Thursday, April 26, 2018

Accepted in a colloquium (2)


Mathematics
A philosophical approach to Fermat Last Theorem

Fermat did not make public his proof for two reasons :
it was only an outline of a proof and he was not satisfied with it (some case missing),
it would have been considered blasphemous by the Church because of some infinite not existing.
He would have had the same problem as Galileo Galilei.
Equation of Fermat :
Considering the equation
with infinite products
z.z.z.z………=x.x.x.x……………+y.y.y.y……….
and considering
countable axiom of choice for at most y elements sets
we assume 5<=x<=y
C(2 through y)
something which exists equal to something which does not exist (z.z.z…..z…..)
the equation has no solution.
Does it imply that the finite equations for n>= 5 have no solutions ?
Intuitively, for me, it is the case.
The relation between finite and infinite has to be investigated according to
Godel theorem that we are always in need of new axioms.
CUSTOM-MADE AXIOM FOR THE PROOF OF FERMAT LAST THEOREM :
The disjoint union of a Cartesian product of a set (of a number of
elements >=5) a number of times >=5 and of another (OF greater
CARDINALITY) set the same number of times being OF THE SAME CARDINALITY
THAN a Cartesian product of a third set the same number of times MAKES :
the equality of the sum of the two cardinalities of the two first infinite Cartesian products with the
cardinality of the third infinite product holds.
(for a set theoretical proof)
Mr Andreas Blass wrote in 2002 about a complication for x,y and z integers less than 5 in :
http://www.math.lsa.umich.edu/~ablass/dpcc.pdf
After centuries of research, we should have thought that a new axiom is needed for Fermat Last Theorem.
The creativity of Fermat should not be doubted.
The proof of Fermat used an equation with infinite products.
It is completed by a set theoretical explanation.
I am the author of « An axiom to settle the continuum hypothesis ? »Logic Colloquium 2004 (by title).

Fermat was reading Aristotle who wrote about the existence of the infinite.
An extrapolation principle enables to go from the equation zn=xn+yn to z.z....z...=x.x....x...+y.y...y....
Mathematicians are waiting for someone to come up with an axiom from an intuition, I did and they do not recognize it.

Excerpt from « All things are numbers (continuation)”, abstract from
the Logic Colloquium 2002 published in the Bulletin of Symbolic Logic :
The equation with infinite products zzz…z…=xx…x…+yy…y… with z>y
has no solution in the universe where only the restricted axiom CC(2
through x) is true.
It is because otherwise the infinite products xx…x…
and yy…y… exist but not zzz…z… and we cannot have a side of the equation
existing and the other not.
What could be a philosophical interpretation of the proof of Fermat Last Theorem
is that God is mysterious and not all powerfulness (the infinite is an attribute of God).
Fermat was reading Descartes about existence.
Fermat was also reading Desargues about the point at the infinite.
I  published in “A philosophy for scientists” Adib Ben Jebara Shield Crest Publishing UK 2015


From my philosophy

About shortcuts
In front of (when
we face) complexity, there are short cuts.
The existence of short cuts is a consequence of the existence of God.
To know histories helps to find shortcuts.
To know where a doubt persists helps to find shortcuts.
We often have to be interested in 1963.
Until 1963 there was no option for extreme specialization.
This theory of short cuts fits in the theory of total quality management.


BRIEF HISTORY OF THE NEGATION OF THE AXIOM OF CHOICE

The story begins with Cantor who was working on the functions theory and
noticing there could not exist any bijection between N and R. A tremendous
work made him able to give shape to the set theory, being encouraged by
Dedekind and criticized by Kroneker .
On about 1900, it happened to him tohit on two problems.
The first is a paradox which make it impossible for the set of all sets to exist.
The second , called continuum problem, is about the proof that there does not
exist a cardinal number between that of N and of R.

Indeed, Cantor was reasoning by taking for obvious what will later be called
the Axiom of choice as he is taking for granted a total order relationship
between cardinal numbers . This last issue made him exhausted.
Zermelo was trying to make the theory of Cantor formalized.
So on 1904, he defines the Axiom of choice as being the assumption
that the cartesian product of an infinite family of sets is always different
from the empty set. Zermelo and Russell gave at about the same time
another statement for the Axiom which is that there exists a function
which associates to each set of the infinite family one element .

The Axiom was called at the beginning the multiplicative Axiom.
Controversies started between mathematicians, particularly, the french
mathematicians Hadamard, Borel and Lebesque while Zermelo was going on with
his formalization work. The polish mathematician Sierpinski undertook the
identification of the theorems which need the axiom. The german mathematician
Fraenkel used the axioms of Zermelo to define as soon as 1922 a model where
the negation of the axiom of choice is an axiom.

Polish mathematicians like Tarski, Mostowski, Lindenbaum studied around the
thirties the negation of the axiom of choice.
However tragic deaths happened after Banach pointed out unrealistic consequences
of the axiom. The french mathematician Herbrand died, a disciple of Russell, also
Lindenbaum died during the second world war and some papers of Mostowski were
lost at that time.

Meanwhile , the reference to Zorn's lemma was taking the place of the reference
to the axiom.
Mostowski was the one who studied most the particular cases of the axiom :
denumerable or non denumerable family and above all, the case where all the
sets have the same number of elements n, whose notation is Cn. He made about
it a work remarkably difficult to access. In the fifties, the swiss Specker
and the french Fraissé studied also the negation of the axiom of choice.

But the coming of forcing on 1963 produced a fashion phenomena which relegated
Fraenkel Mostowski, as they were called henceforth, as of secondary
importance. However, the use of the axiom for the definition of infinite sums
and products was not drawing attention, although it was mentioned in the
“Principia Mathematica” of Russell and Whitehead before the first world war.

However, this use of the axiom was written about in a remarkable paper, from
Sierpinski, on 1918, which did not get the echo it deserved. Gauntt, in the
seventies, was the only one trying to complete the work of Mostowski on the
Cn. May be a contribution from Truss should be also mentioned.

THE IDEA OF BEN JEBARA

Ben Jebara wrote about the possibility that Fermat was a harbinger for the
Theories using the negation of the axiom of choice through the existence of
Infinite products. Fermat was reading Aristotle who wrote about the existence
of the actual infinite and Fermat's intuition could have enabled him to distinguish
between the infinite products existing and the others.

An extrapolation principle which enable to go from the equation zn=xn+yn to z.z....z...=x.x....x...+y.y...y....
Ben Jebara wrote about that in newsgroups in 1994 and in 1999 and was corrected
By Mr Andreas Blass who was helped by Mr Paul Howard.





Accepted in a colloquium (1)


About space and time of elementary particles : Introduction

The subject of space and time of elementary particles is at the intersection of physics,
mathematics and philosophy of science.
It was not approached before because we are in an Age of partioning.
And also because conjectures are not welcome, only evidence is.

What is difficult to understand in the subject is the mathematical axiom of choice of
set theory as it is applied or rather its negation.

The existence of a second component of time at the level of elementary particles
is an idea which did not occur to me directly.
I started by trying to explain the Big Bang in quantum cosmology  by introducing
more mathematics for time and starting from scratch.
That was some 15 years ago while working in a company and corresponding with
Mr Andreas Blass.

The existence of the Big Bang still meets undue skepticism with some people.

The existence of the Big Bang is deduced by me from the existence of a Big Crunch
(collapsing) as the Big Bang follows the Big Crunch.
And my idea of space and time at the level of elementary is checked by the existence
of the Big Bang.

The Big Bang following a Big Crunch is an idea quickly considered in 1930 by Einstein
who did not look for arguments.
My argument is from mathematical modeling or rather mathematical explanation as
space and time are treated as mathematical variables.

The subject can be applied to teleportation of elementary particles and then to groups
of elementay particles.
Teleportation where particles are not replaced by others with the same caracteristics
in the process.
That is the most interesting part for physicists but most of it is still in preparation in
September 2017.

ANNEX TO THE INTRODUCTION
Axiom of choice for a countable family of sets
of number of elements between 2 and m included,
that is CC(2 through m).

For an infinite family of non empty sets,
an equivalent of the axiom of choice :
A1xA2xA3x... is not an empty,
We will see that it could be a set of paths.

Cardinality means number of elements.
 A very important idea is that from what is true in quantum cosmology, we can deduce things in quantum
Mechanics.

SPACE AND TIME ARE DISCONTINIOUS
WHEN SPACE DISAPPEARS AFTER AN INFINITE TIME

After an infinite time, we will see that the set of paths will be the void set.
Physical space would become void, the universe would collapse and a Big
Crunch would happen.

we consider locations as urelements (non sets), elements of U.
Ui is a subset of U with number of elements n.
XiUi is the infinite cartesian product and a set of paths.
If n is greater than m in CC(2through m), countable choice for k elements
sets k=2 through m, the set of paths will be the void set.
n>m

From what is true in quantum cosmology, we can deduce in quantum
mechanics the following :

We can us notice that Newton first law is partly contradicted :
F=0  V constant but the particle does not move indefinitely as there is no
infinite path.
Time is also a set of urelements.
The particle could be using the second component of time.


SPACE AND TIME ARE INFINITE

Because of the Big Bang explained by the
negation of the axiom of choice, space is
a set of urelements of the negation of the
axiom of choice.
Such a set is Dedekind infinite.
Space is a mathematical entity which is infinite.
As a set of urelements, it is discontinuous.
The number of urelements in between particles
can be used to define a distance.
However, we do not have a vector space.

Such a reasoning can be made for time.
To write that time has 2 components is to
make an approximation.

A set which is Dedekind infinite has a cardinality
which is not an aleph of the cantorian infinite.


It is not only that physical space and mathematical space are entwined, it is that
space is seen more with the eye of the mind than with the eye.
People are so much tied up to their
bodies that they cannot see with
the eye of the mind, as if Descartes
and Galileo did not exist.

For time, it is blatant that the eye of the mind should be used,
even more so since the theory of Relativity.



Let us assume as an approximation that CC(2 to m) holds for m<n,
n being the number of locations of particles in the universe at a given
time.
The Big Crunch occurs.
Such an idea could be seen as  looking for the particular axiom of
choice which applies in physics.

About time and indeterminism in the physics of particles

Let Ui be a countable family of non empty sets of urelements (non sets), the
negation of
the axiom of choice implies that the Cartesian Product of the family is
empty.

We know from “A philosophical approach to Fermat Last Theorem” in "A
philosophy
for scientists" Adib Ben Jebara Shield Crest Publishing that only a
particular
case of the axiom of choice is true.

And from "About space and time in quantum mechanics" Adib Ben Jebara
Bulletin of Symbolic Logic September 2008, p. 410., we know that the
negation of the axiom of choice can be applied to particles.
That is a basis for the teleportation of the particle since the particle
will have much
“time” to move without the time at our level being much .
EXCERPT from “About a time not totally ordered
(published in the colloquium brochure WSEAS MCSS’15 Dubai 22 February) :
“For elementary particles, time is a set of urelements of the negation of
the
axiom of choice.

So, time is not totally ordered and there is a lateral time.
In an experiment, if a particle enters a hole twice that must be that it
enters and enters again from the same side in a lateral time.
The second time is perceived at our level as being after the first time
while it is not at the level of the particle.
In another experiment, the particle enters two holes at the same time, the
lateral time appears to be the same time.”
Mechanics theory has a tendency to progress by introducing more mathematics
which may
receive industrial applications after some dozens of years.
We are no more in statistical mechanics, because the 2 coordinates of time
are known, the probability of finding the particle in one place is either
zero or 1.
Addendum : one has to pay attention to the weak structure of time at the
level of elementary particles.
it does not matter so much if fundamental indeterminism exist
because it will be reduced whenever physics progress.
Heisenberg uncertainty principle can be bypassed.

The principle states that the more precisely the position of some particle
is determined, the less precisely its speed can be known, and vice versa
That is if we do not know the orthogonal time for the particle but only the
time at our level.
If we know the orthogonal time, the speed is changed by it and the
uncertainty principle
with the time at our level does not apply.

Let us notice that Newton first law is partly contradicted :
F=0  V constant but the particle does not move indefinitely as there is no
infinite path.
I think that there are too many experiments using  particles accelerators
or cyclotrons or colliders and not enough experiments about beams of
 particles which are not of high energy such as teleportation of a beam of
particles.
In the most general  case, the orthogonal time is different from one
particle to another.

Teleportation is not the same than teleportation of properties
because some other properties may not be taken into account.
About Newton first law with F=0, after an indefinte time (approximately a very long time),
the position and speed of a particle will be not defined.

With the evolution of n locations of space of the particles
in the
whole universe, we cannot distinguish an infinite number of 
occurences between 2 urelements of time,
We have a case where the axiom of choice does not hold at all and that is
enough to have the axiom of choice not hold.
Such a reasoning can be made because the duration towards
the Big Crunch is infinite.
In mathematics (for integers), CC(2 through m) is true but in 
physics the axiom of choice is not true at all.

About entanglement of particles

Entanglement of particles is when one is still influencing
the other after the coupling ended.
Coupled means touching one the other.
The other is taken far away and the spin
of the first is changed, the spin of the
other will change.
The repeating of the effects makes
some causality exist.

There is entanglement when the state of a particule is
Influenced by the state of another particle after the
coupling is over.
The particles are said to be correlated.

The explation could be that the second particle
is still at the moment when it is touching the first
because it has been using its orthogonal time ever since.

It seems that there is no change or
change to expect for the particle in orthogonal time.

How orthogonal time is unlike time at our level ?
We cannot act on the particle during orthogonal time
and may be that is something which can be (may prove) useful.


Beside knowing that a Big Crunch will occur after an infinite
time (not Cantorian infinite), we know that
the axiom of choice is not true at all in physics (the opposite
of what people think).
In mathematics, the countable axiom of choice for sets of
number of elements between 2 and m (m included) is
true.
Besides, the first law of Newton is not true for particles.
The litterature about teleportation and entanglement of
particles is confusing (not clear).


The entanglement of particles was forecasted by Einstein in 1935.