41. MATHEMATICAL LOGIC (code: 314) CONTACT DR PA SYMONDS (M/P8), CREDIT RATING 10. Aims To introduce studentsto Predicate Logic culminating in Gödel’s completeness theorem. http://www2.umist.ac.uk/mathematics/intranet1/Yr3Syllabus/(314) MATHEMATICAL LOG

MATHEMATICAL LOGIC (code: 314) SEMESTER: SECOND CONTACT: DR P A SYMONDS (M/P8) CREDIT RATING: Aims: Intended Learning Outcomes: On successful completion of the course students will: Be able to work with a formal language. Understand how to use interpretations and models. Be able to give simple proofs from the axioms. Understand the importance of consistency and completeness. Pre-requisites: Dependent Courses: None Course Description: The course concentrates on one of the most important results of 20th century logic, Gödel's Completeness Theorem for Predicate Logic. This theorem links two fundamental concepts of Mathematics, truth and provability, and provides deep insights into ways of mathematical thinking. Prospective students should enjoy abstract ideas and have the ability to understand mathematical proofs of the type which occur in Pure Mathematics. Teaching Mode: 2 Lectures per week 1 Tutorial per week Private Study: 5 hours per week Recommended Texts: E Mendelson, Introduction to Mathematical Logic, (4th edition), 1997 or earlier edition, Chapman Hall.

42. TLA Notes that are used. A completeness theorem for TLA 17 November 1993 A relativecompleteness theorem for TLA, with its proof. The first http://research.microsoft.com/users/lamport/tla/notes.html

TLA NOTES Last modified 16 April 1996 This is a collection of material about TLA (Temporal Logic of Actions) and specification in general that may be of interest, but has not appeared in a real paper. These notes are rough and half-baked; they probably contain many errors. But, they provide the only available information on several important topics. The notes marked "LaTeX/ASCII" can be read in ASCII or run through LaTeX to get a somewhat more readable version. To run them through LaTeX, you need the style file spec92.sty . You can click here for an explanation of the ASCII conventions that are used. A Completeness Theorem for TLA 17 November 1993 A relative completeness theorem for TLA, with its proof. The first part was distributed to the TLA mailing list. For intrepid souls only. LaTeX/ASCII Types Considered Harmful Leslie Lamport 23 December 1992 A brief explanation of how to do mathematics without types. (10 pages) Postscript DVI LaTeX Using Tense Logic in Specification and Verification Peter Ladkin 5 August 1993 These are Peter Ladkin's comments on the question of using first-order logic rather than TLA. (Sent to TLA mailing list.)

43. QUAIL '97 -- Daily Questions Godel's completeness theorem has to do with firstorder logic. Godel's CompletenessTheorem showed that a complete proof procedure exists for FOL. http://www-cs-students.stanford.edu/~pdoyle/quail/questions/11_15_96.html

QUAIL '97 (Question of the Day) Back to the Question of the Day Page Patrick Doyle November 18, 1996

44. MTH-3D23 : Mathematical Logic structures. This is Gödel’s completeness theorem. Theorem. Proof of the CompletenessTheorem (Adequacy) for propositional calculus. (5 lectures). http://www.mth.uea.ac.uk/maths/syllabuses/0102/3D2301.html

MTH-3D23 : Mathematical Logic 1. Introduction: The course in concerned with foundational issues of modern pure mathematics. It is a rigorous introduction to first-order logic. Proofs will be given for most of the results discussed. Some degree of mathematical sophistication is called for and familiarity with (and a taste for) mathematical proofs, such as would be seen in a rigorous first-year analysis or algebra course, will be assumed. The prerequisite is Algebra I and there are connections with Discrete Mathematics II. 2. Timetable Hours, Credits, Assessments: 33 one hour lectures; 20 UCU. Assessment: Coursework 20% via assessed homework; 3 hour examination 80%. There will be 4 problem sheets which will make up the coursework component of the unit. Sketch solutions will be distributed and consulting hours arranged. 3. Overview: The final section of the unit is concerned with model theory: the study and classification of mathematical structures in terms of what can be said about them in 1st order languages. 4. Recommended Reading:

45. Works In Progress General logic. S. Valentini, A simple proof of the completeness theorem of theintuitionistic predicate calculus with respect to the topological semantics, http://www.math.unipd.it/~silvio/WorkinProg.html

Works in progress Modal Logic S. Valentini, M. Viale The binary modal logic of the intersection types, Abstract: Looking for a suitable logic for the subtype relation between the types of the intersection types lambda calculus we developed a modal logic with a two-places modality. We present here its main syntactical and semantical properties, that is, the completeness theorem, the finite model property, the cut-elimination theorem and a decision procedure for theoremhood. Formal topology G. Sambin, S. Valentini, Topological characterization of Scott Domains, Abstract: First we introduce the notion of super-coherent topology which does not depend on any ordering. Then we show that a topology is super-coherent if and only if it is the Scott topology over a suitable algebraic dcpo. The main ideas of the paper are a by-product of the constructive approach to domain theory through information bases which we have proposed in a previous work, but the presentation here does not rely on that foundational framework. S. Valentini

46. Computability Complexity Logic Book 350 PART II. completeness theorem 357 1. Derivations and deduction theoremfor 357 sentence logic 2. Completeness of propositional logic. http://www.di.unipi.it/~boerger/cclbookcontents.html

Studies in Logic and the Foundations of Mathematics, vol. 128, North-Holland, Amsterdam 1989, pp. XX+592. CONTENTS Graph of dependencies XIV Introduction XV Terminology and prerequisites XVIII Book One ELEMENTARY THEORY OF COMPUTATION 1 Chapter A. THE MATHEMATICAL CONCEPT OF ALGORITHM 2 PART I. CHURCH'S THESIS 2 1. Explication of Concepts. Transition systems, 2 Computation systems, Machines (Syntax and Semantics of Programs), Turing machines. structured (Turing- and register-machine) programs (TO, RO). 2. Equivalence theorem, 26 LOOP-Program Synthesis for primitive recursive functions. 3. Excursus into the semantics of programs. 34 Equivalence of operational and denotational semantics for RM-while programs, fixed-point meaning of programs, proof of the fixed-point theorem. 4*. Extended equivalence theorem. Simulation of 37 other explication concepts: modular machines, 2-register machines, Thue systems, Markov algorithms, ordered vector addition systems (Petri nets), Post calculi (canonical and regular), Wang's non-erasing half-tape machines, word register machines. 5. Church's Thesis 48

47. PlanetMath: Models Constructed From Constants (The extended completeness theorem) A set of formulas of is consistent if andonly if it has a model (regardless of whether or not has witnesses for ). http://planetmath.org/encyclopedia/GAdelCompletenessTheorem.html

Math for the people, by the people. Encyclopedia Books Papers Expositions ... Random Login create new user name: pass: forget your password? Main Menu the math Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports information Docs Classification News Legalese ... TODO List Models constructed from constants (Definition) The definition of a structure and of the satisfaction relation is nice, but it raises the following question : how do we get models in the first place? The most basic construction for models of first-order theory is the construction that uses constants . Throughout this entry, is a fixed first-order language Let be a set of constant symbols of , and be a theory in . Then we say is a set of witnesses for if and only if for every formula with at most one free variable , we have for some Lemma. Let is any consistent set of sentences of , and is a set of new symbols such that . Let . Then there is a consistent set extending and which has as set of witnesses.

48. Completeness Of KT5-models theorems. Theorem 3. (Generalized completeness theorem) Any consistentset of well formed formulas is realizable. Theorem 4. (Completeness http://www-formal.stanford.edu/jmc/model/node10.html

Next: The Puzzle of Unfaithful Up: Kripke-type Semantics Previous: Soundness of KT5-models Completeness of KT5-models As for the completeness of KT5-models, we have the following theorems. Theorem 3. (Generalized Completeness Theorem) Any consistent set of well formed formulas is realizable. Theorem 4. (Completeness and Decidability Theorem) For any is a theorem of KT5 if and only if is valid in all KT5- models whose cardinality , where n is the cardinality of the finite set Yasuko Kitajima Fri Jun 20 13:39:43 PDT 1997

49. Table Of Contents 3.10.2, The proof of the completeness theorem. 3.10.3, Valid arguments revisited. Chapter6, A Proof System for Firstorder Logic, and Gödel's completeness theorem, http://www.thoralf.uwaterloo.ca/htdocs/LMCS/toc.html

Logic for Mathematics and Computer Science Table of Contents Preface Flow of Topics PART I. QUANTIFIER-FREE LOGICS Chapter 1 From Aristotle to Boole Sophistry The contributions of Aristotle The algebra of logic The method of Boole, and Venn diagrams Checking for validity Finding the most general conclusion Historical remarks Chapter 2 Propositional Logic Propositional connectives, propositional formulas, truth tables Defining propositional formulas Truth tables Equivalent formulas, tautologies, contradictions Equivalent formulas Tautologies Contradictions Substitution Replacement Induction proofs on formulas Main result on replacement Simplification of formulas Adequate connectives The standard connectives are adequate Proving adequacy Proving inadequacy Disjunctive and conjunctive forms Rewrite rules to obtain normal forms Using truth tables to find normal forms Uniqueness of normal forms Valid arguments, tautologies, and satisfiability Compactness The compactness theorem for propositional logic Applications of compactness The propositional proof system PC Simple equivalences The proof system Soundness and completeness Derivations with premisses Proving theorems about derivations Generalized soundness and completeness Resolution A motivation Clauses Resolution The Davis-Putnam procedure Soundness and completeness for the DPP Applications of the DPP Soundness and completeness for resolution Generalized soundness and completeness for resolution Horn clauses Graph clauses Pigeonhole clauses Historical remarks The beginnings Statement logic and the algebra of logic

50. Preface The completeness proofs tend to be quite different in the different proof systems,with Gödel's completeness theorem for firstorder logic being the most http://www.thoralf.uwaterloo.ca/htdocs/LMCS/preface.html

Logic for Mathematics and Computer Science Preface Mathematical logic has been, in good part, developed and pursued with the hope of providing practical algorithmic tools for doing reasoning, both in everyday life and in mathematics, first by hand or mechanical means, and later by electronic computers. In this text elementary traditional logic is presented side by side with its algorithmic aspects, i.e., the syntax and semantics of firstorder logic up to completeness and compactness, and developments in theorem proving that were inspired by the possibilities of using computers. Here we are referring to Robinson's resolution theorem proving, and to the KnuthBendix procedure to obtain term rewrite systems, both using the key idea of (most general) unification. Thus we find the choice of topics important as well as accessible to a wide range of students in the mathematical sciences. These topics are rich in basic algorithms, giving the students a desirable hands-on experience. No background in abstract algebra or analysis is assumed, yet the material is definitely mathematical logic, logic for mathematics and computer science that is developed and analyzed using mathematical methods.

51. The System FL_ For Specification Analysis And Its Completeness Theorem The System FL_ m,n for Specification Analysis and its completeness theorem. The completenesstheorem (soundness and completeness) for FL_ m,n is also proved. http://www.ipsj.or.jp/members/JInfP/Eng/0904/article002.html

Last Update¡§Tue Jun 19 17:31:37 2001 Journal of Information Processing Abstract Vol.09 No.04 - 002 HIOSE KEN TAKAHASHI MAKOTO YAMADA SHINICHI Waseda University Text ¢¬Index Vol.09 No.04 Journal of Information Processing Contents Web Members Service Menu Comments are welcome. Mail to address editj@ips j.or.jp , please.

52. Course Notes On First Order Logic installment on semantics. It proves the compactness theorem from thecompleteness theorem for the given rule set. There are many http://www.ai.mit.edu/people/dam/notes/fol.html

Course Notes on First Order Logic This node contains one installment of the course notes for MIT's graduate course on the foundations of artificial intelligence This node contains two installments of the notes describing basic results on first order logic. The first gives the basic syntax and sematics of the language. It gives a complete set of inference rules derived from general rules given in the installment on semantics . It proves the compactness theorem from the completeness theorem for the given rule set. There are many problems at the end of the installment which give applications of the compactness theorem. The second installment contained here gives the basic results of resolution theorem proving. postscript for first order logic postscript for resolution theorem proving The course no longer follows these notes very closely. Lectures now start with the syntax and semantics. They then explain Skolem normal form and clausal normal form for fist order theories. They then given Herbrand's theorem, making an explicit analogy with the free graph theorem for STRIPS planning given in the installment on graph search and STRIPS planning . After presenting Herbrand's theorem we prove the completeness of Robinson's binary resolution rule (with factoring treated as a seperate rule). It is then shown how the completeness of resolution implies the compactness theorem. The importance of the compactness theorem in establishing the expressive weakness of first order logic is emphasized.

53. Tatra Mountains Authors Petr Hájek, David Švejda Title A strong completeness theoremfor finitely axiomatized fuzzy theories Abstract. The aim http://tatra.mat.savba.sk/paper.php?id_paper=437

54. Theory Of Ordinals OmodeoSchwartz Chapter 2, Sections (a) The propositional calculus (bi-iii); Introductionto the predicate calculus; Goedel completeness theorem; Some examples http://www.settheory.com/syllabus_2002.html

G22.3033-005 Computational Logic and Set theory - Fall 2002 Prof. J.T. Schwartz To contact me, call 673-3242 most days or evenings till 8PM Textbook: Concrete set logic: a proof verifier and its application to the basic theorems of analysis. By Domenico Cantone, Eugenio Omodeo, and J.T. Schwartz (Draft manuscript) You can E-mail me at jack@nyu.edu. The course text is on-line at http://www.settheory.com/intro.html There will be no examinations. Grades will be assigned on the basis of a final paper, on a subject related to the course, to be selected in consultation with the instructor. A final project proposal is due in week 8. Finished final projects are due one week after the last class. Students actively interested in Computational Logic, perhaps for a dissertation in this area, may wish to undertake a programming project, to be worked out in consultation with the instructor. Week 1: (September 10) Cantone-Omodeo-Schwartz Chapter 1, Sections (a-d); (e.i.a) The Choice Operator; (e.i.b) Ordinal and Cardinal Numbers in Set Theory. Week 2: (September 17) Cantone-Omodeo-Schwartz Chapter 1

55. Gödel's Completeness Theorem Gödel's completeness theorem. If is a set of Axioms in a firstorderlanguage, and a statement holds for any structure satisfying http://lib4web.lib.msu.edu/crcmath/math/math/g/g193.htm

If is a set of Axioms in a first-order language, and a statement holds for any structure satisfying , then can be formally deduced from in some appropriately defined fashion. See also Eric W. Weisstein

56. Education, Master Class 1988/1999, MRI Nijmegen of the course Incompleteness theorems Lecturer J. van Oosten Prerequisites basicpredicate logic (in particular, the Godel completeness theorem) and basic http://www-mri.sci.kun.nl/education/course_9899.html

Education, Master Class, Master Class 1998/1999, Detailed Course Content Detailed Content of the Courses Course content 1st semester: Model Theory W. Veldman Lambda Calculus H. Barendregt, E. Barendsen Recursion Theory and Proof Theory H. Schellinx Logic Panorama seminar 2nd semester: Type Theory and Applications H. Barendregt, E. Barendsen Incompleteness Theorems J. van Oosten Sheaves and Logics I. Moerdijk Mathematical Logic seminar Courses Name of the course: Model Theory Lecturer: W. Veldman Prerequisites: Some familiarity with mathematical reasoning. Literature: C.C. Chang, H.J. Keisler, Model Theory, North Holland Publ. Co. 1977 W. Hodges, Model Theory, Cambridge UP, 1993 Contents: Model theory studies the variety of mathematical structures that satisfy given formal theory. It may also be described as a study of mathematical structures from the logician's point of view. Model theory at its best is a delightful blend of abstract and concrete reasoning. Among the topics to be treated in this course are Fraisse's characterisation of the notion 'elementary equivalence' (structures A,B are called elementarily equivalent if they satisfy the same first-order-sentences), the compactness theorem and its many consequences, ultraproducts, some non-standard-analysis, Tarski's decision method for the field of real numbers by quantifier elimination and Robinson's notion of model completeness. If time permits, some attention will be given to constructive and recursive model theory.

57. Fuzzy Logic Survey of logical systems with a continuum of truth values; from the Stanford Encyclopdia by Petr Hajek .Category Science Math Fuzzy Logic...... BL. The standard completeness theorem Cignoli et al. (2000b) saysthat a formula is a ttautology iff it is provable in BL. There http://plato.stanford.edu/entries/logic-fuzzy/

version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A B C D ... Z content revised SEP Fuzzy Logic The term "fuzzy logic" emerged in the development of the theory of fuzzy sets by Lotfi Zadeh [ Zadeh (1965) ]. A fuzzy subset A of a (crisp) set X is characterized by assigning to each element x of X the degree of membership of x in A (e.g. X is a group of people, A the fuzzy set of old people in X). Now if X is a set of propositions then its elements may be assigned their degree of truth intermediate connectives truth functions different from probability theory since the latter is not truth-functional (the probability of conjunction of two propositions is not determined by the probabilities of those propositions). Two main directions in fuzzy logic have to be distinguished (cf. Zadeh (1994) Fuzzy logic in the broad sense (older, better known, heavily applied but not asking deep logical questions) serves mainly as apparatus for fuzzy control, analysis of vagueness in natural language and several other application domains. It is one of the techniques of soft-computing , i.e. computational methods tolerant to suboptimality and impreciseness (vagueness) and giving quick, simple and

58. Www.math.niu.edu/~rusin/known-math/99/completeness completeness of the reals In article m33dtei4vg.fsf@localhost.localdomain , LarryMintz kabir@citenet.net wrote Is the completeness theorem which states http://www.math.niu.edu/~rusin/known-math/99/completeness

From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: analysis question Date: 7 Dec 1999 22:06:11 -0500 Newsgroups: sci.math Keywords: more or less equivalent statements of the completeness of the reals In article , Larry Mintz wrote: : :Is the Completeness Theorem which states that :Every Cauchy sequence of real numbers has a unique limit : :and Bolzano-Weirstrass Theorem which states : An infintely bounded sequence of real numbers has a limit point : :stating the same thing ? : :To me they seem verrry close in meaning. :Larry In a way, yes. In the theory of Archimedean ordered fields, they are both "statements of completeness": each of them can be deduced from the supremum property, and each of them implies the supremum property. (It is too late now for me to speculate where Archimedean property can be dropped as a background assumption.) Other "statements of completeness": every monotone bounded sequence has a limit, every absolutely convergent series is convergent, every continuous function on [0,1] is bounded, every continuous function on [0,1] attains its maximum, every continuous function on [0,1] has the Intermediate Value Property, every continuous function on [0,1] is uniformly continuous, [0,1] is a compact set, [0,1] is a connected set, variants of the above with any interval [a, b] where a

59. Syllabuses For MSc In Logic 2002-03, Department Of Mathematics, Univ. Of Manches The module will lead up to a proof of the completeness theorem, a striking resultof Kurt Gódel (1930), which demonstrates the equivalence of a natural notion http://www.ma.man.ac.uk/DeptWeb/MScCourses/Logic/Syllabus/Syllabus.html

MSc in Mathematical Logic Session 2002/03 DEPARTMENT OF MATHEMATICS MT5151 Predicate Logic Credit Rating: Level: M.Sc. Delivery: Semester One Lecturer: Peter Aczel (Room 2.52 of the Computer Science Department, Telephone:56155, email: petera@cs.man.ac.uk). General description: Aims: To introduce students to the formal notions of language, proof, semantics, and completeness with quantificational logic, in order to: improve their understanding and appreciation of the foundations of mathematics and provide the necessary background knowledge for later logic course units. Learning Objectives: On successful completion of the course unit the students will appreciate how arguments involving predicates can be formalised semantically and syntactically and how these are connected (via the Completeness Theorem) in simple cases be able to show that 'A follows from B' both by giving a semantic argument and by constructing a formal proof. in simple cases be able to show that 'A does not follow from B' by using semantics. Prerequisites: Some familiarity with the propositional calculus. (This may be gained by study of a chapter(s) on propositional calculus up to the completeness theorem from one of the many logic textbooks, such as the ones listed below perhaps combined with the first 3 weeks of MT5181).

60. Course Description 2002-03 The highlight of the course unit will be the completeness theorem whichsays that these two characterisations are equivalent. This http://www.ma.man.ac.uk/DeptWeb/UGCourses/Syllabus/Level2/2002/MT2151.html

Last updated 17 Jul 02 DEPARTMENT of MATHEMATICS University of Manchester Course Description for MT2151 Propositional Logic Credit Rating: Level: Second Level Delivery: Semester One Lecturer: Prof J. Paris (Room 15.09, Telephone 55880, email:jeff@ma.man.ac.uk). General Description Logic is the study of arguments, what they are, what it means to say they are sound. As such it is central to Mathematics, Philosophy, and, to an increasing extent in recent years, Computer Science. This course unit will deal with the most basic sort of argument (i.e., in everyday parlance, what we mean by ‘A follows from B’), namely those which depend for their soundness simply on the commonly agreed interpretation of the connectives ‘not’,’and’,’or’, and ‘implies’. Essentially we shall characterise this relation of ‘follows’ in two ways, firstly in terms of preservation of truth (semantically) and secondly in terms of the formal rules it obeys (proof theoretically, or syntactically). The highlight of the course unit will be the Completeness Theorem which says that these two characterisations are equivalent. This is a fundamental result for mathematics, its essence is that if something isn't formally provable then there must be a counter-example. Aims The course unit will introduce the student to the idea of formalising arguments, both semantically and syntactically, and to the fundamental connection between these approaches.

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