Book:Elliott Mendelson/Introduction to Mathematical Logic/Sixth Edition

Subject Matter

 * Mathematical Logic

Contents

 * Preface
 * Introduction


 * 1. The Propositional Calculus
 * Propositional Connectives:Truth Tables
 * Tautologies
 * Adequate Sets of Connectives
 * An Axiom System for the Propositional Calculus
 * Independence: Many-Valued Logics
 * Other Axiomatizations


 * 2. First-Order Logic and Model Theory
 * Quantifiers
 * Parentheses
 * First-Order Languages and Their Interpretations: Satisfiability and Truth: Models
 * First Order Theories
 * Logical Axioms
 * Proper Axioms
 * Rules of Inference
 * Properties of First-Order Theories
 * Additional Metatheorems and Derived Rules
 * Particularization Rule A4
 * Existential Rule E4
 * Rule C
 * Completeness Theorems
 * First-Order Theories with Equality
 * Definitions of New Function Letters and Individual Constants
 * Prenex Normal Forms
 * Isomorphism of Interpretations: Categoricity of Theories
 * Generalized First-Order Theories: Completeness and Decidability
 * Mathematical Applications
 * Elementary Equivalence: Elementary Extensions
 * Ultrapowers: Nonstandard Analysis
 * Reduced Direct Products
 * Nonstandard Analysis
 * Semantic Trees
 * Basic Principle of Semantic Trees
 * Quantification Theory Allowing Empty Domains


 * 3. Formal Number Theory
 * An Axiom System
 * Number-Theoretic Functions and Relations
 * Primitive Recursive and Recursive Functions
 * Arithmetization: Godel Numbers
 * The Fixed-Point Theorem: Godel's Incompleteness Theorem
 * Recursive Undecidability: Church's Theorem
 * Nonstandard Models


 * 4. Axiomatic Set Theory
 * An Axiom System
 * Ordinal Numbers
 * Equinumerosity: Finite and Denumerable Sets
 * Finite Sets
 * Hartogs' Theorem: Initial Ordinals - Ordinal Arithmetic
 * The Axiom of Choice: The Axiom of Regularity
 * Other Axiomatizations of Set Theory
 * Morse-Kelley (MK)
 * Zermelo-Fraenkel (ZF)
 * The Theory of Types (ST)
 * ST1 (Extensionality Axiom)
 * ST2 (Comprehension Axiom Scheme)
 * ST3 (Axiom of Infinity)
 * Quine's Theories NF and ML
 * NF1 (Extensionality)
 * NF2 (Comprehension)
 * Set Theory with Urelements


 * 5. Computability
 * Algorithms: Turing Machines
 * Diagrams
 * Partial Recursive Functions: Unsolvable Problems
 * The Kleene-Mostowski Hierarchy: Recursively Enumerable Sets
 * Other Notions of Computability
 * Herbrand-Godel Computability
 * Markov Algorithms
 * Decision Problems


 * Appendix A: Second-Order Logic
 * Appendix B: First Steps in Modal Propositional Logic
 * Appendix C: A Consistency Proof for Formal Number Theory
 * Answers to Selected Exercises
 * Bibliography
 * Notations
 * Index