h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Two Fatal Defects in Andrew Wiles’ Proof of FLT

1) The field axioms of the real number system are inconsistent; Felix Brouwer and this blogger provided counterexamples to the trichotomy axiom and Banach-Tarski to the completeness axiom, a variant of the axiom of choice. Therefore, the real number system is ill-defined and FLT being formulated in it is also ill-defined. What it took to resolve this conjecture was to first free the real number system from contradiction by reconstructing it as the new real number system on three simple consistent axioms and reformulating FLT in it. With this rectification of the real number system, FLT is well-defined and resolved by counterexamples proving that it is false.

2) The other fatal defect is that the complex number system that Wiles used in the proof being based on the vacuous concept i is also inconsistent. The element i is the vacuous concept: the root of the equation x^2 + 1 = 0 which does not exist and is denoted by the symbol i = sqrt(-1) from which follows that,

i = sqrt(1/-1) = sqrt 1/sqrt(-1) = 1/i = i/i^2 = -i or

1 = -1 (division of both sides by i),

2 = 0, 1 = 0, i = 0, and, for any real number x, x = 0,

and the entire real and complex number systems collapse. The remedy is in the appendix to the paper, The generalized integral as dual to Schwarz distribution.. In general, any vacuous concept yields a contradiction.

Response to the commentaries on FLT and my counterexamples to them.

Since there is noticeable increase in commentaries about FLT, Wiles’ proof and my counterexamples, it is time to respond to some major points, present the foundational basis of my counterexamples and make a rejoinder on FLT.

Constructivist mathematics in my sense has nothing to do with intuitionism. It simply avoids sources of ambiguity and contradiction in the construction of a mathematical system which are: the concepts of individual thought, ill-defined and vacuous concepts, large and small numbers, infinity and self reference. I have given examples in my posts in several websites of how these concepts yield contradictions. A contradiction or paradox in any mathematical collapses a mathematical system to nonsense since any conclusion from it is contradicted by another.

Early in the 20th Century David Hilbert pointed out the ambiguity of individual thought being inaccessible to others and cannot be studied and analyzed collectively; nor can it be axiomatized as a mathematical system. Therefore, to make sense, a mathematical system must consist of objects in the real world that everyone can look at, study, etc., e.g., symbols, subject to consistent premises or axioms. A counterexample to an axiom or theorem of a mathematical system makes it inconsistent.

This important clarification by Hilbert has not been grasped by MOST mathematicians, the reason for the popularity of the equation 1 = 0.99… How can 1 and 0.99… be equal when they are distinct objects? It’s like equating an apple to an orange. A lot of explaining is needed, if at all possible, to make sense out of this nonsense.

It is true that the decimals are nothing new. In fact, they have their origin in Ancient India but until the construction of the contradiction-free new real number system nonterminating decimals were ambiguous, ill-defined. A decimal is defined by its digits and if we do not know those digits it is ambiguous; this is the case with any nonterminating decimal. So is an integer divided by a prime other than 2 or 5; the quotient is ill-defined. Thus, the concept of an irrational number is ambiguous but we did not realize it because all along we relied on traditions and did not realize that previous generations of mathematician could have made a mistake or that the world has changed and what was correct then is no longer so now.

The dark number d* is the well-defined counterpart of the ill-defined infinitesimal of calculus. It is set-valued and a continuum that joins the adjacent predecessor-successor pairs of decimals under the lexicographic ordering into the continuum R*, the new real number system. The decimals, of course, are countably infinite and discrete.

To dismiss difficult mathematics or physical theory is like sticking one’s head into the ground as the ostrich does. New ideas are often difficult initially, especially, when they grate one’s hard-earned achievements as they did in my case. If they are right they will pass the test of time. A number of my papers made it to the list of most downloaded papers at Elsevier Science, Ltd, Science Direct website since 2002. At any rate, I will be happy to clarify specific points in my work right here or my message board at http://users.tpg.com.au/pidro/

With respect to FLT I have recently posted my rejoinder on several websites including Larry Freeman’s False Proof. I post it again here with slight editing to avoid redundancy:

Rejoinder on FLT

1. Since every mathematical system is well-defined only by its axioms, universal rules of inference, e.g., formal logic, are irrelevant since they have nothing to do with the axioms.

2) The choice of axioms is arbitrary and depends on what one wants his mathematical system to do provided they are CONSISTENT since inconsistency collapses a mathematical system to nonsense. However, once the axioms are chosen the mathematical space becomes a deductive system where the truth or validity of the theorems rests solely on them.

3) The trichotomy axiom which is false in the real number system is true in the new real number system, a consequence of its lexicographic ordering.

4) To avoid ambiguity or error every concept must be well-defined, i.e., its existence, behavior or properties and relationship with other concepts MUST BE SPECIFIED BY THE AXIOMS. Thus, undefined concepts are allowed only INITIALLY but the choice of the axioms and the construction of the mathematical system are incomplete until every concept is WELL-DEFINED. Existence is important because vacuous concept often yields contradiction. We give another example of a vacuous concept: the greatest integer. Let N be the greatest integer. By the trichotomy axiom one and only one of the following axioms holds: N < 1, N = 1, N > 1. The first inequality is clearly false. If N > 1, then N^2 > N, contradicting the choice of N. therefore N = 1. This is the original statement of the Perron paradox and it is blamed on the vacuous concept N.

5) There are other sources of ambiguity, e.g., large and small numbers due to limitation of computation and infinite set. The latter is ambiguous because we can neither identify most of its elements nor verify the properties attributed to them.

6) Another source of ambiguity is self-referent statement such as the barber paradox: the barber of Seville shaves those and only those who do not shave themselves; who shaves the barber? A statement is self referent when the referent refers to the antecedent or the conclusion to the hypothesis. Unfortunately, the indirect proof is self-referent.

7) Of course, the real number system and, hence, FLT are ambiguous in view of the counterexamples to the trichotomy axiom by Felix Brouwer and this blogger and to the completeness axiom (a variant of the axiom of choice) by Banach-Tarski.

8) What do all these mean? FLT is nonsense being formulated in the inconsistent real number system. To resolve FLT the real number system must be freed from ambiguity and contradiction by constructing it on CONSISTENT axioms. Then FLT can be formulated in it and resolved.

8) To this end, I constructed the new real number system on the symbols 0, 1 and chose three consistent simple axioms that well-define them; then the integers and the terminating decimals are defined and using the latter the nonterminating decimals are well-defined for the first time.

9) To summarize: (a) the present formulation of FLT is nonsense; (b) to make sense of it the decimals are constructed into the contradiction-free new real number system; (c) then FLT is reformulated in it and (d) shown to be false by counterexamples. The counterexamples are given in a number of papers, especially, The real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 2009, pp. 59 – 84..

References

[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
[3] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[4] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[5] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[6] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[7] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[8] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
[9] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[10]] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier
Science, Ltd.), 2009, Paris.
[11] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
[12] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.

E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
http://users.tpg.com.au/pidro/

I can't say anything about Wiles' proof of Fermat's last theorem, but I can give a short and easy reading direct proof of Fermat's last theorem :

Fermat’s Last Theorem :
« It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into two like powers. I have discovered a truly marvelous proof which this margin is too narrow to contain. »

Abstract :
A property P or notP is attached to all power a^n , P(a^n) or ¬P(a^n) :
P(a^n) = " power a^n is sum or difference of two like powers."
¬P(a^n) = " power a^n isn’t sum or difference of two like powers."

Use of rules of mathematical bivalent logic to establish a property inherited by coprime factors of power equal to sum or difference of two like powers.

Establishment of a reduction rule or "finite descent" method :
Theorem F (F : tribute to Fermat) :
If a power of degree n is sum or difference of two like powers, it is true also for its prime factors.

Abel’s conjecture :
( z, y, x, n € N+, n>2) (< z,y,x >=1, z^n = y^n + x^n)
---> none of z, y, x can be prime-powers.

Theorem A (A : tribute to Abel) :
A prime-power of degree n>2 can't be sum or difference of two like prime-powers.

Conclusion :
A power of degree n>2 can't be sum or difference of two like powers.

happy-arabia.org/FLT proof.pdf

Ahmed Idrissi Bouyahyaoui

Professor Escultura’s doubts about Andrew Wiles ‘proof’ of Fermat’s Last Theorem is indicative of a proof that covers more than 100 pages, for which only a minority have the required understanding to form a reliable opinion as to whether Wiles’s reasoning was correct. By the same token it has a potential difficulty when it comes to confirming or denying whether certain objections are valid or invalid. But Professor Escultura goes further than refuting Wiles. On his personal website, he provides a basis for stating:

“The new real number system has countably infinite counterexamples to FLT and proof of Goldbach’s conjecture. The resolution says that FLT is not only undecidable but also false (the questions of decidability and truth are separate). Undecidable propositions are characterized as ambiguous propositions.”

I regret to say, this statement is unsound. In fact there exists a proof for the cubic, which can be generalised to cover all prime exponents, using only the elementary algebra available to Fermat, and which was discovered before Wiles made his announcement in the mid-1990s. The proof also lays claim to being a “demonstrationem mirabilam”, and was the work of a UK mathematician and College Head of Department, with two decades experience, teaching advanced mathematics, including Oxbridge Entrance Examinations, and Mathematics Olympiads.
The elegance of his proof is that it infallibly shows if A^p +B^p = C^p then each one of the three terms is one half the sum of three smaller terms, a, b, c, each raised to the power p; gcd(a,b,c)=1.
This quickly leads to the final stage in which a^p + pabc2m + b^p = c^p (m is a +positive integer).
A binomial expansion for (a + b = c) raised to p, when factored, becomes a^p + pabcK + b^p = c^p
It can be seen from this that 2m=K. But 2m is an even number; whereas, K is always odd. This has been proved.

For example:
c^3=a^3 +3abc(1) + b^3 K=1 FLT implies c^3 = a^3 +3abc(2m) + b^3
c^5=a^5 +5abc(a^2+ab+b^2) + b^5 (K must be odd). FLT implies c^5=a^5 +5abc(2m) + b^5
c^7=a^7+7abc(a^4+2a^3b+3a^2b^2+2ab^3+b^4) (K must be odd), and c^7=a^7+7abc(2m)+b^7
...

This proof has been known for almost 20 years, and can be seen to cover any odd prime exponent. It seems highly probable that it was this that engaged Fermat’s attention. If so, it would explain why he threw out challenges to prove only the cubic, and never mentioned higher prime powers. He didn’t need to; proof of the cubic automatically leads to all higher prime powers.
Unfortunately, the problem with following the ideas of Fermat is twofold. Firstly, many hundreds of false proofs have been submitted over the years, these have had the effect of poisoning the well. Established mathematicians will no longer condescend to read anything claiming to be a proof. This bears upon the second reason. The consensus of academic opinion downgraded Fermat’s ‘theorem’ to a conjecture: one that he had mistakenly believed he had proved using elementary mathematics. Three and a half centuries of investigative mathematics by many of the world’s leading mathematicians failed to recover that proof. This led to doubt that it ever existed.
The consequences of this have been extreme, and in one sense bizarre. Respected journals of mathematics in both Europe and the US have adopted the same attitude, and automatically decline to consider any material associated with Fermat’s Last Theorem if the submitter is unrecognised. This explains why the proof outlined above is unknown.

David Roper

The first poster in this conversation, calling himself "edescultura", is an idiot. He thinks that because he can use sqrt(-1) to prove that 1 = -1, the concept of sqrt(-1) must be meaningless. Has it not occurred to him that mathematicians better than him have studied and used this concept and found it useful? Isn't it more likely that his simple "proof" has a mistake in it? It does. He assumes arbitrarily that the square root of 1 is 1. Wrong. It can be 1 or -1, and you must choose the appropriate one.

Thanks for your comment, David. As far as I can see, you are confirming the facts that are in the Entry. I don't think you need us to make any changes to the Entry as it stands at the moment, do you?

If you are interested in search for Fermat's proof, look at :

http://viXra.org/abs/1304.0070 click on PDF .

I don't speak French well enough to be able to read mathematics in it. What are you saying?