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 [9]. 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, I think it is time to present the foundational basis of my counterexamples to it 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 system is a powerful statement that says the system is nonsense the reason true mathematicians are sensitive to it.

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. Inconsistency collapses a mathematical system since any conclusion drawn from it is contradicted by another. 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. My critics might try to explain it here and I shall gladly respond.

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. (The early historians were Westerner who did not know the flourishing Ancient Indian civilization 26,000 years ago, its splendor recorded in Vedic writings) 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 laws of nature are transitory. For instance, there was no biological law 7 billion years ago and all laws of nature will vanish as our universe reaches its destiny. Consequently, the language of science which is mathematics can change, too, and mathematicians must be alert to such change.

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.

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 on this blogsite 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 Wikipedia and Larry Freeman’s. 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 is incomplete until every concept is WELL-DEFINED. Existence is important because vacuous concept often yields contradiction. We 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. 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 by Banach-Tarski. 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 ref. [6].

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