Tuesday, June 13, 2006

Another False Proof: E. E. Escultura

I must admit that I have a fascination with false proofs. Perhaps, that is why I am so interested in Fermat's Last Theorem.

Because of the many false proofs out there and because so many people are unable to tell the difference between mathematics and pseudomathematics, I have started a new blog track: False Proofs.

This past week I was honored with a long, blog comment from Mr. E. E. Escultura who believes that he has found a flaw in the real number system. The flaw according to Mr. Escultura is the axiom of trichotomy and the solution is to avoid using nonterminating decimals. If you are interested, you can find the details here as well as links to the many other blogs that have features Mr. Escultura's ideas.

E. E. Escultura said...

The flaw according to Mr. Escultura is the axiom of trichotomy and the solution is to avoid using nonterminating decimals. If you are interested, you can find the details here as well as links to the many other blogs that have features Mr. Escultura's ideas.

Response:

This is incorrect interpretation of my post. I built the decimals on the finite basic integers, 0, 1, ..., 9 to avoid the ambiguity of infinite set. The new real number system includes the well-defined nonterminating decimals plus the new integers and two nonstandrd numbers, the dark number d* and the unbounded number u*.

E. E. Escultura

E. E. Escultura said...

Hi Larry,

Let me introduce another false proof: Gauss' diagonal method to prove that the decimals are non-denumerable. The set of non-diagonal elements generated by this method is countable.

Another false proof is Goedel's proof of his incompleteness theorem since it involves mapping between two distinct mathematical spaces which are independent of each other andwhere each is defined by its own axioms.

E. E. Escultura

E. E. Escultura said...

Response to Critics

1) The most important contribution of David Hilbert is the realization almost a century ago that the concepts of individual thought cannot be the subject matter of mathematics because they are not accessible to others and, therefore, can neither be studied or discussed collectively nor axiomatized. He then proposed that the subject matter of any mathematical system be objects in the real world e.g., symbols that everyone can look at provided their behavior, properties and relationship among themselves are specified by a CONSISTENT set of premises or axioms.

2) Thus, universal rules of inference, e.g., formal logic, are useless since they have nothing to do with the axioms.

3) I agree with Hilbert and critics who disagree have no debate with me.

4) The choice of the axioms depends on what one wants to do with his mathematical system as long as they are CONSISTENT because inconsistency collapses a mathematical system. Once the axioms are chosen the mathematical space becomes a deductive system.

4) To avoid ambiguity/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, we can only admit undefined concepts INITIALLY but the choice of the axioms is incomplete until EVERY CONCEPT is well-defined. Existence is important to avoid vacuous concept that can yield contradiction. Example of a vacuous concept: the root of the equation x2 + 1 = 0. That root has been denoted by i = sqrt(-1) from which one can draw the conclusion i = 0 and 1 = 0.

5) There are other sources of ambiguity, e.g., large and small numbers due limitation of computation and infinite set. Infinite set is ambiguous because we can neither identify most elements and 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?

7) A statement is self referent when the conclusion refers to the hypothesis. Unfortunately, the indirect proof is self-referent.

8) Here is a familiar equation that has generated much controversy: 1 = 0.99…; apologists of this equation must explain in what sense 1 and 0.99… are equal when they certainly are distinct objects. It’s like equating apples and oranges.

9) The 12 field axioms of the real number system are inconsistent; therefore, they do not well-define the real numbers. The trichotomy axiom and the incompleteness axioms (a version of the axiom of choice) are false.

10) What do all these mean? FLT is nonsense being formulated in the inconsistent real number system. Therefore, to resolve FLT the real number system must be freed from its ambiguity and contradictions by constructing it on CONSISTENT axioms. Then FLT can be formulated in it without ambiguity and resolved.

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

12) 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.

E. E. Escultura

E. E. Escultura said...

Regarding Goedel's incompleteness theorem, since independent mathematical systems such as the integers and the propositional calculus are each defined by its own axioms concepts in one are ill-defined in the other. Therefore, the proof of Goedel's incompleteness theorem contains ill-defined, ambiguous concepts and the proof itself is ill-defined.

E. E. Escultura

E. E. Escultura said...

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 and missed what David Hilbert already knew almost a century ago that such concepts are ambiguous being unknown to others. 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). I hope he has seen my article, Two fatal defects of Wiles’ proof of FLT, posted in several blogsites and websites.

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

4) He claims mathematicians (he probably means some mathematicians) are happy with traditional mathematics. I wish them continued bliss of innocence.

5) 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, 30(8), Dec. 1997, 521 – 532; the journal is a
publication of Elsevier Science Ltd. based there in Amsterdam.

6) He claims he can compute with nonterminating decimals. His claim is based on unclear thinking. Can he add sqrt2 and sqrt3 and write the precise sum?

7) 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.

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

Conclusion.

The article is not well thought out and uses rumors and gossips. It quotes Alecks Pabico an amateur journalist who lost his job as a journalist for commenting on an issue he knows nothing about and writing about it that 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

E. E. Escultura said...

Summation of 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.

E. E. Escultura

E. E. Escultura said...

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.. It is posted on this message board.

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:

and the discussion is coming to a close as no new issues are being raised.

E. E. Escultura

E. E. Escultura said...

Needless to say, none of my criticisms of my positions on Wiles’ proof of FLT or my critique of the real and complex number systems have been challenged successfully on this website or 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.

E. E. Escultura

E. E. Escultura said...

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. 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 is inconsistent. Felix Brouwer and myself constructed counterexamples to the trichotomy axiom which means that it is false. Banach-Tarski constructed a contradiction to the axiom of choice, one of the field axioms. One version says that if a soft ball is sliced into suitably little piece and rearranged without distortion they can be reconstituted into a ball the size of Earth. This is a topological contradiction in R^3.

E. E. Escultura

E. E. Escultura said...

4) Vacuous concept generally yields a contradiction. For example, consider this vacuous concept: the root of the equation x^2 + 1 = 0. That root has been 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 the paper, The new mathematics and physics, Applied Mathematics and Computation, 138(1), 127 – 149, says that the rationals and irrationals are separated, i.e., the union of disjoint open sets.
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.

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.

E. E. Escultura

E. E. Escultura said...

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 there does not exist a nondenumerable set; 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.

E. E. Escultura

E. E. Escultura said...

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].

E. E. Esculturaa

E. E. Escultura said...

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.

E. E. Escultura

E. E. Escultura said...

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 ItÃ´, 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
http://users.tpg.com.au/pidro/

E. E. Escultura said...

Being a newcomer in mathematics and science (my first publication is “The solution of the gravitational n-
body problem”, Nonlinear Analysis, Series A, 30(8), Dec. 1997, 521 – 532), I was an underdog in contending ideas with the heavy weights of science and mathematics. Therefore, I had to disseminate my ideas broadly as quickly as possible. The first forum I posted my message in was SciMath, 1997, and the topic was the equality 1 = 0.99… I said that this was really nonsense and I’ll explain why later.

There was a howl of protest and hundreds of messages were posted in protest during the year. Some called me crackpot, lunatic, moron, etc. One even wrote my colleague (who sort of discovered me in mathematics), Prof. V. Lakshmikantham, a famous mathematician who founded the only rapidly expanding field of mathematics today, Nonlinear Analysis, founder and editor-in-chief of several scientific journals, and president and founder of the International Federation of Nonlinear Analysts, to tell him that he was a lunatic for associating with me. There were at least five such guys in SciMath. However, in due course they pulled enough rope to hang themselves with academically and are all quiet now.

Going back to 1 = 0.99…, it was David Hilbert who recognized almost a century ago that the concepts of individual thought, being inaccessible to others, are ambiguous and cannot be discussed, studied and analyzed collectively. Therefore, they cannot be the subject matter of mathematics. The proper subject matter of mathematics must be objects in the real world, e.g., symbols that we also call concepts that everyone can look at provided they are subject to consistent premises or axioms. Clearly, 1 and 0.99… are distinct objects like apple and orange and to say apple = orange is simply nonsense.

As it will turn out SciMath is the best forum in this category in terms of open participation and, naturally, diversity of ideas. The worst in this category, however, is Wikipedia along with its sister website Wikia. Wikipedia requires consensus on posted topic. In other words, it requires uniformity of thought. Wikia specifically bars original research. In effect, they block the progress of science and mathematics which do not thrive on consensus and their progress stands on original research. Between these two extremes the blogs and websites range from good to excellent in terms of diversity of ideas with the only exception of HaloScan and its sister website, DLMSY, which cannot stand contrary opinion. Consequently, they lose bloggers. I identify a few excellent ones in the category of in-betweens: False Proofs, MathForge, WorldPress and Faces of the Moon. I add Knowledgerush in terms of ease in posting – no no username and password which are easy to forget.

However, there is a forum that is a class by itself in terms of the quality and level of intellectual discussion: ISCID (International Society for Computing and Intelligent Design (?)). I recommend experts to visit this website.

E. E. Escultura
Research Professor
Lakshmikantham Institute for Advanced Studies and Departments of Mathematics and Physics
GVP College of Engineering, JNT University, Visakhapatnam, AP, India
E-mail: escultur36@gmail.com * URL: http://users.tpg.com.au/pidro/

E. E. Escultura said...

What is the difference between music and mathematics?

Answer: Classical music is a treasure but classical mathematics is a trash.

E. E. Escultura said...

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).

E. E. Escultura said...

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.

E. E. Escultura said...

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.

E. E. Escultura said...

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 < d < x.

E. E. Escultura said...

Now, we allow delta to vary steadily in its domain and also the a_js along the basic integers (not simultaneously 0). Then their terms trace the tail digits of all the decimals and as n increases indefinitely they become smaller and smaller and indistinguishable from each other. We call their nonstandard limits dark numbers and denote by d* which is set valued, countably infinite and includes every g-limit of the nonstandard d-sequence (7). To the extent that they are indistinguishable d* is a continuum (in the algebraic sense since no notion of open set is involved). Thus, the tail digits of the nonterminating decimals merge and form the continuum d*.
At the same time, since the tail digits of all the nonterminating decimals form a countable combination of the basic digits 0, 1, …, 9 they are countably infinite, i.e., in one-one correspondence with the integers. In fact, any set that can be labeled by integers or there is some scheme for labeling them by integers is in one-one correspondence with the integers, i.e., countably infinite. It follows that the countable union of countable set is countable. Therefore, the decimals and their tail digits are countably infinite. However, as the nth d-terms of (7) trace the tail digits of the nonterminating decimals they become unidentifiable and cannot be labeled by the integers anymore; therefore, they are no longer countable. In fact they merge as the continuum d*.

E. E. Escultura said...

Like a nonterminating decimal, an element of d* is unaltered if finite g-terms are altered or deleted from its g-sequence. When delta = 1 and a_1a_2…a_k = 1 (7) is called the basic or principal d-sequence of d*, its g-limit the basic element of d*; basic because all its d-sequences can be derived from it. The principal d-sequence of d* is,

(0.1)^n , n = 1, 2, … (8)

obtained by the iterated difference,

N – (N – 1).99… = 1 – 0.99... = 0 with excess remainder of 0.1;
0.1 – 0.09 = 0 with excess remainder of 0.01;
0.01 – 0.009 = 0 with excess remainder of 0.001;
………………………………………………… (9)

Taking the nonstandard g-limits of the left side of (9) and recalling that the g-limit of a decimal is itself and denoting by d_n the d-limit of the principal d-sequence on the right side we have,

N – (N – 1).99… = 1 – 0.99... = d_n. (10)

Since all the elements of d* share its properties then whenever we have a statement “an element d of d* has property P” we may write “d* has property P”, meaning, this statement is true of every element of d*. This applies to any equation involving an element of d*. Therefore, we have,

d* = N – (N – 1).99… = 1 – 0.99... (11)

Like a decimal, we define the d-norm of d* as d* > 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

Ahmed Idrissi Bouyahyaoui said...

THE FERMAT’S LAST THEOREM :
« z, y, x, n Ïµ N*, n>2 : z^n ≠ y^n + x^n »
DIRECT PROOF :
Resolution method : recurrence by induction.

The proposition :
(1) ( y, x, n Ïµ N*, gcd(y,x)=1, n>2 : (1^n ≠ ^y^n + x^n))˄
( y, x, n Ïµ N*, gcd(y,x)=1, n>2 : (1^n ≠ y^n - x^n))˄
( y, x, n Ïµ N*, gcd(2,y,x)=1, n>2 : (2^n ≠ y^n + x^n))˄
( y, x, n Ïµ N*, gcd(2,y,x)=1, n>2 : (2^n ≠ y^n - x^n))˄
( y, x, n Ïµ N*, gcd(3,y,x)=1, n>2 : (3^n ≠ y^n + x^n))˄
( y, x, n Ïµ N*, gcd(3,y,x)=1, n>2 : (3^n ≠ y^n - x^n))˄
( y, x, n Ïµ N*, gcd(4,y,x)=1, n>2 : ((3+1)^n ≠ y^n + x^n))˄
( y, x, n Ïµ N*, gcd(4,y,x)=1, n>2 : ((3+1)^n ≠ y^n - x^n)) ,
gives the first three and the fourth elements of a recurrence.

By hypothesis : z^n ≠ y^n + x^n for z, y, x, n Ïµ N* and n>2, the three numbers z^n ≠ y^n + x^n, y^n ≠ z^n - x^n and x^n ≠ z^n - y^n with z^n > y^n > x^n ≥ 1 .
So, it needs at least three consecutive elements to determine if the series with first elements (1) is inductive or not.

Proof of (1) :
As any integer n>2 is multiple of 4 or odd prime, it suffices to prove
the Fermat’s last theorem for n=4 and for each odd prime.

For n=4 :
2^4 = y^4 - x^4 = (y^2-x^2)( y^2+x^2) , impossible equality, the factor (y^2+x^2) , greater than 4, isn’t power of 2 (y^2+x^2 ≡ 2 mod 4).
3^4 = y^4 - x^4 = (y^2-x^2)( y^2+x^2) , impossible equality, the factor (y^2+x^2) , greater than 3, isn’t power of 3 (y^2+x^2 ≡ 2 mod 3).

For n=p , p odd prime :
2^p = y^p – x^p
2=y-x mod p → y-x≠1 , 2^p = (y-x)[(y^p – x^p)/(y-x)] ,
the two factors of the second member are necessarily coprime, therefore the equality is impossible.
3^p = y^p – x^p
3=y-x mod p → y-x≠1 , 3^p = (y-x)[(y^p – x^p)/(y-x)] ,
for p≠3, the two factors of the second member are necessarily coprime, therefore the equality is impossible.
For p=3 :
3^3 = (y-x)[(y^3 – x^3)/(y-x)]= (y-x)(y^2+y*x+x^2), impossible equality, since
y^2+yx+x^2 > (y-x)^2 > y-x ≥ 3 → y-x = 3 , (y-x)^2= 32, y^2+y*x+x^2 > 3^2 .

Given the proposition (1), suppose that for an integer r > 4 and for all integer z, 1 ≤ z ≤r, we have :
(2) ( y, x, n Ïµ N*, n>2 : (z^n ≠ y^n + x^n))˄
( y, x, n Ïµ N*, n>2 : (z^n ≠ y^n - x^n)).
In this hypothesis, we also have :
y, x, n Ïµ N*, n>2 : (r+1)^n ≠ y^n + x^n,
otherwise y, x, n Ïµ N*, n>2 : (r+1)^n = y^n + x^n → y^n = (r+1)^n - x^n contrary to the hypothesis (2),
since (r+1) > y > x → x < y ≤ r → y^n ≠ (r+1n - x^n → (r+1)^n ≠ y^n + x^n.

The principle of recurrence by induction allows to conclude :

The proposition y, x, n Ïµ N*, n>2 : z^n ≠ y^n + x^n is true for z=1, 2, 3, 4, …... , r (r>4) and r+1, it is true for all z Ïµ N* :
(3) z, y, x, n Ïµ N*, n>2 : z^n ≠ y^n + x^n .

Detailed explanation :

Let P(z) = (z^n ≠ y^n + x^n)˄(z^n ≠ y^n - x^n) = P+(z)˄P-(z) .
As P(1) (then P+(1)), P(2) (then P+(2)), P(3) (then P+(3)), P(4) (then P+(4)) are true and, for r>4, P(r) (then P+(r)) implies P+(r+1),
the principle of recurrence by induction allows to conclude :
P+(r) is true for all r : y, x, n Ïµ N*, n>2 : r^n ≠ y^n + x^n .

Ahmed Idrissi Bouyahyaoui

Ahmed Idrissi Bouyahyaoui said...

I'm sorry, the "proof" given above is insufficient.
I propose what I believe to be sufficient proof. This proof has two parts:
1) classical logic,
2) arithmetic

www.happy-arabia.org/FLTproof.pdf

Ahmed Idrissi Bouyahyaoui

E. E. Escultura said...

1) I did not prove FLT; on the contrary, I disproved it by counterexamples.

2) Using standard mathematics I do not find a flaw in your argument.

3) My main point is that the field axioms of the real number system are inconsistent; therefore, this number system is ill-defined; consequently, FLT being formulated in it is also ill-defined, ambiguous and ill-formulated as a problem.

3) Therefore, I constructed the consistent new real number system R* using the elements 0, 1 well defined by the addition and multiplication tables (as the two other axioms).

4) Then I constructed the countable counterexamples to FLT in R*.

Thank you for the comment.

E. E. Escultura

E. E. Escultura said...

Correction to my previous post:

I constructed countably infinite counterexamples to FLT. EEEscultura

Ahmed Idrissi Bouyahyaoui said...

Ahmed Idrissi Bouyahyaoui

E. E. Escultura said...

CALL FOR A GRAND UNIFIED JOINT CELEBRATION

Materialist philosophers of all cultures must have pondered this question: what are the basic constituents of matter? The Greeks answered it with four elements they found in nature: earth, water, fire and air. The Chinese added one more item – wood. Of course, they were not satisfactory and since then the search for the basic constituent of matter was in limbo for 5,000 years until in the 1950s inspired by the exciting development of quantum physics particle physicists renewed the search with vigor by smashing the nucleus of the atom in pursuit of the basic irreducible elementary particles or building blocks of matter. By the 1990s the search was a complete success with the discovery of the +quark (up quark) and quark (down quark) and the electron (discovered in 1897). They are basic as constituents of every atom; a heavy isotope has one more constituent – the neutrino. The particle physicists have, indeed, found what they were looking for – the irreducible building blocks of matter – and whatever they have found beyond these is a bonus for natural science.

In the 1980s dark matter came to the fore with overwhelming evidence of its existence [6,7,8] and, using the new methodology of qualitative modeling that explains nature and its appearances in terms of natural laws [1,5], was established in 1997 [4] as one of the two fundamental states of matter the other being ordinary or visible matter [2,5]. That same year the building block of dark matter, the superstring, was discovered as the crucial factor for the solution of the gravitational n-body problem [4] and development of the grand unified theory (GUT). The latter has been established in a series of papers since 1997 and consolidated in [2]. There is only one basic constituent in view of the non-redundancy and non-extravagance natural principles [3] just as there is only one electron since all electrons have identical structure, properties, behavior and functions and differ only in locations. Moreover, it was also established that the superstring coverts to the basic elementary particles as agitated superstring [1,2,3]. In effect, this established the superstring as the basic constituent of matter, dark and visible
[1,2,3,4,5].

This happy turn of events came without fanfare and even without notice. It is an important milestone for science and calls for a grand unified joint celebration by particle and theoretical physicists to mark these monumental achievements and the threshold of a new epoch for natural science and its applications. It is even worth a world congress of particle and theoretical physicists.

References

[1] Escultura, E. E., The mathematics of the grand unified theory, Nonlinear Analysis,
A-Series: Theory: Method and Applications, 71 (2009)
e420 – e431.
[2] Escultura, E. E., The grand unified theory, Nonlinear Analysis, A-Series: Theory: Method and Applications, 69(3), 2008, 823 – 831.
[3] Escultura, E. E., Qualitative model of the atom, its components and origin in the early universe, Nonlinear Analysis, B-Series: Real World Applications, 11 (2009),
29 – 38.
[4] Escultura, E. E., The solution of the gravitational n-body problem, Nonlinear Analysis, A-Series: Theory, Methods and Applications, 38(8), 521 – 532.
[5] Escultura, E. E., Superstring loop dynamics and applications to astronomy and biology, Nonlinear Analysis, A-Series: Theory: Method and Applications, 35(8), 1999, 259 – 285.
[6] Astronomy (a) August 1995, (b) January 2001, (c) June 2002.
[7] Science, Glow reveals early star nurseries, July 1998.
[8] Science, (a) Starbirth, gamma blast hint at active early universe, 282(5395), December, 1998, 1806; (b) Gamma burst promises celestial reprise, 283(5402),
January 1999; (c) Powerful cosmic rays tied to far off galaxies, 282(5391), Nov. 1998, 1969 – 1971.

kSauga said...

Wow... What an awesome example of a crank. I have never seen someone go on such a diatribe about absolute nonsense. I have posted concrete arguements against Esculturas number system and 'disproof' of FLT. He fails to respond to any onf them, but then says nobody has ever argued convicingly against him. I am far from the only one, there are many blogs with irrefutable proof that Escultura is incorrect. However, instead of arguing against their points, he just posts a seemingly never ending series of comments where he does nothing but repeat himself over and over, never once giving any proof that their arguments against him are incorrect.

What in insecure little man!

E. E. Escultura said...

You are rather late, Keven, but before you pull out something from the flat of your foot read the original version of my counterexamples to FLT in:

Escultura, E. E., The new real number system and discrete computation and calculus, J. Neural, Parallel and Scientific Computations (Dynamic Publishers), 2009, Vol. 17, pp. 59 – 84.

Read and refute my arguments point by point.

Unknown said...

FERMAT's LAST THEOREM

X ^n + Y^n = Z^n , X,Y,Z and n are Natural Positive ( N+) , and n >2

MAROSZ' s conditions ( couterexample )

X= [2^(n/n)] , Y = [2^n/n] , Z = [2^n+1/n] and n=3

n=3 > 2 [ok ]

2^(n/n) = 2 it is ( N+) , [ok ]

2^(n+1)/n = 2^(4/3) = 16^(1/3) = 4^(1/2) = 2 it is N+ [ok]

Fermat's equation and Marosz's conditions

basic equation : [2^(n/n)]^n + [2^n/n]^n = [2^n+1/n]^n

LEFT
[2^(n/n)]^n = 2^ [(n/n) *n] = 2^n

2^n + 2^n = 2*(2^n) , n=3 2* (2^3) = 16

RIGHT
[2^((n+1)/n)]^n = [2^(4/3)]^3 = 2^4 = 16

authore
http://maroszmaciej.blogspot.com/

E. E. Escultura said...

Reaply to Maciej Ma

FLT is a deeper problem that what it seems. It is posed as a problem in the system of integers or the real number system that includes the former as a subspace neither of which is well defined. In fact, the real number system is inconsistent, counter examples to its trichotomy and completeness axioms having been constructed. Therefore FLT is ambiguous and cannot be resolved.
For complete discussion of these issues and the resolution of FLT see, Escultura, E. E., The new real number system and discrete computation and calculus. J. Neural, Parallel and Scientific Computations, 17, 2009, pp. 59 - 84Moreover, complete discussion of these issues are posted on this wibsite. E. E. Escultura

E. E. Escultura said...

Correction.

FLT cannot be resolved in the real number system but has been re-formulated and resolved in the new real number system.