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.