Gdel's Work in Set Theory - Case Study Example

 «Gödel's Work in Set Theory» Impact of Gödel’s Incompleteness Theorems Gödel’s Incompleteness Theorems consist of theorems that hold a lot of mathematical logic. The theorems establish a lot of inherent limitations to the most trivial axiomatic system. Among his works, two theorems are highly celebrated both by the mathematicians and the non-mathematicians. Both the completeness and incompleteness theorems have had a significant impact on the set theory used in modern mathematics. The Incompleteness Theorems In his work, Gödel mentioned the likelihood of the insolvability of some questions on the reels already contained in his thesis of 1929. In it, he argued in contradiction of the formalist principle by Hilbert's which stated that consistency is a benchmark for existence. In the argument, he gave finitary proof on the consistency of analysis, a key desideratum. Accordingly, it was the turn to these questions that led him to work on the two incompleteness theorems (Raatikainen & Edward, 2013). The First Incompleteness Theorem offers a counter example to completeness by presenting an arithmetic account which is neither attestable nor refutable in the Peano arithmetic. The second Incompleteness Theorem comes after the first incompleteness theorem (Partee, et al., 1990, p. 230). As stated, the uniformity of arithmetic cannot be substantiated in arithmetic itself. Therefore, Gödel's theorems validated the infeasibility of the Hilbert program (Awodey & Carus, 2010). Godel proved the reliability of analysis using the finitary methods. In 1931, Gödel indicated that in a varied class of formal theories including the PM, existent statements that could be proved true of numbers may lack any evidence within the given theory. This forms Gödel’s first incompleteness theorem (Shankar, 1997, p. 17). The First Incompleteness Theorem In his Logical Journey, Wings publications indicate that Gödel’s works began in 1930 when he started studying the consistency problems of classical analysis (Wang, 1996). At the time, there had been no rigorous justifications and explanations on the rigorous mathematics (Feferman, et al., 2003, p. 339). This study got its motivation from Hilbert’s works. Hilbert had been working towards the provision of a directly consistent analysis to the finitary methods. The problems that this work had formed the driving force to his study. Through this, Gödel’s wanted to prove the constancy of number theory by a finitary numeral theory (Barbara, et al., 1990). He also wanted to prove the dependability of analysis by number theory. He represented real numbers by the predicates in number theory. In so doing, he found out that he had to use the truth concept in order to verify the axioms of the analysis. He came with an enumeration of symbols, sentences and verifications within the specified order. In so doing, he discovered that impression of arithmetic truth cannot be given a defined form in arithmetic. He observed that if a way to define the truth within a system existed, it would lead to a liar paradox (Rahman, et al., 2008). This would show that the system is inconsistent with what is being studied. These arguments were later formalised so that they bring meaning to the existence of undecidable propositions without quoting any individual occurrences. It is observable that Gödel tried to reduce the consistency problem to that of arithmetic for ease of solving. At this point, he temporarily changed the direction with the aim of intruding another element. The element would prove an illumination solution to Liar Paradox (Winterburn, 2012, p. 47). This appeared to require the truth definition for the arithmetic. This, in turn, resulted to paradoxes, like the Liar paradox to mean that the sentence is a false one. Gödel then discerned paradoxes of this form would not necessarily come in existence if truth were to be replaced with provability. The Gödelzed liar is seen to be constructible within logic. The consequences, in it, do not forcibly induce dilemmas of inconsistency since the predicates ‘false’ have been replaced (Dov & Guenthner, 2004, p. 116). The proof of the First Incompleteness Theorem It is imperative to note that these descriptions form the basis for the proof of the two theorems. They all lie in formulating Gödel's outcomes in Peano arithmetic. It can be seen that Gödel used systems related to the one defined in Principia Mathematica, although it contains Peano arithmetic. The presentation of both the First and Second Incompleteness Theorems has a reference to the Peano arithmetic denoted as P; this follows Gödel's notation. He define the notion of ω-consistency as follows, Theorem: P is ω-consistent provided that P ⊢ ¬φ (n) for all implied P ⊬ ∃xφ (x). Certainly, this infers to consistency. In addition, it follows the supposition that natural numbers gratify the axioms contained in Peano arithmetic. The main technical tool applied for the proof involves the Gödel numbering, the CH. This is a mechanism that assigns natural numbers to a problem to the formal theory P. There are several steps to be followed in order to have this done. The most mutual relationship is based on the exclusive demonstration of natural numbers in the form of products of powers of prime factors. In Raatikainen & Edward (2013), characters of number theory gets allocated a given positive natural number, #(s). This is done in a fixed, but then in an, arbitrary way, e.g. #(0) = 1 #(=) = 5 #(¬) = 9 #(1) = 2 #( ( ) = 6 #(∀) = 10 #(+) = 3 #( ) ) = 7 #(vi) = 11 + i #(×) = 4 #(∧) = 8 The natural numbers that correspond to a sequence w = < w0… wk. > of the symbols is ⌈w⌉ = 2#(w0) · 3#(w1) · … · pk#(wk), In all the cases, pk is the k+1first prime. This is known as the Gödel number and designated by ⌈w⌉. Following this, Gödel numbers to various formulas can be designed. An indispensable point is that when interpreting a formula in the form of natural number, it follows that the numeral consistent to that natural number can happen as the argument of the formula. This enables the syntax to refer to itself. This allowed Gödel to formularize the original Liar paradox (Dov & Guenthner, 2004). Another idea necessary to carry out the formalization lies in the concept of numeral perspective expressed in the number theoretic predicates. The number-theoretic method φ (n1… nk) is numeral-wise expressible in the P if for each tuple of natural numbers being used (n1… nk): N ⊨ φ(n1, …, nk) ⇒ P ⊢ φ(n1, …, nk) N ⊨ ¬φ(n1, …, nk) ⇒ P ⊢ ¬φ(n1, …, nk) In this case, n is the formal term that denotes the natural number in n. In P, this can be given as S(S(…S(0)…), where s is the number of repetitions of the successor functions applied to the unceasing symbol 0. The principal goal remains expressing the predicate in a numeral form. Prf(x, y): ‘the arrangement with Gödel number, x is the proof of the sentence with which the Gödel number is y’ (Raatikainen & Edward, 2013). Reaching this goal involves defining forty-five different relations. Each of the relations gets defined in terms of the prior ones. These relations get highlighted in steps of various theorems. If P is ω-consistent, it follows that there is a sentence that cannot be proved or refuted from the give P. Proof: By judicious coding of syntax, the formula can be written as the Prf(x, y) of number theory that is representable in the given P, such that, 1). n codes a proof of the φ ⇒ P ⊢ Prf(n, ⌈φ⌉). Additionally, 2). n does not have coded proof of φ ⇒ P ⊢ ¬Prf (n, ⌈φ⌉). Letting Prov(y) represent the formula ∃x Prf(x, y), then, there is a sentence φ which has the property; 3). P ⊢ (φ ↔ ¬Prov (⌈φ⌉). Therefore, if φ is not provable, then, it can be observed that P ⊢ φ, then by (1) seen above, there is n so that P ⊢ Prf(n, ⌈φ⌉), hence P ⊢ Prov(⌈φ⌉), therefore, by theorem (3) P ⊢ ¬φ, P is inconsistent leading to 4. P ⊬ φ Moreover, by (2), and (4) we have that P ⊢ ¬Prf(n, ⌈φ⌉), for all the natural numbers in n. Following the consistency in ω-, it can be observed that P ⊬ ∃x Prf(x, ⌈φ⌉). Consequently (3) offers P ⊬ ¬φ. This shows that if P is the ω-consistent, then φ is self-governing of P. The proof to the first theorem demonstrated its constructive manner. This implies that it has been proved in an intuitionistic and unobjectionable manner. He highlights that all the existential declarations form their basis on this theorem. This numeral-wise expressibility of the primitive recursive relations, which is intuitionistic are unobjectionable (Gödel 1986, p. 171). The Second Incompleteness Theorem This theorem establishes the improvability, in number theory, of the consistency of number theory. It is written to establish the consistency of the axiom i.e. letting Con (P) be the ¬Prov (⌈0 = 1⌉). If P is dependable, then Con(P) is not taken as provable from P. The Proof: if we let φ be as in (3), the reasoning applied to infer ‘if P⊢ φ, and then P⊢ 0 ≠ 1‘does not go to the limit elementary number theory, this can be represented in other formats. However, to formalize P, a lot of effort is needed. The result becomes: P⊢ (Prov(⌈φ⌉) → ¬Con(P)), hence by (3), P⊢ (Con(P) → φ). Since P⊬ φ, it follows that P⊬Con (P). This is the second incompleteness theorem that is deceptively simple because it avoids formalizations. An arduous proof would have to institute the proof of: if P⊢ φ, then it follows that P⊢ 0 ≠ 1’ in P. Neither ω-consistency is needed in the proof of Gödel's Second Incompleteness Theorem nor is ¬Con(P) demonstrable, by the consistency in P as well as the fact that P⊢Prov(⌈φ⌉) implies P⊢ φ. This is now known as Löb's theorem (Gabbay & Woods, 2009, p. 471). Gödel's Work in Set theory Gödel's evidence on the consistency of the continuum hypothesis, with a given axioms of Zermelo-Fraenkel set theory is the greatest achievement in his mathematical endeavors. This is attributed to practically all of the technical machinery employed in the proof that had to be invented. The Continuum Hypothesis (CH) formulated by Georg Cantor was the initial difficulty on Hilbert's list of 23 unsolved sums as given in his well-known address to the International Mathematical Congress held in Paris in 1900. Hilbert stated the problem as follows: Letting A be the set of an infinite set of the real numbers. Then A is taken as either countable, or cardinality 2ℵ0, i.e., A is in a one-to-one correspondence either within the set of natural numbers or set of all the real numbers. Alternatively, stating the continuum hypothesis; the first uncountable infinite, cardinal is given by ℵ1 = 2ℵ0. As early as 1922, Skolem ventured that the CH was an independent variable of the axioms for the set theory given by Zermelo sometime in 1908. Nevertheless Hilbert published a (falsified) proof of the CH in Hilbert 1926. Gödel proved its consistency in 1937, with the axioms of ZF set theory. Hereafter the standard abridgements for Zermelo-Fraenkel set theory, the ZF, and Zermelo-Fraenkel set theory with the Axioms of Choice, ZFC. The dependability of the repudiation of the CH was shown in 1961 by Paul Cohen (Cohen, 1963) and later together with Gödel's outcome; one infers that the CH are self-determining of ZF and ZFC. Cohen invented a new technique called forcing in the course of proving his result; this method builds models of the set theory. Forcing’s work led to a revitalization of formalism amongst set theorists; the plurality of models forming a suggestion of the essential variability in set theory (DeVidi, et al., 2011). This was away from the concept that there is an envisioned model of set theory—a perception Gödel advocated from at least 1947 or even earlier. There are signs that the CH may again be regarded to be a problem to be solved mathematically as attributed to some new evident axioms extending ZF (Woodin & Hugh, 2002). If any of the suggested solutions gain approval, this would confirm Gödel's opinion that the CH would be ultimately decided by judging an evident extension of the ZF axioms used for set theory (Foreman, 1998). The programs associated with this view are referred to as “Gödel's Large Cardinal Program” (Kanamori, 2009, p. 31). In his work, Gödel mentioned the likelihood of the insolvability of some questions on the reels already contained in his thesis of 1929. In it, he argued in contradiction of the formalist principle as raised by Hilbert's; that consistency is a benchmark for existence. In the argument, he gave a finitary proof on the consistency of analysis that was a key desideratum. Accordingly, these questions led to the two incompleteness theorems. Weakness The theorem has been viewed as irrelevant by many who have taken their time to study what the theorem entails. They give the reason that accounts for the irrelevance in Gödel’s theorems. The incompleteness of any sufficiently resilient consistent axiomatic theory established by that theorem concerns only on what may be called the arithmetical component of a theory. Any formal scheme contains such a component. These components occur if it is probable to interpret part of the statements in it; as statements about natural numbers, in a way that the system provides a proof of some basic principles of the arithmetic. Given these ideas and contributions, other theorems were produced using Rosser’s strengthening of Gödel’s theorem in combination with the evidence adduced by the Matiyasevich-Davis-Robinson-Putnam theorem pertaining to the representation ability and presentations of recursively enumerable sets. These productions resulted into the Diophantine equations. In this regard, a unique and profound declaration of the form “a Diophantine equation p(x1... xn)=0 has no solution” is obtained in its undecidable state. This provisions lead to the conclusions that the theorems are consistent. Whereas it is numerically and mathematically a precise striking fact that any sufficiently strong and consistent formal structure is incomplete with respect to this class of declarations, it is highly unlikely to be assumed interesting in a non- mathematical perspective where completeness or consistency is an issue. There is also another different appeal to the first incompleteness theorem away from mathematics. This recognizes that the theorem has to be applied. Conclusion The first incompleteness theorem positions that: no consistent system of any axioms has its theorems listed by a procedure that is very effective. For such systems, there always has to be a statement about the natural numbers seen to be true. However, this numbers cannot be proved within the system. The second incompleteness theorem is an extension of the first theorem in the sense that it has manifestations that systems cannot be demonstrated in their own consistency. It is clear that every sufficiently strong axiomatic theory is either incomplete or inconsistent (Ben-Ar, 2001, p. 136). Many, especially non-mathematicians, find this statement very fascinating and find it easy to apply what they view as the in incompleteness theorem in different circumstances such as in Rand’s philosophy, the Bible and the U.S Constitution (Kent, 2011, p. 63). With the above demonstrations, it becomes very clear that Gödel's work has had a very significant impact. List of References Awodey, S. & Carus, A. W., 2010. "Gödel and Carnap”, in Kurt Gödel: Essays for his Centennial, Solomon Feferman, Charles Parsons & Stephen G. Simpson. eds ed. Cambridge: Cambridge University Press. Ben-Ar, M., 2001. Mathematical Logic for Computer Science. Second ed. London: Springer London. Cohen, . P., 1963. The Independence of the Continuum Hypothesis”, Proceedings of the National Academy of Sciences of the U.S.A., 50: 1143–1148.. s.l.:s.n. DeVidi, D., Hallett, M. & Clarke, P., 2011. Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in Honour of John L.Bell. Waterloo Ontario: Springer. Dov, M. G. & Guenthner , F., 2004. Handbook of Philosophical Logic - Volume 11. s.l.:Springer. Feferman, S. et al., 2003. Collected Works. V: Correspondence H-Z. eds. ed. Oxford: Oxford University Press. Foreman, M., 1998. Generic Large Cardinals: New Axioms for Mathematics. Documenta Mathematica, Extra Volume, Proceedings of the International Congress of Mathematicians, Volume II, p. 11–21 . Gabbay, D. . M. & Woods, J., 2009. Logic from Russell to Church. s.l.:s.n. Kanamori, A., 2009. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Berlin: Springer. Kent, R. C., 2011. Sum of the Parts: The Mathematics and Politics of Region, Place, and Writing. Iowa City: University of Iowa Press.. Partee, B., Alice ter, M. & Wall, R., 1990. Mathematical Methods in Linguistics. Dordrecht Netherllands: Kluwer Academic Publishers. Raatikainen, . P. & Edward, Z. . N., 2013. Gödel's Incompleteness Theorems. The Stanford Encyclopedia of Philosophy URL = ., Winter Edition. Rahman, S., Tulenheimo, T. & l Genot, 2008. Unity, Truth and the Liar: The Modern Relevance of Medieval Solutions to the. Helnsinki: Springer. Shankar, N., 1997. Metamathematics, Machines and Gödel's Proof. Cambridge: Cambidge University Press. Wang, H., 1996. A Logical Lourney: From Gödel to Philosophy (Representation and Mind). Cambridge: MA: MIT Press. Winterburn, M. D., 2012. Secrets of the Paradox: Solving the Liar and other logical problems. leicestershire: Troubador Publishing Ltd. Woodin & Hugh, W., 2002. Correction: ‘The Continuum Hypothesis. II’”, Notices of the American Mathematical Society, 49(1): 46.. s.l.:s.n. Read More