Internet Draft E. Terrell Category: Proposed Standard ETT-R&D Publications Expires October 28th, 2006 April 2006 The Mathematics of Quantification, and the Rudiments Of the Ternary Logical States of the Binary Systems 'draft-terrell-math-quant-ternary-logic-of-binary-sys-04.txt' Status of this Memo Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet-Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." "This document may not be modified, and derivative works of it may not be created, except to publish it as an RFC and to translate it into languages other than English." 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E Terrell Internet Draft [Page 1] The Ternary Logical States of the Binary System October 28, 2006 Abstract This paper, opening with the historical that documents the source of the Binary Enumeration Error, utilizes the proof of 'Fermat's Last Theorem' (Normative References - [1], [2] and [3]), the Mathematics of Quantification, and the Logic of Set Theory, to prove that the Binary System represents an Alternate Mathematical Field, which is Closed and Finite. That is, using the Elementary Laws of Algebra, with the Basic Principles from Analytic Geometry, provides the final clarification simplifying the proof for the correction of the Counting Errors and the Logical Foundation for the New Binary System. And more importantly, this also establishes the basic foundational principles for 3 State Ternary Logic. In other words, using an askew, or mathematically incorrect Binary System, defined as the misinterpretation of ZERO, sustains the Counting Error (an Accumulating Propagation) levying a substantial loss of IP Addresses in the IPv4 IP Specification, affecting as well, the Address Pool Total for the IPv6 Specification. Hence, from the foregoing foundation an unquestionable proof concludes; the Elementary Mathematical 'Resolution of the Counting Error in the Binary System' - [4. IANA Considerations]. E Terrell Internet Draft [Page 2] The Ternary Logical States of the Binary System October 28, 2006 Table of Contents Abstract Introduction 1. The Beginnings of Binary Enumeration 1.1 Gottfried Wilhelm von Leibniz's Binary System 1.2 George Boole's Mathematical Logic 1.3 The Arithmetical Error and the flaw in Binary Enumeration 2. The Unary and The Binary Mathematical Systems 2.1 Two Distributive Laws & The Binary System Proves Fermat's Last Theorem 2.2 The Mathematics of Quantification and Binary Arithmetic System 2.3 The Binary and Ternary Systems and George Boole's Mathematical Logic 3. Security Considerations 4. IANA Considerations - 'Resolution of the Counting Error in the Binary System' 5. References E Terrell Internet Draft [Page 3] The Ternary Logical States of the Binary System October 28, 2006 Introduction The investigation of the origin of the Binary System revealed that Leibniz, its principle author, is responsible for the askew error, because he never understood or actually developed a Binary System of counting. And this is clearly shown to be the handicap that not only resulted in the Loss of available IP Addresses in the IPv4 Specification, but it contributed to the difficulties preventing the development of the Binary and Ternary Relations defined by Boolean Algebra. That is, by clearly showing that this is a Closed Finite Mathematical System, which defines an incremental progression using ' 1's '. This greatly simplified the Boolean Mathematical Relationships for the 'Theory of Three State Logic', and corrected the error in Binary Enumeration, which generated the loss of IP Addresses in the IPv4 Specification. In other words, the proof of 'Fermat's Last Theorem" defines a special case of the Distributive Law, which is defined in the mathematical logic of Set Theory, as the Intersection of the two Universal Sets that represents the Binary and the Unary Systems. And this conclusively proves, that there are only Two logical Systems of Counting, which are mathematically viable. E Terrell Internet Draft [Page 4] The Ternary Logical States of the Binary System October 28, 2006 1. The Beginnings of Binary Enumeration The History of the Binary System has its recorded beginnings starting about the 5th century BC. But, there is a problem with this recorded date, because the historians have not defined, or established an agreement regarding what they mean jointly, or independently, when they are referencing the development of the Binary System. In other words, for many people, specifically mathematicians, when they speak or make reference to the Binary System, they are talking about mathematics. The Binary System, as a Mathematical System actually did not come into fruition until the 1600. That is, from the 5th Century to the 1600, what is thought to be a Binary System for Mathematical Enumeration, was in fact, either a system of Drum Beats for communications, a system of Open and Closed Bars used for counting, or a system for distinguishing musical notes in musical compositions. In any case, each of these so called Binary Systems shared the same flaw; they skew the counting by the misrepresentation of the Binary equivalent of '1'. E Terrell Internet Draft [Page 5] The Ternary Logical States of the Binary System October 28, 2006 1.1 Gottfried Wilhelm von Leibniz's Binary System The general consensus regarding Leibniz would contend that, he made significant contributions to the foundations of Mathematics, Philosophy, and the beginnings of Set Theory. However, because he was indeed, a man of the times, Leibniz was occupied by a broad range of subjects. Nonetheless, while he did make significant contributions to humanity, an investigation of some of his most noted contributions would show that he did not completely finish the work for closure of the proposed subject(s). That is, I am of the opinion that, for most of his life, Leibniz was looking for the pieces of his puzzle, the clues or solution to clarify the concerns involving his ongoing research in the areas of Philosophy, Logic, and Metaphysics (The Laws and Logic of Critical Thinking). Needless to say, my opinion is evinced more clearly by the study of the works from one of his contemporaries, Perrie de Fermat, and the man most profoundly influenced by his research in Metaphysics, George Boole. Nevertheless, while Leibniz correctly translated the symbolisms for enumeration, as presented in the book of I Ching, into a Binary System of counting, which was similar to the Unary System. However, the reality of this accomplishment is that, his only achievement was the 'Ø' and the '1' solution to his problem concerning his Metaphysical Research, which pertained to the Logical Analysis for the presentation of 'The Laws and Logic of Critical Thinking'. In which case, had he either knew, or fully understood that Numerology, or Number Theory in general, involved the Logical Analysis of the Elementary Laws of Mathematics. He probably would have correctly completed his Numbering System, and 'Fermat's Last Theorem' would not have become one of the greatest, from a historical perspective, Mathematical Enigmas of all times. In any case, since 'Fermat's Last Theorem' was not solved until November 1979, there was no logical connection ever established between the works of Fermat and Leibniz. Hence, in the absence of a logical reason for a comparable analysis, there was no reason to question the validity of Leibniz's numerical translation. In other words, the Modern Binary System, as depicted in figure 1, is the direct consequence from the work of E Terrell Internet Draft [Page 6] The Ternary Logical States of the Binary System October 28, 2006 Leibniz, and it remains logically incorrect. This because, the discovery of the solution to the problem that qualified as the logical reason for the comparable analysis questioning his results, from the mathematical perspective, it violates the laws from elementary mathematics, the Field Postulates, the Axioms for Equality, and the logical foundation of Set Theory. Modern Primitive Binary Unary System System 00 0 01 1 10 11 11 111 100 1111 101 11111 110 111111 111 1111111 1000 11111111 1001 111111111 1010 1111111111 1011 11111111111 1100 111111111111 1101 1111111111111 1110 11111111111111 1111 111111111111111 10000 1111111111111111 figure 1 E Terrell Internet Draft [Page 7] The Ternary Logical States of the Binary System October 28, 2006 1.2 George Boole's Mathematical Logic The influence of Leibniz upon George Boole is unquestionable, however, Boole's greatest contribution to mathematics overshadow considerably, his retake on objectives of Leibniz's life's work. In other words, Boole's work; "An investigation of the Laws of Thought on Which are founded the Mathematical Theories of Logic and Probability", is a mathematical and logical marvel that clearly renders a rational demystification of the Metaphysical rhetoric encompassing the logic of the 'Ø' and the '1' foundation, which was the hallmark of Leibniz pursuit to resolve 'The Laws and the Logic Foundation of Critical Thinking'. Still, George Boole was unaware of the contributions he made to Mathematics and the Mathematical Sciences, because it was embedded in his most famous work; "An investigation of the Laws of Thought on which are founded the Mathematical Theories of Logic and Probability". Furthermore, while using the principle foundation of the '0'and the '1' concepts created by Leibniz, Boole correctly established an Algebraic and Logical Foundation that was later to have applications throughout the fields Computer Science and Electronics. However, the result from Boole's work was wrongly interpreted as the 'Logic of the Binary System', when in fact, it is actually 'The Logic of the Unary System', because only One State Works, or because only One Stated Condition can be True, as shown in Figure 2. The Truth Relation of Two State Logic Key to the Truth Table The Table on the Right shows the combinations |A = T |A = T | of Truth Values for the two Operands, A and B, |B = T |B = F | in relation to the Truth Function |A = F |A = F | |B = T |B = F | True Or If A Implies B Iff End |T F| |T F| |T F| |T F| |T F| |T F| |T F| |T F| |F T| |T F| |T F| |T F| |T F| |T F| |T F| |T F| Nand Xor Not B Nimp Not A Nif Nor False |T F| |T F| |T F| |T F| |T F| |T F| |T F| |T F| |F T| |T F| |T F| |T F| |T F| |T F| |T F| |T F| E Terrell Internet Draft [Page 8] The Ternary Logical States of the Binary System October 28, 2006 |--------------------------------------------------------------| |Boolean Algebra: Algrbraic and Logical Laws of Two State Logic| | | | AND ( . } Associative Law Communative Law | | 0.0 = 0 (A.B).C = A.(B.C) A + B = B + A | | A.0 = 0 (A + B) + C = A + (B + C) (A.B) = (B.A) | | 1.0 = 0 | | A.1 = A | | 0.1 = 0 Distributive Law Identity Law | | A.A = A (A + B) x C = AC + BC A + A = A | | 1.1 = 1 (AB) + C = (A + C)(B + C) AA = A | | A.A' = 0 | | | | | | | | Or ( + ) Precedence Not ( * ) DeMorgan's Theorem | | 0+0 = 0 AB = A.B 0* = 1 (A.B)' = A' + B* | | A+0 = A A,B + C = (A.B) + C 1* = 0 (A+B)' = A' + B' | | 1+0 = 1 A* = A | | 0+1 = 1 | | A+1 = A | | A+A = A | | A+A'= 1 | | 1+1 = 1 | | | |--------------------------------------------------------------| figure 2 E Terrell Internet Draft [Page 9] The Ternary Logical States of the Binary System October 28, 2006 Nevertheless, given that an argument can be made claiming the existance of Two States, '0' and '1'. However, not until it is realized that Boole's ascribes to a literal usage, using their actual numeral values, it will then become understood that a Unary System is a Two State System, because it is a System of Counting uses '1s' to represent something and a '0' to represent nothing: 'Hence, A Two State System'. So, the question of ponder that one might ask is: 'If the number of States in the Logic of the Modern Binary System equals that of the Unary System. How many States defined by Boolean Relationships does the True Binary System have ...??... Figure 3. States of the Modern Binary System States of a Unary System 00 \ / 0 2 States 01 / \ 1 figure 3 E Terrell Internet Draft [Page 10] The Ternary Logical States of the Binary System October 28, 2006 1.3 The Arithmetical Error and the flaw in Binary Enumeration While it should be quite clear that a fundamental knowledge of Archeagology, Anthropology, and perhaps that of the early Languages, is a necessary prequsite to the study of any anchient Civilization. Still, there should never be any doubts that if there is, or was a Civilization whose first system of counting was a True Binary System, then this would probably be the most advanced Civilization in the Universe. In other words, because of the inherent complexities involved in the meaning and the interpretation of the concept of Zero, the development of a True Binary System by any Ancient or Primitive Civilization boarders on the Highly Unlikely, or the Impossible. In which case, prior to Leibniz's discovery of the Two State Logical System for his Metaphysical Analysis of Critical Thinking, I cannot accept as being possible for any Civilization, before his time, to have either created, or fully understood the Mathematics of the Binary System. The case in point is the mathematical error discovered in 1999, which was an obvious mathematical descrepancy between two different Binary Mathematical Systems. However, it is also quite obvious that knowone since Leibniz, could either rationalize this difference, or understood why a difference occured. And while the most notiable self-rightous and unspoken claims, under the guise of Religin, Politics, Racial, or Economic depervation / discrimination, for every Civilization since mankinds beginnings, has been the horrifyingly tortoureous control and exploits of its people. Yet, even with the persistance of these living conditions today, it is still difficult not wonder, how, or why it is possible for a blunder having such simple a solution, could have lasted for so long. ...???... In other words, the pointed reality of this descrapancy, ask the question: 'Is it possible for a one-to-correspondence of a Set X, with the Set of Integers, I, which yields the count of the total number of members the Set X contains, to have more than one value to represented in the Set I?' {... No!} That is, it is not possible for any one-to-one pairing between the members of two Sets, the Set X, and the Set I, for any member contained in any one of the two Sets to have more than one pairing with the members of the other Set. And this is because, such a pairing establishes a count that can be translated into a equality, when both Sets, given in Table I, are said to represent the same (Identical) method for enumeration. E Terrell Internet Draft [Page 11] The Ternary Logical States of the Binary System October 28, 2006 TABLE I 1 2 3 Modern Modern Primitive Binary Positive Unary System Integers System 00 0 0 01 1 1 10 2 11 11 3 111 100 4 1111 101 5 11111 110 6 111111 111 7 1111111 1000 8 11111111 1001 9 111111111 1010 10 1111111111 1011 11 11111111111 1100 12 111111111111 1101 13 1111111111111 1110 14 11111111111111 1111 15 111111111111111 10000 16 1111111111111111 In any case, to say the very least, it should be quite clear from the examination of Table I, that if a given Binary Number, say, '11111111', has two Intergeral Values, '255' and '256', there is a undeinable problem with the Binary System when it is used as a System of Counting. Still, anyone, and with good reason, could quite easily present the excues; "It is a Typo-Graphical Error!", as a viable opposing argument. However, such an argument would easily fail, because there is absolutely No proof, if {a, b} = {0, 1}, which would now account for the existence of the 4 conditions that must clearly represent a number; Substitution Law for Equality now yields,{a, a}, {a, b}, {b, a}, and {b, b} given in Table II. Especially since, it is evident in this scenario that Zero cannot be equal to E Terrell Internet Draft [Page 12] The Ternary Logical States of the Binary System October 28, 2006 either '0', or the Null Set, (Out of Sight, Out of the Consciencous thought ... Does not exist!) because 'a' references something in the real sence. Furthermore, when comparing the three rows from Table I, it is also evident that there is a common coefficient between different numerical representations that are equal to the same number. But, this accestment is only valid between the members of columns 2 and 3 in Table I, and conditionally valid between the members of columns 1, 2, and 3, in Table II. Note: The unfortunate reality of Table II, is that, the New Binary System impacts Gregor Mendel's work in Genetics. In other words, from an 'A a' and 'B b' paring, {A, a, B, b}, Mendel's results referenced only 6 of the possible 16* combinations; {A, A}, {B, B}, {A, B}, {B, b} {A, a}, and {a, b}. However, while I have not wrote the New Foundation representing Finite Chemistry, the reality of the mathematical results from the Mathematics of Quantification now questions the validity of Mendel's claims. In any case, it has been proven, using the current foundation, that the order of the addition of Chemicals is a vital consideration for the determination of the Chemistry of the resulting Chemical Compound (10 combinations are missing*). Still, what's alarming? Well. ...considering the 'X' and 'Y' Chromosomes that represent this relationship. This also suggest the possibility of an error in the Chromosome Count defining the Base Pairs; A = adenosine, C = cytosine, G = guanine, and T = thymine, given that they current identify 23 + 23 = 46 Chromosomes. That is, from the Mathematics of Quantification this defines, 2^5 + 2^5 = 2^6 = 64 = 8^2 Chromosomes, four pairs of 8 Bit Bases Pairs, or 32 + 32 = 64, that yields about 2^32 = 4,294,967,296 Bases, which translates into two 8^10 pairs of 8 Bit Bases Pairs per Cell of human DNA. (et 2004) E Terrell Internet Draft [Page 13] The Ternary Logical States of the Binary System October 28, 2006 TABLE II 1 2 3 Another Modern Primitive Binary System Positive Unary Representation Integers System 0 0 0 aa 1 1 ab 2 11 ba 3 111 bb 4 1111 baa 5 11111 bab 6 111111 bba 7 1111111 bbb 8 11111111 baaa 9 111111111 baab 10 1111111111 baba 11 11111111111 babb 12 111111111111 bbaa 13 1111111111111 bbab 14 11111111111111 bbba 15 111111111111111 bbbb 16 1111111111111111 E Terrell Internet Draft [Page 14] The Ternary Logical States of the Binary System October 28, 2006 Nevertheless, while studying the analysis from Tables III and IV, recall the former proofs, because it was clearly shown that if '00 = aa = 1', and '01 = ab = 02', and the Exponent 'F = either a Rational or Irrational Number, then the Binary Translation could only equal the Binary Representation for the Number. This meant, the exponent 'F' was not a whole Number. However, when the result from the secquential variable of the exponent having a of base '2' equaled the value of a whole number, and the exponent was also a whole number, then given that 'Multiplication is the Quantified Sum of Addition', the the value of the exponent equaled the sum of the Binary 1's and the Product of the Binary 1's equaled the Binary Number and the Unary Number. That is, because '2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 = 1111111 = 2^7' and '2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 = 11111111 = 2^8', there is clearly a relationship between the columns in Table IV, and since (2 + 2) = (2 x 2), it shall be proven in Part II, not only that the established proof for the New Binary System remains correct. But, that its validity is derived from the proof of 'Fermat's Last Theorem' and the discovery of the 'Distributive Law for Exponental Functions'. Nevertheless, this proves that the differences between Tables III and IV clearly do not represent a Contradiction, the necessary requirement as stated by "Chief Executive Administrator for The Electronic Library of Mathematics", Aleksandar Perovic, when he said: "Mathematicians do not accept claims at truth of any possible, non-selfcontradictory (= consistent) mathematical system". Needless to say, while this difference is not a Contradiction, it is indeed a troubling Inconsistency which at the very least, warrants an investigation. E Terrell Internet Draft [Page 15] The Ternary Logical States of the Binary System October 28, 2006 TABLE III "The Modern Interpretation of the Binary System of Enumeration" Counting, using only "1's" and "0's" Depicting the Results from its current Presentation Exponential Binary Positive Enumeration Representation Integer 1. 0^0 = 0 00000000 = 0 0 2. 2^0 = 1 00000001 = 01 1 3. 2^1 = 2 00000010 = 10 2 4. 2^F = 3 00000011 = 11 3 5. 2^2 = 4 00000100 = 100 4 6. 2^F = 5 00000101 = 101 5 7. 2^F = 6 00000110 = 110 6 8. 2^F = 7 00000111 = 111 7 9. 2^3 = 8 00001000 = 1000 8 . . . 129. 2^7 = 128 10000000 = 10000000 128 . . . 257. 2^8 = 256 100000000 = 100000000 256 E Terrell Internet Draft [Page 16] The Ternary Logical States of the Binary System October 28, 2006 TABLE IV "The Reality of the New Binary System of Enumeration" And the Series Generated when Counting, using the "1's" and "0's", and from the Axioms for Equality, "a's" and "b's" Exponential Binary Positive Enumeration Representation Integer 1. 0^0 = 0 0 0 2. 2^0 = 1 00 = aa 1 3. 2^1 = 2 01 = ab 2 4. 2^F = 3 10 = ba 3 5. 2^2 = 4 11 = bb 4 6. 2^F = 5 100 = baa 5 7. 2^F = 6 101 = bab 6 8. 2^F = 7 110 = bba 7 9. 2^3 = 8 111 = bbb 8 . . . 129. 2^7 = 128 01111111 = bbbbbbb 128 . . . 257. 2^8 = 256 11111111 = bbbbbbbb 256 E Terrell Internet Draft [Page 17] The Ternary Logical States of the Binary System October 28, 2006 2. The Unary and The Binary Mathematical Systems Throughout mankinds beginnings, there has been several different Systems of Counting, several different methods for performing elementary arithmetic, and an equal number symbols for those that were written, as well as the variety of sounds for those that were only spoken. However, only one numbering system, which is nearly complete, survived the trials of mankind's jouney towards civilation; 'The Unary System'. And while the Laws from the Axioms for Equality, the Field Postuletes, and Logic of Set Theory, which are an essenctial part of Unary System, was not developed until long after its discovery, sometime during the early and mid 1800's. Still, it doubtful that anyone before 1979, tested the validity of the Unary System. Needless to say, it should be quite clear now, that every System of Enumeration must comply with the Laws from the Axioms for Equality, the Field Postuletes, and Logic of Set Theory before it can ever be accepted as a valid System of Counting, which conforms to the elementary laws of arithmetic. In other words, the additional requirement, which any civilization must meet to claim the creation or the development of a True Binary System, is one that requires a prior the knowledge of the Unary System. If not, how could anyone justify the use of two objects to account for only one material possession... Hence, to use a Stick to represent the sumation of an arithmetic progression incrimented by the addition of 1, is far simplier than the use or discovery of the 'Stick and a Rock', which is used to represent the same incrimented addition. Clearly, if this were not the case, then the Binary System would not have, after its initial claim of discovery, to wait 2500 years to become a True Binary System. E Terrell Internet Draft [Page 18] The Ternary Logical States of the Binary System October 28, 2006 2.1 Proof of Fermat's Last Theorem; and The Two Distributive Laws It is extremely amazing that it required more than 300 years after 'Perrie de Fermat' composed, before his death in 1665, a riddle involving an elementary algebraic equation, which eluded everyone, including the greatest mathematicians, to find, in 1979, a solution. A joke? Perhaps. But, Fermat was the first to claim, while writing this riddle, that he knew the simple solution. And clearly, if this were true, which I believe that it is, then perhaps, "Fermat's Last Theorem" should rightfully be called; 'the greatest joke of all times'. However, while I accept Fermat's claim, I do not believe that he actually knew, or fully understood, the profound implications of his discovery. Especially since, it may be concluded, as presented below, there are only 3 loghically vaible 'Interconnected Complimentary Solutions' that would solve the riddle regarding why; "There are No solutions in Whole Numbers to the Equation, X^N + Y^N = Z^N, when N > 2". 1. There is no Common Coefficient between the Sum of Two Exponents, the Exponent equaling their Result, and their respective Roots, when 'N > 2' , and 'N' defines the Exponent of the base variables. (Equal Number of Parts Contained in the Whole.) E Terrell Internet Draft [Page 19] The Ternary Logical States of the Binary System October 28, 2006 2. Fermat's Solution defines how he interpreted the problem, which is based upon the current mathematical knowledge known duing his time, Pythagoras Theorem, and the Analytical Geometric solution(s) explaining the Difference regarding (the Geometric Shapes of Objects) 'Why', when 'N = 2'; 'The Sum of the Area of two Perfect Squares Equals the Area of another Perfect Square'. - Or - "The Sum of the Area(s) of TWO Squares having equal Integer Sides, equals the Area of another Square having equal Sides that are Integers." And, when 'N = N', 'The Sum of the Areas of two Perfect Nth Powers is not Equal to the 'ROOT' defining the Area of a Perfect Nth Power'. Nevertheless, this assumption builds an explanation that explains this difference, which it is believed to be the foundation for the proof that Fermat claimed would not fit in the margin of his paper, but would explain why, when 'N > 2', his theorem is true. 3. In Exponental Operations, there is No equal Distribution of Multiplication over Addition when 'N > 2', and 'N' defines the value of the Exponent. (The Discovery of the Distributive Law for Exponental Functions, and the Foundation for the Finite Mathematical Field: "The Rudiments of Finite Algebra; The Results of Quantification".) Note: If this is the Text version of this manuscript, then the imagination of the reader is required to picture the shapes of the Objects being described. But, if this is the PDF version, then all of the figures representing the Objects Shapes and special Mathematical Symbols are included. E Terrell Internet Draft [Page 20] The Ternary Logical States of the Binary System October 28, 2006 X Y X Y +-------+ +---------+ +---+-------+ | | | | | \ | | | + | | = | \ |X | | | | | Z + +-------+ | | + \ | +---------+ | \ |Y | \ | +-------+---+ Figure 4 Nevertheless, proof that it is assumed Fermat was thinking about, would proceed something like this, when 'N = 2': ""If the Length of the Side of a Perfect Square inscribing another Perfect Square is equal to 'X + Y', then the Sum of the Areas of Two Perfect Squares is equal to the Area of the Perfect Square inscribing another Perfect Square, and since the Area of a Square is given by; 1. 'L x W = Area', the Area of the Inscribing Perfect Square, from the Mathematics of Quantification is given as; 2. (X + Y) x (X + Y) = (X + Y)^2 = X^2 + 2XY + Y^2 And if the Length of the Side of the inscribed Perfect Square is equal to 'Z', and the Area of this Perfect Square is given by equation 1, then from Pythagoras Theorem, 'Z' is the Root of the equation given by; 3. X^2 + Y^2 = Z^2 = L x W = Z x Z E Terrell Internet Draft [Page 21] The Ternary Logical States of the Binary System October 28, 2006 Hence, the 'X, Y, and Z' variables, by Pythagoras Theorem now equals the Sides of the 4 Right Triangles forming, or Creating the Boarders of the Inscribing Perfect Square and the Perfect Square it inscribes. That is, if the Length of the Two Sides joining the 90 degree angle of the Right Triangle equals 1/4 the Length of the Parimeter of theInscribing Perfect Square, then the Sum of the X and Y variables defining the Two Sides of the Right Triangle equals the Length of the Side of the Inscribing Perfect Square. And given by equation 4, we have; 4. X + Y = Y + X, which means: If the Sum of the Length of the Two Sides, 'X + Y' of any Right Triangle forming the Right Angeled boarder of any Perfect Square, any Square having Four Equal Sides, is equal to 1/4 the Length of its Parameter, then the Sum of the Length of the Two Sides of the Right Triangle, is equal to the Length of One Side of the Perfect Square. (The Associative Law for Equality; "X + Y = Y + X". ET 2004) Furthermore, if the Sum of the Length of the Two Sides, 'X + Y' of any Right Triangle forming the boarder of any Perfect Sqaure equals the Length of One of its Sides, and if a Perfect Square defines the Closed shape of a figure having the 4 Sides defined by the shape of a Rectangle, but equal, then the boaders of the Perfect Square is defined by Four Equal Right Triangles. Hence, from Pythagoras Theorem, if of the Two Sides of the Right Triangles forming the boarders of the Perfect Square join to form the 90 degree Right Angles connecting the 4 Sides of the Perfect Square, then the Two Sides of the Right Triangles must respectively Equal the Adjance Side and the Side Opposite the Hypotnuse. Therefore, since the Right Triangles join the Sides of the Perfect Square, the connection of the Side forming the Hypotenuse of the Right Triangles must also meet, and be joined at 90 degree angles. And if the Four Right Triangles are equal, then the Length of Hypotenuse equals the Length of One Side of an Inscribed Perfect Square. In other words, this means that; The Sum of the Areas of Two Perfect Squares equals the Area of the Perfect Square Inscribing another Perfect Square, if and only if, The Sum of the Areas of the Four equal Right Triangles forming the E Terrell Internet Draft [Page 22] The Ternary Logical States of the Binary System October 28, 2006 boarders of the Inscribing Perfect Square and the Area of the Perfect Square it Inscribes, equals the Area of the Perfect Square Inscribing another Perfect Square. And from equation 5, the Area of a Triangle is given by; 5. 1/2(b x h) And given that only the Adjancent Side and the Opposite Side of the Right Triangles can, respectively equal the Base, b, and the Height, h, since there a 4 Right Triangles having equal sides, X and Y, by equation 5, the Area of the 4 Right Triangles is given by; 6. 4((1/2(X x Y) = 4/2(XY) = 2XY And from these results, it is easy to descern the equation for Sum of Areas of the 4 Right Triangles, as given by the equation; 7. (X - Y) x (X - Y) = X^2 - 2XY + Y^2, where 8. X^2 + Y^2 = 2XY Hence, the Area of the Perfect Square Inscribing a Perfect Square, which is equal to the Sum of the Areas of Two Perfect Squares, is given by; 9. (X + Y) x (X + Y) = X^2 + 2XY + Y^2 = 2XY + Z^2 Therefore; 10. X^2 + Y^2 = 2XY - 2XY + Z^2 = X^2 + Y^2 = Z^2 E Terrell Internet Draft [Page 23] The Ternary Logical States of the Binary System October 28, 2006 Thus, the equation, X^2 + Y^2 = Z^2, which is defined by Pythagoras Theorem, clearly states that the Sum of the Areas of Two Perfect Squares is equal to the Area of a Perfect Square"". And clearly, from his analysis, Fermat would have concluded the X and Y relations: 11. If X = Y, then X and Y are Two equal Perfect Squares, and If X > Y, or Y > X, then X and Y are Two different equally Perfect Squares. X Y X Y +-------+ +---------+ +---+-------+ / /| / /| / /| / / | / / | / / | +-------+ | + +---------+ | = +---+-------+ | | | | | | | | \ | + | | + | | + | \ |X | | | / | | / | Z + | +-------+ | | / + \ | + +---------+ | \ |Y / | \ | / +-------+---+ Figure 5 And clearly, from this analysis, Fermat would easily conclude that if the length of the Sides of a Perfect Cube are equal to that of a Perfect Square, when 'N = 3', then the Area of Cube is given by; 12. L x W x T = Area E Terrell Internet Draft [Page 24] The Ternary Logical States of the Binary System October 28, 2006 Hence, would have also known that since Area of Cube is given by equation 12, the Sides of a Perfect Cube are equal to that of a Perfect Square, when 'N = 3', must be equal, that the change in equation 12 is given by equation 13; 13. L x W x T = Area = X x Y x R = Z^3 Hence, If the Root of Z^3 is equal to (X + Y), then the Area of a Perfect Cube, which inscribes another Perfect Cube is equal to the equation given by; 14. (X + Y) x (X + Y) x (X + Y) = (X + Y) x (X^2 + 2XY + Y^2) = X^3 + 3YX^2 + 3XY^2 + Y^3 Furthermore, he would have quickly noticed that a Perfect Cube has 8 90 degree Angles forming its boarders, or 4 pairs of 3 dimensional Right triangles, Prisms having 5, 2 dimensional face. This he would have reasoned further, meant that, only a Pyramid could have 4 equal lengths measuring its sides. In other words, Fermat would have quickly concluded that, it is not possible for either any one of the 8, or 4 pairs of Right Triangles forming the boarders of a Perfect Cube, could have equal sides, and still be a Right Triangle. Needless to say, he would have also known that this did not mean that the Sum of the Areas of these 3 dimensional Right Triangles did not equal the Area of a Perfect Cube. Nevertheless, he would continue to follow the logic from the conclusions involving 'N = 2' by first, confirming the formula for the Area of a 3 dimensional Traingle, to determine if the Sum of the Areas of Two Perfect Cubes is equal to the Area of another Perfect Cube. However, he would eventually notice, that while the Volume and the the Area of a Perfect Cube were equal formulas, the Volume and the Area of a 3 dimensional Triangle, or Prism, represented 2 different formulas. Where by, the Area of a 3 dimensional triangles is given by equation 13a, the Volume of the same Triangle is given by equation 13b; E Terrell Internet Draft [Page 25] The Ternary Logical States of the Binary System October 28, 2006 14a. Area of a Prism = A = 2(b^2) + 3b(h), where b^2 = Area of base, 3b = b + b + b = Perimeter of base, and h = Height of the Prism 14b. Volume of Traingle = V = Area of the Base (B^2) x the Height (h) = b^2h = b^2(h) = B^2 x h, V = b^2(h) Clearly, while an argument can be made regarding the difference between the formulas in equations 14a and 14b, which represents the two distinct results that respectively measure the 'Area of a Prism' and the Volume of 3 dimensional Triangle. Even still, Fermat would have probably continued to follow the logical patterns reasoning derived from the conclusions when 'N = 2', because he could quite easily test for the conclusions that would verify either one, or both of these formulas. Thus, following the logical reasoning concluding equations 6, 7, and 8, in an attempt to dervie the results that would conclude the Perfect Cube, which logically concludes results simular to that involving equations 9 and 10. Needless to say, I am hard pressed to imagine, but I seriously doubt that Fermat was supprised by his discovery, when trying to confirm equations 14a and 14b, that there are actually 5 different formulas, which must be used in the logical analysis that would determine the validity of; 'The Sum of the Areas / Volume of Two Perfect Cubes are equal to the Area, or Volume of another Perfect Cube'. In any case, it should be understood that the Cubes of the 'X, Y, and Z' variables must be Positive Integers, because their respective Cube Roots must be a Positive Integer. Where by, given below, we have; 15. [(X + Y) x (X + Y)] X (X - Y) = X^3 + X^2y - XY^2 - Y^3 16. [(X - Y) x (X - Y)] X (X + Y) = X^3 - X^2y + XY^2 + Y^3 E Terrell Internet Draft [Page 26] The Ternary Logical States of the Binary System October 28, 2006 17. [(X - Y) x (X + Y)] X (X - Y) = X^3 - X^2y - XY^2 + Y^3 18. [(X - Y) x (X + Y)] X (X + Y) = X^3 + X^2y - XY^2 - Y^3 And since by Definition; Exponent: Any symbolic representation, 'Q', which is used inconjuction with the Number, 'X', representing a Multiplicand, represents the count of the number of Identical Multiplicands used in the equation representing Product of Q Multiplicands; x^Q = (Xv1 x Xv2 x Xv3 x ... XvQ). Hence, given by equation 19, we have; 19. [(X - Y) x (X - Y)] X (X - Y) = X^3 - 3X^2y + 3XY^2 - Y^3 Clearly, once Fermat realized, upon inspection of equations 14a through 19, that neither the Sum of the Areas, or the Volumes of the Right Angled Prisms forming the Perimeter of the Perfect Cube equaled the factors from equaton 12, '3X^2Y + 3XY^2', whose difference would yield the same conclusions established by equation 18. He would have reasoned that, 'The Sum of either the Area, or the Volume of Two Perfect Cubes did not equal another Perfect Cube', because there is a divergence diminishing the equality between factors in noted equations. And further testing, he would have reasoned, the divergence diminishing the equality between factors increases for every unit of increase of the Exponent, 'N'. Hence, he would finally concluded, since (2 + 2) = (2 x 2), "There are No solutions in Whole Numbers to the Equation, X^N + Y^N = Z^N, when N > 2", because the Operation of Multiplication, M, is equal to the Operation of E Terrell Internet Draft [Page 27] The Ternary Logical States of the Binary System October 28, 2006 Addition, A, M = A, only when the number Variables involved in each of these operations, is equal to TWO. And the translation, or interpretation of this conclusion yields; 'If the Whole Number sought must be equal to the Root of a Square, because a Perfect Square equals the Product of Two Equal Variables. And since an equation of Mutilpication is equal to an equation of Addition only when each of these operations involves two variables, then only an equation equaling Sum of the Two Variables equal to the Two Perfect Squares can equal the Product of the Two equal variables that is equal to a Perfect Square'. In which case, there is no Integer that can equal the Nth Root of the Nth Power that is equal to the equation of the Sum of Two Nth Powers. "In other words, since an equivalency between the Operations of Multiplication and Addition only exists at Power of 2 (denoting the number of Variables involved in both of these operations), then only the Sum of (in this case; Two) Perfect Squares can equal the product of the two equal multiplicands, which is equal to another Perfect Square, and still retain an integer solution for the values of the Variables representing Power of the Exponent and the respective Roots". Note: I investigated the same conditions, in the proof entitled; "The Proof of Fermat's Last Theorem; The Revolution in Mathematical Thought". However, I concluded, from the same data, that "If 'N > 2' in the equation, X^N + Y^N = Z^N, then there are no Whole Number Solutions for the Nth Power of the Sum of Two Nth Powers and their respective Nth Roots. That is, because there is No incremental (Additive) progression using ' 1's ' defined for Fermat's / Pythagoras Equation, the Integer Coefficient, which is the Common Coefficient between the Powers of N and their respective Nth Roots do not exist. Nevertheless, this concludes the rendering of the proof, that I believe, Fermat understood to be True. E Terrell Internet Draft [Page 28] The Ternary Logical States of the Binary System October 28, 2006 Nevertheless, from the analysis of the forgoing conclusions, and the realization that equation 8 and the equation from "Fermat's Last Theorem", represented a special case defining the 'Distributive Law', as given by equations 19 through 24. That I also understood the profound meaning of the proof of "Fermat's Last Theorem". In other words, I reasoned first; 'Any complete proof of "Fermat's Last Theorem" must be founded upon the 'Distributive Law', and conclude with the discovery of a New 'Distributive Property'. And this meant that when 'N > 2' in the equation, X^N + Y^N = Z^N, the Multiplication was not equally Distributed over the operation of Addition. Hence, from the results of equations 19 through 24, it is was easy to conclude, since the Operation of Mutiplication is not equally Distributed over Addition in the case where 'N > 2': There is no Common Coefficient between the Nth Power of the Sum of Two Nth Powers, and their respective Nth Roots. In which case, I concluded that Fermat was correct, and had the knowledge of proof I demonstrated above. However, Fermat's mathematical background lead me to conclude that he did not understand fully the implication of his riddle, because I believe, if he did, he would made a pointed reference. E Terrell Internet Draft [Page 29] The Ternary Logical States of the Binary System October 28, 2006 Special Case of the Distributive Law is the conclusion of Equation 25: 20. (X - Y)^2 = (X - Y) × (X - Y) = X^2 - 2XY + Y^2 21. X^2 + Y^2 = 2XY = XY + XY 22. (X + Y)^2 = (X + Y) × (X + Y) = X^2 + 2XY + Y^2 23. X^2 + 2XY + Y^2 = 2XY + Z^2 24. X2 + Y2 = Z^2 + 2XY - 2XY = X^2 + Y^2 = Z^2 25. Z^2 = 2XY: hence, X^2 + Y^2 = Z^2 X^2 + Y^2 = 2XY X^2 + Y^2 = XY + XY = X(Y + Y) E Terrell Internet Draft [Page 30] The Ternary Logical States of the Binary System October 28, 2006 Furthermore, because the conclusion from the proof and the equation involved in "Fermat's Last Theorem", represented an Algebraic Expression of the Exponential Function concluding the existence of the 'Distributive Law for Exponential / Non-Linear Functions. I knew, or reasoned, since the Distributive Law is also logically valid in 'Set Theory', that an Exponential Expansion of the Mathematical Logic of Set Theory must also sustain logical validity, and conclude the logical support for the conclusions derived from the foregoing proof: The Discovery of a New Distributive Property. Still, the clarification and definition of the Exponent, and the Exponential Operations employed in the Mathematical Logic of Set Theory, required more precise definitions of the familiar operations involving Addition, Subtraction, Multiplication, and Division. In other words, the Exponential Expansion of Set Theory, which also logically sustains only the operations of Addition and Subtraction, nearly mirrors the proof of the 'Distributive for Exponential / Non-Linear Functions'. And the Exponential Expansion of the Field Postulates, concluded the existence of the Mathematics of Quantification, which is defined as a Finite Mathematical Field, conditionally closed over the Set 'R' for the Operations involving Addition, Subtraction, Multiplication and Division. E Terrell Internet Draft [Page 31] The Ternary Logical States of the Binary System October 28, 2006 The Definitions Multiplication: The Quantified Sum of the equal distribution of the Multiplicand, which is equal to the Addend that is used in the Summation of the equal Addends, which are equally distributed by a factor equal to the other Multiplicand that is used in the equation representing a product. "Hence, Multiplication is the Quantified Sum of Addition" 5 x 14 = (14 + 14 + 14 + 14 + 14) = (5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5) = 70 Division: The Quantified Difference of an ever changing Dividend, which becomes the Subtrahend that is used in the repeated Subtractions performed on a Constant, which is the Divisor the becomes the Minuend in the equation. "Hence, Division is the Quantified Difference of the Repeated Subtraction performed on a Constant, which results in the Count of the Total Number of Parts Contained in the Whole. 18/2 = 9, and Nine Subtractions of 2 from 18 equals; '(((((((((18 - 2) -2) -2) -2) -2) -2) - 2) - 2) - 2)' Addition: The mathematical operation representing a Summation, indicating a growth, or an increase in the number of the members contained in the Whole, by the inclusion of new members: The Union of Sets; 'U'. Subtraction: The mathematical operation representing a Difference, indicating a depreciation, or a reduction in the number of the members contained in the Whole, by the exclusion of members: The Disunion of Sets; 'BarU'. E Terrell Internet Draft [Page 32] The Ternary Logical States of the Binary System October 28, 2006 The Definitions Disjoint: If there are two sets, A and B, such that, A and B share no common members, then the two sets are said to be Disjoint; A ñ B, (read; A is not connected to B: 'A ñ B = Ø'. Dis-Union: If A U B = C and C n A = A is true, then the Dis-Union of the Set A from the Set C, C BarU A = B, (read; C dis-union A) is the exclusion of the members from the Set C, which are common to the Sets C and A, iff, A ñ B = Ø. 2. If A Not= C and C n A = B, then C U A = E ñ D. 3. If every Set A is a Sub-Set of itself, and A n A = A, then A U A = Ø. Exponential Cardinal: If for every X, where X E U, there is a condition, such that; X n X = X, X n X n X = X, (Xv1 n Xv2 n Xv3 n ... n XvQ) = X, and X^Q = X is True. Then there is a Exponential Number, Q, called the Exponential Cardinal of X, which is the number that represents the occurrences of X in the equation representing it's Intersection. Set: If a Unit Whole contains a collection of Objects, and each Object defines, one and only one, Part belonging to the Unit Whole, then the Unit Whole defines a Set as a Collection of Objects, iff, each Object defines one and only one Element, or Member, that defines the Part belonging to the Unit Whole. E Terrell Internet Draft [Page 33] The Ternary Logical States of the Binary System October 28, 2006 The Theorems Sub-Set: If every element, E, of a Set B is a E of the Set A, then the Set A is said to contain every E of the Set B, and the Set B is said to be a Sub-Set of the Set A. Hence, every Set is a Sub-Set of itself, iff, A n A = A. Cardinal Number: If it may be concluded that the Multiplicative Identity Law is True, and X x 1 = X, where X does not change, then from Set Theory, X is the Multiplicative Identity of Itself. And if this defines X, when X = X^Q, then X defines the Identity Element as the Unit Base, or the Cardinal Number = 1 defines the Common Coefficient as the Multiplicative Identity Element for all X| X E U. Therefore, if {Uv1 n Uv2 n Uv3 n ... n UvQ} = U^Q = U^Qv{N} = U, and given that Multiplication is the Quantified Sum of Addition, where X^Q = U^Qv{N} is True. Then for all X| X E U = U^Qv{N} = UvN, the Cardinality of any Set UvN, is the Sum or Union of Cardinal Numbers, or UvN = {Xv1 U Xv2 U ... U XvQ} = (1v1 + 1v2 + ... + 1vQ), iff, for all X| X E U, X = 1 defines the Cardinal Number for the Element of every Set as a Sub-Set of I | I = Set of Integers. In which case, the Unary Set, {1}, defines the Cardinal for the Element X of the Set I for all X| X E I, given that I = {X}, when X = 1, and the Cardinal for every Element X of the Set I for all X| X E I, when I = {X, X, X ... X}, and X = 1, I = (1v1 + 1v2 + 1v3 + ... + 1vQ). Hence, the definition of a Cardinal Number is given by: Cardinal Number: The Cardinal Number is the Multiplicative Identity Element for all X| X E I, which represents the Element of the Unary Set that is used to determine the Cardinality of every Set from the Sum or Union of the Multiplicative Identity Element for every E X of the Set I: iff X^Q = X. E Terrell Internet Draft [Page 34] The Ternary Logical States of the Binary System October 28, 2006 Note: This defines the Unit Base X, for all X| X E I as the Element of the Unary Set, because X is the Multiplicative Identity of Itself that defines, X = 1. [The next proof presented, is the interpretation of the Proof, or implications, that Fermat never understood, or could not explain. This is the accepted rationalization because Set Theory, the complete Logical Model of Mathematics, was not finished for nearly 200 years later. However, because he Co-Discovered the Cartesian Coordinate System representing the Mathematics of Analytic Geometry. The mathematical relationships from the foregoing, he should have maintained an above average understanding of the foundational theory of the proof presented. Still, for me, these results initially implied the existence of: the 'Distributive Law for Exponential / Non-Linear Functions'; an alternate Mathematical Field that was Finite and Closed / True as defined by the Axiom for Equality, the Field Postulates, and Set Theory. In which, it was later discovered, actually defined the Binary Set and the {Binary Enumeration & Mathematics} Mathematics of the Binary System. e. Terrell 1983] Nevertheless, since the foregoing conclusions proves that because the 'Multiplicative Identity Element' defines the Universal 'Common Coefficient', which is the same for all Objects, as the element, 1, defined in the Unary Set. And since it may also be concluded that counting is actually the assignment of a '1' to every object to be counted, and then, adding the "1's" that represent the objects, determines the Cardinality of the Set containing the objects being counted. Clearly, if the Set I, the Set of Integers defines the Set of all Symbols used to represent the result of the addition, inclusion, or incremental progression using the element, 1, defined in the Unary Set (given by Table II), then the (Arabic Numerals / Positive Integers) Modern System of Counting is defined by the Unary Set: As a Unary System. E Terrell Internet Draft [Page 35] The Ternary Logical States of the Binary System October 28, 2006 In other words, since the Cardinal Number, by definition, must define the Neutral Multiplicative Identity Element that represents the Unit Base X of X^Q, then any change in the Count of the Number of Members contained in the Set X, must define the Union (or Sum) of the members belonging to the Disjoint Set representing the Set Xv[2 thru N], iff X = X^Q, the Cardinality of the Set equals the Sum of the Cardinal Numbers representing each of the its Members. In which case: If the Unit Base X of X^Q is defined ONLY when X = XvN = X^Q remains valid, and; I. 2 Members in a Binary Set = (A U B)^Q = Xv[2 = (A U B)] = X^Q, or II. 3 Members in a Ternary Set = (A U B U C)^Q = Xv[3 = (A U B U C)] = X^Q, or III. 4 Members in a Quaternary Set = (A U B U C U D)^Q = Xv[4 = (A U B U C U D)] = X^Q, or IV. N Members in a N-nary Set = (A U B U ... U NvN)^Q = Xv[N = (A U B U ... U N)] = X^Q, is TRUE, E Terrell Internet Draft [Page 36] The Ternary Logical States of the Binary System October 28, 2006 THEN: I.a 2 Members in a Binary Set = Xv[2 = (A U B)] = X^2 = X^Q, or II.a 3 Members in a Ternary Set = Xv[3 = (A U B U C)] = X^3 = X^Q, or III.a 4 Members in a Quaternary Set = Xv[4 = (A U B U C U D)] = X^4 = X^Q, or IV.a N Members in a N-nary Set = Xv[N = (A U B U ... U N)] = X^N = X^Q, Must also be TRUE. In other words, the Proof for the existence of any Numbering System involving the Unit Base X of X^Q, would conclude the definition for the existence of another system of counting. And this defines a Unit Base X of X^Q containing more Base elements than Unary System, as the UNION of More than One Element; Confirms Fermat's Last Theorem only for the Binary System for all N > 2. That is, given by the foregoing proof of Fermat's Last Theorem, which is translated into the rigor from the Mathematical Logic of Set Theory, and confirms the Conditions for; E Terrell Internet Draft [Page 37] The Ternary Logical States of the Binary System October 28, 2006 ( A^nN U B^nN ) = (A U B)^nN; given below, we have: X Y X Y +---+-------+ +---+-------+ | \ | / /| | \ |X / / | | Z + +---+-------+ | + \ | | \ | + | \ |Y | \ |X | | \ | | Z + | +-------+---+ + \ | + | \ |Y / | \ | / +-------+---+ Figure 10 If for all X | X E I, X = X for every XvU = X^Q, and when X = XvU there is a XvN | XvN = X^Q, which also True for all X | X E I for every X = XvN when X = X and XvN = (A U B U C U ... U N), then XvN = XvU, if and only if (iff): X^QV{U} = XvU = X^Q = 'X' = X^Q = XvN = X^Qv{N}, or XvN Not= XvU, because X Not= XvN. Proof: Since the Theorem concluding the definition for the Cardinal Number defines the E of Unary Set as the Unit Base X of X^Q for all X | X E I, then the Multiplicative Identity Element for all X | X E I defines XvN = XvU when X = XvU. Therefore, when XvN = XvU, and N = 2 = Q, X n X = X^[Q = 2] = (A U B) n (A U B) X^[Q = 2} = (A U B) n (A U B) = (A n A) U [(A n B) U (A n B)] U (B n B) E Terrell Internet Draft [Page 38] The Ternary Logical States of the Binary System October 28, 2006 And from the Distributive Law; (A n A) U (B n B) = [(A n B) U (A n B)] = (A B) U (A B) = A (B U B) Hence, from the Substitution Law for Equality; X = XvU = (X U Y), equation 25; [(A n B) U (A n B)] = (X n Y) U (X n Y) = (XY) U (XY) = X (Y U Y): which concludes; XvN = XvU, X = (A U B), and the Unit base X of X^[Q = 2] defines X = X, which means, by definition; X n (Y U Y) = X (Y + Y). In other words, this proves Fermat's Last Theorem and confirms the definition of the Cardinal Number, '1', for the Binary Set given by: Cardinal Number: The Cardinal Number is the Multiplicative Identity Element for all X| X E I, which represents the Elements of the Binary Set that is used to determine the Cardinality of every Set from the Sum or Union of the Multiplicative Identity Element for every ELement X of the Set I: iff X^Q = X. And from the foregoing (excluding the rigor from the Mathematical Logic) it can be easily proven that since A, B, C, D, ... N must be Disjoint initially, when defining the elements, E, contained in the Unit Base X of X^Q; by the equations given below, X = X^Q is not valid. In other words, because there is no confirmation by the Distributive Law for XvN = XvU for all X | X = X^Q when Q = N, and N > 2. E Terrell Internet Draft [Page 39] The Ternary Logical States of the Binary System October 28, 2006 II.a 3 Members in a Ternary Set = Xv[3 = (A U B U C)] Not= X^2 Not= X^3 Not= X^Q Not= X, or III.a 4 Members in a Quaternary Set = Xv[4 = (A U B U C U D)] Not= X^2 Not= X^4 Not= X^Q Not= X, or IV.a N Members in a N-nary Set = Xv[N = (A U B U ... U N)] Not= X^2 Not= X^N Not= X^Q Not= X, The Distributive Law for The Distributive Law for Non-Linear Functions Linear Functions Binary Set Unary Set \ / +----|+++|----+ | |+++| | | |+++| | | |+++| | | |+++| | | |+++| | +----|+++|----+ / | \ / | \ +---------------------/ v \--------------------------+ The Position or Point of Intersection between the Two System of Counting (Number Sets) defines a Special Case of the Distributive Law (The Intersection between the Binary and the Unary Sets) for Positive Integers. +------------------------------------------------------------+ Figure 11 E Terrell Internet Draft [Page 40] The Ternary Logical States of the Binary System October 28, 2006 Nevertheless, these conclusions confirm the existence of the Two Systems of counting defining; 'The Unary Set' and 'The Binary Set', they also support the conclusion defining these Sets, by Figure 11, as; 'The Infinite Set = Unary System' and 'The Finite Set = Binary System'. Furthermore, it should be clearly understood: When X = (A U B), X defines the Binary pair {a, b} And reasoned further that if either 'a', or 'b' is equal to the Null Set {Ø}, then the foregoing conclusions would be invalid. Moreover, since the Cardinal Number, the Multiplicative Identity Element of the Unary Set, is same for Binary Set, the Binary pair, {a, b}, must represent, by Figure 12, a unique combination of the Binary Pair incrementing in units of '1', which defines the Cardinality of any Set, also defined by the Unary System. +---------------------------------------------------------+ | The Combinations generated using the Binary Pair; {A,B} | +---------------------------------------------------------+ | Binary Set Unary Set Positive Integers | | | |{A,A} 0r {a, a} = 1 = 1 | | | |{A,B} 0r {a, b} = 11 = 2 | | | |{B,A} 0r {b, a} = 111 = 3 | | | |{B,B} 0r {b, b} = 1111 = 4 | | | +---------------------------------------------------------+ Figure 12 E Terrell Internet Draft [Page 41] The Ternary Logical States of the Binary System October 28, 2006 In other words, from the definition of the Cardinal Number, the Cardinality of the Unary and the Binary Sets represents a 1 : 11 ratio, which denotes the number of Elements each Set contains. Nevertheless, the defining expression representing this relationship given by; 'Unary Set = 1', 'Binary Set = 11', or '1 = 2' - 'Prime Numbers' Note: A 'Prime Number' or 'Prime Integer', is a positive integer, 'p is Greater Than or Equal to 1', that has no positive integer divisors other than itself, 'p', and '1'. And if, from the Substitution Law for Equality; {0, 1} = {a, b}, where '1 = {00}, and {00} is Not Equal to {Ø}', then the correct Binary System and its associated method for enumeration, given by Table IV, confirms '11111111 = 256 = 2^8, because 2^8 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 11111111 = 256'. Hence, the definition of the Cardinal Number, by figure 11, defines the special case of the Distributive Law as the intersection of the Distributive Properties defining the Binary and the Unary Sets, for all X | for every Element of I, the Cardinal Number X, defines the Cardinality of both Sets. 2.2 The Mathematics of Quantification and Binary Arithmetic System It should be clearly understood that the forgoing conclusions, and the new definitions and theorems from the Logic of the Mathematics of Quantification, defines the closure Laws for the operations of Subtraction and Division. And this completes the Set of Laws defining the operations of Addition, Multiplication, Subtraction, and Division, which governs the Mathematics and the Mathematical Logic defined by Set Theory, the Field Postulates, and the Axioms for Equality. That is, given by Table V, we have: E Terrell Internet Draft [Page 42] The Ternary Logical States of the Binary System October 28, 2006 TABLE V +------------------------------------------------------------+ | AXIOMS for EQUALITY | | | | | | Fundamental Law for Equality: A + (-A) = 0 | | Additive Identity Law for Equality: A + 0 = A | | Multiplicative Identity Law for Equa;ity: A = 1 = A | | Common Coefficient Law or Equality: A/1 = A = A x 1 | | Substitution Law for Equality: If A = B, and B + C = D, | | then A + C = D | | Reflective Law for Equality: A = A | | Transitive Law for Equality: A + B = B + A | | | +------------------------------------------------------------+ Table VI (Addition) 111 = 8 1111 = 16 11111 = 32 110 = 7 1010 = 11 10110 = 23 _____ 15 _____ 27 ______ 55 1110 11010 110110 111111 = 64 100 = 5 1000 = 9 101011 = 44 100 = 5 1000 = 9 ______ 108 ____ 10 _____ 18 1101011 1001 10001 Table VII (SUBTRACTION) 111 = 8 1111 = 16 11111 = 32 110 = 7 1010 = 11 10110 = 23 ____ 1 _____ 5 ______ 9 00 = 1 100 1000 111111 = 64 100 = 5 1000 = 9 101011 = 44 100 = 5 1000 = 9 ______ 20 ____ 0 _____ 0 10011 0 0 E Terrell Internet Draft [Page 43] The Ternary Logical States of the Binary System October 28, 2006 2.3 The Binary and Ternary Systems and George Boole's Mathematical Logic It should readily be concluded, because it has been mentioned that the Boolean, or Leibniz, Operators are Unary; they are both logically valid for the Unary and the Binary Systems. Furthermore, since Zero, Ø, or the Null Set, is not defined by the Cardinal Number, which is equal to the Unit Base X of X^Q for all X| X is an Element of I, then Ø, is not an element of the Set of Integers, 'I'. Hence, Binary and Ternary Logic, or 3 State Logic is defined by the Unary Set, and contains the elements {Ø, +1} and {-1, Ø, +1}, which are governed by the Closure Laws. Given by Table VIII, we have; TABLE VIII Logical States of the Unary and the Binary Systems +---------------------------+---------------------------------+ | BOOLEAN BINARY STATES | BOOLEAN TERNARY STATES | +---------------------------+---------------------------------+ | | | | | | | { 0 } { + 1 } | { - 1 } { 0 } { + 1 } | | | | | No Yes | No Nothing Yes | | | | | False True | False Undecided True | | | | | Open Closed | Open By-Pass Closed | | | | | Stop Foward | Reverse Stop Foward | | | | | Two State Switch | Three State Switch | | | | | | Or, Flip-Flop 4 state Switch | | | | +---------------------------+---------------------------------+ Note: It should be understood nevertheless, these conclusions confirms that the Binary System is Finite and Closed over 'R' (not true for all values of the Base Variables over 'R'), and the Unary System is Infinite, and it is also Closed over 'R'. [VIMP - e. terrell Nov. 1979 to Aug 1983] E Terrell Internet Draft [Page 44] The Ternary Logical States of the Binary System October 28, 2006 3. Security Considerations This document, whose only objective was the deliberation of the final explanation of the new foundation for the Binary System, which resulted from the Mathematics of Quantification, does not directly raise any security issues. Hence, there are no issues that warrant Security Considerations. E Terrell Internet Draft [Page 45] The Ternary Logical States of the Binary System October 28, 2006 4. IANA Considerations - 'Resolution of the Counting Error in the Binary System' I. IPv4 Address Loss Table Exponential Binary IPv4 Address Enumeration Representation Specification +--------------------+------------------------+------------------+ | | 1. 0^0 = 0 | 0 | 0 | | 2. 2^0 = 1 | 00 = aa | 0 | | 3. 2^1 = 2 | 01 = ab | 1 | | 4. 2^F = 3 | 10 = ba | 2 | | 5. 2^2 = 4 | 11 = bb | 3 | | 6. 2^F = 5 | 100 = baa | 4 | | 7. 2^F = 6 | 101 = bab | 5 | | 8. 2^F = 7 | 110 = bba | 6 | | 9. 2^3 = 8 | 111 = bbb | 7 . | . | . . | . | . . | . | . 129. 2^7 = 128 | 01111111 = bbbbbbb | 127 . | . | . . | . | . . | . | . 257. 2^8 = 256 | 11111111 = bbbbbbbb | 255 +--------------------+------------------------+------------------+ Totals: 256 | 256 = 256 | 255 +--------------------+------------------------+------------------+ IPv4 Address Loss using an askew Binary System; 256^4 - 255^4 = 66,716,671 IP Addresses +--------------------+------------------------+------------------+ II. Using Extended ASCII CODE & Binary '00' = 1 In the Extended ASCII CODE character Set, True Zero is defined as the Null Set Character, ' Ø '. However, because Binary equivalent of ' 1 ' is ' 00 ', I believe that it would be easier if the Character Set were changed to represent the Binary equivalent of ' 1 ' as ' 0 ', as opposed to '00', because '00' is 2 Bits and '0' is '1' Bit. E Terrell Internet Draft [Page 46] The Ternary Logical States of the Binary System October 28, 2006 Exponential System Binary System Zero of Counting --------------------+-----------------+----------------------- No Definition 0 0^X = 0 = 0EX 1. 00 = aa No Definition 2^0 = 1 = 2E0 2. 01 = ab No Definition 2^1 = 2 = 2E1 3. 10 = ba No Definition 2^F = 3 = 2EF 4. 11 = bb No Definition 2^2 = 4 = 2E2 : : : : : : 8. 111 = bbb No Definition 2^3 = 8 = 2E3 9. 1000 = baaa No Definition 2^F = 9 = 2EF 10. 1001 = baab No Definition 2^F = 10 = 2EF [Given that: E = Exponential Operator; F = Variable Irrational Number; and X = Any Variable defined as a Member of the Real Number Set] III. Equating the Exponent from a Base 2 Exponential Operation to the Binary Translation that Equals the Result * More importantly, when rationalizing these conclusions, their validity becomes even more evident when any mathematical comparison between the 'Bit-Mapped' Lengths, or Displacement of an IP Address, is made with the Equation representing the Total Number of Available IP Addresses - the Address Pool representing the Addressing Specification; e.g. IPv4, or IPv6. That is; If the Bit Length is Equal to 32, in the IPv4 Specification, or 128 Bits in the IPv6 Specification, and their respective Address Pool Totals is given by: IPv4 = 32 Bit Length (Bit-Mapped Displacement) 32 Bit = 2^32 Address Pool Total 2^32 = 4,294,967,296 IP Addresses IPv6 = 128 Bit Length (Bit-Mapped Displacement) 128 Bit = 2^128 Address Pool Total 2^128 = 3.4028236692093846346337460743177e+38 IP Addresses E Terrell Internet Draft [Page 47] The Ternary Logical States of the Binary System October 28, 2006 Then it becomes quite obvious that the Total Number of IP Addresses available in the Address Pool for either the IPv4, or the IPv6 Specification, is a function of the Address's Bit-Mapped Displacement, or Bit Length. In other words, a Bit Length Regression to Progressively smaller Address Bit-Mapped Displacement Units, just as the foregoing conclusions revealed, accounts for the total number of available IP Addresses in the Address Pool - and this also determines, equals, and represents, the exact number of Bits equal to the Number representing the IP Address Pool Total. In other words, this Number or Integer, which equals the Result from an Exponential Base 2 Operation, has a Binary Translation that is equal to the value of the Exponent in the Equation. Hence, Enumerating, or Counting using only the Exponent reveals: 1) An 8 Bit-Mapped Length = 2^8 = 256 IP Addresses = 256 = 11111111 2) A 7 Bit-Mapped Length = 2^7 = 128 IP Addresses = 128 = 1111111 3) A 6 Bit-Mapped Length = 2^6 = 64 IP Addresses = 64 = 111111 4) A 5 Bit-Mapped Length = 2^5 = 32 IP Addresses = 32 = 11111 5) A 4 Bit-Mapped Length = 2^4 = 16 IP Addresses = 16 = 1111 6) A 3 Bit-Mapped Length = 2^3 = 8 IP Addresses = 8 = 111 7) A 2 Bit-Mapped Length = 2^2 = 4 IP Addresses = 4 = 11 8) A 1 Bit-Mapped Length = 2^1 = 2 IP Addresses = 2 = 01 9) A '0' Bit-Mapped Length = 2^0 = 1 IP Address = 1 = 00 E Terrell Internet Draft [Page 48] The Ternary Logical States of the Binary System October 28, 2006 So, how then is it possible for anyone to use an Askew Binary System of Counting, when the Exponent representing the Bit-Mapped Displacement in the Base 2 Exponential Equation, equals the Binary Translation representing the " Equation's " Result? - The Binary Translation Comparison Table - Computer Operating Systems, Electronic..., and Software is Wrong! 4 = 100 - Binary Translation: How...? When 2^2 = 4 = 11 3 = 11 - Binary Translation: How...? When 2^F = 3 = 10 2 = 10 - Binary Translation: How...? When 2^1 = 2 = 01 1 = 01 - Binary Translation: How...? When 2^0 = 1 = 00 E Terrell Internet Draft [Page 49] The Ternary Logical States of the Binary System October 28, 2006 IV. Binary Zero { 00 } Representing an Irrational Number...?? If every Base 2 Exponential Equation representing the Product of 2 or more Identical Multiplicands, defines the Result as a Function of the Square Root of 2 when Binary '00' = 1. Then, from the "Proof of Fermat's Last Theorem, and the Mathematics of Quantification; when "00" = 1, "00" defines an Irrational Number, which is a Member of the "Real Number Set" - Where by; ' IF ' " 00 " = 1 is True, then; X( 0 + 0 ) = ( 0^2 + 0^2 ) = 1 = (2^1/2)/2 [((2^1/2)/2) + (2^1/2)/2)] = (2^0.5)/2 [(2^0.5)/2) + (2^0.5)/2)] = (2^0.5)/2)^2 + (2^0.5)/2)^2 " 1 " = (0.707106)^2 + (0.707106)^2 " 1 " = 0.5 + 0.5 = X( 0 + 0 ) Where, if " 1^0.5 " = " 1 ", and " F = 0 "; then " F = Variable Irrational Number ". Hence; (2^0.5)/2 = 0.70710678118654752440084436210485 0.5 = (0.70710678118654752440084436210485)^2 " 1 " = ( F^2 + F^2 ) = 2^0 = " 00 " [ * - See page 41; Figure 12; [12]; Exponential Cardinal page 32 - Note; 2^0.5 = ' The Square Root of 2 ' ~ 1.4142135623731 ] E Terrell Internet Draft [Page 50] The Ternary Logical States of the Binary System October 28, 2006 Note: Microsoft's Windows Calculator, and others, is wrong. - The Operating Systems and Software of Microsoft, Cisco, IBM, Wolfram, and others, who use the HEX System of Counting, are also wrong; there is No Conversion with the Base 2 Exponential Equations defining the Binary System. - And this includes every Electronic Device / Component - In other words, using the HEX System of Counting does not change anything, because it maps to the current Binary System- it is also an Askew System of Counting. In which case, any measurements derived from its use in any Calculation(s) will be Wrong... And if, the Trial and Error Tests cannot be performed, or fail, lives could be Lost as a direct result... - The Irony? Today’s Authority in Mathematics maintains; Isaac Newton was a great Mathematician who invented Calculus. The truth however, is that; 'There was never a Conflict of Plagiarism, between Newton and Leibniz, which involved the discovery of Calculus: A Ruse. Newton hated Leibniz, because Leibniz proved that the Mathematics involving Newton's Laws of Motion was wrong!' A fact, nearly a 100 years later, that was proven to be true by several noted mathematicians, which includes "Emilie de Breteuil du Chatelet". - 'Time-Travel' and 'Parallel (Nested) Universes': the thoughts of Science Fiction writers, the Beliefs of World renowned Physicists, or the utterances of the disassociated - those who are believed to be Insane, because they do not have a University Affiliation. It does not matter who believes 'what', because; 1) 'Time-Travel' is an impossibility, which would violate the Conservation Laws. In other words, Matter and Energy Cannot be Re-Animated; Created or Destroyed. 2) 'Parallel Universe(s)', just like the existence of more than 3 Dimensions, or any claim that Empty Space defines a 'VOID of Nothingness' having Material Properties: a Physical Impossibility, because it violates the Conservation Laws of Physics. - Clearly, in a Supercilious world controlled by Posturing Charlatan(s), mired by the allegories of Buffoons, only the Insane is believed to be Intelligent... E Terrell Internet Draft [Page 51] The Ternary Logical States of the Binary System October 28, 2006 Work(s) in Progress; Computer Science / Internet Technology: These drafts represent the twelve chapters of the Networking Bible, designing a Network IP Addressing Specification that maintains a 100 Percent backward compatibility with the IPv4 Specification. In other words, this is a design specification developed from the Theory of the Expansion of the IPv4 IP Addressing Specification, which allowed the representation of the Network for the entire World on paper, and the possibility of an Infinite IP Address Pool. Nevertheless, the Internet-Drafts listed below, "Cited as Work(s) in Progress", explain the design Specification for the development of the IPtX (IP Telecommunications Specification) Protocol Addressing System and the correction of the Mathematical Error in the Binary System. 1. http://www.ietf.org/internet-drafts/draft-terrell-logic -analy-bin-ip-spec-ipv7-ipv8-10.txt "Work(s) in Progress" (Foundational Theory for the New IPtX family IP Addressing Specification, and the Binary Enumeration correction) 2. http://www.ietf.org/internet-drafts/draft-terrell-simple -proof-support-logic-analy-bin-02.txt "Work(s) in Progress" (The completion of the 2nd Proof correcting the error in Binary Enumeration) 3. http://www.ietf.org/internet-drafts/draft-terrell-visual -change-redefining-role-ipv6-01.pdf "Work(s) in Progress" (Argument against the deployment of IPv6) 4. http://www.ietf.org/internet-drafts/draft-terrell-schem -desgn-ipt1-ipt2-cmput-tel-numb-02.pdf "Work(s) in Progress" (The foundation of the New IPtX IP Addressing Spec now simular to the Telephone Numbering System) 5. http://www.ietf.org/internet-drafts/draft-terrell-internet -protocol-t1-t2-ad-sp-06.pdf - "Work(s) in Progress" (The IPtX IP Addressing Specification Address Space / IP Address Allocation Table; establishes the visual perspective that actually represents Networking Schematic of the entire World.) E Terrell Internet Draft [Page 52] The Ternary Logical States of the Binary System October 28, 2006 6. http://www.ietf.org/internet-drafts/draft-terrell-iptx-spec -def-cidr-ach-net-descrip-01.pdf - "Work(s) in Progress" (Re-Defining 'CIDR' {Classless Inter-Domain Routing Architecture} for the IPtX Addressing Standard) 7. http://www.ietf.org/internet-drafts/draft-terrell-math -quant-new-para-redefi-bin-math-04.pdf "Work(s) in Progress" (The completion of the 3rd Proof correcting the error in Binary Enumeration) 8. http://www.ietf.org/internet-drafts/draft-terrell-gwebs -vs-ieps-00.pdf - "Work(s) in Progress" Global Wide Emergency Broadcast System) 9. http://www.ietf.org/internet-drafts/draft-terrell-iptx -dhcp-req-iptx-ip-add-spec-00.pdf "Work(s) in Progress" (The development of DHCP {Dynamic Host Configuration Protocol} for the IPTX IP Addressing Spec) 10. http://www.ietf.org/internet-drafts/draft-terrell-iptx -dns-req-iptx-ip-add-spec-03.pdf "Work(s) in Progress" (The development of DNS {Domain Naming Specification} for IPTX IP Addressing Spec) 11. http://www.ietf.org/internet-drafts/draft-terrell-math-quant -ternary-logic-of-binary-sys-04.pdf(Derived the Binary System from the proof of "Fermat's Last Theorem", and Developed the Ternary Logic for the Binary System) 12. http://www.ietf.org/internet-drafts/draft-terrell-cidr-net -descrpt-expands-iptx-add-spc-13.pdf- "Work(s) in Progress" (An application of Quantum Scale Theory, the 2^X : 1 Compression Ratio, the Expansion derived from the 'CIDR Network Descriptor, and the Mathematics of Quantification provided the foundation for the development of the "Intelligent Quantum Tunneling Worm Protocol"; A Routable Mathematical Exponential Expression, BackEnd IP Addressing Protocol that provides an (nearly) Unlimited IP Address Space using the Compression Ratio 2^X : 1.) NOTE: These Drafts has Expired at www.ietf.org Web Site. However, you can still find copies of these Manuscripts posted at Web Sites all over the World. Suggestion; Perform Internet Search using either Yahoo or Google. Keyword: "ETT-R&D Publications"}. E Terrell Internet Draft [Page 53] The Ternary Logical States of the Binary System October 28, 2006 5. Normative References Pure Mathematics: 1. The Proof of Fermat's Last Theorem; The Revolution in Mathematical Thought {Nov 1979} e. terrell 2. The Rudiments of Finite Algebra; The Results of Quantification {July 1983} e. terrell 3. The Rudiments of Finite Geometry; The Results of Quantification {June 2003} e. terrell 4. The Rudiments of Finite Trigonometry; The Results of Quantification {July 2004} e. terrell 5. The Mathematics of Quantification and the Metamorphosis of Pi : Tau {October 2004} e. terrell Informative References 1. G Boole ( Dover publication, 1958 ) "An Investigation of The Laws of Thought" On which is founded The Mathematical Theories of Logic and Probabilities; and the Logic of Computer Mathematics. 2. R Carnap ( University of Chicago Press, 1947 / 1958 ) "Meaning and Necessity" A study in Semantics and Modal Logic. 3. R Carnap ( Dover Publications, 1958 ) " Introduction to Symbolic Logic and its Applications" E Terrell Internet Draft [Page 54] The Ternary Logical States of the Binary System October 28, 2006 Author: Eugene Terrell 3312 64th Avenue Place Oakland, CA. 94605 Voice: 510-636-9885 E-Mail: eterrell00@netzero.net Note: Illinois Institute of Technology, University of Chicago, Northeastern Illinois University, University of Illinois Chicago Circle Campus, Stanford University, UCLA, Kennedy-King College, Canada, United States, Russia, Germany, France, Scientific American, and several other popular magazines received a copy of one, or both, of the proofs are listed above; 1 and 2, the notarized proofs that were sent for review between, 1980 and 1983 (to name, just only a few recipients). "This work is Dedicated to my first and only child, 'Princess Yahnay', because she is the gift of Dreams, the true treasure of my reality, and the 'Princess of the Universe'. (E.T. 2006)" E Terrell Internet Draft [Page 55] The Ternary Logical States of the Binary System October 28, 2006 Copyright (C) The IETF Trust (2006). This document is subject to the rights, licenses and restrictions contained in BCP 78, and except as set forth therein, the authors retain all their rights. This document and the information contained herein are provided on an "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY, THE IETF TRUST, AND THE INTERNET ENGINEERING TASK FORCE DISCLAIM ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. 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The IETF invites any interested party to bring to its attention any copyrights, patents or patent applications, or other proprietary rights that may cover technology that may be required to implement this standard. Please address the information to the IETF at ietf-ipr@ietf.org. Acknowledgement Funding for the RFC Editor function is provided by the IETF Administrative Support Activity (IASA). E Terrell Internet Draft [Page 56] The Ternary Logical States of the Binary System October 28, 2006