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Two fatal defects of Wiles' proof of FLT...

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Edgar E. Escultura...

Posted: Fri Nov 06, 2009 6:03 pm

Guest

Of course, sqrt(-1) has two roots and the reason you choose one of them is to hide the contradiction in it. Your choice, however, does not resolve the fact that sqrt(-1) yields a contradiction. The source of the problem is the vacuous concept i = the root of the equation, x^2 + 1 = 0, among the real numbers which does not exist. Therefore, i is ill-defined, ambiguous, and contradiction usually hides in ambiguity. The full remedy for the complex plane is in the appendix to my paper, The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies. E. E. Escultua

Edgar E. Escultura...

Posted: Fri Nov 06, 2009 6:57 pm

Guest

Welcome Fernando; hope you are a compatriot.

The bottomline is that the field axioms that supposedly define the real numbers are inconsistent. In particular, the trichotomy and completeness axioms are false. The counterexamples to them are noted elsewhere on this thread and the referecences are cited. Consequently, the real number system is neither complete nor a field nor ordered by "<". In fact, it is ill-defined and all those results you cited fall through.

The objects that satisfy x^2 + 1 = 0 are ill-defined. Hamilton tried to build the complex plane as ordered pairs but he did not identify the right consistent axioms that well define them. The remedy is in the appendix to my paper, The generalized integral as dual of Schwartz distribution, in press, Nonlinear Studies.

Cheers,

E. E. Escultura

Edgar E. Escultura...

Posted: Fri Nov 06, 2009 7:25 pm

Guest

Correct me if I'm wrong, you can define C to be R[x]/<x^2+1>, the
quotient ring of the set of polynomials with real coefficients modded
out by the ideal generated by x^2+1. The isomorphism sends x to i.
There is nothing ill-defined about that.
----
The fact the the concepts and spaces you have here are defined in terms of the real number system makes them ill-defined because the latter is, the field axioms that define it being inconsistent. The ideal generated by x^2 + 1 needs to be well defined by a set of consistent axiom.

You can also define C by the set of ordered pairs of real numbers with and multiplication for ordered pairs
of real numbers. Again, nothing ill-defined about that.
-------
Yes, this has been done by Hamilton. But he relied on the real numbers which are ill-defined.

What is really needed here is fix the real number system first which I did in the paper I cited elsewhere on this thread.

Moreover, you seem to be suggesting that since a polynomial equation
has no real roots implies that C is ill-defined. Maybe you can explain
why I'm wrong but that's like saying that since 3x+1=-1 has no roots
among the integers that Q is ill-defined, and like saying that since
x^2 = 2 has no rational roots that the set of irrationals are ill-
defined. There is a progression of number sets that allow for more
polynomial equations to be "solved" (ie, roots found), starting with
N, then Z, Q, R, and finally C. That C is algebraically closed
implies that polynomials will not yield further number sets.

Certainly, the root of a polynomial that has no root is ill-defined, in fact, a contradiction.

What you are suggesting is an extension of the reals that will yield roots to such polynomials. But an extension of any mathematical space belongs to its complement and is not covered by its axioms assuming that they are consisent. Therefore, you need a new set of consistent axioms for them.

Congratulations. You have the most advanced commments so far in contrast to the name callers whose posts come from the flat of their foot because the top is quite empty.

Cheers.

E. E. Escultura

fernando revilla...

Posted: Fri Nov 06, 2009 8:13 pm

Guest

E.E. Escultura wrote:

[quote]The bottomline is that the field axioms that
supposedly define the real numbers are inconsistent.
[/quote]
In that case, and from those field axioms you have to provide
a well formed formula A such that A and (~ A) are theorems.

Regards.

Edgar E. Escultura...

Posted: Fri Nov 06, 2009 10:55 pm

Guest

An axiomatic system is completely well defined by the axioms including the rules of inference. Therefore, formal logic does not apply since the axioms have nothing to do with it.

Cheers.

E. E. Escultura

Edgar E. Escultura...

Posted: Fri Nov 06, 2009 10:58 pm

Guest

The counterexamples prove the inconsistency of the field axioms.

Cheers.

E. E. Escultura

Brian...

Posted: Sat Nov 07, 2009 9:35 am

Guest

I vaguely recall something about a way to define real numbers as
equivalence classes of sequences of rational numbers where the
equivalence relation between two rational sequences is that two
sequences are considered equivalent if the terms in the sequence get
arbitrarily close to one another (ie, the tails of the two sequences
differ by an arbitrarily small rational number). For example, the
square root of 2 is defined to be the equivalence class of the
sequence {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...}. The addition
and multiplication of two real numbers (ie, two equivalence classes of
rational sequences) is done by taking the equivalence class of the
sequence whose nth term is the sum or product of the nth term of
representative sequence1 with nth term of representative sequence2.
(The sum and product need to be shown independent of which
representative sequence you use, of course.) From those definitions,
you can define subtraction and division.

Between that and the Dedekind cut construction of the real numbers,
what is inconsistent?
http://en.wikipedia.org/wiki/Dedekind_cut#The_cut_construction_of_the_real_numbers

[quote]What is really needed here is fix the real number system first which I did in the paper I cited elsewhere on this thread.
[/quote]
At any rate, you say you've fixed the real number system. Why can't
the complex numbers be defined as ordered pairs of Escultura-type real
numbers?

[quote]Therefore, formal logic does not apply since the axioms have nothing to do with it.
[/quote]
Please explain!

fernando revilla...

Posted: Sat Nov 07, 2009 10:37 pm

Guest

Edgar E. Escultura...

Posted: Sat Nov 07, 2009 11:06 pm

Guest

Brian Tenneson:

I vaguely recall something about a way to define real numbers as
equivalence classes of sequences of rational numbers where the
equivalence relation between two rational sequences is that two
sequences are considered equivalent if the terms in the sequence get
arbitrarily close to one another (ie, the tails of the two sequences
differ by an arbitrarily small rational number). For example, the
square root of 2 is defined to be the equivalence class of the
sequence {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...}. The addition
and multiplication of two real numbers (ie, two equivalence classes of
rational sequences) is done by taking the equivalence class of the
sequence whose nth term is the sum or product of the nth term of
representative sequence1 with nth term of representative sequence2.
(The sum and product need to be shown independent of which
representative sequence you use, of course.) From those definitions,
you can define subtraction and division.

Between that and the Dedekind cut construction of the real numbers,
what is inconsistent?
http://en.wikipedia.org/wiki/Dedekind_cut#The_cut_construction_of_the_real_numbers
----

The Dedekind cut or its equivalent, the completeness axiom of the field axioms of the real numbers do not apply apply to infinite set such as the digits of a nonterminating decimal because ot the latter's ambiguity since not all its digits are known. Any statement about ambiguous set or concept is ambiguous and is not admissible as an axiom for it erodes the validity of any theorme. The other source of inconsistency of the real numbers is the trichotomy axiom to which Brouwer and myself has constructed counterexamples.

[quote]What is really needed here is fix the real number system first which I did in the paper I cited elsewhere on this thread.
[/quote]
At any rate, you say you've fixed the real number system. Why can't
the complex numbers be defined as ordered pairs of Escultura-type real
numbers?
-------

That would be an improvement. However, there must be a reason people do not use this Hamiltonian scheme and resort to the standard notation. Cumbersome, perhaps? At any rate there is simple remedy by looking a i as on operator on the Euclidean plane vectors and e^itheta. This is found in the appendix to my paper, The generalized integral as dual to Schwartz distribution, in press, Nonlinear Studies.

[quote]Therefore, formal logic does not apply since the axioms have nothing to do with it.
[/quote]
Please explain!

Since a mathematical systems is completely well defined solely by its axioms reasoning within it, i.e., the rules of inference must follow from the axioms. In fact, in practice mathematicians do not use formal logic.

Thank you for the most advanced post on this thread so far. E. E. Escultura

fernando revilla...

Posted: Sat Nov 07, 2009 11:08 pm

Guest

E.E. Escultura wrote:

[quote]The counterexamples prove the inconsistency of the
field axioms.
[/quote]
Well, we need only one. We can analyze it.

Regards.

Edgar E. Escultura...

Posted: Mon Nov 09, 2009 5:21 pm

Guest

MMM: Double idiot squared, plus infinity, plus one, squared, tied in a sack, and
thrown over the back of a donkey. Times two.
---

Racist needs frustrate surgery to remove sour grapes and placate inadequacy. E. E. Esscultura

Edgar E. Escultura...

Posted: Mon Nov 09, 2009 5:24 pm

Guest

Fernando: Well, we need only one. We can analyze it.
---

I agree but it's nice to know that there are, in fact, countably infinite counterexamples.

Cheers.

E. E. Escultura

Edgar E. Escultura...

Posted: Mon Nov 09, 2009 6:13 pm

Guest

Brian Tenneson wrote:

[quote]At any rate, you say you've fixed the real number
system. Why can't
the complex numbers be defined as ordered pairs of
Escultura-type real
numbers?
[/quote]
Right. I also thought that way.
-----

I replied to Brian’s post elsewhere and said that this will be an improvement but that in practice mathematicians revert back to the standard notation and bring back the concept i because of the inconvenience of computing with ordered pairs. My remedy is to replace i by the operator j that rotates a vector in the Euclidean plane positively by pi/2 and i^theta by the operator h_theta that rotates the unit vector with initial point at the terminal point of the given vector positively by theta. Detail is in the appendix to my paper, The generalized integral as dual to Schwarz distribution, Neural, Parallel and Scientific Computations.

Cheers.

E. E. Escultura

Edgar E. Escultura...

Posted: Tue Nov 10, 2009 1:30 am

Guest

On Usenet he can write whatever he wants about his current
title, but it looks like the institute forced him to include the
Emeritus qualifier on his website :-)

Yes, people like this are allowed on the street.

Dirk Vdm
---------
When the top is empty mathematics becomes transparent and only minor things catch attention. Moreover, racist needs frustrate surgery to remove sour grapes and placate inadequacy. E. E. Escultura

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