Computational Complexity: Counterexample to Fermat's Last Theorem Found!!!

Computational Complexity: Counterexample to Fermat's Last Theorem Found!!!

THE FINAL STRETCH IN THE CONSTRUCTION OF THE NEW REAL NUMBER SYSTEM R*: WELL DEFINING THE NONTERMINATING DECIMALS (for the first time)

First we note that since a decimal is defined by its digits the only well defined decimals are the terminating ones. Nonterminating decimals are ill-defined or ambiguous because not all their digits are known. Therefore, the concept rational (and also irrational) is ambiguous because it is impossible to verify if its decimal representation is periodic since we cannot check all its digits being infinite. However, ambiguity can be contained by approximating it with certainty, e.g., by a terminating decimal (which has no ambiguity); such approximation is valid if the margin of error is known and can be made small as desired. Thus, while nonterminating decimals cannot be well defined we can contain its ambiguity to the point where we do algebraic operations with them and approximate the result with desired margin of error. Now we introduce the generating or g-sequence and its g-limit, a nonterminating decimal which has contained ambiguity (approximable by certainty).

A sequence of terminating decimals of the form,

N.a_1, N.a_1a_2, …, N.a_1a_2…a_n, … (5)

where N is integer and the a_ns are basic integers, is called standard generating or g-sequence. Its nth g-term, N.a_1a_2…a_n, defines and approximates its g-limit, the nonterminating decimal,

N.a_a_2…a_n,…, (6)

at margin of error 10^-n. The g-limit of (5) is nonterminating decimal (6) provided the nth digits are not all 0 beyond a certain value of n; otherwise, it is terminating. As in standard analysis where a sequence converges, i.e., tends to a specific number, in the standard norm, a standard g-sequence, converges to its g-limit in the g-norm where the g-norm of a decimal is itself. Note that a nonterminating decimal is well defined by its g-sequence although it is ambiguous.

Since addition and multiplication and their inverse operations subtraction and division are defined only on terminating decimals computing nonterminating decimals is done by approximation each by its nth g-terms (called n-truncation) and using their approximation to find the nth g-term of the result as its approximation at the same margin of error. (Note that the g-nth term is a terminating decimal whose last digit is the nth digit) This is standard computation, i.e., approximation by decimal segment at the nth digit. Thus, we have retained standard computation but avoided the contradictions and paradoxes of the real numbers. We have also avoided vacuous statement, e.g., vacuous approximation, because nonterminating decimals are g-limits of g-sequences which belong to R*. Moreover, we have contained the inherent ambiguity of nonterminating decimals by approximating them by their nth g-terms which are not ambiguous being terminating decimals. In fact, the ambiguity of R* has been contained altogether.

As we raise n, the tail digits of the nth g-term of any decimal recedes to the right indefinitely, i.e., it becomes steadily smaller until it is unidentifiable. While it tends to 0 in the standard norm it never reaches 0 and is not a decimal since its digits are not fixed; ultimately, they are indistinguishable from the similarly receding tail digits of the other nonterminating decimals. In iterated computation when we are trying to get closer and closer approximation of a decimal the tail digits may vary but recede to the right indefinitely and become steadily smaller leaving fixed digits behind that define a decimal. We approximate the result by taking its initial segment, the nth g-term, to desired margin of error.
.

Consider the sequence of decimals,

(delta^n(a_1a_2…a_k), n = 1, 2, …, (7)

where delta is any of the decimals, 0.1, 0.2, 0.3, …, 0.9, a_1, …, a_k, basic integers (not all 0 simultaneously). We call the nonstandard sequence (7) d-sequence and its nth term nth d-term. For fixed combination of delta and the a_j’s, j = 1, …, k, in (7) the nth term is a terminating decimal and as n increases indefinitely it traces the tail digits of some nonterminating decimal and becomes smaller and smaller until we cannot see it anymore and indistinguishable from the tail digits of the other decimals (note that the nth d-term recedes to the right with increasing n by one decimal digit at a time). The sequence (7) is called nonstandard d-sequence since the nth term is not standard g-term; while it has standard limit (in the standard norm) which is 0 it is not a g-limit since it is not a decimal but it exists because it is well-defined by its nonstandard d-sequence. We call its nonstandard g-limit dark number and denote by d. Then we call its norm d-norm (standard distance from 0) which is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 no matter how large n is so that if x is a decimal, 0 < delta =" 1" a_k =" 1" n =" 1," 09 =" 0" 009 =" 0"> 0.

We state some theorems about R*.
Theorem. The d-limits of the indefinitely receding (to the right) nth d-terms of d* is a continuum that coincides with the g-limits of the tail digits of the nonterminating decimals traced by those nth d-terms as the aks vary along the basic digits.
Theorem. In the lexicographic ordering R* consists of adjacent predecessor-successor pairs (each joined by d*); therefore, the g-closure R* of R is a continuum [9].
Corollary. R* is non-Archimedean and non-Hausdorff in both the standard and the g-norm and the subspace of decimals are countably infinite, hence, discrete but Archimedean and Hausdorff.
Theorem. The rationals and irrationals are separated, i.e., they are not dense in their union (this is the first indication of discreteness of the decimals) [7].
Theorem. The largest and smallest elements of the open interval (0,1) are 0.99… and 1 – 0.99…, respectively [6].
Theorem. An even number greater than 2 is the sum of two prime numbers.

Remark. Gauss’ diagonal method proves neither the existence of nondenumerable set nor a continuum; it proves only the existence of countably infinite set, i.e., the off-diagonal elements consisting of countable union of countably infinite sets. The off-diagonal elements are not even well-defined because we know nothing about their digits (a decimal is determined by its digits). We state the following corollaries from our discussion: (1) Nondenumerable set does not exist; (2) Only discrete set has cardinality; a continuum has none.

(This article is excerpted from Escultura, E. E., The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations 17 (2009), 59 – 84)

E. E. Escultura
Research Professor
GVP - V. Lakshmikantham Institute for Advanced Studies
and Departments of Mathematics and Physics
GVP College of Engineering, JNT University
Madurawada, Visakhapatnam. AP, India

CLARIFICATION ON THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM
By E. E. Escultura

Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding. Let me, therefore, make the following clarification:

1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers [8] and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers [3]. The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2.

2) From the definition of d* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].

3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
x^n + y^n = z^n, (F)

for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.

4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [8]. They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture.

The following references include references used in the consolidated paper [8] plus [2] which applies [8]

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] Corporate Mathematical Society of Japan , Kiyosi Itô, Encyclopedic dictionary of mathematics (2nd ed.), MIT Press, Cambridge, MA, 1993
[4] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[5] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[6] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[7] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[8] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[9] 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
[10] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[11] Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
[12] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
[13] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
[14] 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
Madurawada, Vishakhapatnam, AP, India
http://users.tpg.com.au/pidro/

SUMMATION ON THE DEBATE ON THE NEW REAL NUMBER SYSTEM AND THE RESOLUTION OF FERMAT’S LAST THEOREM by E. E. Escultura

The debate started in 1997 with my post on the math forum SciMath that says 1 and 0.99… are distinct. This simple post unleashed an avalanche of opposition complete with expletives and name-calls that generated hundreds of threads of discussion and debate on the issue. The debate moved focus when I pointed out the two main defects of Andrew Wiles’ proof of FLT and, further on, the discussion shifted to the new real number system and the rationale for it. Naturally, the debate spilled over to many blogs and websites across the internet except narrow minded ones that accommodate only unanimous opinions, e.g., Widipedia and its family of websites as well as websites that cannot stand contrary opinion like HaloScan and its sister website, Don’t Let Me Stop You. SciMath stands out as the best forum for discussion of various mathematical issues from different perspectives. There was one regular at SciMath who did not debate me online but through e-mail. We debated for about a year and I learned much from him. The few who only had expletives and name-calls to throw at me are nowhere to be heard from.

There was one unsigned feeble attempt from the UP Mathematics Department to counter my arguments online. But it wilted without a response from the science community because it lacked grasp of what mathematics is all about.

The most recent credible challenge to my positions on these issues was registered by Bart van Donselaar in the online article, Edgar E. Escultura and the Inequality of 1 and 0.99…, to which I responded with the article, Reply to Bart van Donselaar’s article, Edgar E. Escultura and the inequality of 1 and 0.99…; a website on the Donselaar’s paper has been set up:

http://www.reddit.com/r/math/comments/93n3i/edgar_e_escultura_and_the_inequality_of_1_and/

and the discussion is coming to a close as no new issues are being raised. Needless to say, none of my criticisms of Wiles’ proof of FLT or my critique of the real and complex number systems have been challenged successfully across the internet. In peer reviewed publications there is not even a single attempt to refute my positions on these issues.
We highlight some of the most contentious issues of the debate.
1) Consider the equation 1 = 0.99… that almost everyone accepts. There are a number of defects here. Among the decimals only terminating decimals are well-defined. The rest are ill-defined or ambiguous. In this equation the left side is well-defined as the multiplicative identity element while the right side is ill-defined. The equation, therefore, is nonsense.
2) The second point is: David Hilbert already knew almost a century ago that the concepts of individual thought cannot be the subject matter of mathematics since they are unknown to others and, therefore, cannot be studied collectively, analyzed or axiomatized. Therefore, the subject matter of mathematics must be objects in the real world including symbols that everyone can look at, analyze and study collectively provided they are subject to consistent premises or axioms. Consistency of a mathematical system is important, otherwise, every conclusion drawn from it is contradicted by another. In order words, inconsistency collapses a mathematical system to nonsense. Consider 1 and 0.99…; they are certainly distinct objects like apple and orange and to write apple = orange is simply nonsense.
3) The field axioms of the real number system are inconsistent. Felix Brouwer and I constructed counterexamples to the trichotomy axiom which means that it is false. Banach-Tarski constructed a contradiction to the axiom of choice of the field axioms, a variant of the completeness axiom. One version says: if a soft ball is sliced into suitably little pieces and rearranged without distortion they can be reconstituted into a ball the size of Earth, a topological contradiction.
4) Vacuous concept generally yields a contradiction. For example, consider this vacuous concept: the root of the equation x^2 + 1 = 0. That root does not exist and yet it is denoted by i = sqrt(-1). The notation itself is a problem since sqrt is a well-defined operation in the real number system that applies only to perfect square. Certainly, -1 is not a perfect square. Mathematicians extended the operation to non-negative numbers. However, the counterexamples to the trichotomy axiom show at the same time that an irrational number cannot be represented by a sequence of rationals. In fact, a theorem in my paper, The new mathematics and physics, Applied Mathematics and Computation, 138(1), 127 – 149 [4], says that the rationals and irrationals are separated, i.e., the union of disjoint open sets (see also [6].
At any rate, if one is not convinced of the mischief that vacuous concept can play, consider this:
i .= sqrt(-1) = sqrt1/sqrt(-1) = 1/i = -i or i = 0. 1 = 0, and both the real and complex number systems collapse. To hide this contradiction some mathematicians invented two supposed roots, the other i = -sqrt(-1) neither of which exists but choose the latter which does not yield a contradiction. There is no logical or mathematical reason for this choice other than to hide the contradiction. Both of them are nonsense anyway since they are both ill-defined.

5) With respect to Andrew Wiles’ proof of FLT it has two main defects: a) Since FLT is formulated in the inconsistent real number system it is nonsense and, naturally, the proof is also nonsense. The remedy is to first remove the inconsistency of the real number system which I did and reformulate FLT in the consistent number system, the new real number system. b) The use of complex analysis deals another fatal blow to Wiles’ proof. The remedy for complex analysis is in the appendix to the paper, The generalized integral as dual to Schwarz Distribution, in press, Nonlinear Studies [5].

6) By reconstructing the defective real number system into the contradiction-free new real number system and reformulating FLT in the latter, countably infinite counterexamples to it have been constructed showing the theorem false and Wiles wrong.

7) In the course of making a critique of the real number system some new results have been found: a) Gauss diagonal method of proving the existence of nondenumerable set only generates a countably infinite set; b) as of this time no nondenumerable set exists; c) only discrete set has cardinality, a continuum has none..

8) The new real number system is a continuum, countably infinite, non-Hausdorff and Non-Archimedean and the subset of decimals is also countably infinite but discrete, Hausdorff and Archimedean. The g-norm simplifies computation considerably.

Finally, we note that all the issues about the new real number system, my critique of Wiles’ proof of FLT and my counterexamples to FLT to prove it false have been debated thoroughly in cyberspace during the last 12 years and ALL resolved COMPLETELY in my favor. Not a single hole has been punched on my entire work.

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, Nonlinear Studies 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, Neural, Parallel and Scientific Computations 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., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
[11] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
[12] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
[13] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.

E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
and Departments of Mathematics and Physics
GVP College of Engineering, JNT University

BACKGROUNDER ON THE GRAND UNIFIED THEORY (GUT) SOME BASIC INFORMATION by E. E. Escultura

Why do problems in mathematics and physics defy solution or resolution for a long time? In mathematics the most famous unsolved problem was the 360-year-old Fermat’s conjecture known as Fermat’s last theorem (FLT) and in physics it was the 200-year-old Laplace or gravitational n-body problem. The author posed this question in 1988 after a 17-year absence from his mathematical career. Given that both problems appear to be very clearly stated he came to the conclusion that the difficulty lies in the inadequacy and other defects of their underlying fields. He then proceeded to first make a thorough critique of the underlying fields of FLT, namely, foundations, number theory and the real number system and he found, among others that the real number system is inconsistent and is, therefore, ill-defined or ambiguous. Consequently, FLT being formulated in it is also ambiguous and cannot be resolved. He proceeded to construct the consistent new real number system on three simple axioms and reformulate FLT in it to make it clear and open to resolution. Indeed, FLT has countably infinite counterexamples in the new real number system.

What has FLT to do with GUT? The first major theorem in its resolution was the characterization of undecidable (unprovable) propositions that says, essentially, that a proposition is unprovable if it is ambiguous, i.e., involves ambiguous or ill-defined concepts. Being “ill-defined” is the negation of “well-defined” and a concept is well-defined if its existence, properties or behavior and relationship with other concepts are specified by the axioms of the given mathematical space. To avoid ambiguity and contradiction (the latter often hides in the former) every concept in a mathematical space must be well-defined and in its construction the choice of the axioms is not complete until this requirement is achieved. When we have two distinct mathematical spaces every concept in one is ill-defined in the other since each mathematical space is well defined only by its axioms. A physical theory is a mathematical space whose axioms are laws of nature. In a mathematical space the axioms are man-made and have nothing to do with the laws of nature.

In the present methodology of physics called quantitative modeling (formerly called mathematical modeling) natural phenomena are described mathematically and a physical problem is modeled by a mathematical problem so that the solution of the latter is attributed to the solution of the physical problem. Reasoning is purely by analogy since there is no causal relation between the physical and mathematical spaces concerned. This is the reason for the existence of long-standing unsolved problems and unanswered fundamental questions of physics like what the basic constituent of matter and the structure of the electron are.

The remedy for this inadequacy of methodology is qualitative or non-quantitative modeling (formerly called dynamic modeling) that explains nature or natural phenomena in terms of the laws of nature. While quantitative modeling describes the appearances of nature mathematically, qualitative modeling explains its internal dynamics and interactions including its appearances in terms of its laws. The former is based on computation, measurement and intuition, the latter on qualitative mathematics, rational thought and analysis. Qualitative mathematics includes the following routine activity of the mathematician or scientist:
Making conclusions, visualizing, abstracting, thought experimenting, engaging in creative activity, intuition, imagination and trial and error to sift out what is more appropriate, negating what is known to gain some insights into the unknown, altering premises to draw out new conclusions, thinking backwards and all other techniques that yield results.

Qualitative modeling alters the task of the scientist from computation and measurement to the search for the laws of nature. It was used for the first time to solve the gravitational n-body problem in 1997. The solution required the discovery of the basic constituent of matter, the superstring. It required 11 laws of nature to accomplish both. They where the initial laws of nature of GUT known as the flux theory of gravitation then.

At present particle physicists are still smashing the nucleus of the atom in search of the basic constituent of matter, the superstring, which has been going on for over half a century. Actually, the superstring has been staring at us since 1811 when Ernest Rutherford discovered the electron. The electron is an agitated superstring. A non-agitated superstring is dark, i.e., its size is less than 10^(-10) meters. It is the basic constituent of dark matter, one of the two fundamental states of matter, the other being visible or ordinary matter. Dark matter is not observable with present technology and is known only by its impact on visible matter. When suitably agitated by cosmic waves the superstring expands to a primum, unit of visible matter such as the electron or positive or negative quark. These three prima are called basic prima because they are constituents of every atom. They are converted from dark matter at staggering rate in the Cosmos and in the cells of living things – plants or animals. In the Cosmos alone the prima form cosmic dust that get entangled into cosmological vortices and collect at their cores at the rate of one star per minute.

References

[1] Escultura, E. E., The solution of the gravitational n-body problem, Nonlinear Analysis, Series A: Theory, Methods and Applications, 30(8), Dec. 1997, 521 – 532.
[2] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[3] Escultura, E. E. (1999) Superstring loop dynamics and applications to astronomy and biology, J. Nonlinear Analysis, 35(8), 259 – 285.
[4] Escultura, E. E. (1999) Recent verification and applications, Proc. 2rd International Conf.: Tools for Mathematical Modeling, St. Petersburg, vol. 4, 74 – 89.
[5] Escultura, E. E. (2001) From macro to quantum gravity, J. Problems of Nonlinear Analysis in Engineering Systems, 7(1), 56 – 78.
[6] Escultura, E. E. (2001) Quantum gravity, Proc. 3rd International Conference on Dynamic Systems and Applications, Atlanta, 201 – 208.
[7] Escultura, E. E. (2001) Turbulence: theory, verification and applications, J. Nonlinear Analysis, 47(2001), 5955 – 5966.
[8] Escultura, E. E. (2001) Vortex Interactions, J. Problems of Nonlinear Analysis in Engineering Systems, Vol. 7(2), 30 – 44.
[9] Escultura, E. E. (2001) Chaos, turbulence and fractal, Indian J. Pure and Applied Mathematics, 32(10), 1539 – 1551.
[10] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[11] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[12] Escultura, E. E. (2003) Macro and quantum gravity and the dynamics of cosmic waves, J. Applied Mathematics and Computation, 139(1), 23 – 36.
[13] Escultura, E. E., (2003) Dynamic Modeling and Applications, Proc. 3rd International Conference on Tools for Mathematical Modeling, State Technical University of St. Petersburg, St. Petersburg.
[14] Escultura, E. E., (2004) Problems and Unanswered Questions of physics and their resolution, Nonlinear Analysis and Phenomena, I(1), 1 – 26.
[15] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[16] Escultura, E. E., (2005) Dynamic Modeling of Chaos and Turbulence, Proc. 4th World Congress of Nonlinear Analysts, Orlando, June 30 – July 7, 2004; Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, e519-e532.
[17] Escultura, E. E., (2005). The theory of everything, Nonlinear Analysis and Phenomena, II(2), 1 – 45.
[18] Escultura, E. E., (2006) Foundations of Analysis and the New Arithmetic, Nonlinear Analysis and Phenomena, January 2006.
[19] Escultura, E. E., The Pillars of the new physics and some updates, Nonlinear Studies, 14(3), 2007, 241 – 260.
[20] Escultura, E. E., The physics of the mind, accepted, The Journal of the Science of Healing Outcome.
[21] Escultura, E. E., The cosmology of our universe, submitted, Problems of Nonlinear Analysis in Engineering Systems.
[22] Escultura, E. E., (2007) Dynamic Modeling and the new mathematics and physics, Neural, Parallel and Scientific Computations, 15(4), 2007, 527 – 538.
[23] Escultura, E. E., The grand unified theory, contribution to the Felicitation Volume on the occasion of the 85th birth anniversary of Prof. V. Lakshmikantham: Nonlinear Analysis: TMA, 69(3), 2008, 823 – 831.
[24] Escultura, E. E. The mathematics of the grand unified theory, in pres, Nonlinear Analysis.
[25] Escultura, E. E. Dynamic and mathematical models in physic, Proc. 5th International Conference on Dynamic Systems and Applications, June 30 – July 5, 2007, Atlanta, 164 – 169.
[26] Escultura, E. E. (2004) Dynamic Modeling of Chaos and Turbulence, NA, TBA, 63(5-7), e519 – e532.
[27] Escultura, E. E. The basic concepts and dynamics of quantum gravity with applications, in press, Nonlinear Studies
[28] Escultura, E. E., Qualitative model of the atom, its components and origin in the early universe, in press, Nonlinear Analysis: Real World Applications.

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. (Main reference: Escultura, E. E., The new real new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 17 (2009), 59 – 84).

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 press, Nonlinear Studies.

Another example of a vacuous concept is the greatest integer. Let N be the greatest integer. By the trichotomy axiom one and only one of the following axioms holds: N < n =" 1,"> 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. In general, any vacuous concept yields a contradiction.

E. E. Escultura

Reply to Bart van Donselaar’s article, Edgar E. Escultura and the inequality of 1 and 0.999...

1) The reason Bart van Donselaar cannot see why 1 and 0.99… are distinct is he looks at them as concepts in one’s mind. He missed what David Hilbert already knew almost a century ago that such concepts are ambiguous and unknown to others. Therefore, they cannot be the subject matter of mathematics. 1 and 0.99.. are distinct objects in the real world like orange and apple and to write the equation orange = apple is simply nonsense.

2) He could not understand why I “claim” that FLT is false and Wiles’ proof is incorrect since he says the proof is admired Worldwide (actually only four or five mathematicians do). Well, an error is an error and I hope he has seen my article, Two fatal defects of Wiles’ proof of FLT, posted in several blogsites and websites.

3) He relies on dictionary definitions of concepts which is quite inappropriate in mathematics. Constructivism in my sense has nothing to do with intuitionism. It simply avoids sources of ambiguity and contradiction.

4) He claims that constructivists have not found hard evidence of defects in standard mathematics. The evidences is just under his nose: Felix Brouwers’ counterexample to the trichotomy axiom, Putnam and Benacerraf, Philosophy of Mathematics, Cambridge University Press, 1985; I also have my own version in, The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computation, 17(2009), 59 – 84.

5) He thinks mathematicians (he probably means some mathematicians) are happy with traditional mathematics for there is nothing wrong with it. Well, I wish them continued bliss of innocence.

6) He doubts that I have solved the gravitational n-body problem. I did in the paper, The solution
of the gravitational n-body problem, Nonlinear Analysis, Series A: Theory, Methods and Applications,
30(8), Dec. 1997, 521 – 532; the journal is a publication of Elsevier Science Ltd. based there in
Amsterdam.

7) He claims he can compute with nonterminating decimals. Such computation depends on the digits and most of the digits of a nonterminating decimal are unknown. His claim is based on imprecise thinking. At any rate, I would like to see how he did this impossible feat. Can he add sqrt2 and sqrt3 and write the sum precisely? We can only approximate a nonterminating decimal or result of computation with nonterminating decimals.

8) He also cannot understand why it is impossible to verify whether a nonterminating decimal is periodic or nonperiodic. Clue: the digits are infinite and we cannot look at all of them to check.

9) He chastises me for writing difficult mathematics and physical theory. New ideas are initially difficult but if they are correct they will pass the test of time. Initial critics of my work had a hilarious time calling me a crackpot, lunatic, moron, etc., but where are they now? My posts had been picked up by many blogs and websites and my papers have been used by renowned publications such as the Encyclopedic Dictionary of Mathematics and Elsevier Science. A number of them made it to the top 25 most downloaded papers published by Elsevier Science, online at Science Direct archives. Only Wikipedia Encyclopedia have barred my posts entirely because the administrator explains that it requires unanimity of ideas. Therefore, only Wiles’ proof is published there and kept in its archives. HaloScan and DLMSY also have rejected my posts but continue to publish criticisms of my work without my response because they cannot stand contrary opinion.

10) I notice lately, that Wiles’ supporters have done massive promotion of his proof including publication of some books about it. Unless they address point blank my specific criticisms of the proof, it will not prosper.

Conclusion.

The article is not well thought out and uses rumors and gossips. For example, it quotes Alecks Pabico an amateur journalist who lost his job as a journalist for commenting on an issue he knows nothing about or writing about it which he posted in blogsites and websites across the internet.

Bart is unsure of his ideas, makes claims he cannot verify and resorts to name-dropping which makes me doubt if he, like Alecks, understands what he is writing about.

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