 4 --100%--
Charles H. Giffen
>
> ------------------
>
> In article <375BF2FC...@virginia.edu>,
>
> Charles H. Giffen <ch...@virginia.edu> wrote:
>
> >Alexander Abian wrote:
> >[snip]
> >> PLUTONIUM ARITHMETIC
> >>
> >> To develop the arithmetic of Plutonium integers the very first
> >> step is to define the addition and multiplication of two
> >> Plutonium integer.
> >>
> >> For the sake of brevity I will let "ap" stand for "Archimedes Plutonium"
>
> >
> >Perhaps the biggest problems with ap-integers for real applications
> >are these:
> >
> >(1) The ap-integers have no good ordering: the
> >negative of ...32875092 is ...67124908. Since their
> >sum is ...00000000 (= ap-zero), if there were a good
> >ordering, one of these would be positive and one would be
> >negative -- but which is which?
>
> Abian answers:
>
>
> If A and B are two ap-integers (i.e., Plutonium integers)
> then one can define
>
> A < or = B iff for every final finite segment fA of A
> there is a final finite segment fB of B such that
> fA is < or = in the ordinary arithmetic.
>
> There is nothing wrong with that there are many examples of
> such orders.
>
It appears from your definition of <= ("< or =") that
A = ...12121212 and B = ...21212121
have the property that A <= B and B <= A, unless you mean
that A <= B provided for each n = 0,1,2,... the n digit
tails must satisfy
t_n(A) <= t_n(B) .
As it stands, it appears that the tails ("final segments") you
specify may be of different length. Assumming that you mean
equal length for tails, then the examples A and B given
are incomparable.
By "good order", I meant one that is translation invariant
( A <= B iff A + C <= B + C ), at the very least -- and
presumably also has a set P of "positive elements" such
that (1) ap-zero is not in P, (2) for any non-ap-zero A,
either A is in P or -A is in P, (3) if A is in P and if
B <= C, then AB <= AC, etc.
>
> The fact that the sum of two nonzero ap-integers is the zero
> ap-integers or even the fact that every ap-integer has a
> additive inverse only makes them to have richer structure.
>
> In Boolean rings every element is its own additive inverse.
>
> Same goes for the existence of divisors of zero in Plutonium
> Arithmetic.
>
I guess this depends upon what you mean by "rich" structure.
[snip]
The reliance upon 2 and 5 seems to have bothered A.P., and
he wonders in another posting in this thread, in response to
a comment from A.Abian (included as >=:
>= Note to Archimedes Plutonium.
>= ____________________________
>=
>= Plutonium intrgers and their Arithmetic HAVE NO DIRECT
>= CONNECTION with p-adic numbers and their arithmetic.
>
> But can they be linked to the p-adic numbers? Can they be made to be
>isomorphic? Can these Plutonium Integers be so devised as to get rid of
>the base dependency of p-adic integers? And that one can then visualize
>all the p-adics at once, rather than disjoint union of 2-adics, 3-adics
>etc.
>
Some comments are in order which should shed considerable light
on the ap-integers and A.P.'s query. I apologize for the length
of this posting, but the comments are really a brief exposition
of some well-know mathematical ideas that shed light on this
thread and generalize it in an attractive way.
(1) Consider the m-adics Z_m = inv lim( Z/m^k ) . For each
prime p dividing m, there is a reduction map (surjective
homomorphism of rings with unit)
r_p : Z_m --> Z_p .
To see this, suppose p^s divides m but p^{s+1} does not.
then the natural surjective reduction homomorphisms
Z/m^k --> Z/p^{sk}
induce the map of inverse limits
r_p : Z_m --> inv lim( Z/p^{sk} ) = inv lim( Z/p^j ) = Z_p,
where the middle isomorphism follows from cofinality of the
sequence p^{sk}Z in p^jZ .
Thus, if m = p[1]^s[1] p[2]^s[2] ... p[t]^s[t] is the
prime factorization of m, we have the homomorphism into the
product induced by the r_p[i]'s :
r : Z_m --> Z_p[1,...,t] = Z_p[1] x Z_p[2] x ... x Z_p[t] .
This homomorphism is an imbedding of Z_m as a closed subset
(each of Z_m, Z_p[i] is a compact Hausdorff space, being
the inverse limit of a sequence of finite sets; the r_p and
hence r are continuous). And, in fact we have the well-known
result:
LEMMA. Z_m, as a subset of Z_p[1,...,t], is the closure of
the ordinary integers Z in Z_p[1,...,t] .
Thus the ap-integers Z_10 are the closure of the (diagonal
copy) of the integers sitting in Z_2 x Z_5 . I'm not sure
what you mean by the statement that the Plutonium integers
(ap-integers) Z_10 and their arithmetic having no direct
connection with p-adic numbers and their arithmetic -- since
the above discussion precisely illustrates just such a
connection between the ap-integers and the 2- and 5-adics.
More generally, the above Lemma and the discussion preceding
it demonstrate a connection between any Z_m and the p-adic
integers Z_p for any prime p|m .
(2) Here is a method for capturing the essence of all the
p-adic completions of the integers at once. Again, it is
well-known classical mathematics. The replacement for
Z_10 (or any Z_m ) is:
Z^ = inv lim( Z/m! ),
ie. the inverse limit of the infinite sequence of surjective
ring homomorphisms
... --> Z/(m+1)! --> Z/m! --> ...
... --> Z/24 --> Z/6 --> Z/2 --> Z/1 = 0 .
Recall that an element of Z_10 may be regarded as an infinite
sequence {x[i]} of integers with
0 <= x[i] < 10^i and x[i+1] mod 10^i = x[i]
for i = 1,2...; or, more generally, an element of Z_m may
be regarded as an infinite sequence {x[i]} of integers with
0 <= x[i] < m^i and x[i+1] mod m^i = x[i]
for i = 1,2,... . For Z^ there is a somewhat similar
interpretation -- an element of Z^ is an infinite sequence
{x[i]} of integers with
0 <= x[i] < i! and x[i+1] mod i! = x[i]
for i = 1,2,... . Notice that, as with Z_10 or Z_m,
any x[i] with i > 1 determines all the x[j] uniquely
for 1 <= j < i (since x[j] is x[i] mod <something> ).
Once again, there are surjective reductions homomorphims
r_p : Z^ --> Z_p
for every prime p. Also, the resulting map into the product
r : Z^ --> Z_2 x Z_3 x Z_5 x ... A^
is an imbedding of Z^ as a closed subset of A^, and we have:
PROPOSITION. As a subset of A^, Z^ is the closure of the
(diagonal copy of) the integers Z in A^.
Z^ is usually known and referred to as the profinite completion
of the integers. It arises in the study of algbraic number
theory, algebraic topology, class-field theory -- capturing
"mod m" phenomena for all m at once, as it were.
(3) The p-adic integers Z_p, being a compact local ring,
have a locally compact field of quotients Q_p, usually called
the p-adic numbers, and the inclusion of Z in Z_p extends
to give an inclusion of the ordinary rational numbers
Q in Q_p -- indeed, Q_p may viewed as the completion of
Q with respect to the p-adic valuation on Q. Thus we have
a bicartesian square of inclusions:
Z ---> Q
| |
| |
V V
Z_p ---> Q_p
known as the p-adic "arithmetic square". In a problem in
which only the prime p is of interest (for instance, because
p-torsion is the only torsion present), this square often
tells one how to analyze a problem "integrally" by looking at
it both "rationally" and "p-adically" and comparing these
two views "p-adic rationally".
When there are several (possibly infinitely many, eg. all)
primes "present" in a problem, there are the evident diagonal
inclusions
Z --> Z_2 x Z_3 x Z_5 x ... = A^
and
Q --> Q_2 x Q_3 x Q_5 x ... = B^,
and may analyze general problems may making use of the resulting
square:
Z ---> Q
| |
| |
V V
A^ ---> B^
However, it is just as traditional (and often more convenient)
to do the analysis in terms of the related bicartesian square
Z ---> Q
| |
| |
V V
Z^ ---> Q^
where of course Q^ is the closure of the diagonal copy of
the rationals Q in B^. This latter square is sometimes
referred to as the arithemtic square, and, while Z^ is
referred to as the profinite completion of the integers,
Q^ is usually called the ring of adeles (at least in other
similar number theoretic or class-field theoretic contexts --
where this procedure may be mimicked).
(4) A little insight into arithmetic in Z^: If x = {x[i]}
and y = {y[i]} are in Z^, then
(x+y)[i] = (x[i] + y[i]) mod i!
and
(xy)[i] = (x[i] y[i]) mod i!
yield the sum and product operations. Note: a similar device
produces the sum and product in Z_m for any m .
An ordinary integer n in Z is viewed as an element of Z^
by defining n[i] = n mod i! for each i - 1,2,3,... . If
n in Z is an ordinary integer and x in Z^, then nx in Z^
is unambiguously defined, since
(n x[i]) mod i! = (n[i] x[i]) mod i! .
Now we can describe elements of Q^: First, recall that
a rational number r in Q is an equivalence class of
ordered pairs (m,n) of integers m,n in Z with n
non-zero, where (m,n) ~ (m',n') iff nm' = n'm -- the
equivalence class of (m,n) is usually written m/n .
Well, an element of Q^ is an equivalence class of
ordered pairs (x,n), where x in Z^ and n in Z with
n non-zero. Equivlence of pairs is given by
(x,n) ~ (x',n') iff nx' = n'x
-- ie. (n x'[i]) mod i! = (n' x[i]) mod i! for i = 1,2,3... .
And the equivalence class of (x,n) may be denoted x/n .
Addition and multiplication of elements of Q^ are given by
the rules
(x/n) + (y/m) = (mx + ny)/(mn)
(x/n) (y/m) = (xy)/(mn)
and make Q^ a locally compact Z^-algebra.
EXERCISE. What goes wrong if one tries to form "fractions"
of the form x/y with *both* x,y in Z^ ? Hint: The
same problem (and reason) occur when one tries to form
"fractions" x/y with both x,y in Z_10 .
I hope these comments, though long, will help to clarify the
situation rather than cloud the issues.
Sincerely,
--Chuck Giffen
Karlo Takki
DeLaney) wrote:
> If Abian decides there was a BIG SUCK associated with Kibo, I'm not
> sure I want small children reading sci.physics to be exposed to it.
> Look at what happened to poor Hanna-Maria, after all!
Kibo'S BIG SUCK must be REORBITED to a NEAR-PEROT ORBIT
TO BECOME a PEROT-LIKE GIANT SUCKING-SOUND!
k.
Archimedes Plutonium
"Charles H. Giffen" <ch...@virginia.edu> writes:
> where of course Q^ is the closure of the diagonal copy of
> the rationals Q in B^. This latter square is sometimes
> referred to as the arithemtic square, and, while Z^ is
> referred to as the profinite completion of the integers,
> Q^ is usually called the ring of adeles (at least in other
> similar number theoretic or class-field theoretic contexts --
> where this procedure may be mimicked).
If we consider each Adic, starting with 2-adics and going up, 3-adics,
5-adics, 6-adics, ad infinitum.
Also consider the negative adics of -2-adics, -3-adics, -5-adics, ad
infinitum.
Combine the positive adic with its respective negative adics, such as
+2-adics combined with -2-adics, and +3-adics combined with -3-adics,
ad infinitum
Now, the +2-adics is one cone of a conic-section-system and the
-2-adics is the other cone of a conic section system. The zero point of
+2-adics is the same as the zero point of -2-adics. Likewise for the
other adics.
Picture all of these Adics with their zero points as the origin in 3d
Euclidean Space. All the Adics have the same zero point. But their
cones are all different, such that the +3-adics cone is different from
the +2-adics cone. They share the origin or zero point. But what other
points do the +3-adics and +2-adics share? Of course the +3-adics and
the +6-adics share many points
John R Ramsden
Plutonium) wrote:
>
> Karl Heuer back in 1993-4 suggested that the p-adics are
> geometrically described as a "tree". I never liked that description.
> And although I have not proved it, I intuit that the p-adics are
> geometrically nested sphere surfaces or ellipses. P-adics are
> Riemannian geometry in other words.
I think you can represent the p-adic integers (for a given value
of p) by a disk. If I'm not mistaken Poincare studied this.
BTW, did you know that under a non-Archimedian (p-adic) valuation
(measure of distance) every point inside a circle is a centre, i.e.
an equal distance from the circumference?
Cheers
John R Ramsden (j...@redmink.demon.co.uk)
Archimedes Plutonium
ab...@iastate.edu (Alexander Abian) writes:
= In ZF classical set theory, the natural numbers 0, 1, 2, 3, ....
= are (and are called) finite ordinals which are the same as finite
= cardinals. The smallest transfinite ordinal (which is also the
smallest
= transfinite cardinal) is "omega"
=
= Now, if you present a set-theoretician with a picture, say, like
=
= (1) .....256341765439800654
Dr. Abian, I am going to take issue and to explore these ideas some
more, although I would have preferred to do this next Autumn when the
students go back to school. What I have now that I did not have in past
years is the idea that the Conic-Section-Systems have an enormous role
in this.
I reject the idea of omega and any transfinite infinities.
Occam's razor says to accept the most simpliest explanation.
If you imagine, believe, that Finite Integers when endless do not
merge into Infinite Integers, then you come up with this Cantor
plethora of infinities. However, if you say that when plucking out all
of the so called Finite Integers and that they do indeed merger into
Infinite Integers, then, then, the beautiful result occurs that no
Cantor transfinites exist.
Now, also in this imagination, I ask you the question Dr. Abian et
al. Would it not be nicer for logic sake, that a small part of the
Reals should not be more "numerous" than the larger part of the Reals.
For example the small interval of all the Reals in the open or closed
set of [0,1] corresponding to the larger interval of the set
{0,1,2,3,4,5,.....}. Would it not be a nicer logical picture if any
infinite set of Reals should all be equinumerous?? My answer is yes,
out of logical beauty.
So, here I have two logical conundrums of which both point towards a
flaw in the Cantor system.
The first conundrum, or blemish, or flaw, is that out of Occam's
Razor Argument, it would and should be nicer if the set
{0,1,2,3,4,5,.....} were equinumerous with the set [0,1]. And to make
it equinumerous we only need to make a slight adjustment in
mathematics. What we need to say is that when you ask for this set
{0,1,2,3,4,5,.....} what you really are asking for is the Infinite
Integers. That to ask for All Natural Numbers implies that the set
never ends and hence merges into the Infinite Integers. When you accept
that idea, you banish from the mathematical world all Cantor
Transfinites. They were a mirage, a fantasy of the imagination. So, the
first Logical Beautiful Picture is an application of Occam's Razor. And
ask yourselves this pragmatic question. We have had Cantor transfinites
for over a century now, and has there ever been any utility or science
application of transfinites? The answer is clearly no. That the Cantor
transfinites were a dead end topic the day Cantor created them. That
smells of fakery. When something is true, it blossoms in many other
places. The Cantor transfinites never blossomed, but rather, they died
the moment they were born.
Occam's Razor is a first logical application. The second logical
application is an application of CommonSense. Let us ask ourselves
these questions. If you are given a set of numbers the Reals, would it
make more commonsense to think that a small interval of these Reals
would be more numerous than the larger interval of these Reals? The
example above was the small interval [0,1] compared to the large
interval {0,1,2,3,4,5,.....} Commonsense would say that the larger
interval should always be larger, but if worse comes to worse, they
would be equal. Like saying that the number of beetles living in a tree
in Yellowstone is more numerous than every other beetle counted up in
all of Earth. Or like saying the number of people in your house is more
numerous than all the other people in all the other houses added up on
Earth. Commonsense says the larger should be more numerous or equal to
the smaller, but never smaller than the smaller.
What is wrong and how can it be corrected? What is wrong is that
people think that when you want to talk about this set of Reals
{0,1,2,3,4,5,.....} that although infinitely long, that each member is
well behaved and never becomes an Infinite Integer. But, ohh, how the
imagination can be so tricky and deceitful.
What defines that set {0,1,2,3,4,5,.....} is an endless adding of 1.
That process is the same process as the definition of p-adic integers,
they too are an endless adding of 1. Thus, although for a small number
of these so-called Finite Integers say a trillion of them or a trillion
trillion of them, they look well behaved, but when you ask for all of
these numbers such as in Fermat's Last Equation, well, the mind has
played a trick on you. They no longer are well behaved but have hopped
into the Infinite Integers.
What I have new to add to the discussion is the Conic Section System.
I have geometry to add. Before I ran into the trouble and difficulty of
saying that this set {0,1,2,3,4,5,.....} goes into the Infinite
Integers, or p-adic integers. The trouble I had with that is how can
Euclidean geometry of 3 dimensional space go into Infinite Integers.
This implies that the Cartesian Coordinate system would have to bend
around like a large sphere and come back to the origin point. So, I had
huge immense difficulty on the geometry side of the house of
mathematics by stating that:
Natural Numbers (positive Whole Reals) = p-adic
Integers.
The trouble I had with that is that the Cartesian 3-d would
eventually curve back around like some huge sphere. Where the set
{0,1,2,3,4,5,.....} would come back around to numbers like these
....99999999997, then .....99998, then .....99999999 and then 0 again.
But in the last months I have connected p-adics with Conic Section
Systems and that helps me out of the jam above. I can picture that the
quadrants of Cartesian 3-d are Conic Section Systems. Easier in
2-dimension with its four quadrants. Thus, quadrant I and III and
quadrants II and IV form two distinct conic section systems.
Thus Euclidean Geometry with Infinite Integers is not destroyed but
rather is a conic section system and retains its Euclidean nature.
More later
David Harden
(Archimedes Plutonium) wrote:
> Karl Heuer back in 1993-4 suggested that the p-adics are
>geometrically described as a "tree". I never liked that description.
>And although I have not proved it, I intuit that the p-adics are
>geometrically nested sphere surfaces or ellipses. P-adics are
>Riemannian geometry in other words.
>
> Well, the irony occurred last night shortly after I typed in the
>above reply to Abian. The irony I had after writing the conic section
>system asking Abian for a reference, is that I wrote that p-adics are
>nested balls.
>
> Would it not be ironic, very ironic that when this entire subject or
>subjects are settled in the future, say a couple centuries in the
>future. That the p-adics when described geometrically are not nested
>balls but rather are conic section systems!!!
>
> In other words, God, 231Pu, brought Abian to post that, and for me to
>reply as such in order for God to bring together the idea that the
>p-adics are geometrically the conic sections
p-adics cannot be geometrically embedded in any finite number of
dimensions of Euclidean space. There are several fundamental
differences between Euclidean and p-adic topology and geometry:
1. In p-adic geometry, all triangles are isoceles or equilateral.
(proof: assume a triangle is scalene, take the longest side and draw a
contradiction with the strong triangle inequality)
2. In p-adic geometry, the uncountably infinitely many points in all
space are separated by only a countably infinite set of possible
distances between any two of them.
These are all that I feel like listing now. However, your "God" seems
to be incapable of recognizing a few things like that (never mind
being destroyed by radioactive decay) and so seems very limited.
Perhaps you should reconsider.
---- David Harden
Archimedes Plutonium
w_ha...@bellsouth.net (David Harden) writes:
> p-adics cannot be geometrically embedded in any finite number of
> dimensions of Euclidean space. There are several fundamental
> differences between Euclidean and p-adic topology and geometry:
>
To the contrary, they can all be embedded in Euclidean 3rd dimension,
where each of the quadrants serves as a cone of a conic-section-system
> 1. In p-adic geometry, all triangles are isoceles or equilateral.
> (proof: assume a triangle is scalene, take the longest side and draw a
> contradiction with the strong triangle inequality)
>
False. You apply definitions from Euclidean Geometry and thus assume
falsehoods.
> 2. In p-adic geometry, the uncountably infinitely many points in all
> space are separated by only a countably infinite set of possible
> distances between any two of them.
>
There exists no such thing as a "countably infinite set".
All infinite sets are uncountable.
When you say that this set 1,2,3,4,.... does not merge into p-adic
integers then you have a plethora of infinities.
When you say that this set does merge into the p-adic integers, then,
the world has only one type of infinity. Occam's Razor choses the
merger. And besides, the very Peano axioms of the endless adding of 1
is the exact same definition as the p-adic integers of a series
definition.
> These are all that I feel like listing now. However, your "God" seems
> to be incapable of recognizing a few things like that (never mind
> being destroyed by radioactive decay) and so seems very limited.
> Perhaps you should reconsider.
You need the Atom Totality to be radioactive, otherwise you have no
time itself. And a Big Bang of the last atom totality of the Uranium
Atom Totality created for us the Plutonium Atom Totality. Your mind
seems to be uncomfortable with new ideas
Archimedes Plutonium
j...@redmink.demon.co.uk (John R Ramsden) writes:
> I think you can represent the p-adic integers (for a given value
> of p) by a disk. If I'm not mistaken Poincare studied this.
>
False. You are thinking of Lobachevskian geometry.
> BTW, did you know that under a non-Archimedian (p-adic) valuation
> (measure of distance) every point inside a circle is a centre, i.e.
> an equal distance from the circumference?
Blurred. What you are trying to describe is any point in Riemannian
Geometry.