It’s all nonsense Ravikiran. Isn’t that clear from that report itself? You could have put an exclamation mark at least.
I might be wrong here, but isn’t the Wiles letter in the article dripping with sarcasm? “I’d like to have the address of the guy who let you get a PhD 30 years ago”? That sounds more like a cheap insult than an admission of failure.
And I really don’t think a person like Wiles would be making cheap insults such as this one. The letter is most likely a forgery or a prank.
And I would suggest that people wait for other mathematicians to back Escultura before believing that the proof is wrong. Too often in science and mathematics, sensational stuff like this one ends up amounting to nothing.
What can I say?
I am red faced.
(Disclaimer: I have not read Escultra’s “work”)
Escultura epitomizes the definition of a crank. Here are a few suggestive sentences from the article:
—
“He took the position that the failure to resolve the problem for over 360 years reveals the inadequacy and defects of foundations, number theory and the real number system. ” – FLT (not Shimura-Taniyama!!) has as little to do with the real numbers as George Bush with intelligence. FLT is a statement about the non-existence of *RATIONAL* solutions; there are tons of real solutions.
“Specifically, two of its axioms (the trichotomy and completeness axioms, for those who took basic algebra in high school and college) are false.”
– How can axiom be wrong? At worst, it can be inconsistent with the theory, but it can’t be wrong.
“The result is a new real number system that is free from defects and contradictions, finite and enriched with new numbers that have important applications for physics.”
– No comment.
“I read all of it just yesterday and let me tell you I respect you”
– If Wiles actually wrote this, it’s a disguised way of saying Escultra’s work is not deep at all. But knowing the kind of excitement FLT generates among amateurs (damm you Pierre de Fermat and your small margins), I’m pretty sure Wiles has learnt to ignore such emails (also, he usually signs differently).
“Escultura was a former math and science editor and columnist of The Manila Times.”
– It gets fishier.
—
Just FYI, there are a few hilarious comments on this issue over at fark.com
Cheers,
Bhargav
The partial list of publications that discusses the counterexamples to Fermat’s last theorem follows:
[18] Escultura, E. E. (1996) Probabilistic mathematics and applications to dynamic systems including Fermat’s last theorem, Proc. 2nd International Conference on Dynamic Systems and Applications, Dynamic Publishers, Inc., Atlanta, 147 – 152.
[20] Escultura, E. E. (1998) FTG VII. Exact solutions of Fermat’s equation (Definitive resolution of Fermat’s last theorem, J. Nonlinear Studies, 5(2), 227 – 254.
[29] Escultura, E. E. (2002) FTG V. The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[31] Escultura, E. E. (2003) FTG XVII: The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[35] Escultura, E. E. FTG XV. The new nonstandard analysis and the intuitive calculus, submitted.
[36] Escultura, E. E. (2002) FTG VI. The philosophical and mathematical foundations of FLT’s resolution, rectification and extension of underlying fields and applications, accepted, International J Nonlinear Differential Equations.
[37] Escultura, E. E. (2002) FTG XXII. Extending the reach of computation, accepted, International J Nonlinear Differential Equations.
[43] FRG XXVII – XXVIII. The new frontiers of mathematics and physics. Part II. The new real number system: Introduction to the new nonstandard analysis, Nonlinear Analysis and Phenomena, II(1), January, 15 – 30.
[45] Escultura, E. E. FTG. XXVI (2005). The resolution of Fermat’s last theorem and applications, accepted, Nonlinear Analysis.
Take your comments from the top, not from the flat of your foot.
Cheers.
E. E. Escultura
For those interested in the countably infinite counterexamples to FLT, here they are:
Let x = (0.99…)^T, where T = 1, 2, …, is an ordinary integer.
y = d* = 1 – 0.99…, called dark number.
z = 10^T.
For n > 2,
x^n + y^n = z^n.
This countably infinite set of new integers provides countably infinite counerexamples to FLT that prove this conjecture false. Moreover, the the countably infinite triples (kx,ky,kz), k = 1, 2, …, are also counterexamples to FLT since
(kx)^n + (ky)^n = (kz)^n.
These counterexamples are constructed from the new integers, namely, d* and numbers of the form N.99…, where N = 0, 1, … is an ordinary integer. They form a subspace of the contradiction-free new real number system.
The mapping,
N -> (N-1).99…, N = 1, 2, …,
is an isomorphic embedding of the ordinary integers into this contradiction-free mathematical space as integral parts of the decimals. This resolves a fundamental flaw of number theory that the integers are ill-defined. In other words, until this embedding, the integers were nonsense.
For details of the construction, visit my websites.
E. E. Escultura
For the benefit of the experts I am sharing my strategy for the capture of FLT.
1) I wondered why this conjecture could not be resolved for centuries and I concluded that present mathematics is defective or, at least, inadequate. So I embarked on a critique of the its underlying fields, namely, foundations, number theory and the real number system.
2) In foundations here is what I found:
a) I agreed with David Hilbert that the concepts of individual thought cannot be the subject matter of a mathematical space because they are not accessible to others and can neither be studied collectively nor axiomatized. His remedy was to represent them by symbols well-defined by a consistent set of axioms. A symbol or concept is well-defined if its EXISTENCE, properties and relationship with other symbols are specified by the axioms. I stress existence because vacuous concepts are contradictory as the following definition shows: A triquadrilateral is a plane figure with three vertices and four edges.
b) Other ambiguous or uncertain concepts or proposition, which are sources of contradiction, include: large or small number (depending on context), ill-defined concepts, infinite set and self-reference. An example of self-reference is Richard paradox: The barber of Seville shaves those and only those who do not shave themselves; who shaves the barber? Incidentally, the indirect proof is a self-referent is not valid. The remedy is constructive proof. We may introduce some ambiguity in a mathematical system provided it can be approximated by certainty. For example a nonterminating nonperiodic decimal is inherently ambiguous but we can approximate it by a segment up to the nth digit and the margin of error will be 10^-n.
c) Any proposition about ambiguous set (e.g., infinite set) involving the universal or existential proposition is not verifiable and cannot be an axiom of a mathematical space.
d) The following are required for a contradiction-free mathematical space:
i) It must be well-defined by consistent set of axioms.
ii) Every concept must be well-defined
iii) A decimal is well-defined if every digit is known or computable, i.e., there is some algorithm or rule for determining it uniquely.
iv) The rules of inference must be specific to and well-defined by its axioms.
3) The real number system does not satisfy the requirements of a contradiction-free mathematical space and is, therefore, nonsense. Consequently, FLT presently formulated is nonsense. Here are some of the defects of the real number system:
a) Most of its concepts are ill-defined.
b) The trichotomy axiom that says, given two real numbers x, y, one and only one of the following holds: x y, is false. A counterexample to it is a pair of normal numbers. A normal number is constructed by taking every digit from the basic integers, 0, 1, …, 9, at random.
c) The completeness axiom is unverifiable because it involves the universal and existential quantifiers.
4) The remedy for the real number system is to reconstruct it as decimal numbers R* subject to the following simple axioms R*:
a) R* contains the basic integers.
b) The addition table.
c) The multiplication table
Then FLT, reformulated in R*, has countable counterexample posted elsewhere on this website.
5) With respect to number theory its main flaw is that the integers, its subject matter, have no valid axiomatization. The remedy is to embed them into R* isomorphically as the integral parts of the decimals.
6) To clarify a previous post, the isomorphic embedding of the ordinary integers into the contradiction-free new real number system as the integral parts of the decimals well-defines the ordinary integers. The mapping 0 -> d*, N -> (N-1).99…, N = 1, 2, …, establishes an isomorphism between the ordinary integers and the new integers.
E. E. Escultura
Correction:
The mapping 0 -> d*, N -> (N-1).99…, where N = 1, 2, …, is an integer establishes an isomorphism between the integers and the new integers, showing that they have almost identical properties, the only difference being that d* > 0.
The appropriate mapping of the integers into the integral parts of the decimals also establishes isomorphism between them and embeds the former isomorphically into the new real number system providing it a valid axiomatization. This rectifies the major flaw of number theory, lack of valid axiomatization of the integers.
E. E. Escultura
I am lost when it comes to understanding anything, especially Fermat’s Last Theorem – And clearly, this is true, since I am not a member of any political group preaching some form of Bias – In other words, Andrew Wiles never proved anything – Look People; Ask yourselves what the equation is saying – or what the equation is talking about?
X^2 + Y^2 = Z^2
Quite simply, the equation is saying;
THE SUM OF TWO SQUARES IS EQUAL TO ANOTHER SQUARE –
And Fermat is asking:
When is the “SUM OF TWO SQUARES EQAUL TO ANOTHER SQUARE HAVING EQUAL SIDES THAT REPRESENTS AN INTEGER” ? Or more specifically: How it possible to determine the Area of Object having an Infinite Number of Sides, when the Object is the SUM of TWO Objects having the same Number of Sides?
In other words, this is a simple problem dealing with the calculation of Areas –
People – All of you had better know, if I am right, then everything, and I do mean everything – Mathematics, Chemistry, and Physics – is wrong!
The viewers might be interest on the latest about the new real numbers:
They are countably infinite, discrete, consistent, have natural ordering (the reals have none), complete ordered semi-field, enriched beyond the reals by d*, u* and the new integers; non-Archimedean and non-Hausdorff but the complement of d* and u* in R* is; moreover, Cauchy convergence induces the Cauchy metric and topology.
This will be part of my keynote address at the World Congress of Nonlinear Analysts on the subject, The mathematics of the Grand Unified Theory, July 2 – 9, 2008, Orlando Florida
Update on the new real number system.
The following is lifted from a keynote address for July 2008, The mathematics of the grand unified theory, at the 5th World Congress of Nonlinear Analysis, July 2 – 9, 2008, Orlando, Florida.
1) The new real number system R* is the closure of the terminating decimals in the g-norm. The g norm of a decimal is the decimal itself, e.g., g-norm of pi = pi.
2) The set-valued d* is a continuum.
3) The decimals consists of adjacent predecssor-sucessor pairs in the lexicographic ordering. Adjacent decimals differ by d*, e.g., 4.3800… and 4.399… are adjacent. The average of adjacent pairs is the smaller one, e.g., the average of 1 and 0.99… is 0.99…
4) R*, the union of these adjacent pairs plus d* is a continuum, non-Archimedean and non-Hausdorff but its subset of decimals is countabbly infinite and discrete, Archimedean and Hausdorff.
5) R* does not contain nondenumerable set.
6) The only set that has cardinality is discrete; the continiuum has none.
7) Nondenumerable set does not exist.
Update III.
WELL-DEFINING THE NONTERMINATING DECIMALS FOR THE FIRST TIME
This update is lifted from my lecture at a plenary session (distinct from the keynote) of the 5th World Congress of Nonlinear Analysts, Orlando, Florida, July 2 – 9, 2008, entitled, The terminating decimals and their g-closure.
At this time, only the terminating decimals are well-defined; this is our legacy from the Ancients. A nonterminating decimal is simply a meaningless infinite array of digits. In fact, we compute by cutting segments (which are terminating decimals) from them since they are the only decimals we know. Now, we build them on the terminating decimals R and make the latter our point of reference for any further extensions.
A sequence of terminating decimals of the form,
N.a_1, N.a_1a_2, …, N.a_1a_2…a_n, …, (1)
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_1a_2…a_n,…, (2)
at margin of error 10?n. The nth term of (1) is called the nth g-term, a terminating decimal, and its g-limit a nonterminating decimal provided the nth digits are not all 0 beyond a certain value of n.
THE g-NORM AS NATURAL NORM FOR COMPUTATION WITH ADVANTAGES OVER STANDARD NORM
In Update II we introduced the g-norm of a decimal, i.e., itself. It has several advantages over the standard norm:
(a) It avoids indeterminate forms.
(b) Since the g-norm of a decimal is itself, computation yields the answer directly as decimal, digit by digit, and avoids the intermediate approximations of standard computation. This means significant savings in computer time for large computations.
(c) Computation by the g-norm avoids radicals altogether and yields the result directly, digit by digit.
(d) The value or limit of function f(x) at s or as x ? s is a decimal obtained by iterated computation of f(x) along successive refinement of sequence x_j that tends to s, as j tends to infinity; again, it yields the result directly as decimal, digit by digit to any desired level of accuracy measured by the number of digits computed.
(e) Approximation by the nth g-term or n-truncation contains or limits the ambiguity of nonterminating decimals.
(f) Calculation of distance between two decimals is direct, digit by digit, and requires no square root. For example, define the nth distance d_¬n between two decimals a, b as the numerical value of the difference between their nth g-terms, a_n, b_n, and their distance is the g-limit of d_n which does not even require square root. In general, computation by the g-norm does not involve roots or radicals.
Theorem. The tail digits of the nonterminating decimals merges into the continuum d*.
Finally, the closure of R in the g-norm or R* is a continuum, non-Archimedean and non-Hausdorff and includes the nonterminating decimals and the dark number d*. However, the system of decimals is a countably infinite subspace of R*, hence, discrete (digital), Archimedean and Hausdorff.
E. E. Escultura
For the latest on the new real number system, it is in the new ebook, The Grand Unified Theory, to be published by Bentham Science Publishers with E. E. Escultura as editor-contributor.
The new real number system will also be a chapter in an upcoming book, The Grand Hybrid Unified Theory, by
E. E. Escultura and V. Lakshmikantham.
Here are some important points about the
new real number system.
1) In both the real and new real number
systems the only well-defined decimals are
the terminating ones; the nonterminating
decimals are simply arrays of digits
most of which are unknown.
2) In the new real number system the
nonterminating decimals are defined, for the
the first time, in terms of the terminating
decimals R as follows:
a) Consider the sequence of terminating
decimals of the form,
N.a_1, N.a_1a_2, …, N.a_1a_2…a_n, …; (1)
the sequence (1) is called standard
generating or g-sequence. Its nth g-term,
N.a_1a_2…a_n, which is a terminating decimal,
defines and approximates the g-limit, the
nonterminating decimal,
N.a_1a_2…a_n…, (2)
at margin of error (maximum error) 10^–n.
b) If the nth digit of the g-limit (2) is not 0
for all n beyond a certain integer k then (2)
defines a nonterminating decimmal.
Note that the nth g-term repeats all the
previous digits of the decimal in the same
order so that if finite terms of the g-sequence
are deleted, the nonterminating decimal it
defines, i.e., its g-limit, remains unaltered.
c) In analysis we define limit in terms of
some norm. We define the g-norm of a
decimal as the decimal itself so that the
g-limt is also defined in terms of the g-norm.
Computation with the g-norm has advantages
one of which being that the result is obtained
directly as a decimal digit by digit so that the
intermediate steps of approximation is avoided.
3) Consider the sequence of decimals,
(d^n)a_1a_2…a_k, n = 1, 2, …, (3)
where d is any of the decimals,
0.1, 0.2, 0.3, …, 0.9, and a_1, …, a_k
finite basic integers (not all 0 simultaneously).
For each combination of d and the a_js,
j = 1, …, k, in (3) the nth term, which
we now refer to as the nth d-term of
this nonstandard d-sequence, is not a
decimal since the digits are not fixed.
As n increases indefinitely it traces the
tail digits of some nonterminating
decimal (note that the nth g-term recedes
to the right with increasing n), becomes
smaller and smaller until it becomes
indistinguishable from the tail digits of the
other decimals. We call the sequence (3)
nonstandard d-sequence since the nth term
is not a standard g-term but has a standard
limit, i.e., limit in the standard norm, which
is 0. Like the g-limit, the d-limit exists since
it is defined by its nonstandard d-sequence
of terminating decimals; we call it a dark
number d, the d-limit of the nonstandard d-
sequence (3). Moreover, while the nth term
becomes smaller and smaller with increasing
n it is greater than 0 no matter how large n is
so that if x is any decimal, 0 < d < x. The set
of d limits of all nonstandard d-sequences is
the set-valued dark number d*
4) We state some important results:
Theorem. The d-limits of the tail digits of
all the nonterminating decimals traced by
the nth d-terms of the d-sequence (3) form
the continuum d*.
Theorem. In the lexicographic ordering R
consists of adjacent predecessor-successor
pairs of decimals (each joined by d*) so
that the closure R* in the g-norm is a
continuum.
Note that the trichotomy axiom follows
from the lexicographic ordering of R*
which is not defined on the real numbers
since nonterminating decimals are not
well-defined there.
Corollary. R* is non-Archimedean and
non-Hausdorff but the subspace of decimals
are Archimedean and Hausdorff in the
standard norm.
Theorem. The rationals and irrationals are
separate, i.e., they are not dense in their union
(this is the first indication of discreteness
of the decimals).
Whenever d* appears in an equation or expression
it means that the equation or expression holds for
each element d of d*.
Theorem. The largest and smallest elements
of R* in the open interval (0,1) are 0.99… and
d* = 1 – 0.99…, respectively.
The following theorem used to be a conjecture.
It now has a proof in R*.
Theorem. An even number greater than 2
is the sum of two prime numbers.
(This post is excerpted from my keynote
address at the 5th World Congress of
Nonlinear Analysts, The Mathematics of
the Grand Unified Theory, July 5, 2008,
Orlando, Florida, now in press at Nonlinear
Analysis, Series A, Theory, Methods and
Applications)
E. E. Escultura
GVP Institute for Advanced Studies and
Departments of Mathematics and Physics,
JNT University, Visakhapatnam, India
Rejoinder on FLT
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 axioms is arbitrary and depends on what one wants to do provided they are CONSISTENT since inconsistency collapses a mathematical system. Once the axioms are chosen the mathematical space becomes a deductive system.
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. Example of a vacuous concept: the root of the equation x^2 + 1 = 0. That root has been denoted by i = sqrt(-1) from which one can draw the conclusion i = 0 and 1 = 0 and, for any real number x, x = 0.
5) There are other sources of ambiguity, e.g., large and small numbers due 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?
7) 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.
8) Consider this 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 because the trichotomy and incompleteness axioms (a version of the axiom of choice) are false.
Therefore, they do not well-define the real numbers.
10) 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 contradictions by constructing it on CONSISTENT axioms. Then FLT can be formulated in it and resolved.
11) To this end I 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 are defined and using them 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
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It’s all nonsense Ravikiran. Isn’t that clear from that report itself? You could have put an exclamation mark at least.
I might be wrong here, but isn’t the Wiles letter in the article dripping with sarcasm? “I’d like to have the address of the guy who let you get a PhD 30 years ago”? That sounds more like a cheap insult than an admission of failure.
And I really don’t think a person like Wiles would be making cheap insults such as this one. The letter is most likely a forgery or a prank.
And I would suggest that people wait for other mathematicians to back Escultura before believing that the proof is wrong. Too often in science and mathematics, sensational stuff like this one ends up amounting to nothing.
What can I say?
I am red faced.
(Disclaimer: I have not read Escultra’s “work”)
Escultura epitomizes the definition of a crank. Here are a few suggestive sentences from the article:
—
“He took the position that the failure to resolve the problem for over 360 years reveals the inadequacy and defects of foundations, number theory and the real number system. ” – FLT (not Shimura-Taniyama!!) has as little to do with the real numbers as George Bush with intelligence. FLT is a statement about the non-existence of *RATIONAL* solutions; there are tons of real solutions.
“Specifically, two of its axioms (the trichotomy and completeness axioms, for those who took basic algebra in high school and college) are false.”
– How can axiom be wrong? At worst, it can be inconsistent with the theory, but it can’t be wrong.
“The result is a new real number system that is free from defects and contradictions, finite and enriched with new numbers that have important applications for physics.”
– No comment.
“I read all of it just yesterday and let me tell you I respect you”
– If Wiles actually wrote this, it’s a disguised way of saying Escultra’s work is not deep at all. But knowing the kind of excitement FLT generates among amateurs (damm you Pierre de Fermat and your small margins), I’m pretty sure Wiles has learnt to ignore such emails (also, he usually signs differently).
“Escultura was a former math and science editor and columnist of The Manila Times.”
– It gets fishier.
—
Just FYI, there are a few hilarious comments on this issue over at fark.com
Cheers,
Bhargav
The partial list of publications that discusses the counterexamples to Fermat’s last theorem follows:
[18] Escultura, E. E. (1996) Probabilistic mathematics and applications to dynamic systems including Fermat’s last theorem, Proc. 2nd International Conference on Dynamic Systems and Applications, Dynamic Publishers, Inc., Atlanta, 147 – 152.
[20] Escultura, E. E. (1998) FTG VII. Exact solutions of Fermat’s equation (Definitive resolution of Fermat’s last theorem, J. Nonlinear Studies, 5(2), 227 – 254.
[29] Escultura, E. E. (2002) FTG V. The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[31] Escultura, E. E. (2003) FTG XVII: The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[35] Escultura, E. E. FTG XV. The new nonstandard analysis and the intuitive calculus, submitted.
[36] Escultura, E. E. (2002) FTG VI. The philosophical and mathematical foundations of FLT’s resolution, rectification and extension of underlying fields and applications, accepted, International J Nonlinear Differential Equations.
[37] Escultura, E. E. (2002) FTG XXII. Extending the reach of computation, accepted, International J Nonlinear Differential Equations.
[43] FRG XXVII – XXVIII. The new frontiers of mathematics and physics. Part II. The new real number system: Introduction to the new nonstandard analysis, Nonlinear Analysis and Phenomena, II(1), January, 15 – 30.
[45] Escultura, E. E. FTG. XXVI (2005). The resolution of Fermat’s last theorem and applications, accepted, Nonlinear Analysis.
Take your comments from the top, not from the flat of your foot.
Cheers.
E. E. Escultura
For those interested in the countably infinite counterexamples to FLT, here they are:
Let x = (0.99…)^T, where T = 1, 2, …, is an ordinary integer.
y = d* = 1 – 0.99…, called dark number.
z = 10^T.
For n > 2,
x^n + y^n = z^n.
This countably infinite set of new integers provides countably infinite counerexamples to FLT that prove this conjecture false. Moreover, the the countably infinite triples (kx,ky,kz), k = 1, 2, …, are also counterexamples to FLT since
(kx)^n + (ky)^n = (kz)^n.
These counterexamples are constructed from the new integers, namely, d* and numbers of the form N.99…, where N = 0, 1, … is an ordinary integer. They form a subspace of the contradiction-free new real number system.
The mapping,
N -> (N-1).99…, N = 1, 2, …,
is an isomorphic embedding of the ordinary integers into this contradiction-free mathematical space as integral parts of the decimals. This resolves a fundamental flaw of number theory that the integers are ill-defined. In other words, until this embedding, the integers were nonsense.
For details of the construction, visit my websites.
E. E. Escultura
For the benefit of the experts I am sharing my strategy for the capture of FLT.
1) I wondered why this conjecture could not be resolved for centuries and I concluded that present mathematics is defective or, at least, inadequate. So I embarked on a critique of the its underlying fields, namely, foundations, number theory and the real number system.
2) In foundations here is what I found:
a) I agreed with David Hilbert that the concepts of individual thought cannot be the subject matter of a mathematical space because they are not accessible to others and can neither be studied collectively nor axiomatized. His remedy was to represent them by symbols well-defined by a consistent set of axioms. A symbol or concept is well-defined if its EXISTENCE, properties and relationship with other symbols are specified by the axioms. I stress existence because vacuous concepts are contradictory as the following definition shows: A triquadrilateral is a plane figure with three vertices and four edges.
b) Other ambiguous or uncertain concepts or proposition, which are sources of contradiction, include: large or small number (depending on context), ill-defined concepts, infinite set and self-reference. An example of self-reference is Richard paradox: The barber of Seville shaves those and only those who do not shave themselves; who shaves the barber? Incidentally, the indirect proof is a self-referent is not valid. The remedy is constructive proof. We may introduce some ambiguity in a mathematical system provided it can be approximated by certainty. For example a nonterminating nonperiodic decimal is inherently ambiguous but we can approximate it by a segment up to the nth digit and the margin of error will be 10^-n.
c) Any proposition about ambiguous set (e.g., infinite set) involving the universal or existential proposition is not verifiable and cannot be an axiom of a mathematical space.
d) The following are required for a contradiction-free mathematical space:
i) It must be well-defined by consistent set of axioms.
ii) Every concept must be well-defined
iii) A decimal is well-defined if every digit is known or computable, i.e., there is some algorithm or rule for determining it uniquely.
iv) The rules of inference must be specific to and well-defined by its axioms.
3) The real number system does not satisfy the requirements of a contradiction-free mathematical space and is, therefore, nonsense. Consequently, FLT presently formulated is nonsense. Here are some of the defects of the real number system:
a) Most of its concepts are ill-defined.
b) The trichotomy axiom that says, given two real numbers x, y, one and only one of the following holds: x y, is false. A counterexample to it is a pair of normal numbers. A normal number is constructed by taking every digit from the basic integers, 0, 1, …, 9, at random.
c) The completeness axiom is unverifiable because it involves the universal and existential quantifiers.
4) The remedy for the real number system is to reconstruct it as decimal numbers R* subject to the following simple axioms R*:
a) R* contains the basic integers.
b) The addition table.
c) The multiplication table
Then FLT, reformulated in R*, has countable counterexample posted elsewhere on this website.
5) With respect to number theory its main flaw is that the integers, its subject matter, have no valid axiomatization. The remedy is to embed them into R* isomorphically as the integral parts of the decimals.
6) To clarify a previous post, the isomorphic embedding of the ordinary integers into the contradiction-free new real number system as the integral parts of the decimals well-defines the ordinary integers. The mapping 0 -> d*, N -> (N-1).99…, N = 1, 2, …, establishes an isomorphism between the ordinary integers and the new integers.
E. E. Escultura
Correction:
The mapping 0 -> d*, N -> (N-1).99…, where N = 1, 2, …, is an integer establishes an isomorphism between the integers and the new integers, showing that they have almost identical properties, the only difference being that d* > 0.
The appropriate mapping of the integers into the integral parts of the decimals also establishes isomorphism between them and embeds the former isomorphically into the new real number system providing it a valid axiomatization. This rectifies the major flaw of number theory, lack of valid axiomatization of the integers.
E. E. Escultura
I am lost when it comes to understanding anything, especially Fermat’s Last Theorem – And clearly, this is true, since I am not a member of any political group preaching some form of Bias – In other words, Andrew Wiles never proved anything – Look People; Ask yourselves what the equation is saying – or what the equation is talking about?
X^2 + Y^2 = Z^2
Quite simply, the equation is saying;
THE SUM OF TWO SQUARES IS EQUAL TO ANOTHER SQUARE –
And Fermat is asking:
When is the “SUM OF TWO SQUARES EQAUL TO ANOTHER SQUARE HAVING EQUAL SIDES THAT REPRESENTS AN INTEGER” ? Or more specifically: How it possible to determine the Area of Object having an Infinite Number of Sides, when the Object is the SUM of TWO Objects having the same Number of Sides?
In other words, this is a simple problem dealing with the calculation of Areas –
People – All of you had better know, if I am right, then everything, and I do mean everything – Mathematics, Chemistry, and Physics – is wrong!
Now read my Proof:
http://www.ietf.org/internet-drafts/draft-terrell-math-quant-ternary-logic-of-binary-sys-01.pdf
Note: This is not political… This is the survival of Humanity!
Eugene Terrell
(ETT-R&D Publications)
Engineering Theoretical Technologies
R&D Publications
The viewers might be interest on the latest about the new real numbers:
They are countably infinite, discrete, consistent, have natural ordering (the reals have none), complete ordered semi-field, enriched beyond the reals by d*, u* and the new integers; non-Archimedean and non-Hausdorff but the complement of d* and u* in R* is; moreover, Cauchy convergence induces the Cauchy metric and topology.
This will be part of my keynote address at the World Congress of Nonlinear Analysts on the subject, The mathematics of the Grand Unified Theory, July 2 – 9, 2008, Orlando Florida
Update on the new real number system.
The following is lifted from a keynote address for July 2008, The mathematics of the grand unified theory, at the 5th World Congress of Nonlinear Analysis, July 2 – 9, 2008, Orlando, Florida.
1) The new real number system R* is the closure of the terminating decimals in the g-norm. The g norm of a decimal is the decimal itself, e.g., g-norm of pi = pi.
2) The set-valued d* is a continuum.
3) The decimals consists of adjacent predecssor-sucessor pairs in the lexicographic ordering. Adjacent decimals differ by d*, e.g., 4.3800… and 4.399… are adjacent. The average of adjacent pairs is the smaller one, e.g., the average of 1 and 0.99… is 0.99…
4) R*, the union of these adjacent pairs plus d* is a continuum, non-Archimedean and non-Hausdorff but its subset of decimals is countabbly infinite and discrete, Archimedean and Hausdorff.
5) R* does not contain nondenumerable set.
6) The only set that has cardinality is discrete; the continiuum has none.
7) Nondenumerable set does not exist.
Update III.
WELL-DEFINING THE NONTERMINATING DECIMALS FOR THE FIRST TIME
This update is lifted from my lecture at a plenary session (distinct from the keynote) of the 5th World Congress of Nonlinear Analysts, Orlando, Florida, July 2 – 9, 2008, entitled, The terminating decimals and their g-closure.
At this time, only the terminating decimals are well-defined; this is our legacy from the Ancients. A nonterminating decimal is simply a meaningless infinite array of digits. In fact, we compute by cutting segments (which are terminating decimals) from them since they are the only decimals we know. Now, we build them on the terminating decimals R and make the latter our point of reference for any further extensions.
A sequence of terminating decimals of the form,
N.a_1, N.a_1a_2, …, N.a_1a_2…a_n, …, (1)
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_1a_2…a_n,…, (2)
at margin of error 10?n. The nth term of (1) is called the nth g-term, a terminating decimal, and its g-limit a nonterminating decimal provided the nth digits are not all 0 beyond a certain value of n.
THE g-NORM AS NATURAL NORM FOR COMPUTATION WITH ADVANTAGES OVER STANDARD NORM
In Update II we introduced the g-norm of a decimal, i.e., itself. It has several advantages over the standard norm:
(a) It avoids indeterminate forms.
(b) Since the g-norm of a decimal is itself, computation yields the answer directly as decimal, digit by digit, and avoids the intermediate approximations of standard computation. This means significant savings in computer time for large computations.
(c) Computation by the g-norm avoids radicals altogether and yields the result directly, digit by digit.
(d) The value or limit of function f(x) at s or as x ? s is a decimal obtained by iterated computation of f(x) along successive refinement of sequence x_j that tends to s, as j tends to infinity; again, it yields the result directly as decimal, digit by digit to any desired level of accuracy measured by the number of digits computed.
(e) Approximation by the nth g-term or n-truncation contains or limits the ambiguity of nonterminating decimals.
(f) Calculation of distance between two decimals is direct, digit by digit, and requires no square root. For example, define the nth distance d_¬n between two decimals a, b as the numerical value of the difference between their nth g-terms, a_n, b_n, and their distance is the g-limit of d_n which does not even require square root. In general, computation by the g-norm does not involve roots or radicals.
Theorem. The tail digits of the nonterminating decimals merges into the continuum d*.
Finally, the closure of R in the g-norm or R* is a continuum, non-Archimedean and non-Hausdorff and includes the nonterminating decimals and the dark number d*. However, the system of decimals is a countably infinite subspace of R*, hence, discrete (digital), Archimedean and Hausdorff.
E. E. Escultura
For the latest on the new real number system, it is in the new ebook, The Grand Unified Theory, to be published by Bentham Science Publishers with E. E. Escultura as editor-contributor.
The new real number system will also be a chapter in an upcoming book, The Grand Hybrid Unified Theory, by
E. E. Escultura and V. Lakshmikantham.
Here are some important points about the
new real number system.
1) In both the real and new real number
systems the only well-defined decimals are
the terminating ones; the nonterminating
decimals are simply arrays of digits
most of which are unknown.
2) In the new real number system the
nonterminating decimals are defined, for the
the first time, in terms of the terminating
decimals R as follows:
a) Consider the sequence of terminating
decimals of the form,
N.a_1, N.a_1a_2, …, N.a_1a_2…a_n, …; (1)
the sequence (1) is called standard
generating or g-sequence. Its nth g-term,
N.a_1a_2…a_n, which is a terminating decimal,
defines and approximates the g-limit, the
nonterminating decimal,
N.a_1a_2…a_n…, (2)
at margin of error (maximum error) 10^–n.
b) If the nth digit of the g-limit (2) is not 0
for all n beyond a certain integer k then (2)
defines a nonterminating decimmal.
Note that the nth g-term repeats all the
previous digits of the decimal in the same
order so that if finite terms of the g-sequence
are deleted, the nonterminating decimal it
defines, i.e., its g-limit, remains unaltered.
c) In analysis we define limit in terms of
some norm. We define the g-norm of a
decimal as the decimal itself so that the
g-limt is also defined in terms of the g-norm.
Computation with the g-norm has advantages
one of which being that the result is obtained
directly as a decimal digit by digit so that the
intermediate steps of approximation is avoided.
3) Consider the sequence of decimals,
(d^n)a_1a_2…a_k, n = 1, 2, …, (3)
where d is any of the decimals,
0.1, 0.2, 0.3, …, 0.9, and a_1, …, a_k
finite basic integers (not all 0 simultaneously).
For each combination of d and the a_js,
j = 1, …, k, in (3) the nth term, which
we now refer to as the nth d-term of
this nonstandard d-sequence, is not a
decimal since the digits are not fixed.
As n increases indefinitely it traces the
tail digits of some nonterminating
decimal (note that the nth g-term recedes
to the right with increasing n), becomes
smaller and smaller until it becomes
indistinguishable from the tail digits of the
other decimals. We call the sequence (3)
nonstandard d-sequence since the nth term
is not a standard g-term but has a standard
limit, i.e., limit in the standard norm, which
is 0. Like the g-limit, the d-limit exists since
it is defined by its nonstandard d-sequence
of terminating decimals; we call it a dark
number d, the d-limit of the nonstandard d-
sequence (3). Moreover, while the nth term
becomes smaller and smaller with increasing
n it is greater than 0 no matter how large n is
so that if x is any decimal, 0 < d < x. The set
of d limits of all nonstandard d-sequences is
the set-valued dark number d*
4) We state some important results:
Theorem. The d-limits of the tail digits of
all the nonterminating decimals traced by
the nth d-terms of the d-sequence (3) form
the continuum d*.
Theorem. In the lexicographic ordering R
consists of adjacent predecessor-successor
pairs of decimals (each joined by d*) so
that the closure R* in the g-norm is a
continuum.
Note that the trichotomy axiom follows
from the lexicographic ordering of R*
which is not defined on the real numbers
since nonterminating decimals are not
well-defined there.
Corollary. R* is non-Archimedean and
non-Hausdorff but the subspace of decimals
are Archimedean and Hausdorff in the
standard norm.
Theorem. The rationals and irrationals are
separate, i.e., they are not dense in their union
(this is the first indication of discreteness
of the decimals).
Whenever d* appears in an equation or expression
it means that the equation or expression holds for
each element d of d*.
Theorem. The largest and smallest elements
of R* in the open interval (0,1) are 0.99… and
d* = 1 – 0.99…, respectively.
The following theorem used to be a conjecture.
It now has a proof in R*.
Theorem. An even number greater than 2
is the sum of two prime numbers.
(This post is excerpted from my keynote
address at the 5th World Congress of
Nonlinear Analysts, The Mathematics of
the Grand Unified Theory, July 5, 2008,
Orlando, Florida, now in press at Nonlinear
Analysis, Series A, Theory, Methods and
Applications)
E. E. Escultura
GVP Institute for Advanced Studies and
Departments of Mathematics and Physics,
JNT University, Visakhapatnam, India
Rejoinder on FLT
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 axioms is arbitrary and depends on what one wants to do provided they are CONSISTENT since inconsistency collapses a mathematical system. Once the axioms are chosen the mathematical space becomes a deductive system.
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. Example of a vacuous concept: the root of the equation x^2 + 1 = 0. That root has been denoted by i = sqrt(-1) from which one can draw the conclusion i = 0 and 1 = 0 and, for any real number x, x = 0.
5) There are other sources of ambiguity, e.g., large and small numbers due 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?
7) 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.
8) Consider this 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 because the trichotomy and incompleteness axioms (a version of the axiom of choice) are false.
Therefore, they do not well-define the real numbers.
10) 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 contradictions by constructing it on CONSISTENT axioms. Then FLT can be formulated in it and resolved.
11) To this end I 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 are defined and using them 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