The Explanatory Power of a New Proof: Henkin's Completeness Proof
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“The Church-Turing “Thesis” As a Special Corollary of Gödel's
“The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known. -
Journal Symbolic Logic
THT, E JOURNAL : OF SYMBOLIC LOGIC •' . X , , ' V. ;'••*• • EDITED B\Y \ ' ALONZO CHURCH S. C. KLEENE ALICE A. LAZEROWITZ Managing Editor'. ALFONS BOROERS Consulting Editors'. i W. ACKERMANN ROBERT FEYS ANDRZEJ MOSTOWSKJ G. A. BAYLIS FREDERIC B. FITCH R6ZSA PETER PAUL BERNAYS CARL G. HEMPEL BARKLEY ROSSER G. D. W. BERRY LEON HENKIN THORALF SKOLEM MARTIN DAVIS JOHN G. KEMENY A. R. TURQUETTE VOLUME 19 NUMBER 4 DECEMBER 1954 1 s PUBLISHED QUARTERLY BY THE ASSOCIATION FOR SYMBOLIC LOGIC, INC. WITH THE AID OF SUBVENTIONS FROM EDWARD C. HEGELER TRUST FUND HARVARD UNIVERSITY RUTGERS UNIVERSITY INSTITUTE FOR ADVANCED STUDY SMITH COLLEGE UNIVERSITY OF MICHIGAN 1_—; , Copyright igss by the Association for Symbolic Logic, Inc. Reproduction by photostat, photo-print, microfilm, or like process by permission only r • A Downloaded from https://www.cambridge.org/core. IP address: 170.106.51.11, on 05 Oct 2021 at 08:48:31, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022481200086618 TABLE OF CONTENTS The formalization of mathematics. By HAO WANG 24.1 The recursive irrationality of n. By R. L. GOODSTEIN 267 Distributivity and an axiom of choice. By GEORGE E. COLLINS . 275 Reviews . ! t 278 List of officers and members of the Association for Symbolic Logic . 305 The JOURNAL OF SYMBOLIC LOCIC is the official organ of the Association for Symbolic Logic, Inc., published quarterly, in the months of March, June, September, and December. The JOURNAL is published for the Association by N.V. Erven P. Noordhoff, Publishers, Gronin- gen, The Netherlands. -
5 Propositional Logic: Consistency and Completeness
5 Propositional Logic: Consistency and completeness Reading: Metalogic Part II, 24, 15, 28-31 Contents 5.1 Soundness . 61 5.2 Consistency . 62 5.3 Completeness . 63 5.3.1 An Axiomatization of Propositional Logic . 63 5.3.2 Kalmar's Proof: Informal Exposition . 66 5.3.3 Kalmar's Proof . 68 5.4 Homework Exercises . 70 5.4.1 Questions . 70 5.4.2 Answers . 70 5.1 Soundness In this section, we establish the soundness of the system, i.e., Theorem 3 (Soundness). Every theorem is a tautology, i.e., If ` A then j= A. Proof The proof is by induction the length of the proof of A. For the Basis step, we show that each of the axioms is a tautology. For the induction step, we show that if A and A ⊃ B are tautologies, then B is a tautology. Case 1 (PS1): AB B ⊃ A (A ⊃ (B ⊃ A)) TT TT TF TT FT FT FF TT Case 2 (PS2) 62 5 Propositional Logic: Consistency and completeness XYZ ABC B ⊃ C A ⊃ (B ⊃ C) A ⊃ B A ⊃ C Y ⊃ ZX ⊃ (Y ⊃ Z) TTT TTTTTT TTF FFTFFT TFT TTFTTT TFF TTFFTT FTT TTTTTT FTF FTTTTT FFT TTTTTT FFF TTTTTT Case 3 (PS3) AB » B » A » B ⊃∼ A A ⊃ B (» B ⊃∼ A) ⊃ (A ⊃ B) TT FFTTT TF TFFFT FT FTTTT FF TTTTT Case 4 (MP). If A is a tautology, i.e., true for every assignment of truth values to the atomic letters, and if A ⊃ B is a tautology, then there is no assignment which makes A T and B F. -
Alfred Tarski and a Watershed Meeting in Logic: Cornell, 1957 Solomon Feferman1
Alfred Tarski and a watershed meeting in logic: Cornell, 1957 Solomon Feferman1 For Jan Wolenski, on the occasion of his 60th birthday2 In the summer of 1957 at Cornell University the first of a cavalcade of large-scale meetings partially or completely devoted to logic took place--the five-week long Summer Institute for Symbolic Logic. That meeting turned out to be a watershed event in the development of logic: it was unique in bringing together for such an extended period researchers at every level in all parts of the subject, and the synergetic connections established there would thenceforth change the face of mathematical logic both qualitatively and quantitatively. Prior to the Cornell meeting there had been nothing remotely like it for logicians. Previously, with the growing importance in the twentieth century of their subject both in mathematics and philosophy, it had been natural for many of the broadly representative meetings of mathematicians and of philosophers to include lectures by logicians or even have special sections devoted to logic. Only with the establishment of the Association for Symbolic Logic in 1936 did logicians begin to meet regularly by themselves, but until the 1950s these occasions were usually relatively short in duration, never more than a day or two. Alfred Tarski was one of the principal organizers of the Cornell institute and of some of the major meetings to follow on its heels. Before the outbreak of World War II, outside of Poland Tarski had primarily been involved in several Unity of Science Congresses, including the first, in Paris in 1935, and the fifth, at Harvard in September, 1939. -
Completeness of the Propositions-As-Types Interpretation of Intuitionistic Logic Into Illative Combinatory Logic
University of Wollongong Research Online Faculty of Engineering and Information Faculty of Engineering and Information Sciences - Papers: Part A Sciences 1-1-1998 Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic Wil Dekkers Catholic University, Netherlands Martin Bunder University of Wollongong, [email protected] Henk Barendregt Catholic University, Netherlands Follow this and additional works at: https://ro.uow.edu.au/eispapers Part of the Engineering Commons, and the Science and Technology Studies Commons Recommended Citation Dekkers, Wil; Bunder, Martin; and Barendregt, Henk, "Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic" (1998). Faculty of Engineering and Information Sciences - Papers: Part A. 1883. https://ro.uow.edu.au/eispapers/1883 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic Abstract Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic prepositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. -
A Bibliography of Publications in American Mathematical Monthly: 1990–1999
A Bibliography of Publications in American Mathematical Monthly: 1990{1999 Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 14 October 2017 Version 1.19 Title word cross-reference NF [737]. !(n) [82]. p [619, 149, 694, 412]. P2 [357]. p ≡ 1 (mod 4) [34]. φ [674]. φ(30n + 1) [947]. Φpq(X) [618]. π 0 y − z 2 [105]. 1 [21]. (logx N) [333]. (x +1) x = 1 [845]. 0 [740, 684, 693, 950, 489]. π 2 3 [495, 1]. 1 [495, 1]. 1=p [511]. 10 [140]. 168 Qc(x)=x + c [399]. R [35, 226].p R [357].p n n R [62, 588]. S6 [315]. σ [19]. −1 [995]. 2 [538]. 2 [335]. $29.50 [792]. 2 :n! [1003]. p p p P1 n × 2 × 2 [26]. 3 [828, 583]. 4 − 2 [748]. A [435]. [473]. 2 3= 6 [257]. n=0 1=n! [619]. A∗A = B∗B [607]. AB [620]. BA [620]. 2n tan(k) z [160]. } [512]. x [859]. x=(sin x) n 0 [260]. mod5 [982]. C1 [832]. cos(2π/n) [322]. [755]. x = f(x ) [832]. x2 + ym = z2n [7]. d [844]. dy=dx [449]. ex [859]. f(x; y) [469]. x2 + ym = z2n [65]. x5 + ax + b [465]. xn =1 − [235]. Z3 [975]. -
Notes on Mathematical Logic David W. Kueker
Notes on Mathematical Logic David W. Kueker University of Maryland, College Park E-mail address: [email protected] URL: http://www-users.math.umd.edu/~dwk/ Contents Chapter 0. Introduction: What Is Logic? 1 Part 1. Elementary Logic 5 Chapter 1. Sentential Logic 7 0. Introduction 7 1. Sentences of Sentential Logic 8 2. Truth Assignments 11 3. Logical Consequence 13 4. Compactness 17 5. Formal Deductions 19 6. Exercises 20 20 Chapter 2. First-Order Logic 23 0. Introduction 23 1. Formulas of First Order Logic 24 2. Structures for First Order Logic 28 3. Logical Consequence and Validity 33 4. Formal Deductions 37 5. Theories and Their Models 42 6. Exercises 46 46 Chapter 3. The Completeness Theorem 49 0. Introduction 49 1. Henkin Sets and Their Models 49 2. Constructing Henkin Sets 52 3. Consequences of the Completeness Theorem 54 4. Completeness Categoricity, Quantifier Elimination 57 5. Exercises 58 58 Part 2. Model Theory 59 Chapter 4. Some Methods in Model Theory 61 0. Introduction 61 1. Realizing and Omitting Types 61 2. Elementary Extensions and Chains 66 3. The Back-and-Forth Method 69 i ii CONTENTS 4. Exercises 71 71 Chapter 5. Countable Models of Complete Theories 73 0. Introduction 73 1. Prime Models 73 2. Universal and Saturated Models 75 3. Theories with Just Finitely Many Countable Models 77 4. Exercises 79 79 Chapter 6. Further Topics in Model Theory 81 0. Introduction 81 1. Interpolation and Definability 81 2. Saturated Models 84 3. Skolem Functions and Indescernables 87 4. Some Applications 91 5. -
Logic and Proof
Logic and Proof Computer Science Tripos Part IB Lent Term Lawrence C Paulson Computer Laboratory University of Cambridge [email protected] Copyright c 2018 by Lawrence C. Paulson I Logic and Proof 101 Introduction to Logic Logic concerns statements in some language. The language can be natural (English, Latin, . ) or formal. Some statements are true, others false or meaningless. Logic concerns relationships between statements: consistency, entailment, . Logical proofs model human reasoning (supposedly). Lawrence C. Paulson University of Cambridge I Logic and Proof 102 Statements Statements are declarative assertions: Black is the colour of my true love’s hair. They are not greetings, questions or commands: What is the colour of my true love’s hair? I wish my true love had hair. Get a haircut! Lawrence C. Paulson University of Cambridge I Logic and Proof 103 Schematic Statements Now let the variables X, Y, Z, . range over ‘real’ objects Black is the colour of X’s hair. Black is the colour of Y. Z is the colour of Y. Schematic statements can even express questions: What things are black? Lawrence C. Paulson University of Cambridge I Logic and Proof 104 Interpretations and Validity An interpretation maps variables to real objects: The interpretation Y 7 coal satisfies the statement Black is the colour→ of Y. but the interpretation Y 7 strawberries does not! A statement A is valid if→ all interpretations satisfy A. Lawrence C. Paulson University of Cambridge I Logic and Proof 105 Consistency, or Satisfiability A set S of statements is consistent if some interpretation satisfies all elements of S at the same time. -
Satisfiability 6 the Decision Problem 7
Satisfiability Difficult Problems Dealing with SAT Implementation Klaus Sutner Carnegie Mellon University 2020/02/04 Finding Hard Problems 2 Entscheidungsproblem to the Rescue 3 The Entscheidungsproblem is solved when one knows a pro- cedure by which one can decide in a finite number of operations So where would be look for hard problems, something that is eminently whether a given logical expression is generally valid or is satis- decidable but appears to be outside of P? And, we’d like the problem to fiable. The solution of the Entscheidungsproblem is of funda- be practical, not some monster from CRT. mental importance for the theory of all fields, the theorems of which are at all capable of logical development from finitely many The Circuit Value Problem is a good indicator for the right direction: axioms. evaluating Boolean expressions is polynomial time, but relatively difficult D. Hilbert within P. So can we push CVP a little bit to force it outside of P, just a little bit? In a sense, [the Entscheidungsproblem] is the most general Say, up into EXP1? problem of mathematics. J. Herbrand Exploiting Difficulty 4 Scaling Back 5 Herbrand is right, of course. As a research project this sounds like a Taking a clue from CVP, how about asking questions about Boolean fiasco. formulae, rather than first-order? But: we can turn adversity into an asset, and use (some version of) the Probably the most natural question that comes to mind here is Entscheidungsproblem as the epitome of a hard problem. Is ϕ(x1, . , xn) a tautology? The original Entscheidungsproblem would presumable have included arbitrary first-order questions about number theory. -
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. -
An Introduction to First-Order Logic
Outline An Introduction to First-Order Logic K. Subramani1 1Lane Department of Computer Science and Electrical Engineering West Virginia University Completeness, Compactness and Inexpressibility Subramani First-Order Logic Outline Outline 1 Completeness of proof system for First-Order Logic The notion of Completeness The Completeness Proof 2 Consequences of the Completeness theorem Complexity of Validity Compactness Model Cardinality Lowenheim-Skolem¨ Theorem Inexpressibility of Reachability Subramani First-Order Logic Outline Outline 1 Completeness of proof system for First-Order Logic The notion of Completeness The Completeness Proof 2 Consequences of the Completeness theorem Complexity of Validity Compactness Model Cardinality Lowenheim-Skolem¨ Theorem Inexpressibility of Reachability Subramani First-Order Logic Completeness The notion of Completeness Consequences of the Completeness theorem The Completeness Proof Outline 1 Completeness of proof system for First-Order Logic The notion of Completeness The Completeness Proof 2 Consequences of the Completeness theorem Complexity of Validity Compactness Model Cardinality Lowenheim-Skolem¨ Theorem Inexpressibility of Reachability Subramani First-Order Logic Completeness The notion of Completeness Consequences of the Completeness theorem The Completeness Proof Soundness and Completeness Theorem Soundness: If ∆ ⊢ φ, then ∆ |= φ. Theorem Completeness (Godel’s¨ traditional form): If ∆ |= φ, then ∆ ⊢ φ. Theorem Completeness (Godel’s¨ altenate form): If ∆ is consistent, then it has a model. Subramani First-Order Logic Completeness The notion of Completeness Consequences of the Completeness theorem The Completeness Proof Soundness and Completeness (contd.) Theorem The traditional completeness theorem follows from the alternate form of the completeness theorem. Proof. Assume that ∆ |= φ. It follows that any model M that satisfies all the expressions in ∆, also satisfies φ and hence falsifies ¬φ. -
An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics
STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters 10.1090/stml/087 An Introduction to Ramsey Theory STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Rosa Orellana Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10, 03-01, 03E10, 03B10, 03B25, 03D20, 03H15. Jan Reimann was partially supported by NSF Grant DMS-1201263. For additional information and updates on this book, visit www.ams.org/bookpages/stml-87 Library of Congress Cataloging-in-Publication Data Names: Katz, Matthew, 1986– author. | Reimann, Jan, 1971– author. | Pennsylvania State University. Mathematics Advanced Study Semesters. Title: An introduction to Ramsey theory: Fast functions, infinity, and metamathemat- ics / Matthew Katz, Jan Reimann. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Student mathematical library; 87 | “Mathematics Advanced Study Semesters.” | Includes bibliographical references and index. Identifiers: LCCN 2018024651 | ISBN 9781470442903 (alk. paper) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | AMS: Combinatorics – Extremal combinatorics – Ramsey theory. msc | Mathematical logic and foundations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Mathematical