DOCSLIB.ORG

Incompleteness – a Very Rich Dessert

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Incompleteness – a Very Rich Dessert Incompleteness – A very rich dessert 2 Knights and Knaves A tribute to Raymond Smullyan Raymond Merill Smullyan (born 1919). American logician, ma- thematician, concert pianist, Taoist philosopher, and magician. Many books on logic puzzles, among them: What is the Name of This Book? (1978), Forever Undecided (1987). First-Order Logic (1968), Set Theory and the Continuum Problem (1996), Gödel’s Incompleteness Theorems (1992), . Suppose you are in Smullyan-land, where knights say always the truth and knaves always lie. You meet someone who tells you “I am not a knight”. What kind of person is she? As you will (hopefully) see soon: This puzzle contains the essence of Gödel’s (first) incompleteness theorem! 3 Another Gödelian Puzzle (with nods to Tarski) Imagine a computing machine that can print strings based on the alphabet: ¬, P, N, (, ) A string is called printable, if the machine can print it. The machine is programmed to print all printable eventually. Def. 1: The norm of string w is the string w(w). Def. 2:A sentence is a string of form P(w), PN(w), ¬P, or ¬PN(w). Def. 3: P(w) is called true iff w is printable. ¬P(w) is called true iff w is not printable. PN(w) is called true iff the norm of w is printable. ¬PN(w) is called true iff the norm of w is not printable. Presuming that the machine never prints non-true sentences, can it print all true sentences? Note: Defs. 1 and 2 are purely syntactic; Def. 3 concerns semantics. “true” is a precisely defined, technical term here: check yourself by replacing “true” by “grmph”! 4 Enter the heroes . Kurt Gödel (1906-1978). Austrian-American logician, ma- thematician and philosopher. Established the completeness of first-order logic in his PhD thesis (1929). Proved his two fa- mous incompleteness results in 1931. Many other important contributions to logic and philosophy, e.g.: consistency of the continuum hypothesis in ZF set theory. Alfred Tarski (1901-1983). Polish-American mathematician and logician. Best known for his work on model theory, me- tamathematics, and algebraic logic. Important contributions also to abstract algebra, topology, geometry, measure theory, set theory, and analytic philosophy. Note: The modern conception of logic, in particular as used in computer science today, is largely to due to Tarski and Gödel (– who built on Hilbert, Frege, Skolem, Gentzen, Boole, Bolzano, and many other “heroes”). 5 Semi-formal statements of Gödel’s and Tarski’s Theorems First Incompleteness Theorem (Gödel [Rosser]): Every [ω-]consistent and reasonably expressive system of arithmetic contains sentences that are neither provable nor refutable. First Incompleteness Theorem (with shades of Tarski): Every correct and reasonably expressive system of arithmetic contains true, but unprovable sentences. Second Incompleteness Theorem (Gödel): No [ω-]consistent and sufficiently strong system of arithmetic can prove its own consistency. Undefinability Theorem (Tarski): No correct and reasonably expressive system of arithmetic can define the set of (arithmetically) true sentences. Important: To be handled with care due to remaining informality! 6 An abstract form of Gödel’s and Tarski’s Theorems Definition. A system Σ is a set containing the following components: E ... expressions of Σ, S ... sentences of Σ (S ⊆ E), T ... true sentences of Σ (T ⊆ S), P ... provable sentences of Σ (P ⊆ S), R ... refutable sentences of Σ (R ⊆ S), H ... predicates of Σ (H ⊆ E), a function Φ: E × N 7→ E: If E ∈ H then Φ(E, n) = E(n) ∈ S. By a number-set A we mean any set of natural numbers (A ⊆ N ). Its complement (w.r.t. N ) will be denoted by A. Definition. A predicate H ∈ H expresses a number-set A if for every n ∈ N : n ∈ A ⇐⇒ H(n) ∈ T I.e., A is expressible if A = {n | H(n) ∈ T } for some predicate H. 7 Expressibility, correctness, completeness Note: Expressibility only concerns “truth”, but not “provability”. This should be contrasted with the related notion of representability of A by H within a (proof) system, defined by n ∈ A ⇐⇒ H(n) ∈ P. We will work with countable, even finite alphabets and consequently the language/system is always countable. Therefore it is clear that not all number sets can be expressed. (Remember Cantor’s diagonal argument that demonstrates the uncountability of R!) Definition (Correctness [aka. soundness]). A system is correct iff all provable sentences are true (P ⊆ T ). Definition (Completeness). A system is complete iff all true sentences are provable (T ⊆ P). Note: It is trivial to present sound systems or complete systems for, e.g., arithmetic. However Gödel (in Tarski’s interpretation) proved: No (sufficiently strong) system for arithmetic is sound and complete. 8 Gödel numbering and diagonalization Note (on self-reference): Strings of a language can always be coded as natural numbers. Therefore every concrete arithmetic system (also) “talks about itself”. Definition (Gödel number). A 1-1-function p·q : E 7→ N is called Gödel numbering. For any expression E ∈ E, pEq is called its Gödel number. We write En if n = pEq. (Thus, pEnq = n.) Definition (Diagonalization). By the diagonalization of an expression En we mean En(n)[= E(pEq)]. The function d : N 7→ N that maps any n into pEn(n)q is called diagonal function of the system. For any number-set A we denote d−1(A) = {n | d(n) ∈ A} by A∗. – Remember: if En is a predicate, then En(n) is a sentence. – A∗ collects the Gödel numbers of exactly those expressions, who’s diagonalizations are named (by their Gödel number) in A. Notation: We will use P to denote {pSq | S ∈ P}. 9 An abstract form of Gödel’s (first) incompleteness theorem Theorem GT (After Gödel with a shade of Tarski) ∗ If Σ is correct and P is expressible in it, then Σ is incomplete. ∗ Proof. Suppose H expresses P ; let h = pHq. Let G be the diagonalization of H, i.e. G is the sentence H(h). We show that G is true but not provable in Σ. ∗ For all n ∈ N : H(n) is true (i.e., ∈ T ) iff n ∈ P . In particular ∗ (by “diagonalizing”): G = H(h) is true iff h ∈ P . Now observe: ∗ h ∈ P ⇐⇒ d(h) ∈ P ⇐⇒ d(h) 6∈ P. But d(h) = pH(h)q = pGq. Therefore d(h) ∈ P means (via Gödelization) “G is provable in Σ” and d(h) 6∈ P means “G is not provable in Σ”. Summing up: G is true ⇐⇒ G is not provable in Σ. Since Σ is correct, G cannot be non-true and provable. Therefore we have obtained a sentence that witnesses the incompleteness of Σ: G is true but unprovable in Σ. QED. 10 Towards concrete systems: expressiveness conditions According to Theorem GT, we can establish the incompleteness of a ∗ correct system Σ by verifying the hypothesis that P is expressible in Σ. This can be broken down to verifying three conditions: G1: For any A: A is expressible in Σ =⇒ A is expressible in Σ. ∗ G2: For any A: A is expressible in Σ =⇒ A is expressible in Σ. G3: P is expressible in Σ. Remark: Proving G1 will turn out to be trivial. Proving G2 is relatively straightforward. Proving G3 is extremely laborious. (But, with hindsight, we get a “free ride” from computability theory!) 11 Gödel sentences, Diagonal Lemma Definition (Gödel sentences). S is a Gödel sentence for a number-set A if: S ∈ T ⇐⇒ pSq ∈ A. Lemma D (A Diagonal Lemma) If A∗ is expressible in Σ, then there exists a Gödel sentence for A in Σ. ∗ Proof. Suppose H expresses A ; let h = pHq. Thus d(h) = pH(h)q. H(n) ∈ T ⇐⇒ n ∈ A∗, in particular H(h) ∈ T ⇐⇒ h ∈ A∗. Since h ∈ A∗ ⇐⇒ d(h) ∈ A, H(h) is a Gödel sentence for A. QED. Note: Condition G2 implies that Lemma D can be applied. Lemma D thus straightforwardly implies Theorem GT. P and R are irrelevant for Lemma D. A Gödel sentence of P can be read as “I am unprovable”. 12 An abstract form of Tarski’s Undefinability Theorem Notation: We use T to denote {pSq | S ∈ T }. Theorem T (After Tarski) ∗ (1) T is not expressible in Σ. (2) If G2 holds, then T is not expressible in Σ. (3) If G1 and G2 hold, then T is not expressible in Σ. Proof. First note: There cannot be a Gödel sentence GT for T since GT were true ⇐⇒ pGT q is not the Gödel number of a true sentence, which is clearly is absurd. ∗ (1) If T were expressible in Σ, then by Lemma D, there would be a Gödel sentence GT for T in Σ. ∗ (2) If G2 holds, the expressibility of T would imply that of T . (3) If also G1 holds, then the expressibility of T would imply that of T . Thus we have reduced (3) to (2), (2) to (1), and (1) to the initial observation about the non-existence of a Gödel sentence for T . QED. 13 Back to Gödel: a (more) syntactic form of incompleteness Note: So far R (set of refutable sentences) played no role. Definition (Undecidable sentences, Gödel-incompleteness) Relative to a system Σ, a sentence S is called (formally) undecidable if S 6∈ P and S 6∈ R. Σ is called Gödel-incomplete if it contains formally undecidable sentences. Theorem GI (Gödel-Incompleteness) ∗ If Σ is correct and P is expressible, then Σ is Gödel-incomplete. Definition (Consistency) Σ is consistent if there is no sentence that is both provable and refutable. (I.e., P ∩ R = ∅.) Note: Correctness implies consistency, but not vice versa. To establish purely syntactic incompleteness — consistency =⇒ Gödel-incompleteness — more specific properties are needed (Rosser).
Load more

Recommended publications
  • Infinite Time Computable Model Theory
    INFINITE TIME COMPUTABLE MODEL THEORY JOEL DAVID HAMKINS, RUSSELL MILLER, DANIEL SEABOLD, AND STEVE WARNER Abstract. We introduce infinite time computable model theory, the com- putable model theory arising with infinite time Turing machines, which provide infinitary notions of computability for structures built on the reals R. Much of the finite time theory generalizes to the infinite time context, but several fundamental questions, including the infinite time computable analogue of the Completeness Theorem, turn out to be independent of ZFC. 1. Introduction Computable model theory is model theory with a view to the computability of the structures and theories that arise (for a standard reference, see [EGNR98]). Infinite time computable model theory, which we introduce here, carries out this program with the infinitary notions of computability provided by infinite time Turing ma- chines. The motivation for a broader context is that, while finite time computable model theory is necessarily limited to countable models and theories, the infinitary context naturally allows for uncountable models and theories, while retaining the computational nature of the undertaking. Many constructions generalize from finite time computable model theory, with structures built on N, to the infinitary theory, with structures built on R. In this article, we introduce the basic theory and con- sider the infinitary analogues of the completeness theorem, the L¨owenheim-Skolem Theorem, Myhill’s theorem and others. It turns out that, when stated in their fully general infinitary forms, several of these fundamental questions are independent of ZFC. The analysis makes use of techniques both from computability theory and set theory. This article follows up [Ham05].
    [Show full text]
  • Notes on Incompleteness Theorems
    The Incompleteness Theorems ● Here are some fundamental philosophical questions with mathematical answers: ○ (1) Is there a (recursive) algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is true? ○ (2) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of Peano or Robinson Arithmetic? ○ (3) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of pure (first-order) logic? ○ (4) Is there a complete (even if not recursive) recursively axiomatizable theory in the language of first-order arithmetic? ○ (5) Is there a recursively axiomatizable sub-theory of Peano Arithmetic that proves the consistency of Peano Arithmetic (even if it leaves other questions undecided)? ○ (6) Is there a formula of arithmetic that defines arithmetic truth in the standard model, N (even if it does not represent it)? ○ (7) Is the (non-recursively enumerable) set of truths in the language of first-order arithmetic categorical? If not, is it ω-categorical (i.e., categorical in models of cardinality ω)? ● Questions (1) -- (7) turn out to be linked. Their philosophical interest depends partly on the following philosophical thesis, of which we will make frequent, but inessential, use. ○ Church-Turing Thesis:A function is (intuitively) computable if/f it is recursive. ■ Church: “[T]he notion of an effectively calculable function of positive integers should be identified with that of recursive function (quoted in Epstein & Carnielli, 223).” ○ Note: Since a function is recursive if/f it is Turing computable, the Church-Turing Thesis also implies that a function is computable if/f it is Turing computable.
    [Show full text]
  • LOGIC I 1. the Completeness Theorem 1.1. on Consequences
    LOGIC I VICTORIA GITMAN 1. The Completeness Theorem The Completeness Theorem was proved by Kurt G¨odelin 1929. To state the theorem we must formally define the notion of proof. This is not because it is good to give formal proofs, but rather so that we can prove mathematical theorems about the concept of proof. {Arnold Miller 1.1. On consequences and proofs. Suppose that T is some first-order theory. What are the consequences of T ? The obvious answer is that they are statements provable from T (supposing for a second that we know what that means). But there is another possibility. The consequences of T could mean statements that hold true in every model of T . Do the proof theoretic and the model theoretic notions of consequence coincide? Once, we formally define proofs, it will be obvious, by the definition of truth, that a statement that is provable from T must hold in every model of T . Does the converse hold? The question was posed in the 1920's by David Hilbert (of the 23 problems fame). The answer is that remarkably, yes, it does! This result, known as the Completeness Theorem for first-order logic, was proved by Kurt G¨odel in 1929. According to the Completeness Theorem provability and semantic truth are indeed two very different aspects of the same phenomena. In order to prove the Completeness Theorem, we first need a formal notion of proof. As mathematicians, we all know that a proof is a series of deductions, where each statement proceeds by logical reasoning from the previous ones.
    [Show full text]
  • Arxiv:1902.05902V2 [Math.LO] 5 Aug 2020 Ee Olnrfrsnigm I Rfso 2,2]Pirt P to Prior 28] [27, of by Drafts Comments
    GODEL’S¨ INCOMPLETENESS THEOREM AND THE ANTI-MECHANIST ARGUMENT: REVISITED YONG CHENG Abstract. This is a paper for a special issue of the journal “Studia Semiotyczne” devoted to Stanislaw Krajewski’s paper [30]. This pa- per gives some supplementary notes to Krajewski’s [30] on the Anti- Mechanist Arguments based on G¨odel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4-6 in Kra- jewski’s [30] and classify some misunderstandings of G¨odel’s incomplete- ness theorem related to Anti-Mechanist Arguments. In Section 4 and 5, we give a more detailed discussion of G¨odel’s Disjunctive Thesis, G¨odel’s Undemonstrability of Consistency Thesis and the definability of natu- ral numbers as in Section 7-8 in Krajewski’s [30], describing how recent advances bear on these issues. 1. Introduction G¨odel’s incompleteness theorem is one of the most remarkable and pro- found discoveries in the 20th century, an important milestone in the history of modern logic. G¨odel’s incompleteness theorem has wide and profound influence on the development of logic, philosophy, mathematics, computer science and other fields, substantially shaping mathematical logic as well as foundations and philosophy of mathematics from 1931 onward. The im- pact of G¨odel’s incompleteness theorem is not confined to the community of mathematicians and logicians, and it has been very popular and widely used outside mathematics. G¨odel’s incompleteness theorem raises a number of philosophical ques- tions concerning the nature of mind and machine, the difference between human intelligence and machine intelligence, and the limit of machine intel- ligence.
    [Show full text]
  • Interpreting True Arithmetic in the Local Structure of the Enumeration Degrees
    INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES. HRISTO GANCHEV AND MARIYA SOSKOVA 1. Introduction Degree theory studies mathematical structures, which arise from a formal no- tion of reducibility between sets of natural numbers based on their computational strength. In the past years many reducibilities, together with their induced de- gree structures have been investigated. One of the aspects in these investigations is always the characterization of the theory of the studied structure. One distin- guishes between global structures, ones containing all possible degrees, and local structures, containing all degrees bounded by a fixed element, usually the degree which contains the halting set. It has become apparent that modifying the under- lying reducibility does not influence the strength of the first order theory of either the induced global or the induced local structure. The global first order theories of the Turing degrees [15], of the many-one degrees [11], of the 1-degrees [11] are com- putably isomorphic to second order arithmetic. The local first order theories of the 1 0 computably enumerable degrees , of the many-one degrees [12], of the ∆2 Turing degrees [14] are computably isomorphic to the theory of first order arithmetic. In this article we consider enumeration reducibility, and the induced structure of the enumeration degrees. Enumeration reducibility introduced by Friedberg and Rogers [4] arises as a way to compare the computational strength of the positive information contained in sets of naturals numbers. A set A is enumeration reducible to a set B if given any enumeration of the set B, one can effectively compute an enumeration of the set A.
    [Show full text]
  • Gödel Incompleteness Revisited Grégory Lafitte
    Gödel incompleteness revisited Grégory Lafitte To cite this version: Grégory Lafitte. Gödel incompleteness revisited. JAC 2008, Apr 2008, Uzès, France. pp.74-89. hal-00274564 HAL Id: hal-00274564 https://hal.archives-ouvertes.fr/hal-00274564 Submitted on 18 Apr 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journees´ Automates Cellulaires 2008 (Uzes),` pp. 74-89 GODEL¨ INCOMPLETENESS REVISITED GREGORY LAFITTE 1 1 Laboratoire d’Informatique Fondamentale de Marseille (LIF), CNRS – Aix-Marseille Universit´e, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France E-mail address: [email protected] URL: http://www.lif.univ-mrs.fr/~lafitte/ Abstract. We investigate the frontline of G¨odel’sincompleteness theorems’ proofs and the links with computability. The G¨odelincompleteness phenomenon G¨odel’sincompleteness theorems [G¨od31,SFKM+86] are milestones in the subject of mathematical logic. Apart from G¨odel’soriginal syntactical proof, many other proofs have been presented. Kreisel’s proof [Kre68] was the first with a model-theoretical flavor. Most of these proofs are attempts to get rid of any form of self-referential reasoning, even if there remains diagonalization arguments in each of these proofs.
    [Show full text]
  • Empiricism, Probability, and Knowledge of Arithmetic: a Preliminary Defense
    Empiricism, Probability, and Knowledge of Arithmetic: A Preliminary Defense Sean Walsh Department of Logic and Philosophy of Science 5100 Social Science Plaza University of California, Irvine Irvine, CA 92697-5100 U.S.A. Abstract The topic of this paper is our knowledge of the natural numbers, and in par- ticular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemol- ogy, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege’s concern about indiscernibility and circu- larity (§ 2.1), while other are more recent, such as Baker’s concern about the unreliability of small samplings in the setting of arithmetic (§ 2.2). Another family of objections suggests that we simply do not have access to probabil- ity assignments in the setting of arithmetic, either due to issues related to the ω-rule (§ 3.1) or to the non-computability and non-continuity of prob- ability assignments (§ 3.2). Articulating these objections and the responses to them involves developing some non-trivial results on probability assign- ments (Appendix A-Appendix C), such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated (Theorem 4 Appendix B).
    [Show full text]
  • Beyond First Order Logic: from Number of Structures to Structure of Numbers Part I
    Bulletin of the Iranian Mathematical Society Vol. XX No. X (201X), pp XX-XX. BEYOND FIRST ORDER LOGIC: FROM NUMBER OF STRUCTURES TO STRUCTURE OF NUMBERS PART I JOHN BALDWIN, TAPANI HYTTINEN AND MEERI KESÄLÄ Communicated by Abstract. The paper studies the history and recent developments in non-elementary model theory focusing in the framework of ab- stract elementary classes. We discuss the role of syntax and seman- tics and the motivation to generalize first order model theory to non-elementary frameworks and illuminate the study with concrete examples of classes of models. This first part introduces the main conceps and philosophies and discusses two research questions, namely categoricity transfer and the stability classification. 1. Introduction Model theory studies classes of structures. These classes are usually a collection of structures that satisfy an (often complete) set of sentences of first order logic. Such sentences are created by closing a family of basic relations under finite conjunction, negation and quantification over individuals. Non-elementary logic enlarges the collection of sentences by allowing longer conjunctions and some additional kinds of quantification. In this paper we first describe for the general mathematician the history, key questions, and motivations for the study of non-elementary logics and distinguish it from first order model theory. We give more detailed examples accessible to model theorists of all sorts. We conclude with MSC(2010): Primary: 65F05; Secondary: 46L05, 11Y50. Keywords: mathematical logic, model theory. Received: 30 April 2009, Accepted: 21 June 2010. ∗Corresponding author c 2011 Iranian Mathematical Society. 1 2 Baldwin, Hyttinen and Kesälä questions about countable models which require only a basic background in logic.
    [Show full text]
  • 1610.0216V1.Pdf
    TOPOLOGY OF THE BRAIN FUNCTION A summary of our published and unpublished papers Arturo Tozzi Center for Nonlinear Science, University of North Texas 1155 Union Circle, #311427 Denton, TX 76203-5017, USA, and Computational Intelligence Laboratory, University of Manitoba, Winnipeg, Canada Winnipeg R3T 5V6 Manitoba ASL NA2 Nord [email protected] James F. Peters Department of Electrical and Computer Engineering, University of Manitoba 75A Chancellor’s Circle, Winnipeg, MB R3T 5V6, Canada and Department of Mathematics, Adıyaman University, 02040 Adıyaman, Turkey, Department of Mathematics, Faculty of Arts and Sciences, Adıyaman University 02040 Adıyaman, Turkey and Computational Intelligence Laboratory, University of Manitoba, WPG, MB, R3T 5V6, Canada [email protected] This manuscript encompasses our published and unpublished topological results in neuroscience. Topology, the mathematical branch that assesses objects and their properties preserved through deformations, stretching and twisting, allows the investigation of the most general brain features. In particular, the Borsuk-Ulam Theorem (BUT) states that, if a single point projects to a higher spatial dimension, it gives rise to two antipodal points with matching description. Physical and biological counterparts of BUT and its variants allow an inquiry of the brain activity. The opportunity to treat the nervous system as a topological structure makes BUT a universal principle underlying neural phenomena and brain function. CONTENT The fourth dimension of brain activity and novel correlated neurotechniques 1) Tozzi A, Peters JF. 2016. Towards a Fourth Spatial Dimension of Brain Activity. Cognitive Neurodynamics 10 (3): 189–199. doi:10.1007/s11571-016-9379-z. 2) Peters JF, Tozzi A. Ramanna S. 2016. Brain Tissue Tessellation Shows Absence of Canonical Microcircuits.
    [Show full text]
  • The Gödel Incompleteness Phenomenon
    Journees´ Automates Cellulaires 2008 (Uzes),` pp. 74-89 GODEL¨ INCOMPLETENESS REVISITED GREGORY LAFITTE 1 1 Laboratoire d'Informatique Fondamentale de Marseille (LIF), CNRS { Aix-Marseille Universit´e, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France E-mail address: [email protected] URL: http://www.lif.univ-mrs.fr/~lafitte/ Abstract. We investigate the frontline of G¨odel'sincompleteness theorems' proofs and the links with computability. The G¨odelincompleteness phenomenon G¨odel'sincompleteness theorems [G¨od31,SFKM+86] are milestones in the subject of mathematical logic. Apart from G¨odel'soriginal syntactical proof, many other proofs have been presented. Kreisel's proof [Kre68] was the first with a model-theoretical flavor. Most of these proofs are attempts to get rid of any form of self-referential reasoning, even if there remains diagonalization arguments in each of these proofs. The reason for this quest holds in the fact that the diagonalization lemma, when used as a method of constructing an independent statement, is intuitively unclear. Boolos' proof [Boo89b] was the first attempt in this direction and gave rise to many other attempts. Sometimes, it unfortunately sounds a bit like finding a way to sweep self-reference under the mathematical rug. One of these attempts has been to prove the incompleteness theorems using another paradox than the Richard and the Liar paradoxes. It is interesting to note that, in his famous paper announcing the incompleteness theorem, G¨odelremarked that, though his argument is analogous to the Liar paradox, \Any epistemological antinomy could be used for a similar proof of the existence of undecidable propositions".
    [Show full text]
  • Kurt G\U00f6del and the Foundations of Mathematics: Horizons of Truth
    This page intentionally left blank Kurt Godel¨ and the Foundations of Mathematics Horizons of Truth This volume commemorates the life, work, and foundational views of Kurt Godel¨ (1906–1978), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency – with the other widely accepted axioms of set theory – of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances, and ideas for future directions not only in the foundations of mathematics and logic but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology, and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Godel¨ personally, providing some interesting insights into his life. By putting his ideas and life’s work into the context of current thinking and perceptions, this book will extend the impact of Godel’s¨ fundamental work in mathematics, logic, philosophy, and other disciplines for future generations of researchers. Matthias Baaz is currently University Professor and Head of the Group for Computational Logic at the Institute of Discrete Mathematics and Geometry at the Vienna University of Technology. Christos H. Papadimitriou is C. Lester Hogan Professor of Electrical Engineering and Computer Sciences at the University of California, Berkeley, where he has taught since 1996 and where he is a former Miller Fellow. Hilary W. Putnam is Cogan University Professor Emeritus in the Department of Philosophy at Harvard University. Dana S. Scott is Hillman University Professor Emeritus of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University in Pittsburgh.
    [Show full text]
  • Arxiv:2009.04887V2 [Math.LO] 14 Dec 2020 ...Telmtof Limit the 4.3.2
    Contents 1. Introduction 2 2. Preliminaries 4 2.1. Definitions and notations 4 2.2. Logical systems 7 3. Proofs of G¨odel’s incompleteness theorems 11 3.1. Introduction 11 3.2. Overview and modern formulation 12 3.2.1. Three steps towards G1 and G2 12 3.2.2. Properties of G1 14 3.2.3. Between truth and provability 15 3.2.4. Properties of G2 17 3.3. Proofs of G1 and G2 from mathematical logic 18 3.3.1. Rosser’s proof 18 3.3.2. Recursion-theoretic proofs 19 3.3.3. Proofs based on Arithmetic Completeness 19 3.3.4. Proofs based on Kolmogorov complexity 21 3.3.5. Model-theoretic proofs 22 3.4. Proofs of G1 and G2 based on logical paradox 22 3.4.1. Introduction 22 3.4.2. Berry’s paradox 22 3.4.3. Unexpected Examination and Grelling-Nelson’s Paradox. 24 3.4.4. Yablo’s paradox 25 3.4.5. Beyond arithmetization 26 3.5. Concrete incompleteness 26 3.5.1. Introduction 26 3.5.2. Paris-Harrington and beyond 27 3.5.3. Harvey Friedman’s contributions 28 4. The limit of the applicability of G1 29 4.1. Introduction 29 4.2. Generalizations of G1 beyond PA 30 4.3. Generalizations of G1 below PA 31 4.3.1. Generalizations of G1 via interpretability 31 4.3.2. The limit of G1 w.r.t. interpretation and Turing reducibility 34 5. The limit of the applicability of G2 35 5.1. Introduction 35 arXiv:2009.04887v2 [math.LO] 14 Dec 2020 5.2.
    [Show full text]