Gödel’s Incompleteness Theorem Formally Undecided Principles.pdf

Koffi1 Leslie Koffi Philosophy 399 Gödel’s Incompleteness Theorem Kurt Gödel proved, in a famous paper published in 1931, that in any true (consistent) axiomatic theory sufficiently rich to enable the expression and proof of basic arithmetic propositions, it will be possible to construct an arithmetical proposition G such that neither G, nor its negation, is provable from the given axioms. Hence the system must be incomplete. Gödel developed a technique now known as Gödel numbering, in an effort to prove this which codifies formal expressions as natural numbers. The incompleteness theorems of 1930 are contrary results in Hilbert’s meta-mathematical program as the first incompleteness theorem shows the impossibility of obtaining a complete axiomatization of arithmetic. Since the extent of Gödel’s mathematical work is firmly rooted in philosophical inquiry, his use of metamathematics was his tool for finite reasoning. He viewed mathematical logic as a continual reflection on the nature of mathematical knowledge. This provided a powerful measure of addressing core epistemological concerns. Because the work of meta-mathematics encompasses the very nature of mathematics itself, its roots within the principles of set-theory has shown to be subjected within its own limitation. The first incompleteness theorem showed there are true propositions about whole numbers that cannot be proved from the axioms. Hence, formal systems which, such as the following, are consistent are not complete: “Any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic)” (Ferreirós, 2007). Historical Context: The Foundation’s Crisis In the early twentieth century, set theory, the concept of the continuum, the role of logic and the axiomatic method versus the role of intuition were heavily debated among mathematicians and philosophers. As these debated concepts contributed to the foundations crisis in the 1920’s, it questioned Koffi2 the philosophical basis of mathematics where Hilbert devised a plan to investigation philosophical theories by exploring the concerning nature of mathematics. The following viewpoints including logicism, formalism and intuitionism questioned the basis of foundational math given that the status of mathematical knowledge in this period led to the historical debate known as the foundations crisis. The foundations crisis was rather a localized event in the 1920’s where the debates between the partisans of “classical” (late-nineteenth century) mathematics, led by Hilbert, and their critics, led by Brouwer, an intuitionist, advocated for stronger revisions of foundational math (Ferreirós, 2007). Logicism, which Hilbert endorsed in 1899, was a unique reaction to the rise of modern mathematics as the set-theoretic approach was an intellectual reaction to the concept of the continuum. “Logicism was the thesis that the basic concepts of mathematics are definable by means of logical notions, and that the key principles of mathematics are deducible from logical principles alone” (Ferreirós, 2007). However, over time this thesis has become unclear because it seemed to be based on an immature conception on logical theory, which became to be known as naïve set theory, confirmed later by the paradoxes. Hilbert’s endorsement of logicism was rather a self-conscious endorsement of certain modern methods in which he followed Richard Dedekind’s understanding of mathematics. This rise of modern methods emerged in the nineteenth century and was associated with Göttingen mathematics. Some of the modern approaches included the following: (i) (ii) (iii) a wholehearted acceptance of infinite sets and the higher infinite a preference “to put thoughts in the place of calculations” (Dirichlet), and to concentrate on “structures” characterized axiomatically a reliance on “purely existential” methods of proof (Ferreirós, 2007) Dedekind’s approach to the algebraic number enabled proven results such as the fundamental theorem of unique decomposition as the proof was conceptually abstract. Dedekind insisted in the importance of a general, conceptual theory. Hence, “he proved in full generality within any ring of algebraic integers, ideals possess a unique decomposition into prime ideals” (Ferreirós, 2007). However, Leopold Kronecker, a German mathematician who worked on number theory and algebra complained that Dedekind’s proofs did not enable calculations and thus the proof was purely existential. Kronecker’s view was that these Koffi3 modes of abstract methodology on algebraic properties were too remote for algorithmic treatment. However, with Dedekind’s modern approach, “thoughts were in place of calculations”, as this principle was also emphasized by Riemann’s theory. Hilbert was rather critical of Kronecker’s methodological condemnation for mathematics. Certainly, Hilbert’s program can be seen, in many ways, as an attempt to dismiss Kronecker’s ideas. Kronecker also complained about the Balzano-Weistrass theorem, where every bounded sequence has a convergent subsequence, because the theorem rests entirely on the completeness axiom for ℝ. Since real numbers cannot be constructed in an elementary way from the rational numbers, one has to make use of infinite sets, such as the set of all possible Dedekind cuts. Hence, Kronecker noted that the accumulation point in the Bolzano–Weierstrass theorem cannot be constructed by elementary operations from the rational numbers hence contributing to the concept of generality (Ferreirós, 2007). Cantor and the Paradoxes The idea of the set of ℝ or “the continuum” are non-constructive elements in modern mathematics. Georg Cantor’s proofs in set theory also became essential examples of the modern methodology of existential proof. An enumerable set ∑ can be listed off in some numerical order and the list may be infinite, hence we now introduce the idea of Cantor’s Diagnolization argument. Cantor's theorem about the indenumerability of the set of total functions from ℕ to ℕ uses the argument that if we assume that all functions from ℕ to ℕ are denumerable—i.e. countable, then we should be able to form a new function that is a bijection from the original one. Now Cantors theorem states that, “for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself” (Ferreirós, 2007). His defense of the higher infinite and modern methods in a paper he wrote in 1883, prompted attacks from Kronecker and in 1887, where he published an attempt to elaborate his foundational views. However, Dedekind replied with a detailed set-theoretic and logistic theory of the natural numbers in 1888. Thus, this early round criticism led to the victory for the modernist. Other powerful allies such as Felix Klein, Koffi4 Giuseppe Peano, Hilbert and more join this new movement of modern thinking. Hence, during the 1890’s, the modernist viewpoint for logic in particular expanded. However, the dramatic change came with the logical paradoxes which were discovered by Bertrand Russell, Ernst Friedrich Zermelo, Abraham Fraenkel and more. Their arguments showed the assumption that certain existing sets do in fact lead to contradictions. These were later known as the settheoretic paradoxes which, in effect, destroyed attractive developments proposed by logicism. Russell’s paradox states, “ the set of all sets are not elements of themselves”. Hence, the set is both a member of itself and not a member of itself. Russell’s paradox relies on the naïve comprehension principle (∃A) (∀x)(x ∈ A ≡ φ). Let φ be that x is not a member of itself. If A is a member of A, the condition φ does not apply and A is not a member of A. However, if A is not a member of A, then condition φ does apply and A is a member of A. Therefore, he showed that formal systems of arithmetic are not complete. Zermelo– Russell tried to get out of Russel’s paradox by showing that the comprehension principle is contradictory, by formulating a property that seems to be rooted in logical theory in stating that (∀A)( ∃B)(∀x)(x ∈ B ≡ ((x ∈ A) & φ) which also led to a contradiction. Let φ be that x is not a member of itself and let A be a universal set. If B is a member of B, then B is a member of the universal set and condition φ applies, so B is not a member of itself. However, if B is not a member of B, then condition φ applies, thus B cannot be a member of the universal set i.e. the universal set does not exist. This lead to another paradox. Before the paradoxes, logicism was embraced by both mathematicians and philosophers, however, by the 1920’s, logicism was of more interest to philosophers. This also led to more questions on the theory of the rationale of foundational math itself as it became uncertain in that set theory was now shown to be unstable. Paris Problem As Hilbert made the ideological transition from logicist to formalist, he founded the "formalist" approach in Philosophy of Mathematics; Hilbert advocated in 1921 that the primary aim should be to Koffi5 establish mathematics on a solid and provably consistent foundation of axioms. Therefore in principle, all mathematical truths could be deduced by standard methods of first order or "predicate" logic. Hilbert’s goal from this point forward would be to establish what mathematics should be in the twentieth century by presenting his famous list of mathematical problems at the Paris International Congress of Mathematics in 1900. He discussed Cantor’s continuum problem, and proposed the problem of whether every set can be well-ordered. His second problem was to establish the consistency of the idea being the set ℝ of real numbers. Hilbert then employed his fellow colleague Zermelo to show that ℝ (the continuum) can be well-ordered and to establish the axiom of choice (Ferreirós, 2007). Predicativity and the Axiom of Choice Before the axiom of choice (AC) was formulated, Jules Henri Poincaré had critical viewpoint points against logicism and formalism. His analysis of the paradoxes led him to formulate an important new concept of predicativity, and maintain that impredicative definitions should be avoided in mathematics. In an informal sense, a definition is impredicative when it introduces an element by reference to a totality that already contains that element. A typical example is Dedekind’s definition of the set ℕ with a sequence beginning with 1, with each number n associated with its successor (Ferreirós, 2007). With the predicativity approach to foundational mathematics, all mathematical objects must be introduced by explicit definitions. Hence, one begins with the idea of definable real numbers, for example. Predicativity refers only to totalities that have already been established. Zermelo, however, argued that this notion was limited in arguing that impredicative definitions were often used unproblematically, not only in set theory but also in classical analysis (Ferreirós, 2007). The least upper bound in real analysis is an example of the impredicative definition as the real numbers are not introduced separately but rather as a completed whole. Russell, who incorporated the theory of types, is a system of higher-order logic, with quantification over properties or sets, over sets of sets and etc. It is based on the idea that the elements of any set should Koffi6 always be objects of a certain similar or uniform type. For example, we can have individual sets such as {a, b}, or sets of sets of individuals, such as {{a}, {a, b}}, but never a mixed set such as {a, {a, b}} (Ferreirós, 2007). This system, together with axioms of infinity and choice, served as the development of set theory and the number systems as it became the logical basis for the renowned Principia Mathematica by Whitehead and Russell. They were considered in the development of the foundation for mathematics. The main idea of Principia Mathematica was to develop an acceptance of the theory of the natural numbers as they were considerably developed using classical logic. A predicative foundation for mathematics lies within the foundation of modern methodology and it is one of several approaches that do not fit into the conventional and not outdated forms of logicism, formalism, and intuitionism. The axiom of choice (AC), is the principle that, given any infinite family of disjoint nonempty sets, there is a set, known as a choice set, that contains exactly one element from each set in the family. In Hilbert’s proposition of well-ordered sets, the essential well-ordering of ℝ “exists” where ideally, the completeness axiom assure that ℝ has no gaps and is non-empty. It seemed clear that this proposition would be completely out of reach from any constructivist perspective. Thus, the axiom of choice aggravated obscurities in previous conceptions of set theory, which forced mathematicians to introduce clarifications. Moreover, AC was an explicit statement of prior interpretations about arbitrary subsets which inherently set the stage was set for deep debate among critics. However ironically, AC in less evident ways proved theorems in analysis (Ferreirós, 2007). The ZFC system stands for Zermelo–Fraenkel with choice. It considerably categorized key traits of modern mathematical methodology, offering a satisfactory framework for the development of mathematical theories and the conduct of proofs. By allowing impredicative definitions and arbitrary functions, this system of AC had strong existence principles. ZFC system asserted that axioms have strong existential assumptions which sufficiently derived all of classical mathematics and Cantor’s theory of the higher infinite. Koffi7 Intuition Brouwer’s philosophical version of constructivism presented his unique metaphysical views on the foundation for mathematics in 1907. His philosophy of “intuitionism” derived from the view that individual consciousness is the one and only source of knowledge. Hence, his proposition insisted on the absolute independence of mathematics from language and logic. Brouwer brought new ideas about questioning traditional ways of reasoning the natural numbers and classical logic. For instance, the use of quantifiers and his famous rejection of the principle of the excluded middle (PEM). Thus his critiques developed an alternative theory of analysis that was more radical than his counterparts. Figures such as Klien and Poincare believed intuition has a necessary role to play in mathematical knowledge. Hence the importance of logical properties in proofs and in the development of mathematical theory infers that mathematics cannot be reduced to pure logic. Therefore, proofs and theories are certainly organized in a logical sense, but their basic principles, axioms, are grounded in intuition. Hence, there was a dramatic shift at this point of the foundations between Hilbert’s attempts to justify “classical” mathematics vs. Brouwer’s reconstruction of intuitionistic mathematics. There were initially high hopes that the development of intuitionism would end in an elegant and yet simple demonstration of pure mathematics. However, Brouwer’s reconstruction developed in the 1920s became clearer at some point that intuitionistic analysis was extremely complicated and extraneous (Ferreirós, 2007). Hilbert Program The goal of Hilbert’s program was to eliminate all the skeptics by establishing the acceptability of the classical theories of mathematics. This new perspective relied profoundly on formal logic that are provable from given formulas (the axioms). It included the following:  Finding suitable axioms and primitive concepts for a mathematical theory T, such as that of the real numbers. Koffi8    Finding axioms and inference rules for classical logic, which makes the passage from given propositions to new propositions a purely syntactic, formal procedure. Formalizing T by means of the formal logical calculus, so that propositions of T are just strings of symbols, and proofs are sequences of such strings that obey the formal rules of inference. A finitary study of the formalized proofs of T that shows that it is impossible for a string of symbols that expresses a contradiction to be the last line of a proof. (Ferreirós, 2007) An Overview of Gödel’s First Incompleteness Result An extremely clever development of metamathematical methods, the arithmetization of metamathematics, allowed Gödel to produce contrary results to Hilbert’s program, thus proving that systems such as axiomatic, set theory or Dedekind–Peano arithmetic are incomplete. Gödel’s first incompleteness theorem states, in any language L strong enough and consistent to express 0, the successor function S, +, <, and ∗ can express all primitive recursive functions, and thus no program can prove or disprove all statements. In particular the language of the ℕ, Peano arithmetic, cannot have its statements neither proved nor disproved with the system. Gödel’s Theorem proves in a corollary of Proposition VI, that “there are arithmetical propositions which are undecidable (i.e. neither provable nor disprovable) within their arithmetic system. Namely, the proposition that g expressed by the formula is an arithmetic proposition. However, the proposition that g is undecidable within the system is not an arithmetic proposition, since it is only concerned with provability within an arithmetic system” (Gödel, 1962). This is a metamathematic principle and therefore not an arithmetic concept. Hence Gödel results are not mathematical but rather metamathematical. Fundamental notions of mathematics exhibited by Frege and Peano, Whitehead and Russell’s Principia Mathematica including arithmetic as a deductive system starting from a limited number of axioms. Each theorem is shown to be deducible by the axioms and theorems which precede it are in accord with a limited number of rules of inference. Hence a represented deductive systems will entail a sequence of a formulae, whereby a calculus. The initial formulae will express the axioms of the deductive system and Koffi9 each of the other formulae. The purpose is to express the theorems which are obtained from the initial formulae by a sequence of symbolic manipulations (Gödel, 1962). Gödel had to be precise and detailed about the exact deductive system of arithmetic in order to prove the undecidability of some arithmetical propositions. Hence for Gödel a proof representation is a series of formulae is an immediate consequence of the formulae proceeding it. His proof schema included Gödel numbering, in which the objective was to provide a unique decoded number for every possible wellformed sentence hence a formal language. The reason for this initiative was he wanted to be able to treat any well-formed sentence as part of any other wff and hence wanted to establish proofs which required this. Godel assigned to every character allowable in the well-formed formula a number, a large number construction from multiplying together pre-selected prime numbers. Prime numbers are special in the following way in that, every number in existence can be factored down to the set of primes which when multiplied together will generate that number and hence a prime number cannot be factored further down than itself and 1. Gödel speaks of his rule as establishing a “one-to-one” correspondence as his rule of arithmetization ensures that in every class of strings, there exist a correspondence in a unique class of Gödel numbers, and vice versa (Gödel, 1962). Interestingly he defines a gradual sequence of arithmetical concepts which correspond, according to his rule of arithmetization. This is thus the metamathematical concepts expressed by the same arguments. For example, he defines an arithmetical binary operation ∗ upon two numbers x and y. The result of performing this operation is the Gödel number of the string obtained by taking the string whose Godel number is x and placing the string whose Godel number is y immediately after it (Gödel, 1962). The idea of recursion played an essential role in matamathematics because of Gödel’s clever development. Hence, the method of the recursive definition is an extension of the method of Koffi10 “mathematical induction” by which the natural numbers are define, step by step. Now mathematical induction is not a rule of inference of predicate logic but is sometimes considered as a rule of "metalogic". Thus, it is a rule we can use to prove properties about axiomatic systems in a more generalized from rather than just proving statements solely within axiomatic systems. “A recursive definition is the specification of the each number sequence of numbers by means of a specification of the first number and of a rule which specified the (k+1)-th number in terms of the k-th number and of k itself” (Gödel, 1962). Hence an arithmetical function is recursive if it is the last term in a finite sequence. With each function being recursively defined by a rule where functions precede it, the recursive structures of other arithmetic concepts is defined by means of a concept of a recursive function. For metamathematics, the importance of recursiveness in general lies with reason that recursive definition enables every number in a recursively defined infinite sequence to be constructed according to a rule. Let Q(x,y) be the recursive relationship between the G-numbers of x and y, equivalent to Q’ (x,y(u)) by the modified arithmetization and x is a string whose G-number is x. Hence we have the following proposition: Proposition VI: If the formal system P satisfies certain condition of ‘consistency’, then there is at least one recursive class-sign r(v) in P such that neither v Gen r(v) nor Neg[v Gen r(v)] is provable within P (Gödel, 1962). Conclusion Despite the fact that almost all of Gödel’s proofs were explicitly finitary, he wanted to emphasize that the “objectivistic conception of mathematics and meta-mathematics in general, and of transfinite reasoning in particular, was fundamental to my other work in logic.” Gödel kept his focus on fundamental questions, and with a concrete and deeply satisfying conception of incompleteness, he had the remarkable capability to enhance our philosophical understanding. However, Hilbert was a more absolute and consummate mathematician, believing with an unbounded faith in the ability of mathematics to solve all Koffi11 problems. However, Gödel’s incompleteness result showed that mathematics is inexhaustible and thus incomplete, and hence no matter which set of axioms is chosen as a foundation, one can always find questions that those axioms cannot answer. Koffi12 An Introduction to Gödel’s Theorems (Chap. 1-21) i) Truth vs. Provability What is the meaning of truth vs. provable in mathematics? Why does the context of provability matter if the truth of a given mathematic property of natural numbers hold for each of its sequences? The notion of obvious truths in arithmetic, for example, provides us with a definite structure of sequential certainties. However, that does not directly infer that its properties are automatically provable. Because axioms contain enough information to enable the truth-value of any basic arithmetic proposition, any such theorem β will be logically deduced to proclaim whether β is true or its negation – β is false. In this particular realm of mathematics, the general idea that mathematically truth preserved structures can be either logically deducible from axiomatic properties to be not only true or false but also provable or not is the very core of what will be explored. The work of Gödel provides us with an outlook to notice that certain arithmetic properties integrated within a truth based structure cannot be provable thus it is incompleteable. Hence, most axiomatic systems have inherent limitations in proving arithmetic. Gödel’s First Incompleteness Theorem shows that the entire natural idea that we can give a complete theory of basic arithmetic with a tidy set of axioms is wrong. In this expedition of incompleteness, the set of natural numbers will be our main consideration. A set T has a nice set of true axioms and a reliably truth-preserving deductive logic--i.e. everything T proves must be true, therefore T is a sound theory. Because the importance of sound or tautological sets denote its inexplicable truth in every calculated sequence, it is critical to note that its properties do not automatically imply that it is necessarily complete. An important understanding of what it means for a sound property to be complete also means that its property is provable. Although Gödel provides us with a sentence ℊ couched in the language of basic arithmetic such that assuming T is really sound, we can show a sentence ℊ can’t be derived in T, and therefore ℊ must be true. A sentence � encodes the claim that that very sentence is unprovable (Smith, 2013). Hence a theory T is either sound + provable = a complete theory, or sound theory + un-provable = an incompleatable theory. Another interesting remark Koffi13 to note is that the foundation of any such property/theory implying its provability should always meet the basic need of being sound. Therefore, it’ll never be a question of how to justify that theory T is sound or not but rather if it is provable. Hence, T being deductively sound however not provable in ℊ, means T is not negation complete-i.e. provable to be true or provable to be false, nor a complete theory. Therefore T is consistently true (sound) yet incompleatable. ii) Functions and Enumerations Among the most powerful forms of relating a set of components to another set resides in functions. Given a formal definition, we state that a function is a relation for which each value from the set of the first components of ordered pairs is associated with exactly one value from the set of second components of the ordered pair. The consideration of functions are critical to our understanding of numerical descriptions as it denotes whether a given property holds true or false. For example given a function � � = � 2 we will denote that if any given � produces an even number, it will be considered true for some property of even numbers and false if otherwise. This is not to be confused with sound theory as every enumeration of a function still produces a tautological sequence however if we limit our capacity to which a certain set δ satisfies a given criterion then we are bounded within a particular property—e.g. an even or odd result, that denotes whether it is true or false. This directly leads to the concept of a characteristic functions which states that a numerical property/relation P is the one place total function cp(n)=0 or 1 where P is a property of being either true or false (0, true) (1, false). Because characteristic functions provide a descriptive result—i.e truth or otherwise, we note that total functions map each and every element of its domain to some unique corresponding value in its codomain i.e. range. Examples of total functions are subjective (onto), injective (one-to-one), and bijective (one-to-correspondence. As previously mentioned, every enumeration of a function produces a tautological sequence where a set ∑ is enumerable iff its members can be listed off in some numerical order and the list may be infinite hence the set ∑ surjective where f: ℕ→ ∑ (ℕ maps to ∑)—i.e every codomain has at least one matching domain. Koffi14 iii) Cantors Diagnolization Argument Because an enumerable set ∑ can be listed off in some numerical order and the list may be infinite, we now introduce the idea of Cantor’s Diagnolization argument. Now Cantor's theorem about the indenumerability of the set of total functions from ℕ to ℕ uses the argument that if we assume that all functions from ℕ to ℕ are denumerable—i.e. countable, then we should be able to form a new function that is a bijection from the original one. Recall section: Cantor and the Paradoxes. How does this relate to the idea of natural numbers? Let’s suppose we have a finite set where we could simply count the elements of some set £ and find the cardinality—i.e. the number of elements in the set and now let’s suppose we have an infinite set. With an infinite set we are not able to simply list its number of elements because it is unlimited. However a different way to compare their sizes would be to use the concept of a total function—i.e. a bijection. We recall that a bijection is a one-to-one correspondence where we compare one set of elements between sets—i.e. we compare one set of elements to another and vice versa. The result of a bijection between two sets will yield the conclusion that they have the same size—i.e. they are countable; otherwise, one set will be bigger or more infinite than the other. For example, a set of ℕ say Ҩ = {1,3,5} will directly correspond with set Ҕ={2,4,6} provided we have a function say � � = � + as its inverse function that will map the elements in Ҕ back to Ҩ. Provided we have a bijection between two sets, we denote they are countably infinite, however let’s look at a set that is uncountable—i.e. Cantor’s Argument. We will use the argument of real numbers ℝ. Let S be a set which contains every infinite sequence of 0 and 1—i.e. a decimal expansion of a real number. Suppose S is countable, so we must be able to order them. As we present different layers of S1= {0 1 0 1 0 0 1 1….}, S2= { 1 1 0 1 0 0 1….} and so on, we form a diagonal from an x number of layers where we have a set Sp= { 1 0 0 1 0 1 1….} where the flipped diagonal yields a different set of strings Sn= { 0 1 1 0 1 0 0….} which is now different every other sequence in the ordering—i.e. we cannot formulate a one-to-one correspondence between the two sets. However, our original assumption was the claim that Koffi15 every element of S is in the set Sp, and Sp is surely within the set S but it is not a bijection with the other subsets, thus we have a contradiction where S is uncountable and therefore infinitely uncountable. We notice that we have the same occurrence with any set of ℕ’s, where we can create a power set Ƥ (ℕ) which contains all the subsets of all ℕ which are also uncountable—i.e. infinitely uncountable. Hence, taking the power set always results in a bigger set, and we conclude that there has to be infinitely many sizes to infinity, and therefore indenumerable. iv) Effectively Computable and Effectively Enumerable As previously mentioned, the consideration of functions are critical to our understanding of numerical descriptions as it denotes whether a given property holds true or false. Hence, a set can be enumerated by an effectively computable function. The central idea of an effective computable function involves the execution of an algorithm which successfully terminates and hence involves an entirely determinate sequence of discrete step-by-step procedures. The set ∑ is effectively enumerable (e.e.) iff either ∑ is empty or there is an effectively computable function that enumerates it. For example, an e.e. infinite set, the computable function � � = n2 effectively enumerates the natural numbers which are perfect squares. The importance the word effective merely signifies that there exist an algorithm where given enough time, it would successfully terminate. Hence, we denote that a total functions f: £→ β is effectively computable iff there is an algorithm which can be used to calculate in a finite number of steps. This is particularly important within the study of incompleteness because the central idea resides in whether there is an algorithm that will successfully terminate any theorem in arithmetic as will see that it does not. Since algorithmic sequences not only compute functions but also decides whether a property holds, we begin to actually answer if a given property is either true or false. The term decidability holds a significant weight to numerical functions because it will be seen how important property/relation applies to any appropriately given item. Therefore, a numerical property or relation is effectively decidable iff its characteristic function is effectively computable. This directly leads to the theorem that if is an effectively decidable set of numbers, it is effectively enumerable, however its converse is not true because any Koffi16 effectively enumerable function will successfully terminate however it will not automatically infer that a given property is decidable true or false—e.g. even vs. odd determinates of a function where a function is decidable to be true only if it is even, in this case. v) Effectively Axiomatized Theories Within the assessment of complex arguments we need to develop a formalized structure that is completely free from any obscurity where a suitable artificial language reveals a logical structure. The infusion between mathematical and logical structure aims to show perfect clarity and honest inferences as a theory regimented by a tidy set of axioms will lead to the idea of an axiomatized formal theory—i.e. a theory built in a formalized language. Within the correctness of a stated proof, we want to show that it is explanatory—i.e. clear with no ambiguity. Because our study of incompleteness entails the notion of an incomplete theory, Gödel notes the absolute importance of a deducible and formalized language. Since we have defined a formal language L as being a pair <ф, µ>, where ф is a syntactically defined system of expressions (words) and µ gives the interpretation (image) of these expressions, its semantics effectively tells us the condition under which a given sentence is true or false. Hence, if we think of a language strictly from a logical perspective, a claim or statement written within a formal language follows that there is an effective way of conveying whether a condition actually holds and whether the sentence really is true—i.e. valid or sound. An effectively axiomatized theory being effectively enumerated is not to say that the theory is decidable. This is very interesting because the concept of a decidable property holds whether a property is true or not but effective enumeration of an axiomatized theory does not automatically guarantee that a property will automatically be decidable to be true or false. However, a theory built in a formalized language will be deducible to generalize any mechanical method for any theorem eventually. This leads to one the most important theorems which states, Any consistent, effectively axiomatized, negation-complete theory T is effectively decidable. Because T, in this case, is negation-complete, its theory is therefore either true of false. So to decide whether a sentence Ҩ of T’s language is a T-theorem—i.e. where Ҩ is a Koffi17 theorem (hypothesis) of T, we effectively list theorems until we capture either Ҩ or –Ҩ. It must be noted that every modern mathematical theory begins from a given set of non-logical axioms (i.e. hypothesis) where a developed principle theory could be axiomatized and formalized within the language of logical formulas (Smith, 2013). vi) Expressive vs. Capturing It is noted that a theory T is expressive as it depends on the richness of the theory’s language. Hence, an interpreted formal L is sufficiently expressive iff it can express every effectively computable one-place numerical function. Because an effective computable function successfully terminates we note that the richness of its language depends on whether it can be express T characteristic function. Another important theorem which states, the set of truths of a sufficiently expressive language L is not effectively enumerable, leads to this example. Say K is the set of all even numbers and K’s complement is the set of all odd numbers. Because a function � � = � example only enumerates even outcomes however not odd ones. Although we remember that language L is still sufficiently expressed because it can express every computable one-place numerical function. The central importance of denoting capturabilty entails that fact that if we merely capture a sentence P in a sound theory T, then Ҩ(x) expresses P—i.e. T is provable to be true or false. We therefore have a complete theory. In terms of performing how to succumb to an incomplete theory we must weaken one assumption from soundness to mere consistency, where a formal theory is still sufficiently strong if it captures all effectively decidable numerical properties. However, in terms of an undecidable theorem we use the theorem which states, no consistent, sufficiently strong, effectively axiomatized theory is decidable. Because everything depends, for starters, on whether the ideas of a ‘sufficiently expressive’ arithmetic language and a ‘sufficiently strong’ theory of arithmetic are in good order, Gödel’s proof tells us how to take a consistent effectively axiomatized theory T and actually construct a true but unprovable-in-T sentence. Because Gödel did this by finding a general method that enabled him to take any theory T strong enough to capture a modest amount of basic arithmetic and construct a sentence ℊ, it thereby Koffi18 encoded the claim that the sentence ℊ itself is unprovable in the theory T (p.53, Smith). So ℊ is true if and only if T can’t be prove it. The methods of how Gödel constructs his sentences will be constructed by mapping a set of symbols where a Gödel number will appear and we will see how his construct will include a sentence ℊ. vii) Two Formalized Arithmetics Because the language of a formal system is an essential step of Gödel's proof, it precisely defines and fixes a correspondence of a particular type between the expressions of that language and the system of natural numbers. Hence, a code sequence or Gödel numbering of the language will suffice for wellformed formulas and consequently as sequences of primitive symbols were each are assigned a unique number. BA’s language of non-logical vocabulary LA = < LB, IB >, where LA is the first-order language of basic arithmetic, is the same as that of LA. Non-logical vocabulary signifies a constant that is not a logical constant nor a symbol of grouping i.e. logical variable. For instance, a variable standing for a number is a non-logical constant because it cannot be replace by a statement. Hence, in the context of mathematical logic, a non-logical symbol signifies a symbol (Smith, 2007). Since the language of BA, LB, is a subset of the language LA, they have the same non-logical vocabulary, namely the symbols 0, S, + and ×. Hence, it only knows about the successor, addition and multiplication functions. Although LB lacks the apparatus of quantification i.e. missing numerical variables and quantifiers, it still contains the identity and equality symbol. The corresponding components of LA and the interpretation of LB are identical. Because the inference rules of propositional logic can be used to deductively deduce the language of BA together with rules for the equality predicate, Leibniz's law enables us to infer the sentence obtained by substituting in φ for every occurrence of ρ by τ for example, where the identity ρ = τ, where ρ and τ are terms (Smith, 2007). The natural number sequence and the addition and multiplication functions are defined by BA 6 nonlogical axioms. “Since BA does not contain quantifiers nor variables, each axiom is a template Koffi19 or schema using the Greek letters α and β as placeholders for numerical expressions.” Hence for each generated sequence, a schema generates an infinite number of sentences, where one is a possible substitution of the Greek letter(s). For instance, the first sentences generated below demonstrate the first schema 0 ≠ S0, 0 ≠ SS0, 0 ≠ SSS0, 0 ≠ S(S0 × SS0), etc.” The following are the 6 axiom schemata of BA: Schema 1: 0 ≠ Sα 0 is not the successor of any number Schema 2: (Sα = Sβ) → (α = β) by contraposition, any two distinct numbers have distinct successors Schema 3: α + 0 = α base case for the definition of addition Schema 4: α + Sβ = S(α + β) recursive case for the definition of addition Schema 5: α × 0 = 0 base case for the definition of multiplication Schema 6: α × Sβ = (α × β) + α recursive case for the definition of multiplication As shown, in an axiomatized formal theory, the requirements for BA are met. “Hence, it is decidable whether or not a sentence of LB is an instance of an axiom schema of BA. Hence, Smith proves that BA is negation-complete i.e. for any sentence φ in LB, either BA ⊢ φ or BA ⊢ ¬ φ. In addition, Smith proves that BA correctly decides all sentences in LB.” (Smith) Therefore, we can inevitably infer that BA is decidable, since BA is negation-complete. In conclusion, BA is complete: it can prove every true statement that its language can express. However, its language is rather limited. Specific statements, such as: 3 + 2 = 5, 7 ≠ 2 × 3, can be express, the successor of 0 is not 0, the successor of 1 is not 0, the successor of 2 is not 0, and etc. However, since it lacks the apparatus of quantification, it cannot express general statements, such as: all natural numbers have a successor that differs from 0. Nonetheless, stronger AFTs of arithmetic can be acquired by extending BA. Koffi20 In order to make BA more expressive, its logical vocabulary must allow for variables and quantifiers. By doing this, we merely obtain the interpreted language LA. Because Gödel’s proof requires the expressions of formal language, quantification must be solidified. Hence, variables and quantifiers can substitute the schemata of BA with regular axioms. The formal system of arithmetic is known as QRobinson Arithmetic. By definition, the interpreted language of Robinson Arithmetic is the language of LA plus the interpretation IA. The inference rules of Robinson Arithmetic will be some form of first-order logic which will include the identity. The following are the axioms of Robinson Arithmetic: Axiom 1: ∀x (0 ≠ Sx) 0 is not the successor of any number Axiom 2: ∀x∀y (Sx = Sy → x = y) by contraposition, any two distinct numbers have Axiom 3: ∀x (x ≠ 0 → ∃y (x = Sy)) no element besides zero that does not have a Axiom 4: ∀x (x + 0 = x) base case for the definition of addition Axiom 6: ∀x (x × 0 = 0) base case for the definition of multiplication distinct successors successor Axiom 5: ∀x∀y (x + Sy = S(x + y)) recursive case for the definition of addition Axiom 7: ∀x∀y (x × Sy = (x × y) + x) recursive case for the definition of multiplication Although Q is not negation-complete, letting φ be the sentence that ∀x (0 + x = x) determines that Q is sound since its logic is truth-preserving. However, Q cannot derive φ and Q cannot derive ¬ φ. In proving this statement, an alternative interpretation for LA should make Q's axioms true but φ false because if Q is sound, then it cannot derive φ. Under the original interpretation IA, Q is sound and ¬ φ is false. Koffi21 For Smith, “an extension of IA with two new elements a and b in the domain such that a is its own successor, b is its own successor, and addition and multiplication are extended in such a way that 0 + a is equal to b and 0 + b is equal to a, thereby falsifying φ.” Since the interpretation of φ shows the addition of what Q cannot prove, it can be inferred that Q is a weak theory of arithmetic. However it’ll be proven that Q is sufficiently strong as this property is defined in terms of capturing properties. For instance, Q can derive all of the sentences 0 + 0 = 0, 0 + S0 = S0, 0 + SS0 = SS0, etc. However, it cannot derive the universally quantified sentence φ. Smith mentions that any consistent theory that extends Q will be sufficiently strong and hence negation incomplete. Since any system in which Q can be deduced is essentially incomplete, then any such theory in which Q is interpretable can be proved directly to be incomplete. For example, in set theory, coding formulas and deriving sets i.e. Gödel sets, and proceeding in the same fashion. However, in order to establish the deductibility of Q, the system must be an axiomatizable extension of Q where “a certain amount of elementary arithmetic can be carried out within a system”. (Smith) Hence Q is considered to be a weak theory as it cannot justify statements such as ∀x (0 + x = x). viii) Primitive Recursion The formal theories of arithmetic seen thus far are the successor function, addition and multiplication built in. Primitive recursive functions are considered as a subclass of the effectively computable functions. Primitive recursive properties are also defined in terms relations i.e. those with a p.r. characteristic function which can effectively decide when the relations and properties holds. Since primitive recursive functions can be defined by a chain of definitions by recursion and arrangement. “A full definition for the p.r. function f is a specification of a sequence of functions f 0,f1,f2,...,fk where each fj is either an initial function or is defined from previous functions in the sequence by composition or recursion, and fk = f.” Because every p.r. function has a simple method of proving that every p.r. function has some property P implies that p.r. functions are in fact total functions. “According to Theorem 19: Koffi22 Primitive recursive functions are effectively computable by a series of (possibly nested) bounded ‘for’ loops.” (Smith, 2007) LA can express the following factorial function: (Sx)! = x! × Sx “Hence, this is the p.r. definition for the factorial where It guides us to construct a sequence of numbers 0!, 1!, 2!, . . . , x!” (Smith, 2007). For loops are used to execute a lump of code a specific number of times. They are considered a main programming structure, used to be produce computation by a program where, given a primitive recursive function, its specified value can be determined and every loop iterates a specified number of times. This is very important because this tool is what is needed to show that negation-complete theories are decidable. This process of enumerating Theorem 19 will generate which of ϕ or ¬ϕ leads to the conclusion of a computational decision procedure (Smith, 2014). Because computer languages allow some process for functions to be iterated until a given condition is satisfied, in this case no prior limit was made for a specified number of iterations to be executed. Hence, if searches where in fact made to be unbounded searches as computations then it is plausible that not everything computable will be primitive recursive. Hence we have, “Theorem 21 There are effectively computable numerical functions which aren’t primitive recursive.” ix) Peano Arithmetic In order to derive the statement ∀x(0+x = x) which is beyond Q’s theory, a formal arithmetic that incorporates some stronger axiom(s) for proving quantified wffs will suffice by using Peano Arithmetic. Peano Arithmetic is considered as, “a formal arithmetic that incorporates some stronger axiom(s) for proving quantified wffs.” Since Peano borrows its axioms from Dedekind which states that, “for every Koffi23 partition of all the points… there is a point of one set which lies between every other point of that set and every point of the other set”, the notion of natural numbers is defined "in itself. PA, in this case, it known as the first order arithmetic where its objects that can assume only distinct, separated values i.e. discrete math (Smith, 2007). “PA – First-order Peano Arithmetic – is the result of adding to the axioms of Q the universal closures of all instances of the Induction Schema. Plainly, it is decidable whether any given wff has the right shape to be one of the new axioms, so PA is a legitimate formalized theory” (Smith, 2014). Considering that the wff ∀x(0+x = x) is unprovable in Q, it can however be proven in PA by induction. Assuming the natural style of deduction yields rules from UI (Universal Instantiation), UG (Universal Generalization on ‘arbitrary’ names) and CP (Conditional Proof). Hence, starting with the instance of the Induction Schema where ϕ(x) is replaced by (0+x = x) will help to prove the two conjuncts in the antecedent of that instance in order to conclude ∀x(0+x = x). The formal version of the proof, “makes the axioms of Q true while making ∀x(0+x = x) false has the object as a self-successor.” Hence, what is needed is the absolute fundamental axioms in order to get standard arithmetical results. (Smith, 2014) It’s been noted that in “Theorem 19: Every p.r. function is intuitively computable – moreover it is computable using only ‘for’ loops and not open-ended ‘do until’ loops.” However, not all computable numerical function is primitive recursive (Smith, 2014). x) Capturing Functions The notion of computability in general concludes that, “addition can be defined in terms of repeated applications of the successor function as multiplication can be defined in terms of repeated applications of addition” (Smith, 2014). Hence, the idea of defining one function in terms of repeated application is essentially the idea of defining a function by primitive recursion. Since the idea of repeated applications of the successor function via programming structure executes a lump of code a specific number of times, we can use this to see how functions are captured. Recall that a formal theory is still sufficiently strong if Koffi24 it captures all effectively decidable numerical properties which is equivalent to capturing one place functions. However, capturing a function, in this case, will consist of Q capturing any p.r. function using Σ1 wffs (Theorem 17.2). Therefore, we note that with sufficiently strong theories, functions can be defined in specifying a whole class of primitive recursive functions (Smith, 2007). Hence, we want to show that Q and PA can at best capture all primitive recursive computable functions which will prove sufficient in Gödel’s argument for incompleteness. In using a coding/decoding technique, we create a sequence using the (primitive recursive) decoding function exp(b,i) i.e. Gödel’s βfunction: This single number b can be thought of as encoding the whole sequence k0, k1, k2, ..., kn…for this function returns the exponent of the prime number πi (series of primes) in the factorization of b. By the construction of b, then, exp(b,i) = ki for i ≤ n. Now let’s generalize. We’ll say A twoplace β-function is a function of the form β(b,i) such that, for any finite sequence of natural numbers k0,k1,k2,...,kn there is a code b such that for every i ≤ n, β(b,i) = ki. (Smith, 2007) So the idea essentially is selecting a corresponding code number b in being the primary argument for β, where the functions will interpret it and give a required sequence to the members in an orderly manner. An example could be the use of the division algorithm, � = � + , where the idea of multiplication and addition on the remainder of division can be elementarily. Hence, L A can express all p.r. functions by using the β concept. Koffi25 Gödel’s Incompleteness Theorem Are there contradictions in mathematics? Since the idea of proofs are supposed to establish certainty, a single contradiction would eliminate all certainty. Suppose we have a system of logical rules which tells us we can make deductions. While Hilbert’s ultimate goal was to prove that the foundations of mathematics was free of all contradiction, Gödel’s results would prove otherwise. Hence, in a consistent mathematical system, there are results which are true but not provable. Using a clever coding system by way of representing mathematics by integers, Gödel created a sentence g, stating “this sentence is not provable”. However, if that statement could be proven, we will show in his proof that a contradiction arises and therefore the statement cannot be proven, but it is true. Gödel created his own version of the liar’s paradox and proved that it is impossible to obtaining a complete axiomatization of arithmetic In retrospect, the fundamental logical concepts seen thus far are the following: Systems of Formal Languages The original idea of a formal language is that, Gödel’s proof can be completely and precisely defined. Hence, it requires that every symbol used must be defined in a language which is included but not limited to commas, periods, colons and etc. Proofs residing in a Formal Language We must understand that a formal language, in and of itself, does not prove anything. Thus, we must define exactly what a proof in the formal language means; which will consist of formal sentences, wff’s, decidable properties, capturing functions and etc. In a proof, it is rather evident that sentences interact with other sentences in which there must be some direct association among the sentences being proven. Koffi26 Rules of Inference Rules clearly stating how one sentence can follow from one or more other sentences are call rules of inference. We use formal expressions only as a combination of symbols where the rules are only justified in how to evaluate one or more expressions i.e. combinations of symbols. For example, if we had sentences “le garçon va” and “à l'école”, then we can create a rule saying that you can combine the first three words (symbols) of the first sentence with the second two (words) symbols of the other sentence. Axioms of Formal Systems Since axioms are considered as fundamental expressions of the formal language requiring no proof, every proof has to use at least one of them. Proof Sequences in a Formal System In the beginning of a formal language proof, starting with at least one axiom, a set of rules is then followed to make a new sentence. By doing that several times, we get a new sentence and etc. where every sentence constructed is proved. Hence, every sentence that has been proven will have a list of sentences that go together to prove it. Consistency In order to avoid inconsistencies and contradictions, we must built a formal language where the proofs are well defined, such that one cannot prove a sentence as well as the opposite of that sentence. Hence, Gödel’s proof only applies to formal languages that do not result in contradictions. Can we then use the system to show that there cannot be a contradiction among results we can prove using the system? Gödel would say yes! A logical system can prove that it itself is consistent if and only if it is not inconsistent. Hence, if a system is consistent and it doesn’t contain any contradictions then we cannot prove that it is inconsistent. However, if our system is not inconsistent, then we can prove that it is consistent. Koffi27 Incompleteness and Completeness By defining a consistent formal language referring to numbers and expressions that can be interpreted, the goal is to make propositions about those numbers. Hence, a formal language proof system is considered complete such that every proposition about numbers can be made and interpreted in the formal language, and there will always be a proof of that proposition or of its negation. Gödel’s proof thus shows that any formal language, regardless of the axioms chosen, and proof rules, can always define a formal expression such that the sentence nor its negation can be proven by those axioms. In addition, if Gödel’s proof is correct, then it infers that every formal language proof system must be ‘incomplete’. Gödel’s Numbering/Coding Gödel’s numbering system represents any sentence of the formal language as a number. This implies that in a formal language, every sentence can be represented as a number. Hence, Gödel fixes a correspondence, i.e. diagonalization, of a particular type between the expressions of that language and the system of natural numbers. Gödel’s code sequence of symbols is matched to a unique number, such that no two combinations/arrangements can have the same number and also, no two numbers can match to the same combination. This is equivalent to an earlier statement where a code sequence or Gödel numbering of the language will suffice for well-formed formulas and consequently as sequences of primitive symbols were each are assigned a unique number. The process goes as follows. This list has not been exhausted. ¬ 1 3 5 → 7 ↔ 9 ∀ 11 ∃ 13 = 15 ( 17 ) 19 0 21 S 23 + 25 × 27 x 2 y 4 For example, suppose we take an example of a formal sentence (x + y) where we match number to its symbols: ( 17 x 27 z 6 Koffi28 + 25 y 4 ) 19 The following sequence are numbers, 17, 27, 25, 4, and 19 where putting them together makes 172725419. However, this method could be distorted in which the original sequence could become 1, 7, 2, 72, 54, 19. In order to retrieve the original information, Gödel devised a clever way of converting these sequences of numbers in a way that retains all the original information. Gödel’s system takes the prime numbers in order, so that for example, to encode a sequence of five numbers, we start off with the first five prime numbers – which are 2, 3, 5, 7 and 11. Hence, we have the following: 217, 327, 525, 74, 1119 Multiplying 217 × 327 × 525 × 74 × 1119, we get: which gives the matching Gödel number for the sentence (x + y). In ordering to derive the original numbers 217 × 327 × 525 × 74 × 1119 first divide by the first prime number, which is 2. Continue to divide by 2 until you don’t get an even dividend. Creating a Special Correspondence with Numbers The idea, essentially, is as follows: any formal system can be linked with a Gödel number and thus any formal proof can prove that sentence. Hence, there exist a matching Gödel number for that proof. Gödel’s system of corresponding relationships between that proof and the sentence that it proves is always well defined. Symbols signify a name or meaning for particulars where in this case, we can represent symbols for sentences and variables. For instance, stating names for particular characters such as Jack and Jill is essentially equivalent to saying ‘3 and 5’, where 3 and 5 are names for particular numbers. Names can also signify variables such that saying a goat and a pig is also equivalent to saying a number x and a number y, provided that x and y are variables for numbers. Koffi29 What would then constitute as a Gödel proof can be defined in general terms, for instance: The relationship in the sentence P is the proof of the sentence Q where P and Q are variables for formal sentences is the particular correspondence of interest. Thus, given any two formal sentences, one can substitute them for variables P and Q, where the proposition is either correct or incorrect. Let’s suppose that P is (y + 7), and Q is (3 + x) which gives the expression (y + 7) is the proof of (3 + x). This proposition is wrong however we are simply illustrating a principle. This Gödel proof has a particular relationship with numbers where the relationship between the wellformed formal sentences P and Q means that GN(P) is the G-proof of GN(Q) and it is a relationship corresponding to the expression P is the proof of the sentence Q. Note that in reference to Gödel’s numbering system, we will use GN. F is a free variable meaning the value of any formal symbol sentence, where the free variable of the Gödel numbering system is GN(F). The name GN(F) is simply an expression of a Gödel number that matches to the formal sentence F. In the earlier example (x + y) has the matching Gödel number: The GN expression of this sentence is GN((x + y)) and now we can notice a relationship between a formal sentence and a formal proof. Matching Formal Sentences with Numeric Relationships Any numeric relationship with at least one free variable will always have a matching formal sentence with at least one free variable. That formal sentence will then express the same theory as that numeric relation. Since that formal sentence has a matching Gödel number, that number relation will also have a matching Gödel number. Koffi30 For any pure numeric relationship, it is a rather simple relationship between numbers that is expressed in a language that is not necessarily formal. For instance, there are many different ways of expressing binary relations such as multiplication and addition. For instance, the result of x and x is four, x by x is equal to four, x plus x is four and so on. Therefore, a numeric relationship should be clearly defined such that it can only be translated into the formal language in which the formal sentence has a matching Gödel number. Take the number relationship ‘x plus x is equal to four’ and suppose that in our formal language the matching formal sentence is: (x + x = SSSS0) In retrospect, the formal system 0 = zero, S0 = one, SS0 = two, SSS0 = three, and etc. That means the matching Gödel number for ‘x plus x is equal to four’ or (x + x = SSSS0) is: 217 × 327 × 525 × 727 × 1115 × 1323 × 1723 × 1923 × 2323 × 2921 × 3119 Since a number relationship with a free variable is not a proposition, that variable substituted by some specific value will validate a proposition. Substitution takes one value and gives another matching value. Hence, substituting the free variable by some number and then substituting the free variable from the formal sentence by the same number will yield a certain number relationship that is a proposition about numbers having a matching formal sentence. Substituting variable x by two, for instance, would yield ‘two plus two is equal to four’, and the matching formal sentence would be (SS0 + SS0 = SSSS0). Gödel would then stipulate the following: “If a formal sentence is provable, then the matching number relationship must be true and for any sentence of the formal language, either there’s a formal proof of that sentence or there isn’t.” An Alternative Numbering System Koffi31 At this point in the game, a new function is needed such that it is similar to the Gödel numbering function, which will be used in Gödel’s proof. Gödel’s numbering function is not a numeric relationship, meaning that the symbols of the formal language are represented as the symbol for equals, plus, and etc. Let’s suppose we would like to use Gödel’s numbering function but only for symbols of the formal language that in fact are numbers. Consequently, we’d like to call Gödel’s numbering function GN(x), such that substituting x only by symbols of the formal language would represent numbers. However, this still wouldn’t infer that GN(x) is a numeric relationship since the definition of the Gödel numbering function still refers to symbols rather than numbers. Hence, in order to establish a working proof for Gödel, we would need a function that is similar to the GN(x) function, but one which is purely a number relationship. Therefore, we need a new function which will only take numeric values for free variable, provided that entering only numbers would mean that the function would only result in number values. The goal would be to get a new function that could do the exact same thing as the Gödel numbering function. Since this new function refers to numbers and variables representing numbers, then it is a number relationship. Let’s call the new function a Simple Numbering function, or SN(x), where x is a free variable. In Gödel’s original proof (relation 17), Gödel calls this function Z(n). This implies that for whichever symbols would represent symbols for a number, the SN function would gives us the correct Gödel number. The Proof In retrospect, Gödel’s proof has a particular relationship with numbers where the relationship between the wff sentences P and Q (also having a number relationship) means that GN(P) is the G-proof of GN(Q) ; which leads to the expression, P is the proof of the sentence Q where it is possible that there could be a formal proof of that sentence. Equivalently, if P is the matching Gödel number for a sequence α, and Q is the matching Gödel number for sentence β, there is matching relationship saying that, P is the proof of the sentence Q and there is possibly a number that is the Koffi32 G-proof of the sentence that is either true or false. However, if a code sequence of a formal sentence isn’t a Gödel number, then there cannot be a G-proof of that sentence. Recall that substitution takes one value and gives another matching value. An example of substitution could be the sentence (x + 3) where substituting x by 5 produces the sentence (x + 3). This expression took two values, one for the sentence to be replaced and the other for the number that’s used for the substitution. This would result in two free variables where we can call it Substitution (Sentence, Number). Hence, if the sentence has only one free variable, then we’re provided a specific substitution. For our example, we have: Substitution ((x + 3), 5) where the value of that is (5 + 3). Gödel’s formulation of his proof involves the concept of substitution; where taking a number relationship, and substituting it by its own matching Gödel number would then generate with a new relationship which appears to self-reference its own Gödel number. First, a relationship of substitution G-Substitution (Numeral , Numeral ) needs to be defined. A relationship that isn’t a numeric relationship would imply there is no G-proof of G-Substitution (x, GN(x)). Because we want a relationship that is equivalent, we define another relationship in which its equivalent statement also says there is no G-proof of G-Substitution (x, SN(x)). Recall that SN is the Simple Number function which is a numeric relationship, defined in terms of numbers and variables only. Also, since we have a numeric relationship, there will always be a corresponding formal sentence for this relationship. Hence, we can call SN a formal sentence FS. Finally we say that the formal sentence FS will be a corresponding Gödel number GS. We can see that all of SN, FS, and GS are self-referential however, SN and FS both have a free variable where GS is only a number. The trick Gödel uses at this point is to substitute the free variable of the Simple Numeric relationship SN by the Gödel Sentence GS. This would result in another relationship stating the following: there is no G-proof of G-Substitution (GS, SN(GS)), equivalently SN(GS), where Koffi33 there is a corresponding formal language/sentence FS(GS). At this particular point we want to show if there is a proof in the formal language FS(GS) or not. Suppose that there is a proof of FS(GS) in the formal language such that the corresponding numeric relationship SN(GS) states, there is no G-proof of G-Substitution (GS, SN(GS)) must be true! However, if that is the case that there cannot be a formal sentence where the free variable of the formal sentence FS is replaced by the GS number by virtue of GS being only a number then we have a contradiction! There cannot in fact be any proof of the formal sentence FS(GS) that would mean the free variable of the formal sentence FS is replaced by the number GS. Therefore, we can say this relationship is true by semantics, but not provable and thus incomplete. Koffi34 Works Cited Ferreirós, José. "II.7 The Crisis in the Foundations of Mathematics." The Princeton Companion to Mathematics (2007): n. pag. Web. Gödel, Kurt. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. New York: Basic, 1962. Print. Meyer, James R. "A Simplified Explanation of Gödel's Proof - Part 8." LL: Gödel Simplified: 8. Web. 04 Feb. 2016. Smith, Peter. "Gödel Without (Too Many) Tears." Logic Matters (n.d.): n. pag. Http://www.logicmatters.net/. 20 Feb. 2014. Web. Smith, Peter. An Introduction to Gödel's Theorems. Cambridge: Cambridge UP, 2007. Print. Smith, Peter. An Introduction to Gödel's Theorems. Cambridge, England: Cambridge UP, 2013. Print. "Summer of Gödel." Summer of Gödel. N.p., n.d. Web. 26 Jan. 2016. . Koffi35