DOCSLIB.ORG

Godel's " "On Formally Undecidable Propositions ••• " •

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Godel's CALIJ:i'ORNIA STATE UNIVERSITY, NORTHRIDGE GODEL'S " "ON FORMALLY UNDECIDABLE PROPOSITIONS ••• " • A thesis submitted in partial satisfaction of the requirements for the Degree of Master of Sciences in Mathematics, Option II by Mary Margaret Pepe June, 1976 Pepe is approved: Date ' Dr. Donald H. Potts Date yalifornia State University, Northridge ii PREFACE Any formal system of arithmetic encompassing the addition and multi­ plication of positive integers and zero contains arithmetical proposi­ :tions which can neither be proved or disproved within the system. Kurt Godel, a young Austrian mathematician who had imigrated to the United States and became a member of the Institute for Advanced Study at Princeton, announced his discovery to the Vienna Academy of Sciences ·in 1930. He published his detailed proof in his paper, "Uber formal ,unentscheidbare Satze der Principia Mathematica und verwandter Systeme ,I," in the Monatshefte Fiir Mathematik und Physik, Volume 38, pages 173- :198 (Leipzig, 1931). Godel intended to write a second part to his paper, but illness prevented him from publishing it. Godel's theorem is sometimes regarded as the most decisive result 1 in mathematical logic. In the paper which follows we shall familiarize ;ourselves with the mathematics (metamathematics) required to understand •this theorem. iii TABLE OF CONTENTS Page(s) PREFACE iii TABLE OF CONTENTS iv-v ABSTRACT vi-vii I. INTRODUCTION l A. THE STATE OF MATHEMATICS @ 1930 l B. THE CREATION OF METAMATHEMATICS - 3 PROOF THEORY II. SUPPORTING DETAILS FOR THE PROOF 8 A. GODEL NUMBERING - ARITHMETIZATION OF 8 METAMATH11MATICS B. SELF-REFERENTIAL STATEMENTS 15 1. Formulation of the Richard Paradox 15 2. The Richard Paradox 16 3. Resolving the Paradox 17 C. THE FORMAL SYSTEM 19 1. Basic Symbols 20 2. Well-Formed Formulas (WE'F' s) 21 3· Axioms (Initial Formulae) 23 4. "Immediate Consequence of" 24 D. PRIMITIVE RECURSIVE FUNCTIONS 24 1. Significance 24 2. Definitions 25 3. Definition by Induction - Recursive 27 Definition 4. Primitive Recursive 28 iv TABLE OF CONTENTS ~age(s) 5. Examples of Primitive Recursive Functions 31 6. Historical Utilization 33 III. GODEL' s PROOF 34 A. STATEMENT OF THE THEOruMS 34 B. TARSKI' S EXAMPLE :;4 C. GOOEL 1 S ASSUMPTIONS 35 D. LEMMA 1 38 E. DEFINITION: ¢ (x,y) 38 F. EXPRESSIBILITY 39 G. "Bew(x)" 39 H. LEMMA 2 4o I. UMMA 3 4o J. GODEL Is SECOND THEOREM 41 K. CONCLUSIONS 41 IV. EPILOGUE - TRUTH IN MATHEMATICS 43 ANNOTATED BIBLIOGRAPHY 45 INDEX OF DEFINITIONS 48 Photograph of Kurt GOdel by Orren J. Turner viii •rables: Table 1 - Godel Numbers of Constant Signs 9 Table 2 - Godel Numbers of Type 1 Variables 9 Table 3 - Godel Numbers of Type 2 Variables 10 Table 4 - Godel Numbers of Type 3 Variables 10 v ABSTRACT GODEL'S 11 0N FORMALLY UNDECIDABLE PROPOSI'l'IOOS ••• 11 by Mary Margaret Pepe Master of Sciences in Mathematics, Option II Godel's theorem is composed of two parts: Theorem 1: For suitable formal systems L, there are undecidable pro+ I positions in L; that is propositions F such that neither F nor not F is provable. Theorem 2: For suitable L, the simple consistency of L cannot be proved in L. Part One of the thesis examines the state of mathematics @ 1930 and !the creation of proof theory - metamathematics. I I Part Two presents the supporting details for the proot including: l) GOdel numbering - the establishing of a one-to-one correspon~ dence between the finite expressions of the system and the natural numbers - is presented. I 2) Belt-referential statements - statements which talk about them The Richard Paradox is formulated, I selves - are described. l vi examined, and resolved. -----· ··-· --~ 3) The Formal System L is described - its basic signs, well-formet· formulas, axioms, and the relation "Immediate consequence of." 4) Primitive Recursive Functions are defined, examples shown, and historical utilization described. Part Three presents a Rosser version of the proof of theorems 1 and 2 stated above. Reference is made to primitive recursive functions and numeralwise expressibility. Part Four, an epilogue, considers ramifications regarding truth in mathematics. An index of definitions and an annotated bibliography follow the thesis. ____ _jI vii I. INTRODUCTION 1 I A. THE STATE OF MATHEMATICS @ 19.30 At the second mathematical congress held in Paris in 1900, David I ! Hilbert posed twenty-three problems which he thought would be among I those occupying the attention of the twentieth-century mathematicians. I ! This paper is in partan answer to Hilbert's second query as to whether: the axioms of arithmetic are consistent. I I Before we continue we need to have a common understanding of what i is meant by consistent, complete, incomplete, decidable, undecidable, I provable, and disprovable. We will give naive definitiorJs and when I I necessary make these definitions more explicit when we an9.lyze the I metamathematics of the formal system. I Con:::t:::tem of axioms is said to be consistent if a finite number of I logical steps based on the axioms would never lead to a contradiction. ~.~! I Inconsistent Not consistent. I Complete I in :~:·:::t:: :~: ::g::.:o::::::u::c:v::yt::-:.:::::ment expressible I Incomplete j A system is said to be incomplete if ?£t eve!Y tru~ statement of the! system is deducible from the axioms. Decidable A formula P written in the vocabulary of a given set of axioms is (1) (2) said to be decidable if there is an effective method (see page 5) for determining whether P or not P is deducible from the axioms. Undecidable A formula P is said to be undecidable if neither P nor not P is a theorem. Provable A formula P is provable, i.e., is a theorelii, iff there is a formal proof of the formula ending in P. A formal proof of a formula P is a finite column of formulas each of whose lines is an axiom or may be inferred from the preceding lines by specified rules of inference (such as modus ponE:ms : A, A--") B, ;. B). Unprovable Cannot be proved. Bertrand Russell and Alfred North Whitehead (1861-1947) had made an elaborate e.ttempt to develop the fundame.ntal notions of arithmetic / from a precise set of axioms. Their text was in the tradition of Leibniz, Boole, and Frege and was based on Peano's axioms. Russell and Whitehead intended to prove that all of pure mathematics could be derived from a small number of fundamen-tal logical principles. If they had succeeded their proof would have supported Russell's view that mathematics is indistinguishable from logic. In 1926 Finsler published his paper "Formal proofs and undecid- ability." Finsler presented a proposition that although false is formally undecidable. In 1930 Presburger published a decision pro- cedure applicable to every proposition of a system of arithmetic which used only addition, but not multiplication. Presburger proved that every one of its propositions is decidable - either provable or unprovable. Hilbert had hoped to prove the consistency of elementary number theory with the possibility of reducing other portions of classical mathematics to that of the natural numbers via models. Work along these lines carne to somewhat of a halt in 1931 when GOdel demonstrated the impossibility of proving the consistency of I any formal theory by constructive methods formalizable within the theory itself. This includes the formulas of the Natural numbers. Godel proved that if the fot'mal .system is consistent then it is incomplete. Nineteenth-century mathematicians had hoped that the axiomc; for ari thmetio· were complete or could be made complete by the .. addition of a finite number of further axioms. ~1e discovery that this.·. was not so was more of a shock than Godel 1 s demonstrating the existence of undecidable propositions. B. THE CREATION OF METAMATHEMATICS - PROOF THEORY Metamathematics, which is the study of rigorous proof in mathe­ matics, was developed by Hilbert during the 1920's-1930's. Metamath­ ematics looks at mathematics from the outside. It is concern~;d with the interpretations of signs and rules and not with the operatlons of arithmetic. Godel's theorem is a result which belongs to metamathematics. Godel put the notion of a formal system at the very center of his investigations~ We thus need to define the following formal notions: (4) Formal When we use the word formal we are referring only to the forms of certain expressions and not to their meaning. •Formal Theory A formal theory is a completely symbolic language built according 'to certain rules from an alphabet of specified primitive symbols. String A finite sequence of formal symbols. Object Language The symbolic ( f\)nnal) language in which the statements of the formal! i :theory are written. A statement in the object language is a statement !.2! the theory. I I I iHetalanguage - Syntax Language i I The language used to present the formal theory. A statement in i the metalanguage is a statement about the theory. It is used to discus$ i the fonnal theory, which includes defining its syntax, specifying its .axioms and rules of inference, and analyzing its properties. Meta theorem A theorem about a formal theory (written in English). A theorem of the theory is written jt1 the symbolism of the theory. A metatheorem 'is a statement that is proved in the metalanguage (English) about the· theory. 'Metatheorz Metatheory is the theory of formal languages and systems and their interpretations. It takes formal languages and systems and their in- :terpretations as objects of study, and consists in the body of truths (5) and conjectures about these objects. Among its main problems are problems about the consistency, completeness, decidability and i!!,­ _?ependence of sets of formulas.
Load more

Recommended publications
  • Epistemological Consequences of the Incompleteness Theorems
    Epistemological Consequences of the Incompleteness Theorems Giuseppe Raguní UCAM - Universidad Católica de Murcia, Avenida Jerónimos 135, Guadalupe 30107, Murcia, Spain - [email protected] After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main episte- mological consequences of the first incompleteness Theorem for the two funda- mental arithmetical theories are shown: the non-mechanizability for the truths of the first-order arithmetic and the peculiarities for the model of the second- order arithmetic. Finally, the common epistemological interpretation of the second incompleteness Theorem is corrected, proposing the new Metatheorem of undemonstrability of internal consistency. KEYWORDS: semantics, languages, epistemology, paradoxes, arithmetic, in- completeness, first-order, second-order, consistency. 1 Semantics in the Languages Consider an arbitrary language that, as normally, makes use of a countable1 number of characters. Combining these characters in certain ways, are formed some fundamental strings that we call terms of the language: those collected in a dictionary. When the terms are semantically interpreted, i. e. a certain meaning is assigned to them, we have their distinction in adjectives, nouns, verbs, etc. Then, a proper grammar establishes the rules arXiv:1602.03390v1 [math.GM] 13 Jan 2016 of formation of sentences. While the terms are finite, the combinations of grammatically allowed terms form an infinite-countable amount of possible sentences. In a non-trivial language, the meaning associated to each term, and thus to each ex- pression that contains it, is not always unique. The same sentence can enunciate different things, so representing different propositions.
    [Show full text]
  • John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected]
    John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected] LOGIC & PHILOSOPHICAL METHODOLOGY Introduction For present purposes “logic” will be understood to mean the subject whose development is described in Kneale & Kneale [1961] and of which a concise history is given in Scholz [1961]. As the terminological discussion at the beginning of the latter reference makes clear, this subject has at different times been known by different names, “analytics” and “organon” and “dialectic”, while inversely the name “logic” has at different times been applied much more broadly and loosely than it will be here. At certain times and in certain places — perhaps especially in Germany from the days of Kant through the days of Hegel — the label has come to be used so very broadly and loosely as to threaten to take in nearly the whole of metaphysics and epistemology. Logic in our sense has often been distinguished from “logic” in other, sometimes unmanageably broad and loose, senses by adding the adjectives “formal” or “deductive”. The scope of the art and science of logic, once one gets beyond elementary logic of the kind covered in introductory textbooks, is indicated by two other standard references, the Handbooks of mathematical and philosophical logic, Barwise [1977] and Gabbay & Guenthner [1983-89], though the latter includes also parts that are identified as applications of logic rather than logic proper. The term “philosophical logic” as currently used, for instance, in the Journal of Philosophical Logic, is a near-synonym for “nonclassical logic”. There is an older use of the term as a near-synonym for “philosophy of language”.
    [Show full text]
  • Truth-Conditions
    PLIN0009 Semantic Theory Spring 2020 Lecture Notes 1 1 What this module is about This module is an introduction to truth-conditional semantics with a focus on two impor- tant topics in this area: compositionality and quantiication. The framework adopted here is often called formal semantics and/or model-theoretical semantics, and it is characterized by its essential use of tools and concepts developed in mathematics and logic in order to study semantic properties of natural languages. Although no textbook is required, I list some introductory textbooks below for your refer- ence. • L. T. F. Gamut (1991) Logic, Language, and Meaning. The University of Chicago Press. • Irene Heim & Angelika Kratzer (1998) Semantics in Generative Grammar. Blackwell. • Thomas Ede Zimmermann & Wolfgang Sternefeld (2013) Introduction to Semantics: An Essential Guide to the Composition of Meaning. De Gruyter Mouton. • Pauline Jacobson (2014) Compositional Semantics: An Introduction to the Syntax/Semantics. Oxford University Press. • Daniel Altshuler, Terence Parsons & Roger Schwarzschild (2019) A Course in Semantics. MIT Press. There are also several overview articles of the ield by Barbara H. Partee, which I think are enjoyable. • Barbara H. Partee (2011) Formal semantics: origins, issues, early impact. In Barbara H. Partee, Michael Glanzberg & Jurģis Šķilters (eds.), Formal Semantics and Pragmatics: Discourse, Context, and Models. The Baltic Yearbook of Cognition, Logic, and Communica- tion, vol. 6. Manhattan, KS: New Prairie Press. • Barbara H. Partee (2014) A brief history of the syntax-semantics interface in Western Formal Linguistics. Semantics-Syntax Interface, 1(1): 1–21. • Barbara H. Partee (2016) Formal semantics. In Maria Aloni & Paul Dekker (eds.), The Cambridge Handbook of Formal Semantics, Chapter 1, pp.
    [Show full text]
  • Solving the Boolean Satisfiability Problem Using the Parallel Paradigm Jury Composition
    Philosophæ doctor thesis Hoessen Benoît Solving the Boolean Satisfiability problem using the parallel paradigm Jury composition: PhD director Audemard Gilles Professor at Universit´ed'Artois PhD co-director Jabbour Sa¨ıd Assistant Professor at Universit´ed'Artois PhD co-director Piette C´edric Assistant Professor at Universit´ed'Artois Examiner Simon Laurent Professor at University of Bordeaux Examiner Dequen Gilles Professor at University of Picardie Jules Vernes Katsirelos George Charg´ede recherche at Institut national de la recherche agronomique, Toulouse Abstract This thesis presents different technique to solve the Boolean satisfiability problem using parallel and distributed architec- tures. In order to provide a complete explanation, a careful presentation of the CDCL algorithm is made, followed by the state of the art in this domain. Once presented, two proposi- tions are made. The first one is an improvement on a portfo- lio algorithm, allowing to exchange more data without loosing efficiency. The second is a complete library with its API al- lowing to easily create distributed SAT solver. Keywords: SAT, parallelism, distributed, solver, logic R´esum´e Cette th`ese pr´esente diff´erentes techniques permettant de r´esoudre le probl`eme de satisfaction de formule bool´eenes utilisant le parall´elismeet du calcul distribu´e. Dans le but de fournir une explication la plus compl`ete possible, une pr´esentation d´etaill´ee de l'algorithme CDCL est effectu´ee, suivi d'un ´etatde l'art. De ce point de d´epart,deux pistes sont explor´ees. La premi`ereest une am´eliorationd'un algorithme de type portfolio, permettant d'´echanger plus d'informations sans perte d’efficacit´e.
    [Show full text]
  • Lecture 1: Tarski on Truth Philosophy of Logic and Language — HT 2016-17
    Lecture 1: Tarski on Truth Philosophy of Logic and Language — HT 2016-17 Jonny McIntosh [email protected] Alfred Tarski (1901-1983) was a Polish (and later, American) mathematician, logician, and philosopher.1 In the 1930s, he published two classic papers: ‘The Concept of Truth in Formalized Languages’ (1933) and ‘On the Concept of Logical Consequence’ (1936). He gives a definition of truth for formal languages of logic and mathematics in the first paper, and the essentials of the model-theoretic definition of logical consequence in the second. Over the course of the next few lectures, we’ll look at each of these in turn. 1 Background The notion of truth seems to lie at the centre of a range of other notions that are central to theorising about the formal languages of logic and mathematics: e.g. validity, con- sistency, and completeness. But the notion of truth seems to give rise to contradiction: Let sentence (1) = ‘sentence (1) is not true’. Then: 1. ‘sentence (1) is not true’ is true IFF sentence (1) is not true 2. sentence (1) = ‘sentence (1) is not true’ 3. So, sentence (1) is true IFF sentence (1) is not true 4. So, sentence (1) is true and sentence (1) is not true Tarski is worried that, unless the paradox can be resolved, metatheoretical results in- voking the notion of truth or other notions that depend on it will be remain suspect. But how can it be resolved? The second premise is undeniable, and the first premise is an instance of a schema that seems central to the concept of truth, namely the following: ‘S’ is true IFF S, 1The eventful story of Tarski’s life is told by Anita and Solomon Feferman in their wonderful biography, Alfred Tarski: Life and Logic (CUP, 2004).
    [Show full text]
  • Conceptual Semantics and Natural Semantic Metalanguage Theory 413
    1 Conceptual semantics and natural semantic 2 metalanguage theory have different goals 3 4 5 RAY JACKENDOFF 6 7 8 9 10 11 12 My interview with Istvan Kecskes, ‘‘On Conceptual Semantics (OCS),’’ 13 o¤ers some brief remarks on the relation of my approach to Anna Wierz- 14 bicka’s theory of semantic primitives, Natural Language Metalanguage 15 Theory (NSM). Wierzbicka’s response, ‘‘Theory and Empirical Findings: 16 A Response to Jackendo¤’’ (henceforth TEF) calls for some further 17 commentary. 18 My remarks began, ‘‘Although I think Wierzbicka has o¤ered many in- 19 sightful discussions of word meaning, I do not think that her approach 20 ultimately responds to the goals of Conceptual Semantics.’’ TEF makes 21 it clear that Conceptual Semantics does not respond to the goals of 22 NSM either. That is, the two theories, although they overlap on some 23 issues of word meaning, are ultimately asking di¤erent questions and set- 24 ting di¤erent standards for answers. 25 Conceptual Semantics is concerned not only with encoding word mean- 26 ings but also with accounting for (a) the combination of word meanings 27 into phrase and sentence meanings, (b) the character of inference, both 28 logical and heuristic, and (c) the relation of linguistic meaning to nonlin- 29 guistic understanding of the world, including the aspects of understanding 30 provided through the perceptual systems. Conceptual Semantics in turn 31 is embedded in a theory of the organization of language, the Parallel Ar- 32 chitecture (Jackendo¤ 2002, Culicover and Jackendo¤ 2005), which ad- 33 dresses the relation of semantics to syntax, phonology, and the lexicon.
    [Show full text]
  • The Strength of Mac Lane Set Theory
    The Strength of Mac Lane Set Theory A. R. D. MATHIAS D´epartement de Math´ematiques et Informatique Universit´e de la R´eunion To Saunders Mac Lane on his ninetieth birthday Abstract AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and S Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke{Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening KPP + AC of KP + MAC, and the Forster{Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and KPP ; we show, again using ill-founded models, that KPP + V = L proves the consistency of KPP ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret
    [Show full text]
  • Metalanguage of Race”
    608 y Symposium: “The Metalanguage of Race” Difference, Power, and Lived Experiences: Revisiting the “Metalanguage of Race” Dayo F. Gore first encountered Evelyn Brooks Higginbotham’s “African-American I Women’s History and the Metalanguage of Race” (1992) as a graduate student in a women’s history course in 1996, and the article has traveled with me ever since. I have assigned the article in at least one of my courses every year, even as I’ve taught across a number of fields, including African American history, feminist theory, and social movement history. Moreover, it has been one of the key theoretical texts I have employed to frame my own research and writing. Although I came to the article after it had been in circulation for four years, it resonated with me as a powerful and much- needed intervention in feminist theory and women’shistory.Indeed,“Meta- language” stands as one of those rare articles that continues to be relevant in ongoing debates about gender, sexuality, and race, and serves as a useful guidepost for my own scholarship. Higginbotham’s article provides an incisive response to debates among a broad spectrum of feminist scholars about how to be attentive to gender across difference and differently lived experiences, and it represented an im- portant addition to the theorization of what would become popularly known as intersectionality. Black feminist legal scholar Kimberlé Crenshaw is most often credited with coining the term “intersectionality” as she details it in two key articles, “Demarginalizing the Intersection of Race and Sex,” pub- lished in 1989, and “Mapping the Margins,” a law review article published in 1991.
    [Show full text]
  • Propositional Calculus
    CSC 438F/2404F Notes Winter, 2014 S. Cook REFERENCES The first two references have especially influenced these notes and are cited from time to time: [Buss] Samuel Buss: Chapter I: An introduction to proof theory, in Handbook of Proof Theory, Samuel Buss Ed., Elsevier, 1998, pp1-78. [B&M] John Bell and Moshe Machover: A Course in Mathematical Logic. North- Holland, 1977. Other logic texts: The first is more elementary and readable. [Enderton] Herbert Enderton: A Mathematical Introduction to Logic. Academic Press, 1972. [Mendelson] E. Mendelson: Introduction to Mathematical Logic. Wadsworth & Brooks/Cole, 1987. Computability text: [Sipser] Michael Sipser: Introduction to the Theory of Computation. PWS, 1997. [DSW] M. Davis, R. Sigal, and E. Weyuker: Computability, Complexity and Lan- guages: Fundamentals of Theoretical Computer Science. Academic Press, 1994. Propositional Calculus Throughout our treatment of formal logic it is important to distinguish between syntax and semantics. Syntax is concerned with the structure of strings of symbols (e.g. formulas and formal proofs), and rules for manipulating them, without regard to their meaning. Semantics is concerned with their meaning. 1 Syntax Formulas are certain strings of symbols as specified below. In this chapter we use formula to mean propositional formula. Later the meaning of formula will be extended to first-order formula. (Propositional) formulas are built from atoms P1;P2;P3;:::, the unary connective :, the binary connectives ^; _; and parentheses (,). (The symbols :; ^ and _ are read \not", \and" and \or", respectively.) We use P; Q; R; ::: to stand for atoms. Formulas are defined recursively as follows: Definition of Propositional Formula 1) Any atom P is a formula.
    [Show full text]
  • Carnap's Contribution to Tarski's Truth
    JOURNAL FOR THE HISTORY OF ANALYTICAL PHILOSOPHY CARNAP’S CONTRIBUTION TO TARSKI’S TRUTH VOLUME 3, NUMBER 10 MONIKA GrUBER EDITOR IN CHIEF KEVIN C. KLEMENt, UnIVERSITY OF MASSACHUSETTS In his seminal work “The Concept of Truth in Formalized Lan- guages” (1933), Alfred Tarski showed how to construct a for- EDITORIAL BOARD mally correct and materially adequate definition of true sentence GaRY EBBS, INDIANA UnIVERSITY BLOOMINGTON for certain formalized languages. These results have, eventu- GrEG FROSt-ARNOLD, HOBART AND WILLIAM SMITH COLLEGES ally, been accepted and applauded by philosophers and logi- HENRY JACKMAN, YORK UnIVERSITY cians nearly in unison. Its Postscript, written two years later, SANDRA LaPOINte, MCMASTER UnIVERSITY however, has given rise to a considerable amount of contro- LyDIA PATTON, VIRGINIA TECH versy. There is an ongoing debate on what Tarski really said MARCUS ROSSBERG, UnIVERSITY OF CONNECTICUT in the postscript. These discussions often regard Tarski as pu- MARK TEXTOR, KING’S COLLEGE LonDON tatively changing his logical framework from type theory to set AUDREY YAP, UnIVERSITY OF VICTORIA theory. RICHARD ZACH, UnIVERSITY OF CALGARY In what follows, we will compare the original results with those REVIEW EDITORS presented two years later. After a brief outline of Carnap’s pro- JULIET FLOYD, BOSTON UnIVERSITY gram in The Logical Syntax of Language we will determine its sig- CHRIS PINCOCK, OHIO STATE UnIVERSITY nificance for Tarski’s final results. ASSISTANT REVIEW EDITOR SEAN MORRIS, METROPOLITAN STATE UnIVERSITY OF DenVER DESIGN DaNIEL HARRIS, HUNTER COLLEGE C 2015 MONIKA GrUBER CARNAP’S CONTRIBUTION TO TARSKI’S TRUTH couple of years ago. In particular, in addition to the previously studied languages, he decides to investigate the languages the MONIKA GrUBER structure of which cannot be brought into harmony with the principles of the theory of semantical categories.
    [Show full text]
  • CCAM Systematic Satisfiability Programming in Hopfield Neural
    Communications in Computational and Applied Mathematics, Vol. 2 No. 1 (2020) p. 1-6 CCAM Communications in Computational and Applied Mathematics Journal homepage : www.fazpublishing.com/ccam e-ISSN : 2682-7468 Systematic Satisfiability Programming in Hopfield Neural Network- A Hybrid Expert System for Medical Screening Mohd Shareduwan Mohd Kasihmuddin1, Mohd. Asyraf Mansor2,*, Siti Zulaikha Mohd Jamaludin3, Saratha Sathasivam4 1,3,4School of Mathematical Sciences, Universiti Sains Malaysia, Minden, Pulau Pinang, Malaysia 2School of Distance Education, Universiti Sains Malaysia, Minden, Pulau Pinang, Malaysia *Corresponding Author Received 26 January 2020; Abstract: Accurate and efficient medical diagnosis system is crucial to ensure patient with recorded Accepted 12 February 2020; system can be screened appropriately. Medical diagnosis is often challenging due to the lack of Available online 31 March patient’s information and it is always prone to inaccurate diagnosis. Medical practitioner or 2020 specialist is facing difficulties in screening the disease accurately because unnecessary attributes will lead to high operational cost. Despite of acting as a screening mechanism, expert system is required to find the relationship between the attributes that lead to a specific medical outcome. Data mining via logic mining is a new method to extract logical rule that explains the relationship of the medical attributes of a patient. In this paper, a new logic mining method namely, 2 Satisfiability based Reverse Analysis method (2SATRA) will be proposed to extract the logical rule from medical datasets. 2SATRA will capitalize the 2 Satisfiability (2SAT) as a logical rule and Hopfield Neural Network (HNN) as a learning system. The extracted logical rule from the medical dataset will be used to diagnose the final condition of the patient.
    [Show full text]
  • Warren Goldfarb, Notes on Metamathematics
    Notes on Metamathematics Warren Goldfarb W.B. Pearson Professor of Modern Mathematics and Mathematical Logic Department of Philosophy Harvard University DRAFT: January 1, 2018 In Memory of Burton Dreben (1927{1999), whose spirited teaching on G¨odeliantopics provided the original inspiration for these Notes. Contents 1 Axiomatics 1 1.1 Formal languages . 1 1.2 Axioms and rules of inference . 5 1.3 Natural numbers: the successor function . 9 1.4 General notions . 13 1.5 Peano Arithmetic. 15 1.6 Basic laws of arithmetic . 18 2 G¨odel'sProof 23 2.1 G¨odelnumbering . 23 2.2 Primitive recursive functions and relations . 25 2.3 Arithmetization of syntax . 30 2.4 Numeralwise representability . 35 2.5 Proof of incompleteness . 37 2.6 `I am not derivable' . 40 3 Formalized Metamathematics 43 3.1 The Fixed Point Lemma . 43 3.2 G¨odel'sSecond Incompleteness Theorem . 47 3.3 The First Incompleteness Theorem Sharpened . 52 3.4 L¨ob'sTheorem . 55 4 Formalizing Primitive Recursion 59 4.1 ∆0,Σ1, and Π1 formulas . 59 4.2 Σ1-completeness and Σ1-soundness . 61 4.3 Proof of Representability . 63 3 5 Formalized Semantics 69 5.1 Tarski's Theorem . 69 5.2 Defining truth for LPA .......................... 72 5.3 Uses of the truth-definition . 74 5.4 Second-order Arithmetic . 76 5.5 Partial truth predicates . 79 5.6 Truth for other languages . 81 6 Computability 85 6.1 Computability . 85 6.2 Recursive and partial recursive functions . 87 6.3 The Normal Form Theorem and the Halting Problem . 91 6.4 Turing Machines .
    [Show full text]