Book:H. Jerome Keisler/Mathematical Logic and Computability

Subject Matter

 * Formal Languages
 * Symbolic Logic
 * Mathematical Logic

Contents

 * Preface


 * 1. Propositional Logic
 * 1.1 Introduction
 * 1.2 Syntax of Propositional Logic
 * 1.3 Induction on Length of Wffs
 * 1.4 Main Connective
 * 1.5 Semantics of Propositional Logic
 * 1.6 Truth Tables and Tautologies
 * 1.7 Tableaus
 * 1.8 Soundness
 * 1.9 Finished Sets
 * 1.10 Completeness
 * 1.11 Compactness
 * 1.12 Valid Arguments
 * 1.13 Tableau Problems (TAB1)
 * 1.14 Exercises


 * 2. Pure Predicate Logic
 * 2.1 Introduction
 * 2.2 Syntax of Predicate Logic
 * 2.3 Free and Bound Variables
 * 2.4 Semantics of Predicate Logic
 * 2.5 Graphs
 * 2.6 Tableaus
 * 2.7 Soundness
 * 2.8 Finished sets
 * 2.9 Completeness
 * 2.10 Equivalence Relations
 * 2.11 Order Relations
 * 2.12 Set Theory
 * 2.13 Tableaus and Mathematical Proofs
 * 2.14 PREDCALC Problems (PRED2)
 * 2.15 Tableau Problems (TAB3)
 * 2.16 Exercises


 * 3. Full Predicate Logic
 * 3.1 Syntax
 * 3.2 Semantics
 * 3.3 Tableaus
 * 3.4 Soundness
 * 3.5 Completeness
 * 3.6 Theory of Groups
 * 3.7 Peano Arithmetic
 * 3.8 Some Applications of Compactness
 * 3.9 Tableau Problems (TAB4)
 * 3.10 Exercises


 * 4. Computable Functions
 * 4.1 Introduction
 * 4.2 Numerical Functions and Relations
 * 4.3 The Unlimited Register Machine
 * 4.4 RM computability
 * 4.5 Examples of RM-computable Functions
 * 4.6 Gödel Numbers, Extract, and Put
 * 4.7 The Advanced RM
 * 4.8 Closure Theorems
 * 4.9 Universal RM Programs
 * 4.10 Church's Thesis
 * 4.11 The Halting Problem
 * 4.12 Church's Theorem
 * 4.13 Simple Gnumber Problems (GNUM5)
 * 4.14 Advanced Gnumber Problems (GNUM6)
 * 4.15 Exercises


 * 5. The Incompleteness Theorems
 * 5.1 Coding Tableaus
 * 5.2 Definability and Representability
 * 5.3 The Equivalence Theorem
 * 5.4 Computable Implies Representable
 * 5.5 First Incompleteness Theorem
 * 5.6 Gödel's Original Incompleteness Proof
 * 5.7 Gödel-Rosser Theorem
 * 5.8 Provability and Modal Logic
 * 5.9 Modal Systems and Tableaus
 * 5.10 First Incompleteness Theorem Revisited
 * 5.11 Second Incompleteness Theorem
 * 5.12 Modal Tableau Problems (TAB7)
 * 5.13 Exercises


 * A. Sets and Functions
 * A.1 Sets
 * A.2 Boolean Operations
 * A.3 Functions
 * A.4 Composition and Restriction
 * A.5 Identity, One-one, and Onto Functions
 * A.6 Cardinality
 * A.7 Inverses
 * A.8 Cartesian Product
 * A.9 Graphing Functions
 * A.10 Finite Sequences
 * A.11 Permutations
 * A.12 Induction


 * B. Listings
 * B.1 Simple GNUMBER Programs
 * B.2 Advanced RM programs
 * B.3 Pseudocode for PARAM
 * B.4 PARAM.GN listing
 * B.5 Pseudocode for NXSTATE and UNIV
 * B.6 NXSTATE0.GN listing
 * B.7 UNIV.GN listing


 * C. The Logiclab Package


 * D. TABLEAU - Tableau Editor for DOS
 * D.1 Introduction
 * D.2 Getting Started
 * D.3 Title Screen
 * D.4 Hypothesis Mode
 * D.4.1 Commands in Hypothesis Mode
 * D.4.2 Propositional Logic
 * D.4.3 Predicate Logic
 * D.4.4 Moving Within a Formula
 * D.4.5 Size Limit for Formulas
 * D.5 Tableau Mode
 * D.5.1 Moving Within the Tableau
 * D.5.2 Mouse
 * D.5.3 Commands in Tableau Mode
 * D.5.4 Propositional Logic
 * D.5.5 Predicate Logic
 * D.5.6 Predicate Logic with Equality
 * D.5.7 Size Limit for Substitutions
 * D.6 Map Mode
 * D.7 The Modal Logic Option
 * D.8 Changing Directories


 * E. TABWIN - Tableau Editor for Windows (R)
 * E.1 Introduction
 * E.2 File Menu
 * E.3 View Menu
 * E.4 Entering Hypotheses
 * E.5 Viewing Tableaus
 * E.6 Building Tableaus


 * F COMPLETE - Tableau Completer for DOS


 * G COMPWIN - Tableau Completer for Windows (R)
 * G.1 Introduction
 * G.2 File Menu
 * G.3 View Menu
 * G.4 Building a Finished Tableau
 * G.5 Other Commands


 * H. PREDCALC - Predicate Calculator for DOS
 * H.1 Introduction
 * H.2 Getting Started
 * H.3 Title Screen
 * H.4 Display Modes
 * H.5 Goals
 * H.6 The Calculator Pad
 * H.6.1 The Time Counter
 * H.6.2 Moving Within the Calculator Pad
 * H.6.3 The Help Window
 * H.6.4 Mouse
 * H.6.5 Using the Calculator Buttons
 * H.7 The Letter Commands
 * H.8 Changing Directories


 * I. PREDWIN - Predicate Calculator for Windows (R)
 * I.1 Introduction
 * I.2 Goals
 * I.3 The Help Menu
 * I.4 The Calculator Pad
 * I.5 The File Menu
 * I.6 The View Menu
 * I.7 The Options Menu


 * J. GNUMBER - Gödel Numberer for DOS
 * J.1 Introduction
 * J.2 Getting Started
 * J.3 Title Screen
 * J.4 Execution Mode
 * J.4.1 Viewing More Instructions or Registers
 * J.4.2 Execution Mode Commands
 * J.5 Program Mode
 * J.5.1 Moving Within the Screen
 * J.5.2 Commands in the Program Mode
 * J.6 Instruction Editor
 * J.6.l Register Machine Instruction Letters
 * J.6.2 Entering Register Machine Instructions
 * J.6.3 Register Machine Program Files
 * J.6.4 Advanced Instruction Letters
 * J.7 Register Mode
 * J.7.l Moving Within the Registers
 * J.7.2 Entering a Number into a Register
 * J.7.3 Exploring a Register
 * J.7.4 Register Mode Commands
 * J.7.5 Advanced Register Mode Commands
 * J.8 Changing Directories


 * K GNUMWIN - Gödel Numberer for Windows (R)
 * K.1 Introduction
 * K.2 Program Execution
 * K.3 Register Machine Instructions
 * K.4 File Menu
 * K.5 Program Menu
 * K.6 The Registers Menu
 * K.7 Windows Menu
 * K.8 Options Menu
 * K.9 Step Command and Go Menu


 * Bibliography


 * Index