IMPORTANT QUESTIONS (Continued)

I’m back after a grueling week with search and rescue teams in the Ozarks, nursing a few scrapes and bruises. After posting the “rest of the story” concerning our search for a lost camper (who happened to be my son Joshua) with, fortunately, a happy ending, I’m back on the thread of thought I was pursuing before, about the most important questions we can ask as human beings. 

To refresh your memory before going on, you may want to re-read the post “THE MOST IMPORTANT QUESTIONS” posted April 29, 2016 and “WHY MAINSTREAM SCIENCE HASN’T BEEN ABLE TO ANSWER IMPORTANT QUESTIONS” posted May 3, 2016.

Those posts lay the groundwork for discussing important questions like:

• Who are we?

• Where did we come from, and where are we going?

• What is the meaning and purpose of reality and human existence?

Four paradigm-shattering discoveries were mentioned in those posts:

• Relativity

• Quantum Mechanics

• The Incompleteness Theorems

• The Third Form of the Substance of Reality

In order to see how these discoveries have revealed a reality we had only glimpsed vaguely before, and how they have led to a radically new understanding of the nature of reality, we need to take a brief look at three profoundly relevant concepts: existence, logic and infinity. The limited understanding we have of these three concepts, has been divided up in the dualistic and fragmented approach of academic formalism and specialization between the studies of ontology (the nature of reality) and epistemology (our understanding of it). Massive volumes have been written on ontology and epistemology, but with due respect, much of the sophisticated academic thinking is superseded and made obsolete by these new discoveries.

Consider the following positions articulated by some of the most powerful thinkers of the past:

 Leibniz (The Principle of Sufficient Reason) – For every meaningful question there is an answer.

Russell and Whitehead (Principia Mathematica) There are three types of statements: True, False and meaningless.

Gӧdel (Incompleteness Theorems) – There are meaningful statements of questions that cannot be answered within the logical system within which they were stated.

Close-Neppe (The Triadic Distinction Dimensional Paradigm) – The universe is the ultimate logical mathematical system, and the logical structures of basic number theory and geometry reflect the logical structure of Reality.

There are other scientists who have somewhat similar notions about the mathematical nature of the universe, for example, Joseph Brenner, in his book “Logic in Reality” says the fundamental physical structure of the world is logical and mathematical. 

And physicist Max Tegmark, in his book “Our Mathematical Universe, My Quest for the Ultimate Nature of Reality" says: “This means that our physical world not only is described by mathematics, but that it is mathematical (a mathematical structure), making us self-aware parts of a giant mathematical object.”

If you accept that the physical universe, along with whatever reality it is part of, is a logical system, then the evidence of relativity, quantum mechanics and the Incompleteness theorems, taken together, strongly suggest that reality, defined as all that exists, is a logical system, and it is infinite. 

This requires a radically different approach to understanding reality and answering those most important questions. Why? Because current science is fragmented by academic political correctness and departmental specialization, producing a dis-jointed group of logical systems of thought that have conflicts and paradoxes, indicating that something is missing. For example, the mathematics of relativity and quantum mechanics are non-commensurable.

The currently prevalent mathematical tools used by science are inadequate and in some ways inappropriate for effective application in our infinite, quantized reality. As discussed in many earlier posts (search archives for Calculus of Distinctions and related topics) Newtonian calculus, based on the exact opposite of quantization, i.e. continuity of variables of extent, must be replaced by the more comprehensive calculus of distinctions, which also incorporates the actions of consciousness. Once that is done, many things that the current paradigm cannot answer are explained.

To be continued.

SPACE, TIME AND CONSCIOUSNESS TRUTHS

SPACE, TIME AND CONSCIOUSNESS

EVERY THING IS EXPLAINABLE!

Let me explain:

By including consciousness in the equations of science (TDVP), Dr. Vernon Neppe and I have reached the point where we have everything we need to explain every thing. For example, I can explain exactly why it takes three quarks to form a proton or neutron, and why it takes three types of elementary particles to form the stable atoms that make up the physical universe. I can explain why fermions have ½ spin, and why the Cabibbo angle and other particle-mixing angles are what they are.

Does this mean that we have the “Theory of Everything” (TOE) dreamed about by physicists since Einstein hypothesized that there should be a unified field theory that would explain all of the known physical forces? No.

Those who think there is a TOE out there waiting to be discovered have not understood Gӧdel’s Incompleteness Theorems. I will explain this, but first, I must explain the difference between explaining every thing and a theory of everything. The word ‘thing’ refers to a finite physical object or concept. The word ‘everything’ means all that exists. To clarify: when physicists talk about a TOE, they are not actually talking about a theory of everything. They are talking about a theory of every thing, i.e. a complete physical theory. But, as I will explain, even that is problematic in light of Gӧdel’s Incompleteness Theorems.

Before I get into discussing Gӧdel’s Incompleteness Theorems, I will state four fundamental truths that can be proved conclusively with what we now know:

1.     Reality, which includes the physical universe, is an infinite logical system.

2.     The physical universe, available through the physical senses, is only a small fraction (less than 5%) of Reality.

3.     Reality is expanding in all nine finite dimensions, not just the three and one-half dimensions of relativistic space-time.

4.     Reality is more than an infinite logical system, it is an Infinite Intelligence that encompasses space, time and consciousness.

As an independent sentient being, you have the right to agree, disagree or reserve judgement concerning these four statements. But they rest on the solid foundations of proved theorems of axiomatic geometry, mathematics, logic, and physical science. These theorems include the Pythagorean Theorem, Fermat’s Last Theorem, the axiomatic principles of Relativity and Quantum Physics, and Gӧdel’s Incompleteness Theorems.  I am as sure of them as I am that I exist. If I can, I will impart that certainty to you in this presentation.

I have touched on the importance and applications of the Pythagorean Theorem, Fermat’s Last Theorem, the axiomatic principles of Relativity and Quantum Physics, and Gӧdel’s Incompleteness Theorems in these posts, and Dr. Neppe and I have published a book (Reality Begins with Consciousness) and a number of papers dealing with their applications in TDVP, but these foundation theorems are so important to TDVP in general, and the four truths stated above, in particular, that I want to elaborate on them a little more here.

Gӧdel’s Incompleteness Theorems

The detailed proof of Gӧdel’s Incompleteness Theorems is beyond the scope of this discussion, but understanding their meaning, importance and impact on human thought, are not. I highly recommend the book “Gӧdel’s Proof” by Ernest Nagel and James R. Newman to anyone who may want to pursue a deeper understanding than I can provide here. This book was written for the non-mathematician. In their words: “The details of Gӧdel’s proofs in his epoch-making paper are too difficult to follow without considerable mathematical training. But the basic structure of his demonstrations and the core of his conclusions can be made intelligible to readers with very limited mathematical and logical preparation.”

In a nutshell, Gӧdel’s Incompleteness Theorems prove that no internally consistent logical system is ever complete. I emphasize the use of the word prove here, because you need to understand that a theorem is not a theory. If it is a theorem, is has been proved. Another way to put Gӧdel’s proof is: In any system of logic, mathematics or geometry, logically legitimate questions can be raised that cannot be answered within the system.

Let this sink in a little. This explains why questions like why it takes three quarks to form a stable proton or neutron, why the Cabibbo mixing angle for quarks is 13.02 degrees, or why the elementary particles making up normal atomic structure have ½ intrinsic spin, are not explainable in the current scientific paradigm.

Gӧdel’s Incompleteness Theorems do not say that there are questions like this that can never be answered; they only say that there are questions that cannot be answered within the paradigm in which they were formed. They may, however, be answerable in a new, more comprehensible paradigm. This, in fact, describes the nature of science and human thought in general. Science, and any system of logical thought, however internally consistent, is never complete. With this in mind, let’s return to the four fundamental truths stated above.

Reality is a logical system, and as such, it is infinite

Reality is the ultimate logical system, which all conceptual models, scientific, mathematical, geometrical, philosophical, theological, spiritual, or whatever, attempt to model. Gӧdel’s Incompleteness Theorems prove that Reality, as the original logical system, is always incomplete from the perspective of finite beings. No matter how many things we explain, the everything of Reality will always be beyond those explanations. Whenever we expand the logical system of our current understanding to explain something heretofore inexplicable, we will find that Reality presents us with new questions that cannot be explained within even our new paradigm. Thus, Reality is an infinite logical system.

The physical universe is a small fraction of Reality

Application of the principles of relativity and quantum physics imply that our physical senses and extensions of them are reduction valves. That is to say that they function not to bring all of Reality into our physical experience, but actually to keep most of Reality out, and bring only a very small part into our physical experience. In fact, our current knowledge of the central nervous system of sentient beings like us suggests that our brains are capable of handling much more than that delivered to them by the senses and neural pathways. Evidence provided by the Hubble space probe and TDVP TRUE analysis strongly support this conclusion.

Reality is expanding in nine finite dimensions

Dr. Neppe and I have developed a comprehensive model that incorporates nine finite dimensions based on the principles of relativity, quantum physics and number theory. Its implications, are vast and far-reaching. It explains why there are no absolute beginnings or ends, only change. In conjunction with the first Truth, it explains many previously inexplicable experimental observations, and implies, among other things, that consciousness, like mass and energy, obeys the laws of conservation, suggesting that survival of consciousness after the death and destruction of the physical body is not only possible, but probable.  

Reality is an Infinite Intelligence encompassing space, time and consciousness

This concept is considered by most people to be objectively unprovable. Some people, including some scientists, accept it, and some reject it. Many claim to have proof by personal experience. It has been considered by organized science and religion to be a matter of personal belief and faith. This is no longer the case. It is logically proved on the basis of the other three truths. Reality is an infinitely logical system that human beings and man-made models reflect imperfectly in mass, energy and consciousness in three dimensions of space and one of time. Proof of this truth changes the world and our understanding of it profoundly.

As far as I am concerned, these four statements are facts that have been proved beyond doubt and are thus unassailable, absolute Truths. The implications of these four truths are profound and far-reaching in their practical implications.

SEEKING CERTAINTY IN AN UNCERTAIN WORLD FREEWILL VS DETERMINISM

FREEWILL VS DETERMINISM                        ?

It is the very nature of human beings, both individually, and through the disciplines of religion and science to try to seek complete knowledge and certainty. From a religious point of view, if God is all-knowing, then He knows the complete history of every atom and every soul from the beginning until the end of time. The Bible says that we can know much with certainty: “When the Spirit of truth comes, he will guide you into all the truth, for he will not speak on his own authority, but whatever he hears he will speak, and he will declare to you the things that are to come.” - John 16:13. But, how accurate are predictions? We also find: “Ask, and it shall be given you; seek, and you shall find; knock, and it shall be opened unto you:” - Matthew 7:7. But, how much can we, as human beings, know?

As for science, there was a time when scientists believed that the goals of completeness and certainty were attainable. See, e.g., the writings of Descartes and Laplace. They believed that, given the initial conditions of the universe, the laws of physics would allow us to explain everything in the universe and predict the entire future in great detail. This deterministic point of view still exists among leading physicists like Stephen Hawking who embrace atheism and declare that a theory of everything is within our grasp. But, if everything can be known, it is pre-determined. Do we have no freewill?

The last post made the point that proof of Gӧdel’s incompleteness theorems changed the way we can see the world forever. We know now that science can explain any aspect of reality we are capable of conceiving of, but will never explain everything. We also have the fact that, at the quantum level, there is a very small, but finite amount of uncertainty. Heisenberg’s uncertainty principle has been validated as a basic fact of physical reality. But, even a tiny bit of uncertainty can allow outcomes that lead to vastly different futures. See, for example, discussions of the ‘Butterfly Effect’. Together, the incompleteness theorems and the uncertainty principle have put a limit on determinism. The atheist’s dream of a ‘theory of everything’ is unattainable, and the true believer can only know the truth in some perfect realm existing beyond the space and time of the physical universe.

The teachings of the past masters of both scientific and spiritual enlightenment suggest that, as human beings, we can change the world by changing our attitudes, by changing the way we think. It now appears that, at least to some degree, reality is what we choose to make of it. Millions attest to the fact that positive attitude and prayer work, and quantum physics suggests that the very nature of physical reality, and what we can say about it, depend upon conscious choices we make. Perhaps you’ve seen this in your own life. My wife and I have seen what many would call amazing miracles in our lives. We have defied the odds and the wisdom of mainstream medicine a number of times. The important question now becomes: how much do we actually change reality by what we believe and choose to do? If our belief is strong enough, can we bring about results that others might think impossible?

The last post, focused on incompleteness, was linked with two different headings for a purpose: the links were designed to get the attention of different types of thinkers. The response was good, indicating interest in this subject, with numerous shares and over 200 visits to the blog. That may be enough to allow me to conduct a small survey. Following up on these posts, I want to try to get some more feedback. Here are a few questions for those who would like to participate:

After reading the last two posts, please go back to the Facebook link, or message me and reply with ‘T, F or ?’ for ‘True’, ‘False’, or ‘I don’t know’, for each of the first six questions below, and provide a percentage for the seventh. I will publish the results within a few days.

1. Incompleteness and quantum uncertainty reduce scientific laws to statistics, and calculations of future events to probabilistic estimates

2. Some things are predictable with almost complete certainty

3. What I believe affects physical reality

4. Reality consists of much more than the physical universe

5. Reality is created by the accumulative effect of what everyone believes

6. Time is an illusion

7. What percentage of the events of life are predetermined by circumstances beyond our control?

IS INFINITY REAL?

IS INFINITY REAL?

I find it very interesting that when I ask someone this question, I usually get one of three responses. They are not always in these exact words, but the meanings are pretty much the same:

1. Of course! Why would anyone ask such a question? Have you never heard of God?

2. Of course not! Even the universe is finite. Only an idiot would even ask such a question!

3. What? Often followed by a blank stare, or a totally unrelated question like: “Where’d you get that shirt? Or, when do we eat?" 

Could it be that there are three different kinds of people in the world? Believers, Non-believers and ‘Who-cares’ people? Let’s call them Type I, Type II and Type III, respectively. We could say ‘respectfully’ too, because of course, everyone has the right to believe any ridiculous thing they want to, whether it’s true or not. Let's have a closer look:

Type I, II and III people may fall into one of several sub-types:

Type Ia.   Followers: “My father, mother, grandmother, grandfather, priest, minister, etc., say it’s true, so it must be true.”

Type Ib. Hedgers: “I’m hedging my bet: If it’s true and I say I don’t believe, I’m in big trouble; if it’s not true, it won’t make any difference!”

Type Ic: True Believers: “Based on personal experience, I KNOW it’s true! And people who don’t believe in God cause all the problems in the world!”

Type II People:

Type IIa. The Intellectually Arrogant: “No one’s smarter than me, and if I don’t believe it’s true, it can’t be true!”

Type IIb. Agnostics: God has never appeared to me, so, yeah, maybe, but … probably not.”

Type IIc. Atheists: “People who believe in God cause all the problems in the world.”

Type III People:

Type IIIa. Unaware: “Life is complicated enough without looking for something else to think about.”

Type IIIb. Self-Absorbed: “Life is too short to be bothered. I just want to have fun!”

Type IIIc. Innocent: “Hmmm, I haven’t thought about it.”

This of course isn’t all of the possible sub-types; you can have fun making up some of your own.

This question, however, turns out to have a definite answer-it’s not just a matter of belief.

We live in a somewhat well-ordered world, where there are many repeating patterns. If this were not the case, there could be no science or religion, life would be totally random and chaotic, and we wouldn’t last long. Science, philosophy and thinking with some level of certainty exist only because there are certain patterns that repeat according to systems of logic that we can discover and understand. In 1931, mathematician Kurt Gӧdel proved what has become known as the incompleteness theorems, which tell us that no logical system is ever complete. No matter how large and complex a logical system becomes, there is always more. As soon as we understand a portion of the system, we discover that there is yet more. It’s as if the universe is always expanding beyond our understanding, no matter how much we know. And our understanding of time, space and consciousness is limited by the limitations of our physical senses and the limited portion of our brains that we choose to use. Thus, there is proof, even in the physical world, that infinity exists. We just can’t experience it until we transcend the limitations of the physical body.

So does this mean that Type II and Type III people are idiots? No, of course not. Type II people are just wrong, and Type III people, well, some of them probably are idiots, but we love them anyway!

FLT65: FERMAT'S LAST THEOREM PART 4

FERMAT'S LAST THEOREM PART 4

NOTE: This part contains some original mathematics. The theorems stated and proved in this section are used in the explanation that follows, but If math is not your thing, you can read the text to get the gist of it and skip over the math. The references cited will be posted with the last part of this paper.

A NOTE ON THE NATURE OF MATHEMATICAL LOGIC AND PROOF

If FLT65 is a valid proof of Fermat’s Last Theorem, why hasn’t it been recognized in all these years? There are a number of reasons, and they are discussed below, but an important underlying reason is the lack of consensus among reviewers concerning exactly what constitutes mathematical logic and proof. We must, therefore, start by explaining what is meant by those terms in this presentation:

Mathematical logic starts with

(1.) the inductive reasoning process of identifying self-evident axioms underlying the logical patterns of reality, like distinctions of subject and object, separation and unity, enumeration and combination, difference and equivalence, extension and content, dependent and independent variables, functional relationships, and

(2.) formalizing the fundamental operations of substitution, addition, subtraction, multiplication and division. Based on such axiomatic concepts and fundamental operations, the logical structures of mathematical systems are built up by processes of deductive reasoning.

 (3.) Mathematical proof consists of using mathematical operations and substitutions of equivalents to determine whether a given statement or proposed theorem is consistent with accepted mathematical logic and the axioms underlying that system.

This sounds like a boring, rigidly structured affair that might be completed in short order. But that is, in fact, not the case.

The history of FLT65 provides a good example of how mathematics is not the ‘cut and dried’ subject that most people seem to think it is. One reviewer of FLT65, a highly intelligent man with deep interests in science and mathematics and significant math skills— but not a professional mathematician —said, after discussing FLT65 for some time with  the author, something to the effect that, “of all things, a mathematical statement like a proposed proof of FLT is something we should absolutely be able to come to an agreement about in short order, because it’s either true or it’s not.” This reflects an erroneous belief about mathematics that was common, even among professional mathematicians, including the great German mathematician David Hilbert, before Kurt Gödel published his proofs of two theorems now known as Gӧdel’s incompleteness theorems (GIT) (1929 – 1931) ^[6]. Gӧdel’s incompleteness theorems are very relevant to FLT and FLT65 because they explain, in part, why some reviewers have not understood FLT65. Additionally, after Gӧdel’s work became widely known, many mathematicians suspected that FLT might actually be an unprovable statement.

Gӧdel’s work was a bombshell that changed mathematics forever: It destroyed Hilbert’s dream of establishing all mathematical logic upon a finite, consistent foundation of axioms, from which all statements could be proved either true, false or meaningless by deductive reasoning. Finding such an all-encompassing algorithmic system was known as the ‘entscheidungsproblem’ (decision problem) ^[7]. Gödel proved, however, that no system of mathematical logic is ever complete. This means that there can be no complete entscheidung algorithm for all mathematical statements. It means that there can be logically legitimate questions posed within any given axiomatic mathematical system that are meaningful, but unprovable within that system.

In the introduction to his breakthrough paper, Gödel said: “As is well known, the development of mathematics towards greater certainty has led to the formalization of much of it such that you can complete proofs by mechanically following a few basic rules. The most comprehensive current formal axiomatic systems are the system of the Whitehead-Russell Principia Mathematica on the one hand, and the Zermelo-Fraenkel axiom-system of set theory on the other hand. These two systems are so thoroughly developed that one can formalize in them all methods of proof currently in use in mathematics, i.e. you can reduce the methods of constructing proofs to a few axioms and deduction rules. Therefore, it seems plausible to [as Hilbert did][1] that these deduction rules are sufficient to decide all mathematical questions expressible in those systems. We will show that this is not true, but that there are even relatively easy problems in the theory of ordinary whole numbers that cannot be decided from existing axioms.”^[8]

 A mathematical statement that has not been proved is properly called a conjecture. With a conjecture, the question to be decided is whether the conjecture is true or not. After Gödel, some mathematicians believed that Fermat’s conjecture (FLT), a legitimate mathematical question, might be an unprovable statement. This idea, however, was based on a misunderstanding of the incompleteness theorems. The incompleteness theorems do not, as one might suppose, imply that questions that cannot be decided within the axiomatic system they are framed in, are necessarily ultimately unprovable. Gӧdel’s theorems did not prove that there are undecidable questions, they proved that there are some questions for which the mathematical system within which one is operating may be inadequate. The system of mathematical logic within which a conjecture is stated may have to be expanded with one or more new axioms and new definitions and theorems before the statement can be proved to be either true or false.

In order for a reader to understand the applicability of the “Division Algorithm” and its three corollaries as they are used in FLT65, it may be necessary to expand the logic of the mathematical system from which the reviewer is operating to include the following definitions, axiomatic principle and theorems:

Concepts of importance in FLT65

1A   Mutually Co-Dependent Variables.

Variables are mutually co-dependent if they are related only by the form of the algebraic expression or equation they appear in, with no designated difference between them in terms of dependence or independence. For example, in the equation zⁿ - xⁿ = yⁿ, the variables x, y and z are mutually co-dependent. If we designate one of them as the dependent variable and assign values to the others so that we can set up an equation in terms of functions of the dependent variable, the variables are no longer mutually co-dependent. For example, if x and y are replaced by specific types of variables or constants, like X and Y for integer variables, or specific values, X₁ and Y₁, z becomes dependent upon their values and we say the equation is of the nth degree in z.[2]

1B. The Principle of Designation of Dependent and Independent Variables:

Given a polynomial f(x, y, z, … ) a function of several variables, over the field of real numbers, any of the otherwise mutually co-dependent variables, x, y, z, …, may be designated as dependent and/or independent, as appropriate to make the polynomial representative of meaningful information as part of an equation or descriptive statement.

This statement expresses an inherent fundamental feature of algebraic expressions consistent with the axioms of a system of mathematical logic. It is a principle rather than a theorem because, while it depends upon the validity of the axioms and fundamental operations of mathematics, it requires no multi-step proof by deductive reasoning from those axioms. This operational principle allows us to use polynomials to set up potentially solvable equations for application to real problems. This principle is relevant to FLT65 and this discussion. This is so because at several points, the designations of dependent and independent variables are changed as needed to express functional relationships consistent with application of the division algorithm. It was used without comment in FLT65, and its use is made explicit here for more clarity.

2A. Parallel Polynomials:

Parallel polynomials are polynomials of the same degree and the same algebraic form, i.e. with the same number of terms and the same coefficients, but with different variables. For example, a₁xⁿ + a₂x^n-1 + a₃x^n-2 + … + a_n+1 and a₁z ⁿ + a₂z^n-1 + a₃z^n-2 + … + a_n+1 are parallel polynomials, as are Az² + Bzx + Cx² and Ax² + Byx + Cy².

2B. The Theorem of Parallel Factorization:

If a given polynomial, f(z), of degree n > 1, defined over the field of real numbers, is divisible by z –a, then the parallel polynomial f(a) is also divisible by z – a.

This theorem is proved indirectly in FLT65, and is proved formally here for the first time, as far as the author knows. It could also be considered to be a corollary to the division algorithm. Note that the choice of symbols used to represent variables in proofs is arbitrary: The validity of axioms and theorems do not depend upon the symbols used—the use of any consistent set of symbols would produce the same theorems. However, the symbols a, x, y, and z were chosen to correspond to the symbols used throughout this paper to facilitate understanding and application of the theorems to FLT65. In the generalized two-variable algebraic polynomial forms used in the following proofs, x and z were chosen for the same reason.

Proof: Consider the generalized n-degree polynomial:

f(z, x) =Azⁿ +Bz^n-1x +Cz^n-2x² +…+Mxⁿ. …………………………………………………………..       1

By the Axiom of Unrestricted Designation of Variables, we may designate x = X₁, to obtain the single-variable polynomial

f(z) =Azⁿ +Bz^n-1X₁ +Cz^n-2X₁² + … + MX₁ⁿ  ……………………………………………………….  2

If we divide f(z) by z – a, by Corollary II of the Division Algorithm, the remainder,

f(a) = Aaⁿ +Ba^n-1X₁ +Ca^n-2X₁² +…+ MX₁ⁿ. ……………………………………………….  3

Then, f(z) – f(a) = Azⁿ - Aaⁿ + Bz^n-1X₁ - Ba^n-1X₁² + Cz^n-2X₁ ² - Ca^n-2X₁² + …+ LzX₁^n-1 - LaX₁^n-1 + Mxⁿ - MX₁ⁿ = A(zⁿ - aⁿ) + B X₁(z^n-1 - aⁿ) +… + 0 = A(z – a)(z^n-1 + z^n-2a +…+ a^n-1) +BX₁(z – a)(z^n-2 + z^n-3a +…+ a^n-2) +…+ LX₁^n-1(z - a) →f(z) – f(a) = (z – a) {A(z^n-1 + z^n-2a + … + a^n-1) + B X₁(z^n-2 + z^n-3a + … + a^n-2) +…+ LX₁^n-1}.  ………………………………….. 4

Then, since f(z) is divisible by z – a, and f(z) – f(a) = (z – a) {A(z^n-1 + … + a^n-1)+B X₁(z^n-2 + … + a^n-2) +…+ L X₁^n-1}, by Corollary I of the Division Algorithm, f(a) is also divisible by z – a, Q.E.D. ………………………………………………………………………………  5

Use of this theorem shortens the discussion of FLT and FLT65 considerably, as the reader will see in the discussions that follow. Also, proof of the companion theorem for integers given below is important because it removes some of the mystery expressed by reviewers concerning the application of theorems that are true for algebraic polynomials to the subgroup involving integers, namely integer polynomials.

2C. The Theorem of Parallel Factorization for Polynomials of Integer Variables:

 If a given polynomial, f(Z), defined over the ring of integers is divisible by Z –a, the parallel polynomial f(a) defined over the ring of integers is also divisible by Z – a.

This can be inferred from the proof of the theorem of parallel factorization for polynomials over the field on real numbers and the fact that the integers are a subset of the set of real numbers. Or, stated more formally: It follows from the fact that all of the elements of the ring of integers are contained within the set of elements of the field of real numbers. However, importantly, it can be proved independent of the proof for polynomials over the field of real numbers as follows:

Proof:

Given the general polynomial in two variables:

f(z,x) =Azⁿ +Bz^n-1x +Cz^n-2x² +…+Mxⁿ . ………………………………………………. 1

By the Axiom of Unrestricted Designation of Variables, we may designate x = X₁ a specific integer and z = Z, an integer variable, in the general polynomial to obtain f(Z) =AZⁿ +BZ^n-1X₁ +CZ^n-2X₁² +…+MX₁ⁿ. ……………………………………………. 2

And then, when we divide f(Z) by Z – a, by Corollary II of the Division Algorithm, the remainder is the parallel polynomial

f(a) = Aaⁿ +Ba^n-1X₁ +Ca^n-2X₁² +…+MX₁ⁿ. ……………………………………………… 3

 Then, f(Z)–f(a) = AZⁿ - Aaⁿ + BZ^n-1X₁ - Ba^n-1X₁² + CZ^n-2X₁ ² - Ca^n-2X₁² + …+ LZX₁^n-1 - LaX₁^n-1 +Mxⁿ - Mxⁿ =A(Zⁿ - aⁿ) + Bx(Z^n-1- aⁿ) +…+ 0 =A(Z – a)(Z^n-1 +Z^n-2a + …+ a^n-1) +BX₁(Z – a)(Z^n-2 +Z^n-3a +…+ a^n-2) +…+LX₁^n-1(Z - a) → f(Z) – f(a) = (Z – a){A(Z^n-1 + Z^n-2a + … + a^n-1) +BX₁(Z^n-2 + Z^n-3a + … + a^n-2) +…+ LZ^n-1} ………………………………………… 4

Then, If f(Z) is divisible by Z – a, and f(Z) – f(a) = (Z – a){A(Z^n-1 +  … + a^n-1) + BX₁(z^n-2 + … + a^n-2) +…+ LX₁^n-1}, then, by Corollary I of the Division Algorithm, f(a) is also divisible by Z – a, Q.E.D. …………………………………………..………………………………… 5

By incorporating the operational principle, definitions and theorems presented above into the axiomatic system of mathematical logic and applying them to FLT, we will show how  the FLT65 demonstration that f(a), the remainder obtained by dividing f(Z) by Z – a can never equal zero [i.e. f(a) ≠ 0], provides an unavoidable contradiction proving Fermat’s Last Theorem.

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[1] A comment added by the author of this paper.

[2] See Appendix D for detailed notational definitions.