I am a newly converted anti-Cantor kook!
anzau...@hotmail.com
any subtle details but on the first principles, I have a question.
Cantor's proof goes like this:
Given an integer a, let not(a) = 1 - a mod 2.
Given any real number R < 1, let R(i) denote the i'th digit in R's
binary representation.
Let R1, R2, R3, .... be a sequence that claims to enumerate the set of
all real R < 1.
Let S to be the real number defined by S(i) = not(Ri(i)).
Supposed conclusion: S is not part of the sequence R1, R2, R3, ....
But why not? What if there exists an integer k s.t. S(i) = 1 for all i
> k? That would really mess things up in a hurry, woudn't it?
What did Cantor do to get around this roadblock?
anzau...@hotmail.com
Ooops, this can be fixed simply by going to the trinary representation
and by setting
S(i) = 1 if Ri(i) = 0
S(i) = 0 if Ri(i) > 0
That's embarassing....
Cantor is once again rehabilitated. :-)
Virgil
anzau...@hotmail.com wrote:
He didn't use binary representation, but decimal representation, and
made sure that his new number was constructed without using either 0's
or 9's in it, so could not be anything with a dual representation.
Dirk Van de moortel
<anzau...@hotmail.com> wrote in message news:1117312945.7...@g14g2000cwa.googlegroups.com...
I guess Cantor does for sci.math what Einstein does for
sci.physics.*. There's a few differences though:
Not much is written in the popular press about Cantor,
and linear transformations look easier to debunk for a
cook than the diagonal proof.
That probably explains why the percentage of cooks on
sci.math is about 5% as opposed to 75% on sci.physics.*
Dirk Vdm
Daniel W. Johnson
A) That wasn't Cantor's first proof.
http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof
B) The usual ways to handle this are to use a base other than 2 and to
take digits in pairs.
Taking digits in pairs looks like this:
S(2i-1) = not(Ri(2i-1))
S(2i) = not(S(2i-1))
So, there can't be a k such that S(i) = 1 for all i > k. (If s(2k) = 1,
we must have s(2k-1) = 0.)
And we still have a difference between S and any Ri (just at the 2i-1'th
digit instead of the i'th digit).
--
Daniel W. Johnson
pano...@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
Dik T. Winter
> anzaurr...@hotmail.com wrote:
...
> > But why not? What if there exists an integer k s.t. S(i) = 1 for all i
> > > k? That would really mess things up in a hurry, woudn't it?
> >
> > What did Cantor do to get around this roadblock?
I do not know what Cantor did, but:
> Ooops, this can be fixed simply by going to the trinary representation
> and by setting
>
> S(i) = 1 if Ri(i) = 0
> S(i) = 0 if Ri(i) > 0
>
> That's embarassing....
The proof can be done in any other base indeed with similar substitutions.
Only base 2 gives problems.
> Cantor is once again rehabilitated. :-)
Indeed.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Ken Quirici
>>
> B) The usual ways to handle this are to use a base other than 2 and to
> take digits in pairs.
>
> Taking digits in pairs looks like this:
>
> S(2i-1) = not(Ri(2i-1))
> S(2i) = not(S(2i-1))
>
> So, there can't be a k such that S(i) = 1 for all i > k. (If s(2k) = 1,
> we must have s(2k-1) = 0.)
>
> And we still have a difference between S and any Ri (just at the 2i-1'th
> digit instead of the i'th digit).
> --
> Daniel W. Johnson
> pano...@iquest.net
> http://members.iquest.net/~panoptes/
> 039 53 36 N / 086 11 55 W
Your citation points to Cantor's original diagonal argument.
He defines the anti-diagonal of a decimal representation as:
(1) S(i,j) = 4 if R(i,j) = 5
(2) S(i,j) = 5 if R(i,j) <> 5
(3) He also apparently, in the case where a number had two
representations, e.g. 4.9999.... and 5.00000...., always chose
the one with 9's.
What's REALLY embarrassing is I can't figure out why (3) is necessary,
given the rules (1) and (2). Why is it necessary?
Thanks.
Ken
anzau...@hotmail.com
>
> That probably explains why the percentage of cooks on
> sci.math is about 5% as opposed to 75% on sci.physics.*
The greatest number of cooks can be found in rec.food. That place is
crawling with them. Check it out!
anzau...@hotmail.com
> In article <1117312945.7...@g14g2000cwa.googlegroups.com> anzau...@hotmail.com writes:
> > anzaurr...@hotmail.com wrote:
>
> > this can be fixed simply by going to the trinary representation
> > and by setting
> >
> > S(i) = 1 if Ri(i) = 0
> > S(i) = 0 if Ri(i) > 0
>
Thanks. Would have never occurred to me.
anzau...@hotmail.com
I don't think it is. It's just that every real number has two decimal
(or any other base) representations, and for simplicity, you would want
to decide in advance which of the two representations you want to use
for each of them.
fishfry
Like I keep saying. The reason we keep getting these posts is because
Cantor's work is accessible to people who don't understand math.
Funny how we never get any posts around here that say, "I'm outraged.
The classification theorem for finite simple groups has simply got to be
false!"
Virgil
wrote:
> In article <1117312945.7...@g14g2000cwa.googlegroups.com>
> anzau...@hotmail.com writes:
> > anzaurr...@hotmail.com wrote:
> ...
> > > But why not? What if there exists an integer k s.t. S(i) = 1 for all i
> > > > k? That would really mess things up in a hurry, woudn't it?
> > >
> > > What did Cantor do to get around this roadblock?
>
> I do not know what Cantor did, but:
>
> > Ooops, this can be fixed simply by going to the trinary representation
> > and by setting
> >
> > S(i) = 1 if Ri(i) = 0
> > S(i) = 0 if Ri(i) > 0
> >
> > That's embarassing....
>
> The proof can be done in any other base indeed with similar substitutions.
> Only base 2 gives problems.
>
> > Cantor is once again rehabilitated. :-)
>
> Indeed.
Actually, base 3 gives minor problems, too, since you cannot avoid the
possibility having the 'anti-diagonal" ending with an infinite sequence
of 0's 0r 2's, unless you have some weird rule like the following:
if the nth digit of the nth trinary number in the list is
0 or 2, put 1 in the nth place of the "anti-diagonal",
1 and n is even, put a 0 in the nth place of the "anti-diagonal" or
1 and n is odd, put a 2 in the nth place of the "anti-diagonal".
With a base of 4 or larger, one can always replace a 2 with a 1 and
anything else with 2 to avoid problems.
Virgil
"Ken Quirici" <kqui...@yahoo.com> wrote:
It isn't.
The World Wide Wade
<1117326748.5...@g49g2000cwa.googlegroups.com>,
anzau...@hotmail.com wrote:
> It's just that every real number has two decimal
> (or any other base) representations,
No, most reals have just one decimal representation. There are
only countably many with two.
anzau...@hotmail.com
I stand corrected.
Ross A. Finlayson
The universe contains all physical objects. A function, with the
domain and range being physical objects, is itself a physical object.
Thus, via induction, there are infinitely many physical objects because
there are at least two.
Now, as there are infinitely many physical objects, and functions
between them are physical objects, then, the functions would have a
higher cardinality. Yet, they remain in the same physical universe.
As the physical universe is equal to itself, via identity then it has
the same cardinal as itself, yet it would have a different cardinal
than itself, and that implies that aleph_0 is equivalent to aleph_1, 2,
..., or alternatively, and exclusively, there is identity, tautology.
So, the universe exists, with all its objects within it, or, infinite
sets are ever not equivalent, and not both. Infinite sets are
equivalent.
When you replace the word "physical" with "set-theoretical" the same
deduction follows.
ZF is inconsistent in a theory with a universal set, because regularity
would be one of what you've been calling a "false axiom".
The antidiagonal result seems strong. It obviously can't be shown in
binary, in terms of integers and reals, where reals have dual
representation. In binary as the coded powerset of the naturals,
without dual representation, it appears stronger.
In unary, or base one, the only way to represent reals from the unit
interval is as tally marks of iota multiples, so it fails. Obviously
enough that uses a nonstandard notion of real number, that is for all
intents and purposes a real number, a scalar. With the infinite radix,
with an infinite alphabet, again it fails, again, nonstandardly. Such
a set of numbers obviates as well the nested intervals result. In
terms of numbers it is as well possible to consider leading zeros, as
many as you want, except not none. Well order the reals, and then,
iterate them. As we discuss the list as being the same in any radix,
where that is possible as Dedekind/Cauchy may be insufficient, there
are certain structural aspects of a number different from each element
in each base.
With the coded powerset in binary, if infinite sets are equivalent,
then that leads to a variety of considerations. One is the dual
representation of some element. In a theory with ordinals as primary
objects, addressing that is more or less the same thing as
Burali-Forti: the order type of ordinals is an ordinal, or plainly
"infinity plus one equals infinity."
So anyways, the universal set is its own powerset.
Infinite sets are equivalent.
In a respect, I agree.
Ross F.
Dik T. Winter
> The universe contains all physical objects.
Mathematics is not about physical objects. Show me the physical object
that corresponds to 1, and to 2.
Dik T. Winter
> I don't think it is. It's just that every real number has two decimal
> (or any other base) representations, and for simplicity, you would want
> to decide in advance which of the two representations you want to use
> for each of them.
This is wrong. 1/3 has only a single decimal representation. It is only
the rationals with a denominator that has only factors 2 and 5 that have
dual representations.
Robert Kolker
> Here's a notion.
>
> The universe contains all physical objects. A function, with the
> domain and range being physical objects, is itself a physical object.
> Thus, via induction, there are infinitely many physical objects because
> there are at least two.
Some things exist only abstractly. For example sets of individual
indivisible things. If you eliminate all intelligent beings from the
Kosmos there would be no sets.
Bob Kolker
Robert Kolker
> In article <1117331500....@g14g2000cwa.googlegroups.com> "Ross A. Finlayson" <r...@tiki-lounge.com> writes:
> > The universe contains all physical objects.
>
> Mathematics is not about physical objects. Show me the physical object
> that corresponds to 1, and to 2.
Hold up your right hand (if you have one). that corresponds to 1, but it
-isnt- 1. Hold up both your hands (if you have both). That corresponds
to 2 but it is not 2.
Bob Kolker
The Ghost In The Machine
<anzau...@hotmail.com>
wrote
on 28 May 2005 13:42:25 -0700
<1117312945.7...@g14g2000cwa.googlegroups.com>:
I'd go quaternary by picking 2 at a time, but ternary works too,
and is appropriate for a Cantor proof as one of Cantor's sets
can be defined
C = {k/3^n: k in ternary has no digit '1'}
:-)
An alternate proof using sequences (and no digit representations)
is also possible.
http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof
--
#191, ewi...@earthlink.net
It's still legal to go .sigless.
anzau...@hotmail.com
>
> Like I keep saying. The reason we keep getting these posts is because
> Cantor's work is accessible to people who don't understand math.
>
> Funny how we never get any posts around here that say, "I'm outraged.
> The classification theorem for finite simple groups has simply got to be
> false!"
Yep. We get 3 groups of kooks:
1. FLT kooks. Each of these people has 5 easy proofs that FLT is true
and 3 more proofs that it's false. Curiously, none of the latter 3
proofs invlves any couterexamples.
2. Anti-Cantor crowd. These people think that adding 2 more members to
the set of rationals creates a set of a larger cardinality, and at the
same time they have an explicit proof that reals have the same
cardinality as integers.
3. Foundations kooks. These people have brilliant new ideas how
mathematicians don't understand what axioms and definitions are, and
are on a mission to "teach" professional mathematicans how to think
properly.
What unites all of them is that they don't come to sci.math to learn or
to solve new puzzles. They are here to prove to the World that they are
more briliant than professional mathematicians like Cantor and Wiles.
Instead of intellectual curiosity, these kooks crave instant fame and
recognition. They crave front pages of the New York Times. Without
having to work hard at learning.
José Carlos Santos
> 1. FLT kooks. Each of these people has 5 easy proofs that FLT is true
> and 3 more proofs that it's false. Curiously, none of the latter 3
> proofs invlves any couterexamples.
Not true. E. E. Escultura claims that FLT is false and describes how
to obtain countable many counterexamples in his article "Exact Solutions
of Fermat's Equation", Nonlinear Studies, 5(2), 1998, pp. 227 - 254.
> 3. Foundations kooks. These people have brilliant new ideas how
> mathematicians don't understand what axioms and definitions are, and
> are on a mission to "teach" professional mathematicans how to think
> properly.
E. E. Escultura is one of those.
Best regards,
Jose Carlos Santos
quia...@yahoo.com
Or any mathematics at all. Then, suppose intelligent life evolved
again. That life would rediscover sets and all the same mathematics all
over again. Now, where were all the sets and mathematics in the
interim? They must have been somewhere, to emerge again unchanged.
Then, why would you not classify the place where all the math is
stored, waiting to be discovered by intelligent life, as part of the
Kosmos or universe?
--
john
Dirk Van de moortel
<anzau...@hotmail.com> wrote in message news:1117326360.5...@g14g2000cwa.googlegroups.com...
hm... I didn't mean to insult the profession.
Perhaps I should have said "kooks" ;-)
Dirk Vdm
Rainer Rosenthal
"Robert Kolker"
> If you eliminate all intelligent beings from the
> Kosmos there would be no sets.
But sci.math could still work for a while ;-)
Smiling regards
Rainer Rosenthal
r.ros...@web.de
Daniel W. Johnson
> 2. Anti-Cantor crowd. These people think that adding 2 more members to
> the set of rationals creates a set of a larger cardinality, and at the
> same time they have an explicit proof that reals have the same
> cardinality as integers.
Don't forget the one who not only proved that the reals have the same
cardinality as integers, he also proved that the rationals have a larger
cardinality than the integers.
anzau...@hotmail.com
> anzau...@hotmail.com wrote:
>
> > 1. FLT kooks. Each of these people has 5 easy proofs that FLT is true
> > and 3 more proofs that it's false. Curiously, none of the latter 3
> > proofs invlves any couterexamples.
>
> Not true. E. E. Escultura claims that FLT is false and describes how
> to obtain countable many counterexamples in his article "Exact Solutions
> of Fermat's Equation", Nonlinear Studies, 5(2), 1998, pp. 227 - 254.
Why publish a whole article to show that the set of counterexamples is
countable? Any college freshman can prove that. Wiles has proved even
more than that: that this countable set has cardinality zero.
You probably mean that Escultura describes how to obtain INFINITELY
many counterexamples, right? For n > 2, right? And the values of all
variables are integer, right?
That's beautiful! I don't really need an INFINITE set of such examples.
Just give me ONE example. Just tell me what the values of X, Y, Z and n
are, and I'll do the rest.
anzau...@hotmail.com
> <anzau...@hotmail.com> wrote:
>
> > 2. Anti-Cantor crowd. These people think that adding 2 more members to
> > the set of rationals creates a set of a larger cardinality, and at the
> > same time they have an explicit proof that reals have the same
> > cardinality as integers.
>
> Don't forget the one who not only proved that the reals have the same
> cardinality as integers, he also proved that the rationals have a larger
> cardinality than the integers.
Well, that just shows that not all rational numbers are for real.
Ross A. Finlayson
theory.
In the theory of sets, the set-theoretical objects are the objects of
the theory, in the theory of numbers the numbers are the objects of the
theory.
I think there is a theory that is the same theory for each of those
things.
This is where I think the null axiom theory, with various
considerations of the dialetheism of the dually minimal and maximal
ur-element and various natural principles of uniquification and
collection is that theory, as where it is deaxiomatized it can be
complete. Being able to prove each true statement of a theory is a
desirable facet of the theory, because I think true means provable and
provable means true.
This is where provable but untrue, inconsistent, or true but
unprovable, incomplete, deductions are each wrong to have in a theory.
The universe is infinite, and infinite sets are equivalent.
ZF is inconsistent.
Ross F.
anzau...@hotmail.com
>
> The universe is infinite, and infinite sets are equivalent.
They may. That depends on what you persoanlly mean by "equivalent",
doesn't it?
José Carlos Santos
>>>1. FLT kooks. Each of these people has 5 easy proofs that FLT is true
>>>and 3 more proofs that it's false. Curiously, none of the latter 3
>>>proofs invlves any couterexamples.
>>
>>Not true. E. E. Escultura claims that FLT is false and describes how
>>to obtain countable many counterexamples in his article "Exact Solutions
>>of Fermat's Equation", Nonlinear Studies, 5(2), 1998, pp. 227 - 254.
>
> Why publish a whole article to show that the set of counterexamples is
> countable? Any college freshman can prove that. Wiles has proved even
> more than that: that this countable set has cardinality zero.
>
> You probably mean that Escultura describes how to obtain INFINITELY
> many counterexamples, right? For n > 2, right? And the values of all
> variables are integer, right?
Right. I have this tendency to use the word "countable" with the meaning
"with the same cardinal as the set of natural numbers".
> That's beautiful! I don't really need an INFINITE set of such examples.
> Just give me ONE example. Just tell me what the values of X, Y, Z and n
> are, and I'll do the rest.
Well, the problem is that Escultura's integers are not the same ones
that you and I use; they may have an infinite number of digits.
Escultura claims that the real number system, as it is usually used, is
defective. Of course, he also claims to have found a non-defective way
of doing things.
Robert Low
[counter-examples to FLT]
> Well, the problem is that Escultura's integers are not the same ones
> that you and I use; they may have an infinite number of digits.
> Escultura claims that the real number system, as it is usually used, is
> defective. Of course, he also claims to have found a non-defective way
> of doing things.
Isn't this what Archimedes Plutonium used to bang on about?
Jesse F. Hughes
Still does. In fact, he recently complained that Escultura has failed
to give him the credit he's due. There is a thread titled "whether
EEE is intellectual theft[1] of A.P." (Sept. 2004) but it is damn near
unreadable on Google's completely fucked format now[2].
He also wrote the following back in August, 2004.
,----[ <411475C9...@iw.net> ]
| I vaguely remember Mr. Escultura in the late 1990s. And that he was
| pretty much copycat of my work done in the early 1990s. There were
| several who cited Mr. Escultura's as saying that it was a rehash of my
| ideas.
`----
I believe he's complained more recently than that, but I don't feel
like tussling with Google's awful interface .
Footnotes:
[1] No, really. That's what it says.
[2] Great. They "upgraded" the UK Google groups. Is there any
reasonable Usenet archive left anywhere?
--
Jesse F. Hughes
"When you try to kiss a girl, it's hard not to get spit on the girl."
-- Quincy P. Hughes, age 3 (almost 4)
Jesse F. Hughes
He means that any two infinite sets have the same cardinality.
Honest, he does.
But don't take my word for it. Google on Ross A. Finlayson and the
phrase "infinite sets are equivalent".
--
"And I wish some of you would grow past thinking that you've discovered
some extraordinary thing [...] as if you found the Holy Grail or
something, when I acknowledge a mistake. After all, I've had to do it
quite a few times. It's not like it's news." --James S. Harris
anzau...@hotmail.com
> On 30-05-2005 1:07, anzau...@hotmail.com wrote:
>
> >>Not true. E. E. Escultura claims that FLT is false and describes how
> >>to obtain countable many counterexamples in his article "Exact Solutions
> >>of Fermat's Equation", Nonlinear Studies, 5(2), 1998, pp. 227 - 254.
>
> > Just give me ONE example. Just tell me what the values of X, Y, Z and n
> > are, and I'll do the rest.
>
> Well, the problem is that Escultura's integers are not the same ones
> that you and I use; they may have an infinite number of digits.
Well, Fermat posed his question about the kind of integers that you and
I use, not about those with an infinite number of digits.
What is an integer with an infinite number of digits? A real? Sure, FLT
has solutions in reals. Uncountably many of them.
> Escultura claims that the real number system, as it is usually used, is
> defective.
How about the integer number system? If it's not defective - then why
can't he produce counter-examples in the usual integer representation
system?
>Of course, he also claims to have found a non-defective way
> of doing things.
Jose,
Edgar Escultura is a kook. Big time! Total certifiable moron.
Embarassed all of the Philippines as part of his egomaniacal
enterprise. Doesn't even undersand basic sarcasm. Total idiot.
You can read a detailed discussion of him and of The Manilla Times in
our recent thread "Help in answering news story on refutation of
fermat's last theorem":
Dik T. Winter
> Jos=E9 Carlos Santos wrote:
...
> > Well, the problem is that Escultura's integers are not the same ones
> > that you and I use; they may have an infinite number of digits.
>
> Well, Fermat posed his question about the kind of integers that you and
> I use, not about those with an infinite number of digits.
Yes, but EEE maintains that the numbers with a finite number of non-zero
digits are not actually the naturals. That definition is wrong, so he
gives what he thinks is the proper definition of the naturals, and solves
FLT in that.
> > Escultura claims that the real number system, as it is usually used, is
> > defective.
>
> How about the integer number system? If it's not defective - then why
> can't he produce counter-examples in the usual integer representation
> system?
He maintains that the integer number system is also defective.
G. Frege
wrote:
> >
> > Well, Fermat posed his question about the kind of integers that you and
> > I use, not about those with an infinite number of digits.
> >
> Yes, but EEE maintains that the numbers with a finite number of non-zero
> digits are not actually the naturals. That definition is wrong, so he
> gives what he thinks is the proper definition of the naturals, and solves
> FLT in that.
>
I see.
Theorem: 0 = 1.
Proof: We define 0 := 1. Then 0 = 1 (from 1 = 1). qed
Hell!
F.
Ross A. Finlayson
be equivalent while Cantor's arguments are true?
One notion is that no infinite set is regular. Yet, while inductively
there's always one more, inductively the set is regular.
Where the set is irregular, the Cantorian result does not necessarily
hold. Basically, where induction via the transfer principle shows that
infinite sets are eqivalent, then there are more to sets and their most
primitive construction than Cantor knew. At the deep level, as the set
theory's minimal element is the primary element for all possible set
theories, it is found to be the maximal element, and instead of
mathematical logic as tradition, the philosophers hold sway as having
explored such issues further, leading basically away from Cantor and
Godel towards Kant and Hegel, as they offer some more telling
explanations of these things, and to modern philosophers of
mathematical logic, such as myself.
There are a variety of theories where the powerset mappability result
does not hold, for various conditions.
Perhaps the most utilitarian area that is basically proscribed from
existence in Cantor's framework is that of the mappings from integers
to each of a set of an interval of real numbers. Basically ignoring
Cantor-Bernstein and describing the real numbers in a way that
Cantor/Megill or nested intervals does not hold, using the
well-ordering principle and trichotomy, a variety of analytical tools
become available, basically to fall among a basket of extant, useful,
and employed engineering functions, towards perhaps making them
rigorous enough in foundation that they don't offend mathematical
sensibility.
Skolemize, your model is countable. Sets of numbers are measurable.
The universe is infinite. Infinite sets are equivalent. ZF is
inconsistent.
Against those is basically the inductive impasse, the standoff between
expansion and nullity. Luckily, that's also for them, about how the
ur-element, or ur-paradox, which is not a paradox as it applies to
itself, as a tee for the singular and plural.
These are not necessarily, and perhaps necessarily not, totally obvious
conclusions.
Anyways, infinite sets are equivalent, and, your existence in the
physical universe proves that.
In a sense, I agree.
Ross F.
a_plu...@hotmail.com
I do not remember whether I saw that thread you speak of -- Sept 2004,
or
whether I participated in that thread you speak of. The trouble that
Mr. Escultura has which is plainly visible in his posts to the Internet
when he trys to give the examples of solutions to FLT is that they
always are some Adic solutions. They are "infinite integers" no matter
how Mr. Escultura gives them a funny name. He does this because Mr.
Escultura knows I pre-date him and that he is a copycat of my work of
the early 1990s.
It was Archimedes Plutonium with the help of a few individuals such as
Karl Heuer, Will Schneeburger (forgive the spelling) and several
others. Around 1993 I realized that the INfinite Integers easily solve
all FLT problems and these Infinite Integers were p-adics and all of
the Adics. The history of mathematics has a very large blind-spot. Call
it a idiot or stupidity spot much like cars have blind spots. Blind to
the fact that working mathematicians today never raise the issue or
raise the question of "can the Natural Numbers be something other than
what we believe they are?" Can the Natural Numbers be
infinite-integers? And if not-- then how does the Infinite Integers fit
alongside that of Finite Integers?
This question should stir working mathematicians to be very
suspicious-- given the fact that of all the branches of mathematics,
there are more outstanding unsolved problems in mathematics in the tiny
branch of Number theory than in all the other branches of mathematics
combined and the oldest problems all focus on Number theory. This
reality would tell a thinking person that our notion of what a Natural
Number truly is-- is probably wrong. That the cause of all of this
unsolved problems is because the very essence of what a Natural Number
is-- is fundamentally wrong.
So what I did in the time period of 1993 to 1994 was set the stage of
argument that the Peano Axiom System has the Successor Function of
endless adding of 1. That is the essence of the Natural-Numbers equal
to Finite Integers. They are the endless adding of 1. But, and here is
the crux, -- the definition of Infinite Integers or Adics is a series
or sequence definition -- the Adics are the endless adding of 1. Hensel
discovered the p-adics in the early 20th century and realized that
defining p-adics was a series formulation of adding 1 endlessly.
So, here is the problem which every modern mathematician ducks or hides
from except myself and a few others such as Karl Heuer and Will
Schneeburger. How can anyone say the Natural Numbers are
finite-integers when the very same axiom of the Successor Axiom goes to
build the p-adics (infinite-integers).
Modern mathematicians after 1993 have decided to ignore or to call
Archimedes Plutonium bad names, simply because they are not smart
enough to face reality, face the issue in question. How does the Adics
fit alongside the old notion of Finite-Integers?
So it is not surprizing that another person in mathematics-- Mr.
Escultura pops up his head and asks the same questions I asked
previously. Who sees also that the Adics solve all the old Number
theory problems unsolvable to date.
I should be angry with Mr. Escultura for trying to steal my ideas. But
I should not be too angry because he really cannot steal because all of
those ideas of mine are so well recorded on the Internet and that Mr.
Escultura is doing the helpful work of convincing others that the issue
of "What are the Natural Numbers really?" needs a deeper look and
investigation. Yes it would be nice for Mr. Escultura to fully give me
credit since he has no new ideas himself but is a copycat of my 1993
work. But it is to Mr. Escultura's credit that he incites other people
to question their beliefs as to what the Natural Numbers really are.
Archimedes Plutonium
www.iw.net/~a_plutonium
Whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
Jesse F. Hughes
> Skolemize, your model is countable. Sets of numbers are measurable.
> The universe is infinite. Infinite sets are equivalent. ZF is
> inconsistent.
No, no, no. Haikus have a specific form. 5-7-5, dammit.
--
"I don't know why I live in a world with so many supposed
mathematicians who are all so dumb AND rude. Why oh why couldn't
someone like Gauss or Dedekind still be around? Shoot, I'd even take
someone like Hardy at this point." -- James S Harris compromises
Jesse F. Hughes
> Jesse F. Hughes wrote:
>> Robert Low <mtx...@coventry.ac.uk> writes:
>>
>> > José Carlos Santos wrote:
>> > [counter-examples to FLT]
>> >> Well, the problem is that Escultura's integers are not the same ones
>> >> that you and I use; they may have an infinite number of digits.
>> >> Escultura claims that the real number system, as it is usually used, is
>> >> defective. Of course, he also claims to have found a non-defective way
>> >> of doing things.
>> >
>> > Isn't this what Archimedes Plutonium used to bang on about?
>> >
>>
>> Still does. In fact, he recently complained that Escultura has failed
>> to give him the credit he's due. There is a thread titled "whether
>> EEE is intellectual theft[1] of A.P." (Sept. 2004) but it is damn near
>> unreadable on Google's completely fucked format now[2].
>>
>> He also wrote the following back in August, 2004.
>>
>> ,----[ <411475C9...@iw.net> ]
>> | I vaguely remember Mr. Escultura in the late 1990s. And that he was
>> | pretty much copycat of my work done in the early 1990s. There were
>> | several who cited Mr. Escultura's as saying that it was a rehash of my
>> | ideas.
>> `----
>>
>> I believe he's complained more recently than that, but I don't feel
>> like tussling with Google's awful interface .
[...]
>
> I do not remember whether I saw that thread you speak of -- Sept
> 2004, or whether I participated in that thread you speak of.
Google for it. If you didn't participate, then someone was forging
your posts. Which seems somehow, I don't know, pointless?
[Snip long explanation of who really discovered that there are
infinite integers and hence who gets the medal, fame and chicks. I
was never much for history of mathematics, I'm afraid.]
--
"It's good for the economy to charge for intellectual property, so
open source software cannot be good, while Microsoft is the most
far-thinking company around and is doing it all for the good of the
public." -- Linus Torvalds paraphrases Microsoft VP Craig Mundie
Pope Urban Daycamp Thirty Minus XIII
does AP use that formulation?
did Euclid have a fake proof of the infinitude of primes, or not?
Jesse F. Hughes wrote:
> >> > Isn't this what Archimedes Plutonium used to bang on about?
--ils ducs d'Enron!
http://tarpley.ney/bush7.htm
anzau...@hotmail.com
> In article <1117452043.9...@f14g2000cwb.googlegroups.com> anzau...@hotmail.com writes:
> > Jos=E9 Carlos Santos wrote:
> ...
> > > that you and I use; they may have an infinite number of digits.
> >
> > Well, Fermat posed his question about the kind of integers that you and
> > I use, not about those with an infinite number of digits.
>
> > Well, the problem is that Escultura's integers are not the same ones
> > that you and I use; they may have an infinite number of digits.
>
> > What is an integer with an infinite number of digits? A real? Sure, FLT
> > has solutions in reals. Uncountably many of them.
>
> digits are not actually the naturals. That definition is wrong, so he
> gives what he thinks is the proper definition of the naturals, and solves
> FLT in that.
Sure. If you come up with your own definitions of integers, which is
completely different from the old definitions and axioms, then yes,
many equations, unsolvable in the old system, will become trivialy
solvable in the new one.
For example, as I said, if you define integers to be infinite sequences
of digits with a period placed somewhere in that sequence, you will be
easily able to come up with a well-behaved structure that has the same
algebraic properties as do the rational numbers. You can call it
"Escultura integers". I prefer to call them "real numbers". But what's
in the name? A real by any other nmae would be just as real, wouldn't
it?
In any case, in my new definition of "integers" (which corresponds to
traditional real numbers), the equation
X^n + Y^n = Z^n
can be solved by any fifth-grade elementary school student. Simply set
n, X and Y to be any positive numbers you want, let B to be defined as
B = X^n + Y^n
and set
Z = exp(1/n*logB)
Then
Z^n = exp(1/n*logB)^n = exp(1/n*n*logB)= exp(logB) = B = X^n + Y^n
But so what? If Fermat were also dealing with this definition of
"integers", he and his predecessors and successors wouldn't have posed
FLT as a very difficult problem, would they?
What's difficult and interesting in one axiom system is easy and
uninteresting in a different one.
> > > Escultura claims that the real number system, as it is usually used, is
> > > defective.
> >
> > How about the integer number system? If it's not defective - then why
> > can't he produce counter-examples in the usual integer representation
> > system?
>
> He maintains that the integer number system is also defective.
That's a different question. But it has nothing to do with the
difficult question of solving FLT in the traditional integer system.
anzau...@hotmail.com
> anzau...@hotmail.com writes:
>
> > Ross A. Finlayson wrote:
> >>
> >> The universe is infinite, and infinite sets are equivalent.
> >
> > They may. That depends on what you persoanlly mean by "equivalent",
> > doesn't it?
> >
>
> He means that any two infinite sets have the same cardinality.
>
> Honest, he does.
So do at least 10 other people who post to sci.math.
Proginoskes
Jesse F. Hughes wrote:
>[...]
> There is a thread titled "whether EEE is intellectual
> theft[1] of A.P." (Sept. 2004) but it is damn near
> unreadable on Google's completely fucked format now[2].
> I believe he's complained more recently than that, but
> I don't feel like tussling with Google's awful interface .
When Google Groups calls up a thread, you can go to the individial
posts and click on "show options", then "show original", which will
open the original post -- in normal text -- in another window.
--- Christopher Heckman
Dik T. Winter
> Dik T. Winter wrote:
...
> > > What is an integer with an infinite number of digits? A real? Sure, FLT
> > > has solutions in reals. Uncountably many of them.
> >
> > Yes, but EEE maintains that the numbers with a finite number of non-zero
> > digits are not actually the naturals. That definition is wrong, so he
> > gives what he thinks is the proper definition of the naturals, and solves
> > FLT in that.
>
> Sure. If you come up with your own definitions of integers, which is
> completely different from the old definitions and axioms, then yes,
> many equations, unsolvable in the old system, will become trivialy
> solvable in the new one.
Indeed. But that is everything EEE relies on.
> For example, as I said, if you define integers to be infinite sequences
> of digits with a period placed somewhere in that sequence, you will be
> easily able to come up with a well-behaved structure that has the same
> algebraic properties as do the rational numbers.
No, this is not correct. If you define them as infinite sequences of
digits with a period placed firmly on the right, you are close to the
10-adics (and that is what AP uses, and I think EEE also uses that, but
I am not sure, he has never been very prolific). It is extremely easy
to find a counter-example of FLT in the 10-adics. And they do not have
the same algebraic properties as the rational numbers at all. For which
rational can you say that a^2 = a, when a != 0 and != 1? There are two
such in the 10-adics, and, surprise, they add up to 1. A canned formula
to get the counter-example.
> > > How about the integer number system? If it's not defective - then why
> > > can't he produce counter-examples in the usual integer representation
> > > system?
> >
> > He maintains that the integer number system is also defective.
>
> That's a different question. But it has nothing to do with the
> difficult question of solving FLT in the traditional integer system.
EEE and AP maintain that it is wrong to ask such a question in a system
that is defective. But do not argue to me about that.
The worst I ever encountered was TLeko, who maintained that f(z) = z was
not an analytic function. He has even some of that stuff published in
conference proceedings. And I have received a fax with a plot showing
it (and also showing how he did not understand derivatives).
Dik T. Winter
Yeah. A *great* user interface. Especially as it shows some articles
with a barely readable font-size.
Dik T. Winter
I think not. As far as I am able to ascertain, Mueckenheim and Blumschein
are of the opinion that it makes no sense to talk about cardinality, and
so it should be elided from mathematics (there is no physical justification).
Orlow appears to be of the opinion that it makes sense to talk about it, and
that two infinite sets do not necessarily have the same cardinality, but that
the definition of cardinality is wrong. What the opinion of Finlayson is
in this case escapes me completely, although I have read his articles quite
some years now. I think he wants only a single infinity (appears to be
similar to Blumschein). Mueckenheim on the other hand is of the opinion
that infinite sets do not exist. But I think that the justifications of
Blumschein and Finlayson are different.
Take your pick.
tal19...@yahoo.com
> In article <1117490316....@z14g2000cwz.googlegroups.com> anzau...@hotmail.com writes:
> > Dik T. Winter wrote:
>
> > For example, as I said, if you define integers to be infinite sequences
> > of digits with a period placed somewhere in that sequence, you will be
> > easily able to come up with a well-behaved structure that has the same
> > algebraic properties as do the rational numbers.
>
> No, this is not correct.
Ooops! Did I say "rational numbers"?!!!! Wow. How embarassing... Why
do I always manage to write things that I don't mean? I meant "real
numbers". Sorry. I am blushing.
But on second thought, reals do have similar algebraic properties as do
the rational numbers. Both are infinite fields. Both don't contain all
the roots of their polynomials... It's just a matter of degree. :-)
> If you define them as infinite sequences of
> digits with a period placed firmly on the right, you are close to the
> 10-adics (and that is what AP uses, and I think EEE also uses that, but
> I am not sure, he has never been very prolific).
Yep. p-adics seems a much closer neighbor to integers than reals. But I
had just one tiny little 1-semester class in p-adics 28 years ago, so I
no longer have a clue how to solve FLT in them. But I still know how
to solve it in reals. :-)
Ross A. Finlayson
How can I make that different? I like to think I can fit what I have
to say about infinite sets and set theory on a few pages.
The real numbers exist, that's very simple. The complete ordered field
of scalars as they are, they also happen to be a contiguous sequence of
points.
With these mathematical objects, there are some similar phenomena as
what apply in some aspects of explanations of counterintuitive effects
from physics and the mathematical physics of particles and waves. That
has to do with for example, uncertainty, but rephrased to instead
reflect surety in reinterpretation.
Here I have some small luxuries of having said those things, and
knowing why I did, and being able to explain them in large part.
About the integers, and integers as ordinals, there is basically a
proto- or ur-number. It's mutable, changeable, it's basically any
ordinal, which is basically any integer, in the cumulative hierarchy of
ordinals in the null axiom number theory where the successor is order
type, which reinterpreted as a set and back is the powerset. For
information not to be lost in translation, a note is made implicit.
Where functions between physical object and physical objects are
physical objects, the physical universe, which exists as a totality or
collection for anything to exist, are physical objects, then the
universe is infinite and infinite sets are equivalent, because you and
I each exist. So, that fact of existence is an empirical example or
physical proof that infinite sets are equivalent. Replacing the word
"physical" with "set-theoretical" leads to a parallel deduction, but I
repeat myself. ZF is inconsistent. With a variety of types of objects
to consider, eg, sets, numbers, sets of numbers, numbers of sets, and
physical and geometrical objects, some notions that affect any theory,
or theory of anything, affect those.
For the universe to either not equal itself or both equal itself and
not equal itself, eg for excluded middle to not apply to the universe's
existence or tautological identity of its existence, a variety of other
vacuous statements would become true. That risks embracing the
objectivist viewpoint, which is inherently untrue, or false to self.
So, it is not. Instead, the universe is infinite, and infinite sets
are equivalent.
Where infinite sets are equivalent, and the contiguous sequence of
points on the real number line comprise each and every value of the
continuous real number line, then it's possible to consider some
analytical results along the lines of verifying standard or classical
integral analysis with unit scalar infinitesimals, or in a nonstandard
context, and as well exploring some differences between what would
otherwise be indistinguishable points. The uncertainty that I mention,
or "infinitesimals have infinitesimals", basically is to be used to
start getting a grasp on the point particles of modern mathematical
physics, for example the photon: sometimes the infinitesimal point to
the infinitesimal point of the electron, where there are a variety of
infinitesimal points in the points of particle physics, and no
theoretical smallest particle, so the atom is a good one to use, a
smallest indivisible particle. Obviously, with the particle/wave
duality, there are various ways to consider the objects there as one or
both, as the indivua among fields and basically clouds, analytically
exactly the same thing as mathematical real numbers.
Dik, I'm glad to hear that. If I write something and you think it's
unclear or wrong, please feel free to note that and I'll probably
continue.
Ross F.
Jesse F. Hughes
How convenient! Got to hand it to them Google guys. They think of
everything.
Of course, it *still* doesn't show the original. Strings that appear
to be email addresses are mangled in this version as well.
What Google has done to the Usenet archive is a bloody shame.
--
"Come on people!!! The US just blew up a lot of people in Iraq, don't
you realize that a person with my exposure might just end up dead, by
mysterious circumstances?"
--James Harris, on the dangers of "proving" Fermat's last theorem
Archimedes Plutonium
"Dik T. Winter" wrote:
> In article <1117493956....@g44g2000cwa.googlegroups.com> "Proginoskes" <progi...@email.msn.com> writes:
> > Jesse F. Hughes wrote:
> > >[...]
> > > There is a thread titled "whether EEE is intellectual
> > > theft[1] of A.P." (Sept. 2004) but it is damn near
> > > unreadable on Google's completely fucked format now[2].
> > > [...]
> > > I believe he's complained more recently than that, but
> > > I don't feel like tussling with Google's awful interface .
> >
> > When Google Groups calls up a thread, you can go to the individial
> > posts and click on "show options", then "show original", which will
> > open the original post -- in normal text -- in another window.
>
> Yeah. A *great* user interface. Especially as it shows some articles
> with a barely readable font-size.
> --
I admire Google's newest format in that it eliminates the quoted material and shows the new response, and one can
access the quoted stuff if need be. But my complaint of the newest Google format is that one cannot access other
related newsgroups as the old format. I like reading smaller newsgroups like sci.logic rather than sci.math, or
sci.physics.electromag rather than sci.physics. In the old Google Format, I could switch newsgroups quickly but the
new format fences me in to one newsgroup.
Archimedes Plutonium
Archimedes Plutonium
> In article <1117490316....@z14g2000cwz.googlegroups.com> anzau...@hotmail.com writes:
> > Dik T. Winter wrote:
> ...
> > > > What is an integer with an infinite number of digits? A real? Sure, FLT
> > > > has solutions in reals. Uncountably many of them.
> > >
> > > Yes, but EEE maintains that the numbers with a finite number of non-zero
> > > digits are not actually the naturals. That definition is wrong, so he
> > > gives what he thinks is the proper definition of the naturals, and solves
> > > FLT in that.
> >
> > Sure. If you come up with your own definitions of integers, which is
> > completely different from the old definitions and axioms, then yes,
> > many equations, unsolvable in the old system, will become trivialy
> > solvable in the new one.
You do not understand how definitions and axioms work. They work the same for mathematics as they
do for physics, chemistry and other sciences.
And Dik is rusty on this also, judging from his posts of the past. You and Dik seem to think that
in mathematics you can define something that is counter to reality and still work with it.
Let me give you a metaphor example to get my point across. Thousands of people were trialed in a
court of law and found guilty of murder. To Dik, definitions and axioms in mathematics is the same
as a convicted murderer is the true murderer of the crime in question. But reality enters. And many
convicted persons never commited a murder which DNA testing of old crimes has proven in the last
decades. But to you and Dik, a convicted person is a murderer. To me, conviction can be different
from truth and reality of the events.
To you and Dik, as you define Natural-Numbers equal to Finite Integers and Peano Axiom System, you
pretend as if for all time, forever that these definitions and axioms fit the reality and truth. To
me, definitions and axioms are a grasping for what the true Natural-Numbers are and that grasping
often catches only a partial reality and not the total reality of the item in question. A
definition may capture the reality and truth of item in question. However, it is often the case
that the definition does not capture the reality and truth in question. Many convicted murderers
were the true murderer of the crime in question, but sometimes an innocent is convicted for a crime
he never did. To you and Dik, a conviction is a murderer even if truly innocent, but to me, a
definition or axiom is a grasping of what the underlying true thing really is in reality.
In physics we try to capture the essence of a photon with definitions and experiments but we always
leave it open ended in that we may have to change those definitions. The Maxwell Equations are a
sort of Axiom system for the photon. And we are not obtuse and ignorant to think that the Maxwell
Equations are eternal and unchanging on nonrevisable for all time.
But you and Dik seem to believe, from your comments that Natural-Numbers equal to Finite Integers
and begot of Peano Axioms are immutable and unchanging and never need of revision. You have the
muddle headed notion that a definition in mathematics captures the essence of whatever item you are
discussing. You have the misconception that a definition in mathematics does not have to fit
reality or truth in other realms of the world. You have the misconception that the moment you utter
a definition in mathematics that it is eternal and never need of adjustment.
>
>
> Indeed. But that is everything EEE relies on.
>
> > For example, as I said, if you define integers to be infinite sequences
> > of digits with a period placed somewhere in that sequence, you will be
> > easily able to come up with a well-behaved structure that has the same
> > algebraic properties as do the rational numbers.
>
> No, this is not correct. If you define them as infinite sequences of
> digits with a period placed firmly on the right, you are close to the
> 10-adics (and that is what AP uses, and I think EEE also uses that, but
> I am not sure, he has never been very prolific). It is extremely easy
EEE wants to use the adics but to use them would mean that he would have to give credit to
Archimedes Plutonium who discovered that Natural Numbers are the Adics. So EEE never uses the adics
because he does not want to give credit to AP. But this is EEE's dilemna because there is nothing
else he can use to give counterexamples of FLT.
That is why EEE spends 99.9 percent of the time yakkity yakking on why the foundations of
mathematics are screwed up and never gives a counterexample. So I wonder whether EEE is in partners
with Andrew Wiles himself, who has read the posts of Archimedes Plutonium on why the Natural
Numbers are the Infinite Integers.
This would make sense in that Andrew Wiles does not have the courage to announce that his FLT is a
fakery and so he partners with EEE.
In the 1990s I posed a research problem to Dik and others who firmly believed in the Wiles FLT
offering. I asked them, if they believed so strongly in the Wiles FLT that is based on
Natural-Numbers equal to Finite Integers, then how come Wiles's alleged proof of FLT depends on the
p-adics as a covering. Why this dependence on the p-adics? And can the Wiles's alleged FLT dispense
with all uses of the p-adics? If it cannot, it is very suspicious that the dependency on the
p-adics is the reason it is a fakery. If the Natural-Numbers are the Adics, then by logical
reasoning, a fake proof such as the Wiles FLT necessarily has to use the Adics.
Dik has never answered that question, probably because Dik is not intimate with the Wiles FLT.
>
> to find a counter-example of FLT in the 10-adics. And they do not have
> the same algebraic properties as the rational numbers at all. For which
> rational can you say that a^2 = a, when a != 0 and != 1? There are two
> such in the 10-adics, and, surprise, they add up to 1. A canned formula
> to get the counter-example.
The previous writer, Dik, meant to say Reals, not rationals.
This is a trait often found in most people in mathematics, in that they flog to death a mistake
made by others and have a blind spot, inability to recognize people's mistakes. But worse is that
they thence spend hours upon correcting those mistakes.
>
>
> > > > How about the integer number system? If it's not defective - then why
> > > > can't he produce counter-examples in the usual integer representation
> > > > system?
> > >
> > > He maintains that the integer number system is also defective.
EEE points to 2 axioms of Peano, I forgotten which two, but distinctly remember they were not the
Successor axiom. This was from the paid advertisement in Google to posts to sci.math on the topic
of FLT.
I should look up that ad and repost it to this post for details.
Archimedes Plutonium points to one axiom of the Peano Axioms which is the Successor axiom of the
endless adding of 1. This is the flaw in the Peano Axioms. Flaw because the creation of the Adics
is the endless adding of 1. This is where Finite-Integers becomes a mirage or delusion. None of the
other axioms of Peano can prevent the Successor Axiom from entering into Infinite Integers of
Adics.
When you have the very same generator that creates Natural Numbers as the same as what creates
Adics, then the Natural Numbers are the Adics. What is surprizing is that Hensel discovered the
adics around 1903 and it took until 1993, some 90 years later for a single mathematician, me, to
point out that the Natural Numbers have the same generator as the Adics.
Was every other mathematician asleep at the helm for those 90 years?
>
> >
> > That's a different question. But it has nothing to do with the
> > difficult question of solving FLT in the traditional integer system.
>
> EEE and AP maintain that it is wrong to ask such a question in a system
> that is defective. But do not argue to me about that.
Dik, you seem to have toned down your animosity towards me. Maybe its because another mathematician
EEE, has followed in my lead. Dik was close to calling me a kook and badmouthing me for some past
years now. So what is up, Dik, have you become more reason minded? And should I take you out of my
killfile.
Dirk Van de moortel
"Ross A. Finlayson" <r...@tiki-lounge.com> wrote in message news:1117512741....@g44g2000cwa.googlegroups.com...
> Hi Dik,
>
> How can I make that different?
You don't have to.
Dirk Vdm
Dik T. Winter
> Hi Dik,
>
> How can I make that different? I like to think I can fit what I have
> to say about infinite sets and set theory on a few pages.
What are you talking about?
> Dik, I'm glad to hear that. If I write something and you think it's
> unclear or wrong, please feel free to note that and I'll probably
> continue.
It is pretty unclear as I have no idea what you are responding to. Please
supply some context in the future.
Dik T. Winter
> Tue, 31 May 2005 01:03:13 GMT "Dik T. Winter" wrote:
> > > Sure. If you come up with your own definitions of integers, which is
> > > completely different from the old definitions and axioms, then yes,
> > > many equations, unsolvable in the old system, will become trivialy
> > > solvable in the new one.
>
> You do not understand how definitions and axioms work. They work the same
> for mathematics as they do for physics, chemistry and other sciences.
> And Dik is rusty on this also, judging from his posts of the past. You
> and Dik seem to think that in mathematics you can define something that
> is counter to reality and still work with it.
Yup. Let A be the set of ducks with fourteen legs. See, I have defined
something. And I can work with set A.
> Let me give you a metaphor example to get my point across. Thousands
> of people were trialed in a court of law and found guilty of murder.
> To Dik, definitions and axioms in mathematics is the same as a convicted
> murderer is the true murderer of the crime in question. But reality
> enters. And many convicted persons never commited a murder which DNA
> testing of old crimes has proven in the last decades. But to you and Dik,
> a convicted person is a murderer. To me, conviction can be different from
> truth and reality of the events.
An extremely strange metaphor. Moreover, I do not understand the third
sentence.
> To you and Dik, as you define Natural-Numbers equal to Finite Integers
> and Peano Axiom System, you pretend as if for all time, forever that
> these definitions and axioms fit the reality and truth.
That is barely true. What we say is *this is our definition of natural
numbers*. If you want another definition, go ahead. But to avoid
confusion, do not call them *natural numbers*.
...
> EEE wants to use the adics but to use them would mean that he would have
> to give credit to Archimedes Plutonium who discovered that Natural Numbers
> are the Adics.
Ah, yes, I had forgotten. You currently use the adics, which is the
agglomerate of the n-adics for all natural n > 1. Also for the non-finite
natural n? And you have not yet shown how to add say a and b when a is
a 2-adic and b a 3-adic.
> In the 1990s I posed a research problem to Dik and others who firmly
> believed in the Wiles FLT offering. I asked them, if they believed so
> strongly in the Wiles FLT that is based on Natural-Numbers equal to
> Finite Integers, then how come Wiles's alleged proof of FLT depends on
> the p-adics as a covering.
What part of the proof are you talking about?
> Dik has never answered that question, probably because Dik is not
> intimate with the Wiles FLT.
You apparently are, pray answer the question above.
> > to find a counter-example of FLT in the 10-adics. And they do not have
> > the same algebraic properties as the rational numbers at all. For which
> > rational can you say that a^2 = a, when a != 0 and != 1? There are two
> > such in the 10-adics, and, surprise, they add up to 1. A canned formula
> > to get the counter-example.
>
> The previous writer, Dik, meant to say Reals, not rationals.
Indeed. But the 10-adics also do not have the same algebraic properties as
the reals. So it does not matter what he would have written or meant.
For which real can youi say that a^2 = a, when a != 0 and != 1?
> This is a trait often found in most people in mathematics, in that they
> flog to death a mistake made by others and have a blind spot, inability
> to recognize people's mistakes. But worse is that they thence spend
> hours upon correcting those mistakes.
Worse is people that do not see that it does not matter one iota in what
I wrote.
> > EEE and AP maintain that it is wrong to ask such a question in a system
> > that is defective. But do not argue to me about that.
>
> Dik, you seem to have toned down your animosity towards me. Maybe its
> because another mathematician EEE, has followed in my lead.
You are both not mathematicians.
Ross A. Finlayson
Hi,
It was basically a reply to where you said something along the lines of
"I read Ross' posts, and their meaning escapes me."
I don't read all the posts to sci.math, that would be ridiculous,
although it's been done, but I read your post and encourage
understanding.
"What the opinion of Finlayson is
in this case escapes me completely, although I have read his articles
quite
some years now. I think he wants only a single infinity (appears to
be
similar to Blumschein). Mueckenheim on the other hand is of the
opinion
that infinite sets do not exist. But I think that the justifications
of
Blumschein and Finlayson are different.
Take your pick."
So, I was writing in an obsequious way to you to show some kind of
pedantic caring, where actually I am cruel and heartless and just
wanted to seem inspiring to the avid followers of the immediate
discussion, in a cold and calculated manner. Oh, damn, now I have
spoiled that. Also I wanted to get over the point about the direct
physical application.
Basically I see two infinities, infinity and the Big Infinity, which is
the infinity of all number spaces. Basically the Big Infinity is Ord,
and the others are nonstandard scalars. I think there are infinite
ordinals, with infinite ordinal arithmetic. There may as well be
infinite cardinals, but they lose much of their meaning, because as
ordinals the powerset operation is just a mechanical successor
operation for which it is quite well-suited.
The reals are obviously a larger set than the reals or for that matter
the naturals. The rationals are obviously a larger set than the
integers. That is where those are all sets of numbers, and those are
all subsets of the real numbers, with varying densities within the real
numbers, for various reasons, for example that in every interval of the
reals or disc or hyperdisc there are more of one than the other because
they're numbers, with a very explicit and specific construction.
Some people derive the natural integers directly from pure sets. They
exhibit an ordinal construction and say they're exactly the natural
numbers. Another method is to construct from the top-down, from
basically the dually-infinite cumulative hierarchy vector bases,
Euclidean and non-Euclidean, top down instead of bottom up, dual, the
integers as a subset of the rationals in each of the positive and
negative vector bases.
That's where 1 = 1/1 = 1.000... = 1.000...e1, integer, rational, and
real, unit scalar.
The other day, somebody called me a theologian. In fact, I believe it
was Jesse. What does it take to be a mathematician? What does that
mean? Is it a role, office, state of mind, pejorative, or mark of
experience? I'm a philosopher of mathematical logic.
Have a nice day,
Ross F.
anzau...@hotmail.com
> anzau...@hotmail.com writes:
> > Dik T. Winter wrote:
>
> > > Yes, but EEE maintains that the numbers with a finite number of non-zero
> > > digits are not actually the naturals. That definition is wrong, so he
> > > gives what he thinks is the proper definition of the naturals, and solves
> > > FLT in that.
> >
> > Sure. If you come up with your own definitions of integers, which is
> > completely different from the old definitions and axioms, then yes,
> > many equations, unsolvable in the old system, will become trivialy
> > solvable in the new one.
>
> > of digits with a period placed somewhere in that sequence, you will be
> > easily able to come up with a well-behaved structure that has the same
> > algebraic properties as do the rational numbers.
Did I ACTUALLY say THAT?!!!! Reals have the same algebraic properties
as rationals?!!!
Yeh, right. Try to solve
x^2 = 2
in rationals....
I am a kook.
> You can call it
> "Escultura integers". I prefer to call them "real numbers".
>
> No, this is not correct.
Sure it is. You can easily define real numbers as sequences of decimal
digits with a period placed somewhere in that sequence, with the
understanding that Sb999999999... is the same as S(b+1)00000000....,
where S is some finite set of digits which contains a period placed
somewhere in it and b is any digit other than 9.
anzau...@hotmail.com
> ...
> > > > has solutions in reals. Uncountably many of them.
> > >
> > > Yes, but EEE maintains that the numbers with a finite number of non-zero
> > > digits are not actually the naturals. That definition is wrong, so he
> > > gives what he thinks is the proper definition of the naturals, and solves
> > > FLT in that.
> >
> > completely different from the old definitions and axioms, then yes,
> > many equations, unsolvable in the old system, will become trivialy
> > solvable in the new one.
>
>
> digits with a period placed firmly on the right, you are close to the
> 10-adics (and that is what AP uses, and I think EEE also uses that, but
> I am not sure, he has never been very prolific).
>
So, is his point that N-adics are better representations of integers
that the interpretation that we, including Fermat, usually use?
If so, has EEE discovered new facts about N- and p- adics that people
didn't know before him? Or did he reinvent the wheel?
And why would anybody care about Fermat Last Theorem as applied to
N-adics? Fermat sure didn't. He didn't even know about them.
Dik T. Winter
> Dik T. Winter wrote:
...
> > supply some context in the future.
> It was basically a reply to where you said something along the lines of
> "I read Ross' posts, and their meaning escapes me."
Yes, I read quite a bit.
> "What the opinion of Finlayson is in this case escapes me completely,
> although I have read his articles quite some years now. I think he
> wants only a single infinity (appears to be similar to Blumschein).
> Mueckenheim on the other hand is of the opinion that infinite sets do
> not exist. But I think that the justifications of Blumschein and
> Finlayson are different.
...
> Basically I see two infinities, infinity and the Big Infinity, which is
> the infinity of all number spaces. Basically the Big Infinity is Ord,
> and the others are nonstandard scalars. I think there are infinite
> ordinals, with infinite ordinal arithmetic. There may as well be
> infinite cardinals, but they lose much of their meaning, because as
> ordinals the powerset operation is just a mechanical successor
> operation for which it is quite well-suited.
O. So succ(aleph_0) = aleph_1, etc. But that means that you have more
than one infinity. So I do not understand that.
> The reals are obviously a larger set than the reals or for that matter
rationals you mean the second time you write reals.
> the naturals. The rationals are obviously a larger set than the
> integers. That is where those are all sets of numbers, and those are
> all subsets of the real numbers, with varying densities within the real
> numbers, for various reasons, for example that in every interval of the
> reals or disc or hyperdisc there are more of one than the other because
> they're numbers, with a very explicit and specific construction.
But densities do you not give a complete distinction between sets. In
another thread I have shown a set of natural numbers, where the density
varied between 1/3 and 2/3 without ever settling. I think similar sets
can be shown for the rationals and the reals.
> Some people derive the natural integers directly from pure sets. They
> exhibit an ordinal construction and say they're exactly the natural
> numbers. Another method is to construct from the top-down, from
> basically the dually-infinite cumulative hierarchy vector bases,
> Euclidean and non-Euclidean, top down instead of bottom up, dual, the
> integers as a subset of the rationals in each of the positive and
> negative vector bases.
This is just handwaving.
Dik T. Winter
> Dik T. Winter wrote:
...
> > digits with a period placed firmly on the right, you are close to the
> > 10-adics (and that is what AP uses, and I think EEE also uses that, but
> > I am not sure, he has never been very prolific).
> So, is his point that N-adics are better representations of integers
> that the interpretation that we, including Fermat, usually use?
No his point (and AP's) point is that their representations of integers
are the only possible interpretations, and that our interpretations are
just plain wrong.
> If so, has EEE discovered new facts about N- and p- adics that people
> didn't know before him? Or did he reinvent the wheel?
No. Hensel discovered p-adics about 100 years ago (I think). I do not
think serious study has been made about the n-adics (with n non-prime),
at least I never have seen anything. AP (Ludwig Plutonium at that time,
and earlier Ludwig Heuer and Ludwig Schneeberger) started with his
counter-examples in the 10-adics in 1993. His first was: a = 1, b = 10,
c = ...52979382777667001 and n = 3. A nice way to show that 1001 is
a third power in the 10-adics. This particular thing was not known.
What was known was that FLT could have solutions in the p-adics (or at
least some of them). So no new facts. And I think that EEE started
later than AP, at least, his appearance in this newsgroup was much
later. Also I pointed out to AP that his counter-example was not a
counter example in the 9-adics (i.e. when you write numbers base 9),
but apparently he was not impressed.
> And why would anybody care about Fermat Last Theorem as applied to
> N-adics? Fermat sure didn't. He didn't even know about them.
But they argue that Fermat did not know what the naturals actually
are. AP has shifted from the 10-adics to the adics, encompassing all
the n-adics. But he does not even give a reasonable rule how to add
a 2-adic to a 3-adic.
Note: the n-adics, when n is the product of different primes p_i, none
occuring more than once, is the direct sum of the p-adics for those
different primes. I think it was either Robin Chapman or Robert Israel
who first stated something similar.
anzau...@hotmail.com
Dik T. Winter wrote:
> In article <1117568478....@g44g2000cwa.googlegroups.com> anzau...@hotmail.com writes:
> > Dik T. Winter wrote:
> ...
> > > If you define them as infinite sequences of
> > > digits with a period placed firmly on the right, you are close to the
> > > 10-adics (and that is what AP uses, and I think EEE also uses that, but
> > > I am not sure, he has never been very prolific).
> ...
> > So, is his point that N-adics are better representations of integers
> > that the interpretation that we, including Fermat, usually use?
>
> No his point (and AP's) point is that their representations of integers
> are the only possible interpretations, and that our interpretations are
> just plain wrong.
What argument does he use to explain this?
> > If so, has EEE discovered new facts about N- and p- adics that people
> > didn't know before him? Or did he reinvent the wheel?
>
> No. Hensel discovered p-adics about 100 years ago (I think). I do not
> think serious study has been made about the n-adics (with n non-prime),
> at least I never have seen anything.
Neither have I. My only knowledge of applications of p-adics comes from
Dwork's zeta function work. And even that was in my early youth.
> AP (Ludwig Plutonium at that time,
> and earlier Ludwig Heuer and Ludwig Schneeberger) started with his
> counter-examples in the 10-adics in 1993. His first was: a = 1, b = 10,
> c = ...52979382777667001 and n = 3. A nice way to show that 1001 is
> a third power in the 10-adics. This particular thing was not known.
> What was known was that FLT could have solutions in the p-adics (or at
> least some of them). So no new facts.
Aha!
> And I think that EEE started
> later than AP, at least, his appearance in this newsgroup was much
> later. Also I pointed out to AP that his counter-example was not a
> counter example in the 9-adics (i.e. when you write numbers base 9),
> but apparently he was not impressed.
>
> > And why would anybody care about Fermat Last Theorem as applied to
> > N-adics? Fermat sure didn't. He didn't even know about them.
>
> But they argue that Fermat did not know what the naturals actually
> are.
That's hardly the point. Fermat found the FLT statement in some book.
Both Fermat and the author of that book had a very particular view of
what they meant by the word "integer" in the formulation: "Does
X^n+Y^n=Z^n have any integer solutions for n > 2?" and the axioms they
satisfy.
Their axiom set may be a very wrong description of what integers
actually are. But that's what they wanted to find: solutions to the FLT
equation within these given axioms.
Solution to the same equation in any other axiom system is totally
irrelevant. Especially if, as you say, some solutions to this equation
in p-adics are already known.
Ross A. Finlayson
Yes, that should read reals compared to proper subsets of the reals.
It wasn't some obscuring statement that the reals were a proper subset
of themselves.
About the powerset as successor, or order type, and the ordinals,
basically that is bout looking at the ordinals, say the finite ordinals
or naturals, and taking their powerset. The order type of the powerset
is the successor of the order type of the ordinal. Then, where
ordinals may multiple representations, and basically the non-ordinal
composite elements of the ordinals are discarded for containing no new
information, where the finite ordinal is only the iterated successor of
zero, then where the order type of the powerset of the ordinal is the
successor of the order type of the ordinal, in only considering
inductive methods having to do with ordinals, which very specifically
form an inductive chain, the powerset is successor is order type, in
the theory with ubiquitous ordinals, and the mapping between set and
powerset is f(x) = x+1, the successor operation.
About the second notion, with naturals basically constructed last in
the construction of the number system instead of first, I have been
thinking that for some time yet had not bothered to phrase it. That
there is basically what I meant. Obviously, it is more intuitive to
start with the counting numbers instead of the most hypercomplex
numbers.
Please explain that set of numbers with varying or indeterminate
number-theoretic asymptotic density and its construction. We've had
private discussions outside of this newsgroup, I'd be happy to continue
them.
Thank you,
Ross F.
anzau...@hotmail.com
everything crystal clear.
Ross A. Finlayson wrote:
> Hi Dik,
>
> About the powerset as successor, or order type, and the ordinals,
> basically that is bout looking at the ordinals, say the finite ordinals
> or naturals, and taking their powerset. The order type of the powerset
> is the successor of the order type of the ordinal. Then, where
> ordinals may multiple representations, and basically the non-ordinal
> composite elements of the ordinals are discarded for containing no new
> information, where the finite ordinal is only the iterated successor of
> zero, then where the order type of the powerset of the ordinal is the
> successor of the order type of the ordinal, in only considering
> inductive methods having to do with ordinals, which very specifically
> form an inductive chain, the powerset is successor is order type, in
> the theory with ubiquitous ordinals, and the mapping between set and
> powerset is f(x) = x+1, the successor operation.
>
> About the second notion, with naturals basically constructed last in
> the construction of the number system instead of first, I have been
> thinking that for some time yet had not bothered to phrase it. That
> there is basically what I meant. Obviously, it is more intuitive to
> start with the counting numbers instead of the most hypercomplex
> numbers.
>
> Please explain that set of numbers with varying or indeterminate
> number-theoretic asymptotic density and its construction.
Is there one or more verbs missing?
anzau...@hotmail.com
> "Robert Kolker"
>
> > If you eliminate all intelligent beings from the
> > Kosmos there would be no sets.
>
> But sci.math could still work for a while ;-)
Don't forget about tennis either.
Robin Chapman
>
> Note: the n-adics, when n is the product of different primes p_i, none
> occuring more than once,
That is redundant.
> is the direct sum of the p-adics for those
> different primes. I think it was either Robin Chapman or Robert Israel
> who first stated something similar.
I may have mentioned that the only book I know who studies
the n-adics in any detail is Mahler's book on p-adic analysis.
He eventually proved that the n-adics are the direct product of Q_p
for p | n.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Elegance is an algorithm"
Iain M. Banks, _The Algebraist_
Archimedes Plutonium
> In article <429C1161...@iw.net> NOiwEMAIL writes:
> > Tue, 31 May 2005 01:03:13 GMT "Dik T. Winter" wrote:
> ...
> > > > Sure. If you come up with your own definitions of integers, which is
> > > > completely different from the old definitions and axioms, then yes,
> > > > many equations, unsolvable in the old system, will become trivialy
> > > > solvable in the new one.
> >
> > You do not understand how definitions and axioms work. They work the same
> > for mathematics as they do for physics, chemistry and other sciences.
> > And Dik is rusty on this also, judging from his posts of the past. You
> > and Dik seem to think that in mathematics you can define something that
> > is counter to reality and still work with it.
>
> Yup. Let A be the set of ducks with fourteen legs. See, I have defined
> something. And I can work with set A.
I don't think you really believe that. Believe, even, to the extent that the
empty set is nonreality.
This is a problem that Dik and many others have-- they think that mathematics is
above, beyond and superior to physics by the allowance of make-believe or
imagination as workable reality. In a sense, they elevate mathematics as above
physics, when in truth mathematics is begot from physics and is a minor subset of
physics.
In physics some 30 years ago I used to play the game of imagine the world without
the force of gravity or imagine the world without the force of electromagnetism
or imagine the world without the force of strongnuclear etc etc. Which of course
is all poppycock nonsense because if anyone of these forces were altered or
removed that there would be no humanity and no person to make such poppycock
imagining. But to Dik's view of the world that imagination is fair play in
mathematics.
The best way I can explain why Dik is wrong about definitions and axioms is the
history example of the Atomic theory begun by Democritus. Dik would say that the
Democritus Atomic theory is the same as Natural Numbers are Finite-Integers.
However, the Democritus Atomic Theory has had so many huge changes, especially in
the 20th century with Planck to Bohr to the Copenhagen Quantum theory, that when
one reads and understands the Democritus Atomic Theory and then reads and
understands the Atomic Theory with Quantum Mechanics injected, then one would
find it very difficult to say that Democritus Atomic theory is different and
relevant.
Dik would say that the Democritus Atomic Theory, like Natural-Numbers equal to
Finite Integers is a stand alone separate and valid theory compared to the
Quantum-Mechanics-Atomic-Theory, or, Natural-Numbers equal to Infinite-Integers.
To anyone in modern physics would instantly realize that our modern day Quantum
Mechanics Atomic theory was a revision of the old Democritus Atomic theory. They
instantly know that the centuries and milleniums after Democritus had revised,
corrected, updated, added-onto to Democritus's beginning work of the ATomic
theory. But Dik does not seem to understand or want to understand that
Natural-Numbers equal to Finite-Integers could be false and nonreality.
Democritus had many falsehoods about what atoms are and how atoms behave which
the 20th century physicists corrected. Dik seems to think that Natural-Numbers
equal to Finite-Integers are perfect, flawless and blemish proof and are
eternally true. Dik is vastly wrong.
>
>
> > Let me give you a metaphor example to get my point across. Thousands
> > of people were trialed in a court of law and found guilty of murder.
> > To Dik, definitions and axioms in mathematics is the same as a convicted
> > murderer is the true murderer of the crime in question. But reality
> > enters. And many convicted persons never commited a murder which DNA
> > testing of old crimes has proven in the last decades. But to you and Dik,
> > a convicted person is a murderer. To me, conviction can be different from
> > truth and reality of the events.
>
> An extremely strange metaphor. Moreover, I do not understand the third
> sentence.
It is not a strange metaphor. I chose it because mathematics is not superior or
above physics, but rather the opposite in that physics is superior to
mathematics. Physics creates mathematics, not the reverse. So I needed a metaphor
to elucidate that fact.
Dik's view of mathematics and definitions in mathematics would force Dik to say
that every convicted murderer had actually committed the murder event. My view of
mathematics and science is that a definition trys to capture the essence of an
underlying reality and underlying truth but often captures only a partial reality
and a partial truth. So in my view of definitions of mathematics or physics is
that the convicted person has a high degree of probability of having committed
the actual murder but as we well know with DNA testing in the past 50 years that
many convicted murderers were actually innocent of the event.
The trouble with Dik is that he places mathematics above physics and reality. Dik
thinks that when a mathematician sits down and writes a definition that this
definition has some immutable eternity-- Dik is a Platonic.
Physics tells us that all of our definitions, all of our axioms, all of our
theories are a grasping of the underlying reality and that we are constantly in
need of revision, updating and correcting our old views and understanding.
>
>
> > To you and Dik, as you define Natural-Numbers equal to Finite Integers
> > and Peano Axiom System, you pretend as if for all time, forever that
> > these definitions and axioms fit the reality and truth.
>
> That is barely true. What we say is *this is our definition of natural
> numbers*. If you want another definition, go ahead. But to avoid
> confusion, do not call them *natural numbers*.
>
Again, this is like Dik saying-- Democritus Atomic theory should be independent
of Quantum revision and that we should be taught that atoms have edges and atoms
have attributes such as "smooth or rough or sharp" and that modern physics should
teach the Democritus Atomic theory alongside and independent of the quantum
atomic theory.
Amazing how Dik can compute a p-adic string and teach a course in University on
p-adics, and yet Dik cannot understand that a concept of Finite Integers could be
a totally flawed concept. Amazing how people have blindspots.
>
> ...
> > EEE wants to use the adics but to use them would mean that he would have
> > to give credit to Archimedes Plutonium who discovered that Natural Numbers
> > are the Adics.
>
> Ah, yes, I had forgotten. You currently use the adics, which is the
> agglomerate of the n-adics for all natural n > 1. Also for the non-finite
> natural n? And you have not yet shown how to add say a and b when a is
> a 2-adic and b a 3-adic.
I suppose Dik would think that Quantum Mechanics from 1901 to 1931 could have all
been discovered by one person. That we did not need Bohr, Heisenberg, Pauli,
Born, Debroglie, Schrodinger, Dirac and many others. That Planck could have
discovered everything about QM. Perhaps the Natural-Numbers equal to
Infinite-Integers (Adics) is more complex than what QM was from 1901-1931 and
needs more than one person to unravel this complexity. Perhaps the
Infinite-Integers are so complex that it takes not just 30 years to have a
platform of understanding but takes 2 milleniums to reach a platform of
understanding.
I discovered the idea -- Natural Numbers equal to Infinite-Integers (Adics) in
1993. Around 1994, Dik asked over the Internet how to add a and b when a is
2-adic and b is 3-adic. I remember Abian entering the conversation with trying to
help, circa 1995-1996. Abian called them the "Plutonium Integers". Somewhere in
the years 2000-2004, I began to ask myself whether we even need adic addition
across different adics? It maybe the case that Dik's question is mute or
nonsensical.
What makes the Reals? They are a representation of the continuum. Numbers that
fill in all holes and provide a continuum. What makes Integers? Is it not
discreteness. So if I search and hunt for addition between 2-adics and 3-adics,
am I not contradicting myself? Is not Dik's question of how to add 2-adics with
3-adics similar to Dik asking me how to make the REals have holes when the whole
idea, the whole purpose of Reals is holelessness.
That is why I call them Adics-- the conglomeration of p-adics and n-adics. To be
able define addition with 2-adics and 3-adics is impossible because the essence
of Integers is discreteness. Yes they possibly may intersect of one adic with
another adic such as the 2-adics, 3-adics and 6-adics.
But I have to give this all more thought, and I believe at this moment that these
things are so complex that it is going to take many persons after me to make this
all more clear.
>
>
> > In the 1990s I posed a research problem to Dik and others who firmly
> > believed in the Wiles FLT offering. I asked them, if they believed so
> > strongly in the Wiles FLT that is based on Natural-Numbers equal to
> > Finite Integers, then how come Wiles's alleged proof of FLT depends on
> > the p-adics as a covering.
>
> What part of the proof are you talking about?
I am talking of the entire Wiles's offering. Show us that this offering can have
zero usage of p-adics or any other form of adics. If Wiles cannot do that, his
alleged FLT is a fake. And it points out that Wiles at the beginning of FLT or
even Fermat himself and all the other historical persons attempting to prove FLT,
would have or should have started their FLT offering by proving that the Natural
Numbers are not the Adics.
So if Wiles in the year 1989 or 1990 or 1991 had first proven that the Natural
Numbers are equal to Finite Integers and not the Infinite Integers of Adics, then
I would have believed his FLT is not a fake. Instead Wiles assumes
Natural-Numbers equal to Finite Integers. And then Wiles uses p-adics in a
necessary use in his alleged FLT.
Archimedes Plutonium
> In article <1117568478....@g44g2000cwa.googlegroups.com> anzau...@hotmail.com writes:
> > Dik T. Winter wrote:
> ...
> > > If you define them as infinite sequences of
> > > digits with a period placed firmly on the right, you are close to the
> > > 10-adics (and that is what AP uses, and I think EEE also uses that, but
> > > I am not sure, he has never been very prolific).
> ...
> > So, is his point that N-adics are better representations of integers
> > that the interpretation that we, including Fermat, usually use?
>
> No his point (and AP's) point is that their representations of integers
> are the only possible interpretations, and that our interpretations are
> just plain wrong.
Some of that is correct. But it dodges the aspect that Dik has an interpretation of what the
Natural Numbers are. I just know the Natural Numbers are not the Finite-Integers. I know they must
be Infinite Integers but that my interpretation is also only partially true and will await for
future revision itself. Dik seems to take the absolutist platform more than I take any absolute
stance.
I looked up EEE's recent paid ad to Google to try to copy paste on his most recent claims.
Unfortunately he took it down and put up a new paid ad. I hope someone archived EEE's recent
claim. If I remember correctly EEE was pointing at the fault of Natural Numbers and Reals in the
trichotomy axiom and ordering axiom. I have never seen EEE attack the Successor axiom, which is
the brunt of my attack.
But I think EEE is comparing finite-integers to REals because I have compared finite integers to
Adics and Infinite Integers and have taken historical priority. And so, if EEE is in want of
original new ideas, he is looking for a REal to finite-integer comparison.
>
>
> > If so, has EEE discovered new facts about N- and p- adics that people
> > didn't know before him? Or did he reinvent the wheel?
>
> No. Hensel discovered p-adics about 100 years ago (I think). I do not
> think serious study has been made about the n-adics (with n non-prime),
> at least I never have seen anything. AP (Ludwig Plutonium at that time,
> and earlier Ludwig Heuer and Ludwig Schneeberger) started with his
That was Karl Heuer of the Open Software organization who had many conversations, dialogues,
inputs to my threads of 1993-1994 when this new idea that Natural-Numbers equal to
Infinite-Integers (Adics) was discovered.
By the way, it was discovered completely on the Internet and so future historians of mathematics
can follow its discovery from all of my posts starting August of 1993. Karl Heuer is a person and
so is Will Schneeburger of Princeton U. back in 1993-1994 who also contributed heavily to this
discovery. I do not know why Dik now names them "Ludwig".
>
> counter-examples in the 10-adics in 1993. His first was: a = 1, b = 10,
> c = ...52979382777667001 and n = 3. A nice way to show that 1001 is
> a third power in the 10-adics. This particular thing was not known.
> What was known was that FLT could have solutions in the p-adics (or at
> least some of them). So no new facts. And I think that EEE started
> later than AP, at least, his appearance in this newsgroup was much
> later. Also I pointed out to AP that his counter-example was not a
> counter example in the 9-adics (i.e. when you write numbers base 9),
> but apparently he was not impressed.
I think EEE started around 1997 or 1998. I started 1993 and made the discovery that
Natural-Numbers equal to Infinite-Integers (Adics) in late 1993. By 1994 I was on to the idea that
the Successor Axiom of the Peano axiom set was the flaw of the Natural-Numbers equal to
Finite-Integers. The flaw because the generator of finite-integers is the Successor axiom which
endlessly adds 1. But this same generator of endless adding of 1 is the same creator of the Adics
with its "series definition". Adics are built by endlessly adding 1. So if the Natural Numbers and
Adics have the very same and identical creation process, the generator for creating them is
identical, would imply that Natural Numbers are the Adics.
EEE never picked or attacked the Successor axiom.
>
>
> > And why would anybody care about Fermat Last Theorem as applied to
> > N-adics? Fermat sure didn't. He didn't even know about them.
>
> But they argue that Fermat did not know what the naturals actually
> are. AP has shifted from the 10-adics to the adics, encompassing all
> the n-adics. But he does not even give a reasonable rule how to add
> a 2-adic to a 3-adic.
Dik, you must realize that some things are so complex and difficult that it will take more than
one human mind to reach a platform of stable-understanding. As I wrote earlier today, from 1901 to
1931 required hundreds perhaps thousands of physicists to reach the stable platform of
understanding of Quantum Mechanics. Perhaps Planck could have done all of that himself, but I
doubt it. And likewise, Natural Numbers equal to Infinite-Integers (Adics) maybe even more
difficult and complex than the history of QM from 1901-1931.
One of my hardships is that I cannot visualize these animals of Adics in any deep geometrical
visualizations. So maybe the future awaits some human who can more easily visualize p-adics and
n-adics and then make leaps of advanced progress. The best I can do at this moment is to visualize
2-adics, 3-adics etc etc as huge spheres or balls that are discrete but that sometimes intersect
such as the 3-adics and 6-adics. And it is discreteness that we expect of integers just as it is
"continuity" that we expect of Reals.
>
>
> Note: the n-adics, when n is the product of different primes p_i, none
> occuring more than once, is the direct sum of the p-adics for those
> different primes. I think it was either Robin Chapman or Robert Israel
> who first stated something similar.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
There is much important work going on in physics, as I speak, concerning Adics. And I have felt
for a decade now that the day when a physicist reports that the Quantum-Hall-Effect of its bizzare
numbers turns out to be Adics is the day of the deathknell of the old idea that Natural-Numbers
are finite-integers. If there is a single region of physics were Adics are essential, then all of
physics is in either Reals and/or Adics.
Archimedes Plutonium
(snipped)
> Dick, thanks a lot for explaining EEE's views to me so well.
>
> Dik T. Winter wrote:
> > In article <1117568478....@g44g2000cwa.googlegroups.com>
(snipped)
> (and AP's) point is that their representations of integers
> > are the only possible interpretations, and that our interpretations are
> > just plain wrong.
>
> What argument does he use to explain this?
An answer to your question in more general terms for the general case is more interesting.
A line of argument that would validate or lend credence to the theory that Natural-Numbers are
Infinite-Integers (Adics) is an argument from *symmetry*. Symmetry is a extremely useful tool. It is
symmetry that lead Maxwell to complete the Maxwell theory by noticing the missing term of Ampere's
law.
Symmetry says that Reals are infinite strings rightward with a finite portion leftward such as
2.333..... If Natural Numbers are finite-integers then the symmetry would be broken. Mathematics
would be asymmetrical. To complete the symmetry then Mathematics would be a Dual system, a duality
with two number systems that are symmetrical. So if we have a Real such as 2.333..... then we have
its dual other particle of ......3333.2. Thus, to make mathematics algreba a symmetrical system then
the compliment of Reals is Natural-Numbers equal to Adics. Finite-Integers were just a crude and
flawed system that captured only a tiny bit of what the Natural-Numbers really are.
>
>
> > No. Hensel discovered p-adics about 100 years ago (I think). I do not
> > think serious study has been made about the n-adics (with n non-prime),
> > at least I never have seen anything.
>
>
snipped
>
>
> That's hardly the point. Fermat found the FLT statement in some book.
> Both Fermat and the author of that book had a very particular view of
> what they meant by the word "integer" in the formulation: "Does
> X^n+Y^n=Z^n have any integer solutions for n > 2?" and the axioms they
> satisfy.
>
> Their axiom set may be a very wrong description of what integers
> actually are. But that's what they wanted to find: solutions to the FLT
> equation within these given axioms.
I hope Dik reads your above for he has the same mindset as you. Both of you believe and think and
expect that someone such as Fermat is doing real mathematics when he uses a flawed set and that it
still remains as legitimate worthwhile mathematics. Do we call the persons who applied leaches or
drained people of blood in the Middle Ages as doctors and people of medicine? We do not call them
medical doctors because they were not using science, likewise Fermat was no longer a mathematician
when he tried to apply finite-integers to a^n+b^n= c^n, or at best Fermat was a wrongheaded
mathematician.
>
>
> Solution to the same equation in any other axiom system is totally
> irrelevant. Especially if, as you say, some solutions to this equation
> in p-adics are already known.
You, like Dik, do not understand that axiom systems and definitions can be flawed and wrong. And
that most systems and definitions have flaws and blemishes that future people have to revise and
correct.
Dik T. Winter
> Yes, that should read reals compared to proper subsets of the reals.
> It wasn't some obscuring statement that the reals were a proper subset
> of themselves.
Pray provide some of the context. I nearly never read messages that do
not contain context.
Dik T. Winter
> Dik T. Winter wrote:
> > In article <1117568478....@g44g2000cwa.googlegroups.com> anzau...@hotmail.com writes:
...
> > > So, is his point that N-adics are better representations of integers
> > > that the interpretation that we, including Fermat, usually use?
> >
> > No his point (and AP's) point is that their representations of integers
> > are the only possible interpretations, and that our interpretations are
> > just plain wrong.
>
> What argument does he use to explain this?
I do not remember the argument EEE uses. It is a long time ago that he
posted in this group. But basically AP uses the same argument as Tony
Orlow is now using, i.e. adding 1 continuously will lead to an infinite
number, or something very similar. I.e. a flawed argument.
> > > And why would anybody care about Fermat Last Theorem as applied to
> > > N-adics? Fermat sure didn't. He didn't even know about them.
> >
> > But they argue that Fermat did not know what the naturals actually
> > are.
>
> That's hardly the point. Fermat found the FLT statement in some book.
> Both Fermat and the author of that book had a very particular view of
> what they meant by the word "integer" in the formulation: "Does
> X^n+Y^n=Z^n have any integer solutions for n > 2?" and the axioms they
> satisfy.
Yup. Do not argue about that with me.
> Their axiom set may be a very wrong description of what integers
> actually are. But that's what they wanted to find: solutions to the FLT
> equation within these given axioms.
But the point is that they argue that the axiom set actually defines
something different. Now, try to argue with them about that.
a_plu...@hotmail.com
"An extremely strange metaphor. Moreover, I do not understand the
third
sentence."
Let me explain it this way. Your idea of definition or axiom in
mathematics would be similar to the position that every person trialed
by jury in a court of law and found convicted of murder is 100 percent
guaranteed to be the true murderer of the event in question. Your idea
of a mathematical definition or mathematical axiom is tantamount to the
idea that a conviction is the underlying event sequence. Your idea of
definition and axiom leaves no room or possibility that the definition
captures only part of the reality and that the definition can be flawed
and room for revision.
To me, a conviction in a courtroom often captures the reality of the
events in question but sometimes the courts make mistakes or they
capture part of the true events. And some convictions were of innocent
people.
So to Dik, definitions and axioms in mathematics have this glow of
eternal absolutism, unchanging and forever useable. To me, a definition
or axiom of mathematics is a attempt to capture underlying and hidden
reality. Sometimes we capture most of the truth about an item or issue
and the definition remains unchanged and unrevised for a long time.
To Dik' s way of thinking in the year 5005 classrooms would be teaching
Number theory with Natural Numbers as finite-integers as equally valid
to a classroom teaching where Natural Numbers are Adics. To me,
finite-integers are flawed and the classroom of 5005 would only be
teaching NaturalNumbers are Adics. We no longer teach in science
classrooms that witches are real and fly around on broom sticks,
likewise finite-integers are fairy tales of today.
a_plu...@hotmail.com
Archimedes Plutonium wrote:
> Wed, 1 Jun 2005 00:29:10 GMT "Dik T. Winter" wrote:
>
(snipped)
> > But they argue that Fermat did not know what the naturals actually
> > are. AP has shifted from the 10-adics to the adics, encompassing all
> > the n-adics. But he does not even give a reasonable rule how to add
> > a 2-adic to a 3-adic.
>
(snipped)
Dik, perhaps I can give some crude way of defining addition in the
All-Adics. After posting earlier today about symmetry of Adics to Reals
where 2.333.... in Reals becomes a Adic of .....333333.2 and
remembering back to when Abian (you remember Abian) was talking about
what he called the Plutonium Integers as Abian was trying to help me
get a addition in Adics.
Anyway let me give symmetry a test trial run. Say that Dik offers me a
2-adic of say .....101010.1 and then Dik offers a 3-adic of
....121212.1 and then Dik asks me to add those and what I come up with
for a rule of addition in various different Adics.
So my answer, roughly would be to rely on symmetry and play a game such
as what Abian did in the 1990s with Plutonium Integers. And I think
Abian if alive today would smilingly approve.
So, Dik, what I do is to take that 2-adic o ...101010.1 and that 3-adic
of ...121212.1 and flip them around or flip them over into a Real of
1.010101... and 1.212121.... and add them together to achieve
2.22222..... and then flip it back around or over into an Adic of
....22222.2 and ask myself is this sum still a 2-adic or 3-adic?
Sometimes the summation of a 2-adic with 3-adic maybe a 4-adic.
I remember Karl Heuer saying that addition across various different
adics such as 2-adics and 3-adics and multiplication are well-defined
already and that I should not, or others have trouble with defining
adic addition and multiplication.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
Pope Urban Daycamp Thirty Minus XIII
thus quoth:
Dik T. Winter
> So, Dik, what I do is to take that 2-adic o ...101010.1 and that 3-adic
> of ...121212.1 and flip them around or flip them over into a Real of
> 1.010101... and 1.212121.... and add them together to achieve
> 2.22222..... and then flip it back around or over into an Adic of
> ....22222.2 and ask myself is this sum still a 2-adic or 3-adic?
> Sometimes the summation of a 2-adic with 3-adic maybe a 4-adic.
Ok, I give you the 2-adic ...00000010 and the 3-adic ...00000010 and ask
you to tell me what it is. You come back with ...0000020. But now I
say the value of that 2-adic is 2, and the value of that 3 adic is 3,
so their sum is 5, what you came back with is not 5 in any of the n-adics.
Or still better, I have two 2-adics:
a = ...111111110 and b = ...111111100,
and two 3-adics:
c = ...111111110 and d = ...111111100.
(Note: a != c). I ask you to add, a and c and you come back with:
e = ...222222220,
the same with b and d and you come back with:
f = ...222222200,
I ask you to subtract f from e and you come back with 20.
To see that this *must* be wrong consider the following actual values:
a = -2, b = -4, c = -3/2, d = -9/2. a+c = -7/2, b+d = -17/2,
(a+c)-(b+d) = 5.
None of the numbers you return has the correct value in any of the
adics.
> I remember Karl Heuer saying that addition across various different
> adics such as 2-adics and 3-adics and multiplication are well-defined
> already and that I should not, or others have trouble with defining
> adic addition and multiplication.
Yes, indeed, because the 6-adics are the direct sum of the 2-adics and
the 3-adics. So the 2-adics and 3-adics can be embedded in the 6-adics
as direct sum, however, there are two different embeddings when you look
in base 6 notation (I think). And converting a 6-adic in base 6 notation
to to direct sum notation is, eh, also quite messy. Also the properties
of the operations will be quite a surprise. In direct sum notation, the
6-adics are the pairs (p, q) where p is a 2-adic and q is a 3-adic,
and so the first component follows 2-adic operations and the second
follows 3-adic operations. Addition and multiplication are defined
as component-wise addition and multiplication.
Any 2-adic a can be written as (a, 0) and any 3-adic b as (0, b),
so their sum is (a, b). The 1 in the 6-adics is (1, 1), which,
surprise, is the sum of the 1 in the 2-adics and the 1 in the 3-adics.
The product of *any* 2-adic with *any* 3-adic yields 0. You may
map the 2-adic 1 to ...152221350213 in the 6-adics and the 3-adic 1
to ...403334205344 in the 6-adics (or the other way around, I think,
but perhaps there is only one way to map, in that case I do not know
which of the two you need).
But I think you want an embedding that is value preserving for the "finite"
adics (i.e. for the finite adics it is just a change of base). But this
will show that the conversion of the "infinite" adics is problematical.
(I put finite and infinite in quotes because there is no consistent order
relation in any of the n-adics.)
Consider the 2-adics:
a = ...111111110 and b = ...111111100,
in 6-adics there notation is:
a = ...555555554 and b = ...555555552.
And the 3-adics:
c = ...111111110 and d = ...111111100.
become:
c = ...555555554.3 and d = ...5555555551.3.
So
a+c = ...55555552.3 and b+d = ...55555543.3
subtracting:
(a+c)-(b+d) = 5
messy, but correct. But it is not really a change of base.
Dik T. Winter
> Wed, 1 Jun 2005 00:29:10 GMT "Dik T. Winter" wrote:
> > In article <1117568478....@g44g2000cwa.googlegroups.com> anzau...@hotmail.com writes:
> > > Dik T. Winter wrote:
> > ...
> > > > If you define them as infinite sequences of
> > > > digits with a period placed firmly on the right, you are close to the
> > > > 10-adics (and that is what AP uses, and I think EEE also uses that, but
> > > > I am not sure, he has never been very prolific).
> > ...
> > > So, is his point that N-adics are better representations of integers
> > > that the interpretation that we, including Fermat, usually use?
> >
> > No his point (and AP's) point is that their representations of integers
> > are the only possible interpretations, and that our interpretations are
> > just plain wrong.
>
> Some of that is correct. But it dodges the aspect that Dik has an
> interpretation of what the Natural Numbers are.
Not an interpretation. I use the definition.
> I just know the Natural
> Numbers are not the Finite-Integers.
In that case you use a different definition.
Dik T. Winter
> Tue, 31 May 2005 12:12:20 GMT "Dik T. Winter" wrote:
> > Yup. Let A be the set of ducks with fourteen legs. See, I have defined
> > something. And I can work with set A.
>
> I don't think you really believe that. Believe, even, to the extent that
> the empty set is nonreality.
> This is a problem that Dik and many others have-- they think that
> mathematics is above, beyond and superior to physics by the allowance of
> make-believe or imagination as workable reality. In a sense, they elevate
> mathematics as above physics, when in truth mathematics is begot from
> physics and is a minor subset of physics.
Oh. In what way is cryptography related to physics?
anzau...@hotmail.com
> 31 May 2005 17:52:33 -0700 anzau...@hotmail.com wrote:
>
> Symmetry says that Reals are infinite strings rightward with a finite portion leftward such as
> 2.333..... If Natural Numbers are finite-integers then the symmetry would be broken.
The symmetry of WHAT would be broken?
What about finite fields? Should all of them be infinite just for
"symmetry"?
> Mathematics
> would be asymmetrical. To complete the symmetry then Mathematics would be a Dual system, a duality
> with two number systems that are symmetrical.
Why do people want to make integers and reals dual to each other?
Integers aren't even a field!
And what do you mean by "dual"? Is there a natural bilinear form on
their pair? Or is it just a trivial and algebraically unimportant fact
that their digital representations look like opposites?
How about all other popular fields like complex numbers, rationals, the
algebraic closure of rationals, etc? Should there be a lot of other
dual pairs among them?
> So if we have a Real such as 2.333..... then we have
> its dual other particle of ......3333.2. Thus, to make mathematics algreba a symmetrical system then
> the compliment of Reals is Natural-Numbers equal to Adics.
Is there any signiphicant homomorphism between the two that reflects
their basic structures?
> Finite-Integers were just a crude and
> flawed system that captured only a tiny bit of what the Natural-Numbers really are.
>
By "Natural-Numbers", you mean what Dik would call "N-adics". Right?
>
> > That's hardly the point. Fermat found the FLT statement in some book.
> > Both Fermat and the author of that book had a very particular view of
> > what they meant by the word "integer" in the formulation: "Does
> > X^n+Y^n=Z^n have any integer solutions for n > 2?" and the axioms they
> > satisfy.
> >
> > Their axiom set may be a very wrong description of what integers
> > actually are. But that's what they wanted to find: solutions to the FLT
> > equation within these given axioms.
>
> I hope Dik reads your above for he has the same mindset as you. Both of you believe and think and
> expect that someone such as Fermat is doing real mathematics when he uses a flawed set and that it
> still remains as legitimate worthwhile mathematics.
First of all, Fermat's set may be a totally wrong representation of
what integers REALLTY are. But why is it "flawed"? Why is it wrong to
try to solve a polynomial equation in some abstract ring? If he tried
to solve his equation in some finite ring, why would that be "flawed"?
He has some ring. So he wants to solve an equation in it. I see nothing
flawed in his desire.
The point is that THERE IS NOTHING MATHEMATICALLY OR PHYSICALLY
SIGNIFICANT in solving the equation
X^n + Y^n = Z ^n.
The only reason why people have devoted themselves to solving it over
the set of "Fermat "naturals"", is because Frmat himself tried to solve
it over his set of "Fermat "naturals"" and porbably failed.
These are not "integers"? Fine. Call them "fermats". So, the equation
X^n + Y^n = Z^n over the ring of "fermats" has no solution for n>2.
Does it have solutions over other rings? Yes, it does. For example, I
already gave you the pretty much full description of the continuum of
solutions over reals. Woopty doo!
You say true integers are dual to reals under some isomorphism? Great.
That means I just gave you a continuuum of solutions to FLT in your
integers. Without breaking a sweat. And any high school senior could
have don the same.
You say true integers are more like N-adics than like reals? Fine. Dik
tells us that solutions to FLT over p-adics have been known for a long
time.
So, what's EEE's or your original contribution to Fermat's Last
Theorem.
> Do we call the persons who applied leaches or
> drained people of blood in the Middle Ages as doctors and people of medicine? We do not call them
> medical doctors because they were not using science, likewise Fermat was no longer a mathematician
> when he tried to apply finite-integers to a^n+b^n= c^n, or at best Fermat was a wrongheaded
> mathematician.
>
If anything, Fermat was probably "a wrongheaded mathematician" when he
tried to solve a^n+b^n= c^n at all. A totally inconsequential thing to
solve.
But Fermat was not a wrongheaded mathematician. He was a man who liked
puzzles. He saw some ring and he saw a polynomial. So, he tried to
solve this polynomial over that ring. Why? Becuase he liked difficult
posers (puzzles) and solving that particular polynomial over that
particular ring seemed like a fun and challenging intllectual puzzle.
If he had a different ring, it wouldn't have been an interesting puzzle
at all. Even if the lallter ring is the ring of "true" integers. Fermat
would have found solutions in 5 minutes and have forgotten all about
this puzzle.
>
> > Solution to the same equation in any other axiom system is totally
> > irrelevant. Especially if, as you say, some solutions to this equation
> > in p-adics are already known.
>
> You, like Dik, do not understand that axiom systems and definitions can be flawed and wrong. And
> that most systems and definitions have flaws and blemishes that future people have to revise and
> correct.
>
What you don't understand is that nobody gives a serious flying damn
about FLT as applied to any ring other than the "flawed" ring of
"fermats".
And if they did, they would have found solutions in 2 minutes flat. So,
what's YOUR original contribution to the "FLT" puzzle, as "F" has posed
it to us?
Ross A. Finlayson
Hi Dik,
I was asking you how to construct that set of natural numbers with the
varying density.
The set of all sets is its own powerset. In a set theory, there are
only sets and quantification over sets implies a universal set or set
of all sets. ZF is inconsistent. That aside, a set of all sets is a
counterexample to that there exists no bijection between a set and its
powerset because the set of all sets is its own powerset. By assigning
basically an index or ordinal to each set, via well-ordering, that set
is as well orderable by what is generally called the set of natural
integers, via transfer.
A question about the real orders is how to well-order them, where it is
accepted that that exists and nobody has an example in the "standard"
real numbers. One notion that many agree upon is that they have some
natural well-ordering. Because the set of reals is a set of points
where trichotomy holds, it's possible to consider basically those
points in order, not via their Cauchy/Dedekind/etcetera formation from
basically the field of rational numbers but via the alternate
perspective as a contiguous, in being continuous, sequence of points,
or recovery of the Newton infinitesimal, the fluxion. Via essentially
exhaustion, analytical results are discovered to show that something
like an infinite point set not dense in the reals can have a positive,
non-infinitesimal integral. Where that is so, then it should be
possible to exhibit a variety of empirical applications.
I think a theory should prove everything that's true and disprove
everything that's false.
Ross
Barb Knox
"Ross A. Finlayson" <r...@tiki-lounge.com> wrote:
[snip]
>The set of all sets is its own powerset.
Not so. For example, in NFU (which does have a universal set V), 2^V is
clearly a set all of whose elements are SETS (namely all the subsets of V),
but V itself also contains ur-elements (i.e. non-sets). Thus V is strictly
larger than 2^V.
[SNIP]
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
Ross A. Finlayson
Hi Barb,
I think the problem with that is in the parallel between ur-element,
and set, and proper class, and set.
That's why in the null axiom set theory the ur-element is a proper
class, and a set, the dually minimal and maximal element.
I think it's that way, because otherwise it's easier to derive
contradictions, and those are not good things to have in the theory.
Quantification over sets implies a universal set, in a set theory, of
sets.
I think that's the first time I've ever heard anyone say that the
powerset was smaller than the set. That's kind of interesting. I
might use that where I equate Ord to negative one to basically derive a
mechanical predecessor operation for "negative" ordinals as I was
discussing a mechanical successor operation, for ordinals.
Ross
Archimedes Plutonium
Thanks for the above Dik, and I have been playing around with them also today but abandoned the
scheme. I realized I had went down that route, that scheme back in 1994.
At the moment I am firmly impressed with the idea that there is no addition or multiplication
between, say the 2-adics and 3-adics. Firmly impressed that the entire 2-adics and 3-adics form a
sphere where each 2-adic is a point in a 2-adic sphere, likewise 3-adics. Every adic forms a sphere
and some are nested and nonintersecting and some are intersecting such as the 3-adics and 6-adics.
In Reals the essence is all about continuity and with continuity is derived addition and
multiplication but with Adics the essence is discreteness. There is intersections of various
different adic points but there is no addition and multiplication of a isolated 2-adic point with an
isolated 3-adic point. There is however addition and multiplication of we take the entire set of all
of the points of 2-adics (2-adic sphere) with the entire set of all the points of 3-adics (3-adic
sphere). And this is the 6-adics. There is no clear addition and multiplication for individual
points of 2-adics with points of 3-adics but there is multiplication and addition for the entire
complete set of 2-adics and 3-adics.
Note: I see the Reals not as forming a Euclidean plane but instead the Reals form a Lobachevskian
hyperbolic surface such as that of a trumpet musical instrument.
Dik, the Adics are independent and so totally different from the Reals that we cannot expect to
muster the Adics into some conforming or behaving like Reals.
In physics, particles cannot be conformed and behave like waves for the two are distinctly
different, same with Reals versus Adics.
So I think what addition and multiplication boils down to for 2-adic points and 3-adic points
becomes more of a question of intersection rather than addition sum or multiplication product. So
when we add a 2-adic point with a 6-adic point or multiply a 5-adic point with a 10-adic point we
get results because there is a great amount of intersection in these adics. So I think the concept
of add and multiply across different adics becomes more of the concept of intersection across
different adics.
The Reals are like dogs and dogs bark but then when we get a cat we do not expect the cat to bark
(addition) or hunt in packs (multiplication).
The Reals are to mathematics what the Wave is to physics and the Adics are to mathematics what the
Particle nature of reality is in Quantum Mechanics. So trying to add and multiply 2-adics with
3-adics is asking how to make a Particle behave semi-wavish.
Archimedes Plutonium
> Archimedes Plutonium wrote:
> > 31 May 2005 17:52:33 -0700 anzau...@hotmail.com wrote:
> >
> > Symmetry says that Reals are infinite strings rightward with a finite portion leftward such as
> > 2.333..... If Natural Numbers are finite-integers then the symmetry would be broken.
>
> The symmetry of WHAT would be broken?
>
> What about finite fields? Should all of them be infinite just for
> "symmetry"?
>
> > Mathematics
> > would be asymmetrical. To complete the symmetry then Mathematics would be a Dual system, a duality
> > with two number systems that are symmetrical.
>
> Why do people want to make integers and reals dual to each other?
> Integers aren't even a field!
>
> And what do you mean by "dual"? Is there a natural bilinear form on
> their pair? Or is it just a trivial and algebraically unimportant fact
> that their digital representations look like opposites?
>
> How about all other popular fields like complex numbers, rationals, the
> algebraic closure of rationals, etc? Should there be a lot of other
> dual pairs among them?
Focus on one important issue in a post, not 50,000 side questions.
You need to study the history of physics more than you need another detail of mathematics. In fact, at
this moment in human history with the Atom Totality theory amongst us which says that mathematics is a
tiny subdepartment of physics, that a master in mathematics or an expert mathematician knows more about
the history of physics than the history of his own subject of mathematics. In other words, as of
post-1990 a literate mathematician knows the history of physics and an illiterate mathematician does not
know the history of physics.
Mathematics prior to 1990 thought that mathematics was bigger than physics. After 1990 with the Atom
Totality theory it is clear that physics is king and mathematics is a mere chamber maid.
Prior to 1990 the history of number systems was that of extensions. First the counting numbers were
discovered then the Rationals became an extension of the Counting Numbers then the Irrationals a further
extension then Reals then Complex. All extensions. But the history of Physics is not one of extensions is
it. Everything that composes physics was there from the beginning. Forces are not extensions of one
another. EM is not an extension of gravity, nor is weaknuclear and strongnuclear. Quantum theory is not
an extension of Maxwell theory.
Symmetry is the working tool of physics. It was symmetry that discovered Special Relativity in that a
moving magnet in a coil causes current and a moving coil around a stationary magnet causes current. It
was symmetry that discovered Maxwell theory in that Ampere law was broken symmetry. It was symmetry that
discovered the Dirac Equation.
The Numbers of mathematics have existed all along and discovered not as some fabrications of extensions
that are useful.
The Symmetry I speak of is that the highest symmetry in Physics is Particle to Wave.
So do we have a symmetry of particle to wave in mathematics? Yes if we say that there are 2
possibilities-- (1) infinite rightward string, finite leftward portion (2) infinite leftward string,
finite rightward portion.
The Reals fit (1) infinite rightward string. But Natural-Numbers equal to finite-integers are a
nonentity, in fact a fiction. What fills out the symmetry are the Adics because they are infinite
leftward strings with finite rightward portion.
And since in Physics all that really exists is particle and wave, tells us that mathematics really has
only two number systems and every thing else in mathematics is a phony and fiction. The difference
between Finite-Integers and Adics is the difference metaphorically between a fire breathing dragon and a
dinosaur that once actually roamed Earth. One is a fiction of the mind and the other is a genuine
reality.
It is quite true that we have reached a moment in the history of science where a mathematician who
aspires to be a master mathematician is better off learning about the history of physics than learning
about the history of mathematics. And the day is rapidly approaching where a mathematician without a full
knowledge of the history of physics is incompetent.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
a_plu...@hotmail.com
plane. I see this as a consequence of the Symmetry of Reals versus
Adics. Leftward infinite strings symmetrical to rightward infinite
strings. That would mean that the Reals cannot be the intrinsic points
of Euclidean geometry but must have a "built in curvature". The Adics
start with 1 and end up coming back around in full circle to
.....999999. The Adics have a built-in curvature and they form a nested
spheres. So the All-Adics forms a Riemannian geometry. That leaves the
Reals as having also a built-in curvature only it is the reverse of the
Adics in that it is a negative curvature. So the Reals must form a
Lobachevskian curvature geometry and the Reals must be the intrinsic
points of the Lobachevskian geometry. Euclidean geometry is merely a
point geometry that involves one point-- the zero point and otherwise
Euclidean geometry is a defunct geometry. I am searching for a better
term to describe Euclidean geometry besides "defunct". Physics has a
concept of higher dimensions being folded and curled up and hidden and
of non-use. Some term along those lines.
Anyway, is there evidence that the REals have intrinsic curvature and
that of Lobachevskian? Several years back I said that the Riemann
Hypothesis when reconfigured to include the Adics as the Natural
Numbers would have the Natural Numbers on the "1/2 Real Line". But the
Adics have a positive curvature themselves and are not straight line.
So the Riemann Hypothesis was never really proven as a mathematical
theorem because it was never revised with the idea that the
Natural-Numbers were Adics and that the REals themselves did not form a
Euclidean Plane but a Lobacheskian hyperbolic surface of negative
curvature.
So if we revise the Riemann Hypothesis with the Adics as the Natural
Numbers and with the Reals as a Lobachevskian surface then we can
garner a endresult much like what Bernard Riemann imagined in the 19th
century. I say much like in that the positive curvature of Adics
unbends the negative curvature of Reals on the 1/2 Real line to give us
the impression that the Reals have the Adics on the 1/2 Real line.
But the key point of the above is that the Reals have an intrinsic
negative curvature.
And another area of mathematics that suggests the Reals are
Lobachevskian geometry is the mess of the Calculus with its thousands
of definitions of integrals and differentials. This is a mess and
nightmare. And the reason it is a mess is because every mathematician
assumes the Reals are Euclidean geometry. But if the Reals are seen as
negative curvature of a Lobachevskian surface such as a trumpet shape
then the Calculus can dispense with its thousands of different
definitions of integral and differential and have just one definition.
What is creating he mess in Calculus is that the assumption of the
Reals forming Euclidean geometry was incorrect. The Reals have a
intrinsic Lobachevskian geometry.
Now there maybe other areas of mathematics that also imply the negative
curvature of the Reals, but the above are the best two examples I can
think of at this moment.
anzau...@hotmail.com
>
> Focus on one important issue in a post, not 50,000 side questions.
Look, you have manged to avoid answering any of my direct questions.
But I think understand.
You guys have the agenda of promoting your numbers (let's call them
"Adics") as the correct way of viewing "integers".
What you need is world-wide publicity for your ideas. So you chose the
most notorious puzzle in mathematics FLT to promote your views.
Because in your Adics axiom system FLT has solutions that even a
college freshman can find in 5 minutes or less, you have been
bombarding lay newspapers with sensationalist claims that you have
"simple" solutions to FLT and "proofs" that Wiles' proof is "wrong".
A very noble way to promote one's world-views. But that' a side note.
Now let's talk about the essence of your world-view.
> Symmetry is the working tool of physics.
Look, when I was in 6th grade, a mathematician told
us that symmetry is the basis of math and that all main areas of math
have been shown to be derivable from symmetry considerations. So I
still kind of believe him. So I am open to your views.
But for a symmetry (isomorphism) to be important, it shouldn't just
concern itself with the two structures ahving the same number of
elements but with the algebraic and topological structures of the two
systems being isomorphic.
You seem to claim that there exists a very deep and profound "duality"
and "symmetry" between real numbers and your Adic numbers.
The symmetry that I see is that when you try to write down a real
number, you use a decimal or N-cimal representation of it as an
infinite sequence of digits. Similarly, when you try to write down a
10-adic or an N-adic, you also use a similar infinite sequence of
digits.
This shows that the two sets have the same cardinality. Woopty doo!
But to me, this isomorphism is highly superficial. To me, the decimal
(or to any other base) digit representation of real numbers isn't the
only way of representing reals. May not be the best way of representing
rationals if your goal is to perform fast additions, subtractions,
multiplications and divisions. It is certainly easier to do that in the
"fraction" form
1/7 + 1/3 = 10/21
than in the decimal form
0.333333333333333... + 0.142857142857142.... = 0.476190476190476.....
For your isomorphism between reals and adics to be important, it must
preserve a lot of algebraic and topological structure. Does it? What
properties are preserved under your isomorphism?
Please reply to my concrete questions rather than giving me yet another
general sermon.
anzau...@hotmail.com
discrete particle at the same time.
So, you seem to suggest that there exists a physical phenomenon that's
both a real number and an Adic number.
What is this object?
To me, symmetry and isomorphisms are most important. As exemplified by
Galois theory of field extensions. Field extensions are an ultimate
expressions of symmetry. So are algebraically complete fields.
Reals are not algebraically complete.
Are Adics a field to begin with? And if they are - are they
algebraically complete?
If neither system is algebraically complete, then neither is the right
thing to study when concerned with symmetry.
Th natural things to study would be the complex numbers and the field
and algeraic closure of your Adics.
Actually, judging by your post, you cannot even add two Adics
together....
And if you can't add them together, why do you want to solve a^n + b^n
+ c^n in them?
So, if you can't do any arithmetic or algebraic operations, what CAN
you do with your addics? Look at them? Count them to your heart's
delight?
> Prior to 1990 the history of number systems was that of
> extensions. First the counting numbers were
> discovered then the Rationals became an extension of
> the Counting Numbers then the Irrationals a further
> extension then Reals then Complex. All extensions.
Sure they were extensions. When we had only Naturals, we couldn't
subtract. So we added 0 and negatives. Got us a nice commutative ring.
But then we couldn't divide. So we added fractions. Now we got us
rational. But we couldn't solve polynomials. So we added all polynomial
roots. Got us the algebraic closure of rationals. That's the structure
that has every imaginable kind of symmetry in it. If we were to use
your criterion of beauty and symmetry, that's the REAL natural numbers
that you are looking for. Pun not intended.
But that's not what you want. Your problem with the traditional
integers is not that you cannot divide them. No. Your problem is that
you CAN add in them. And you don't like that! You want a structure
where you can't add nor divide nor subtract nor multiply. Making sure
that you can't do any algebra or arithmitic in your system, is yuor
idea of symmetry.
And given that you can't add, you then want us to solve the polynomial
equation "a^n added to b^n equals to c^n".
How piquant!
PS: when you talk about "n", you also mean "n" to be a Adic, don't
you?
Shmuel (Seymour J.) Metz
05/31/2005
at 12:41 PM, anzau...@hotmail.com said:
>So, is his point that N-adics are better representations of integers
>that the interpretation that we, including Fermat, usually use?
No. He's not proposing a new interpretation, but, rather, a new
definition.
>If so, has EEE discovered new facts about N- and p- adics that people
>didn't know before him? Or did he reinvent the wheel?
No. It's the usual crank behavior of solving a different problem from
the one that he claims to havwe solved.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
Pope Urban Daycamp Thirty Minus XIII
of ordering might be the same as his use of successor?
thanks for explaining AP's (or EEE's?) "counterexample,"
although I should have gotten that, for n=3.
did Fermat have 10-adics, and
did Euclid give a fake proof of the primality of infinity?
thus:
the oilcos are in front of the action
on trading credits for carbon dioxide,
which has been online since Feb.12;
why does the Liberal Media
Owned by Conswervatives not say,
what effect this is having on prices?
the Iraq War isn't to *control* the oil, but
to enforce the industry's "Hubbard's Peak," by a)
burning a lot of it for a war, and b)
enflaming the jihadis.
thus:
his conviction for "plumbering" the Weather Underground,
makes him sound like a Wanabi; how convenient
for Woodstein, to butter him up before & after a stroke ...
maybe, the idea *gave* him a stroke.
Linda Tripp was a God-am Nixon mol!
thus:
as plausible as it is, esp. given the RMN quote, below,
I have to say that it's covering for RNC Chairgeorge, and
probably Sir Henry, and possibly Katie Graham
(in drag; according to _AtPM_,
both Strep Throat and Graham drank whisky & chainsmoked,
and this was the *only* physical dyscription
that I could find; now, I read in the papers that
Felt liked whisky, period; I also read that
Liddy and Buchanan scoffed at the idea of Felt, but
none of their reasons, Why!)
thus:
it was Katherine Graham;
this could become a musical. well,
musical Chairs, as with
http://tarpley.net/bush12.htm.
thus quoth:
Nixon: And he has to go, of course. Because it's now obvious,
you see we now have these reports, an interesting thing, I
have these reports, Al, I got them directly, you know, from,
he was leaking to Time magazine, from their attorney. This
was, oh, months ago before I sent Pat Gray's name up. I said,
`Pat, I want you to check these leaks.' He said, `Oh, they
couldn't be from the bureau.' I said, `Yes they are. Some
are.' And I said, `We have on very good authority they're
there from Felt.' `Oh, it couldn't be from Felt.' I said,
`Dammit ... you ought to give him a lie detector test.'
You know, I was very tough. `Oh, we can't do that,' he said.
He said, `But I vouch for Felt.' ... I also raised it with
[former Atty. Gen. Richard] Kleindienst. Kleindienst vouched
for Felt. That shows you how clever Felt is. My point is that
this was three, four months ago that we were onto the son of
a bitch. We had something, we had a lead and it shows you
how important it is that when you and I get leaks like this
in the future, we don't disregard them.
Richard Nixon
Source:White House Tapes, May 12, 1973
http://home.att.net/~howingtons/gop/nixpg.html
thus:
The reason why the Watergate scandal escalated into the overthrow of
Nixon has to do with the international monetary crisis of those years,
and with Nixon's inability to manage the collapse of the Bretton Woods
system and the US dollar in a way satisfactory to the Anglo-American
financial elite.
http://tarpley.net/bush12.htm
thus:
uh, re #2, because the Plumbers were started by Sir Henry
of Kiss.Ass.?... see http://tarpley.net/bush12.htm.
your #8 is interesting, two!
thus:
the paper that I read, yesterday,
had no confirmation from Bernwood; only
from Woodstein, who has been playing this crap
for decades. one of those papers, yesterday, also said
that George was at the UN, when he was at the RNC
-- and on the God-am Cabinet;
see http://tarpley.net/bush12.htm --
while it also noted that the money came from Mexico;
that was the check!
so, What, if it happenned to have been
in Liddy's hot little hand (or was that Hunt's?) ??
Laura Bush murdered her boy friend wrote:
> $$$$$$$$$$$$.
St. Nick Bourbaki writ via Ouija Board (tm):
"Death to the trigon!" --J.Duodieunne (er, spelling?)
Long live the tetrahedron! --myself & Brian
--ils ducs d'Enron!
http://tarpley.net/bush7.htm
http://members.tripod.com/~american_almanac
http://larouchepub.com/other/editorials/2005/3221transact_tax.html
a_plu...@hotmail.com
learning the history of physics rather than some more dull details of
mathematics, because for instance, you do not understand the basic
difference between even that of what-is-duality and what-is-symmetry
and you are even confused as to what isomorphism is and how it is
different from duality and symmetry.
goodbye
Archimedes Plutonium
anzau...@hotmail.com
>
> As I said in my previous post to you, that your time is better spent on
> learning the history of physics rather than some more dull details of
> mathematics, because for instance, you do not understand the basic
> difference between even that of what-is-duality
>
To me, duality is when the vectors in a bilinear function can be
thought of as linear functionals acting on other vectors, and vise
versa. And when vectors in a linear space L can be thought of as linear
functionals that act on the linear space of fuctionals that act on L.
>
> and what-is-symmetry
>
To me, symmetry is the group of automorphisms.
>
> and you are even confused as to what isomorphism is
An isomorphism is a bijective mapping between two structures that
preserves the algebraic, topological and other properties of these
structures.
>
> and how it is
> different from duality and symmetry.
>
Duality is an example of an isomorphism. Symmetry the group of
isomorphisms between A and A itself (i..e, automorphisms).
In any case, I was the only person here who was trying to understnad
your points.
You have just insulted the last person who was trying to listen to you.
Your loss not mine.
And you wonder why others don't accept your views?! Try not to insult
people for a change, ignoramus.
> goodbye
good riddance
Shmuel (Seymour J.) Metz
06/02/2005
at 11:51 AM, anzau...@hotmail.com said:
>Look, you have manged to avoid answering any of my direct questions.
You will never get a sensible response out of AP. Google is your
friend.
>But I think understand.
Don't count on it. AP is a net kook from way back.
Minus XVII
maybe we should all be studying continsuously variable fuzzee logeek.
Minus XVII
to grok it like his namesake
(town in N. California, where they may have deposits
of his other namesake .-)
PS: no; otherwise Euclid's proof was a fake!
thus quoth:
georgie
wrong. If you aren't capable of understanding the simple problems, you
don't expect us to show you the more complicated problems, do you?
georgie
generate it.
georgie
georgie
> is true and 3 more proofs that it's false. Curiously, none of the
>latter 3 proofs invlves any couterexamples.
Don't forget about the kook's with the single simple proof of FLT.
That list must, of course, include Fermat because he never proved FLT.
mensa...@aol.com
But he never claimed that he did in public, did he? Wasn't his
claim a personal note to himself found after his death? Wasn't it
others who are to blame for taking that note seriously (based on
the soundness of his body of public work).
That hardly makes him a kook.
georgie
> unsolvable to date.
There's a difference between proving a conjecture like FLT false, and
it being undecidable.