PLAUSIBLE IDEA FOR WHAT WAS THE PROOF OF MR FERMAT
This text was a letter on May 20th 1995 to Mr Andreas Blass,it  is revised 2
april 1998
There is a link between the negation of the Axiom of choise and Fermat’s
Last Theorem in the following manner  (existence of infinite products of
integers) :
zn = xn + yn implies z.z……z……= x.x….x…….+ y.y….y……
extrapolation principle ?)
First Possible case : x,y,z is even, divide by 2
Other possible case : x,y,z is odd xn + yn = zn would not hold
First possible case : z is odd assuming xn + yn = zn means either x or y is
even.
so either
z.z….z… = x.x….x… +   y.y….y…
exists        exists        Does not exist
or
z.z….z… = x.x….x… +            y.y….y…
Exists      does not exist     exists
Second possible case z is even so :
z.z….z… =                   x.x….x…       +   y.y….y…
Does not exists        exists                exists
Something which exists cannot be equal to something which doesn’t.
This proof takes place in a Frankel Mostowski model in which x.x….x….
exists for x odd and not for x even (that model would be the real
mathematical universe).
The particular case of the axiom of choice (the case where sets are all with
same number of  elements n )  Cn for n odd is the  particular case of the
Axiom of Choice which holds and the negation of Cn for n even ,we use as
well.
The proof would have been found by Fermat himself — he liked the end but
at
the beginning he saw a gap so he did not make an error but found after
awhile there
is a gap. That would be why he did not insist on the proof.
But, maybe someday, someone will find how to fill up the gap or will be able
to explain that it is not really a gap,if we publish the hint we have got.
Besides, the Axiom of Choice is to be shown not to be true in the
mathematical reality but only a particular case of the axiom.
Please let me remind you that,for infinite families of sets of same number
of elements n
and using the negation of the particular case of the axiom of choice (Cn,
there is no bijection between A1xA2x……xAnx……. and
B1xB2x…….xBnx…….as the first cartesian product could be equal to the
empty set and the second product to a set with an infinite number of
elements, and so ,we see why n.n….n….is not well defined  (does not
exist ).
So,our knowledge of the infinite is to be tryed.
For other comments, please check  web site www.tunisinfo.com/jebara
For sending an e-mail message use hightech…@planet.tn  (it is necessary to
write "Attention of  Mr Ben Jebara "). And please give your fax number or
postal adress. "
Bibliographical References :
Sierpinski : 1918 L’axiome de M. Zermelo et son role dans la theorie des
ensembles et l’analyse. Bulletin de l’Academie des sciences de Cracovie
,Classe des Sciences Mathematiques.Serie A
Lautman : Essai  sur l’unite des mathematiques collection 10-18, 1977
Russel and Whitehead : Principia Mathematica,1910,Cambridge University
Press.
Ben Jebara Adib
Apt. F3 Residence Badr Manar 1    2092 TUNIS TUNISIA  Fax Number : +216 1
884 819

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I am looking for some java (or c++) classes which cover the creation and
manipulation of first-order logical expressions. They need to be able to
handle e.g.: creation of conjuntions, disjunctions, implications,
predicates, terms, variables, constants etc, and preferably deal with
variable scoping. A database for conveniently storing these would also
be nice.

Also are there any other newsgroups dealing with formal logic related
issues?

If anyone can help please mail me: mailto://Umair.Wah…@bigfoot.com or
post a reply.

Thanks

Umair

On other newsgroups some noisy discussions tend to
involve matters of basic logic and math, with heated
exchanges.

The following questions are of the essence of many
such exchanges, many being so obvious and elementary
one has to wonder how they can be in question.

The ‘right’ answers are sometimes ‘trues’ and sometimes
‘falses’, and rarely directly answered in the discussions,
adherents to one side or the other resorting to name
calling instead of dealing with the implications.  [Some-
times my opposition agrees with what I say, but only when
they have mistakenly taken me as asserting the opposite
of the correct position.

Innyhoo, please just answer the questions as far as you
are able, DIRECTLY. Meaning: without circumlocution based
on not liking the consequences for you en re some treas-
ured belief.

Write, please, for clarification, if any required.
Or suggest another question for the series, in the
same spirit.

I'll synopsize later, if desired, with or without
explanation, if desired. You'd find the material
very interesting, I believe.

============================================================
SW.1.01: If my premise is A and I conclude not-A, then
I have reasoned correctly.  True or false?

------------------------------------------------------------
SW.1.02: Let F = F(a+b/A+c/B); a, b, c being constants,
and A and B not being f(x) directly or by chain/indirectly.
True or false: dF/dt=(1/A+1/B).

-----------------------------------------------------------
SW.1.03: Let F = F(a+bx/A+bx/B), a,b being constants, A and
B not being functions of x directly or indirectly.  True or
false that dF/dx=(1/A+1/B)?

----------------------------------------------------------
SW.1.04: To conclude something along the lines of "Joe scored
higher than Sam" a good logical process would be to have
Joe take the test for both himself and Sam? True or false?

------------------------------------------------------------
SW.1.05: To conclude that uniformally moving pianos can't
be played as well as stationary pianos, an adequate test
is to have the stationary pianist first play the piano
with respect to which he is at rest, then - still seated
and not moving - play the moving piano as it moves by.
True or false?

-----------------------------------------------------------
SW.1.06: If a=x_0, a and x_0 constants, and F=F[a,y]
then dF/da = 1dF/da. True or false?

———————————————————–
SW.1.07: In an equation such as that of a circle,
x^2+y^2=r^2, considering x and y as cartesian coordinates,
there is an implicit presumption/assumption that the
circle center is located at [0,0]. True or False?

———————————————————–
SW.1.08: Vector notation for the same circle equation
would/could be, with X=X[x,y), |X|^2 = r^2 and this
also presumes/assumes the circle center to be at [0,0].
True or false?

————————————————————-
SW.1.09: In fact, for any such equation, there is a more
general form, right out of Rene Descartes, where one
expresses the equation in a way that does not presume the
location of the centroid: (x-x_c)^2 = r^2. True or false?

———————————————————–
SW.1.10: In vector notation, that would be something like
(with X=X(x,y) and P=P(x_c,y_c) – with P referring to the
point of the/a centroid): |X-P|^2 = r^2. True or false?

————————————————————
SW.1.11: In SW.1.09 and SW.1.10, the x_c and P terms are
just as much spatial (x,y,..) coordinates as x and X.
True or false?

———————————————————–
SW.1.11: Using the generalized Cartesian forms ( with
x-x_c ), one can shift/translate the x axis at will
and the equation functionality is not disturbed.
Example: let s=f(anything), and remember that the
centroid figure, x_c, is an x-value. (x’-x_c’)^2=r^2;
(x-s – x_c+s)^2=r^2; (x-x_c)^2=r^2. True or false?

———————————————————-
SW.1.12: Using the vector notation, let S=S[x,y].
|X’-P’|^2=r^2; |X-S – P+S|^2=r^2; |X-P|^2=r^2.
Thus, the functionality of the equation is in no
way disturbed by the translation S represents. True
or false?

———————————————————–
SW.1.13: In cases such as SW.1.11 and SW.1.12, the math
won’t work if somewhere in the equation there is some
class of, or a particular, expression, g(…). True or
false?

———————————————————-
SW.1.14: In cases such as SW.1.11 and SW.1.12, if we
let s or S=f(v,t) the velocity in the translation
must be equal to any velocity already present in the
equation. Example: x^2=(vt)^2. True or false?

———————————————————-
SW.1.15: In cases such as SW.1.11 and SW.1.12, if we
do let the velocity of the translation equal the
velocity already present in the equation, then the
equation velocity variable becomes zero. Example:
(x-x_c)^2=(vt)^2 becomes (x’-x_c’)=0t^2. True or
false?

————————————————————

Eleaticus

!—?—!—?—!—?—!—?—!—?—!—?—!—?—!—?—!—?
! Eleaticus        Oren C. Webster         ThnkT…@concentric.net  ?
! "Anything and everything that requires or encourages systematic   ?
!  examination of premises, logic, and conclusions"                 ?
!—?—!—?—!—?—!—?—!—?—!—?—!—?—!—?—!—?

I came across the following exercise in "Basic Model Theory" (Kees
Doets):

    54. Show that the following two conditions are equivalent:

        1.  The sentence phi has a logical equivalent of quantifier rank
<= n,
        2.  for every two models A, B s.t. A =n B: if A |= phi, then B
|= phi.

(where =n represents n-equivalence).

I’m stuck trying to go from 2 to 1.  If anyone could give me a push
on this, I would sure appreciate it .

Thanks, Brian

I work for Foresight Science and Technology, Inc, where we are involved in a
long-term project developing AI tools for the area of technology
commercialization.  I am tasked with finding an established ontology
covering several areas of interest to us, specifically:

sciences
applied sciences
industries

I’d be interested in any pointers to ontologies, particularly taxonomic
ontologies, covering these subjects.  We have already rejected the Library
of Congress subject headings, Dewey Decimal, patent classifications, and SIC
codes.  ACS abstracts is excellent, but too narrow.  Any ideas?  We have an
empirical ontology developed specifically for our work, but the powers that
be have rejected it, saying that we have to tie into an existing, accepted,
standard.

-Nick Dallett
Foresight Science and Technology, Inc.
http://www.seeport.com/
n…@foresnt.com

/

I have a question regarding Godel’s theorem.   I’m
not a mathematician, just another one of those
software engineering types.  

Wouldn’t a universe that is based ONLY on the KNOWN
laws of mathematics and physics,  preclude the bottom-up
evolution of a consciousness capable of "discovering" what
we now know as Godel’s theorem?

–
Kevin J
To send me email, you must first remove the
anti-spam filter from my email address below.
kjessup@_SPAM_ME_NOT_nconnect.net

Dale <d…@webeyeNOSPAM.demon.co.uk> wrote:
> Joseph Dunphy wrote:
> >2. With Blackbane, we just have a very old flamewar that has been
> >   entrenched. What you’re seeing out of Blackbane is mockery out of the
> >   regulars. When RBB first showed up, s/he made a humorous post, which
> >   got a very rude reception, and things escalated.
> This is NOT true.  RBB did not make a "humorous" post which got a rude
> reception.  I would advise you to go through DejaNews and find the facts
> before you proclaim to know the facts.

Dale, you are without shame. Not only have I gone through Dejanews,
finding the first post that Raven Blackbane did in this group, and the
subsequent flamewars, but I downloaded those posts, myself, and assembled
them into a post which I made here, a few months ago. I have said as
much in the recent past. Those who doubt this can go into Dejanews
themselves, and look up the threads

      Newcomers: What this Raven Blackbane business is about.

      The railroading of Raven Blackbane.  

                  (and associated threads, from about the same time)

I would advise to you to attempt to create the appearance that you have at
least a shred of integrity, but it may be too late …

> > No right side, here.
> >   RBB was ganged up on, very quickly, so you can understand the
> >   resentment on his/her part, though this resentment has been held onto
> >   for over a year, now. Neither side has made any real effort to put an
> >   end to hostilities.
> Again, this is not true.  RBB has made NO attempt to "put an end to
> hostilities".  RBB’s actions are the complete opposite.

I’ve griped before, about the suggestibility of many online, and the
willingness of some to try to exploit this to their advantage.

This is a classic example. Look at the last sentence, of the very
paragraph excerpted.

     "Neither side has made any real effort to put an end to
      the hostilities".

Now, how mentally challenged would someone have to be, to think that this
was the opposite of saying

     "RBB has made no attempt to put an end to the hostilities."

when the opposite would, in fact, be

     "Both sides have made a real effort to put an end to the
      hostilities." ?

Because the only working options we have here, are a lack of reading
comprehension, on even a Middle School level, or a deliberate attempt by
Dale, here, to lie outrageously about a paragraph that he has left in
clear view of the reader, and hope that he has successfully bluffed the
reader into taking his word about what it has said, and not even reading
it on his own. The choices here are between concluding that Dale is
functionally illiterate, mentally retarded, or is being deliberately
dishonest. I don’t know whether or not it is a kind choice to exclude the
first two possibilities, but it is the one that I am making, tentatively.

Dale, go away, and do not return until you have managed to locate your
conscience. The subject at hand, here, is not a matter of opinion. While
the language has ambiguities, they aren’t broad enough to allow you to
slip this interpretation through.

I have crossposted this over to sci.logic, where people may be able to
explain the concept of negation to you, alt.stupidity for reasons that all
but you will probably grasp, and soc.culture.british, with the question,
"Is this the best that the schools in Liverpool can produce, now ?".
Followups are directed out of this group, to a place where you might find
the help that you need.

Good day.

Hello!

Can anyone tell me where to find
a good article about Jan Lukasiewicz?

Carlos

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How can an expression that describes a set be tested (by a computer or a
TM) to be valid?

for example:
Suppose A is Set,
the definition:
B={a in A| a not in B}
is not a valid definition of a set.

But if we don’t allow recursive definitions, we can’t define some valid
sets (for example the Natural numbers are defined recursively).

So how can (if at all) a computer program determine whether a set
definition is legal?

Summer School in Partial Evaluation: Practice and Theory
partial evaluation, specialization, practice, program transformation
Summer School in Partial Evaluation: Practice and Theory in Denmark June 29 – July 10, 1998

                  DIKU International Summer School ’98

                Partial Evaluation: Practice and Theory

                     from June 29 – July 10, 1998

The DIKU International Summer Schools is a series of summer schools at
DIKU, the department of computer science at the University of Copenhagen,
Denmark. The summer schools are aimed at Ph.D.-students and researchers
from research environments that are leading within the area in
question. Having run for a number of years, the series assumed its present
form in 1997 with successful DIKU International Summer Schools on adaptive
robot behaviour, region-based memory management, and mathematical logic.

Description:
===========

Program specialization, also known as partial evaluation, is an automatic
tool for program optimization, similar in concept to but in several ways
stronger than a highly optimizing compiler. It is a source-to-source
staging transformation: a program p together with partial data s are
transformed into an often faster specialized version p-s by precomputing
parts of p that depend only on s. The possibility, in principle, of partial
evaluation is contained in Kleene’s classical s-m-n theorem.

Specialization is worthwhile when p runs for a long time, and p-s is
significantly faster than p. Suitable problem types include: Highly
parameterised computations that use much time consulting parameters, but
are often run using the same parameter settings; programs with many similar
subcomputations; programs of a highly interpretive nature, e.g. circuit and
other simulators, where specialization removes the time to scan the object
being simulated; database query search algorithms; and meta-programming,
where a problem is solved by designing a user-oriented language and an
interpreter for it.

Partial evaluators have been successfully applied to generate efficient
specialized programs for ray tracing, for the Fast Fourier transform, and
for circuit and planetary simulations. Partial evaluators have also been
used to compile using interpreters for programming languages and to
generate compilers from interpreters.

The DIKU International Summer School ’98 on Partial Evaluation offers a
practical introduction to several existing partial evaluators, including
the opportunity for guided hands-on experience, as well as the presentation
of some more sophisticated theory, systems, and applications. Lectures will
be given by some of the leading researchers in the field.

The summer school runs for two weeks. The first week is practically
oriented, focusing on a number of partial evaluation systems, mainly
developed at DIKU. The second week combines practical experience with more
sophisticated theory, systems, and applications. It is possible to follow
each of the two weeks on its own.

Organizing Committee
====================

Neil D. Jones
Jesper Joergensen
Jens Peter Secher
Morten Heine B. Soerensen (chair, e-mail ra…@diku.dk)

Lectures and topics:
===================

Torben Mogensen:        Introduction, overview, applications

John Hatcliff:          Foundations of partial evaluation and program
                        specialization

Jens Peter Secher:      C-mix

Jesper Joergensen:      Similix

Torben Mogensen:        Inherited limits

Lennart Augustsson:     Partial evaluation for aircraft crew scheduling

Neil Jones:             Lambda-mix

Satnam Singh:           Hardware specialization

Jesper Joergensen and   Multi-level specialization
Robert Gluck:

Morten Heine Soerensen: Supercompilation

John Hughes:            Type specialization

Michael Leuschel:       Logic program specialization

Michael Leuschel:       Advanced logic program specialization

Julia Lawall:           FFT, specialization of an implementation of
                        the Fast Fourier Transform

Jens Palsberg:          Eta-redexes

Olivier Danvy:          Type-directed partial evaluation

Peter Thiemann:         Aspects of the PGG system

Deadline for registration:
==========================

                              May 29, 1998.
                              ============

URL of Summer School:
====================

   http://www.diku.dk/research-groups/topps/activities/pe-sommerschool-98/

–
http://www.diku.dk/students/jpsecher    (I *hate* signatures!)