Preface Introduction 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 First-Order Logic and Model Theory Quantifiers First-Order Languages and Their Interpretations: Satisfiability and Truth Models First-Order Theories Properties of First-Order Theories Additional Metatheorems and Derived Rules 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 Elementary Equivalence: Elementary Extensions Ultrapowers: Nonstandard Analysis Semantic Trees Quantification Theory Allowing Empty Domains Formal Number Theory An Axiom System Number-Theoretic Functions and Relations Primitive Recursive and Recursive Functions Arithmetization: Gödel Numbers The Fixed-Point Theorem: Gödel''s Incompleteness Theorem Recursive Undecidability: Church''s Theorem Nonstandard Models Axiomatic Set Theory An Axiom System Ordinal Numbers Equinumerosity: Finite and Denumerable Sets Hartogs'' Theorem: Initial Ordinals--Ordinal Arithmetic The Axiom of Choice: The Axiom of Regularity Other Axiomatizations of Set Theory Computability Algorithms: Turing Machines Diagrams Partial Recursive Functions: Unsolvable Problems The Kleene-Mostowski Hierarchy: Recursively Enumerable Sets Other Notions of Computability 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 ies: Completeness and Decidability Elementary Equivalence: Elementary Extensions Ultrapowers: Nonstandard Analysis Semantic Trees Quantification Theory Allowing Empty Domains Formal Number Theory An Axiom System Number-Theoretic Functions and Relations Primitive Recursive and Recursive Functions Arithmetization: Gödel Numbers The Fixed-Point Theorem: Gödel''s Incompleteness Theorem Recursive Undecidability: Church''s Theorem Nonstandard Models Axiomatic Set Theory An Axiom System Ordinal Numbers Equinumerosity: Finite and Denumerable Sets Hartogs'' Theorem: Initial Ordinals--Ordinal Arithmetic The Axiom of Choice: The Axiom of Regularity Other Axiomatizations of Set Theory Computability Algorithms: Turing Machines Diagrams Partial Recursive Functions: Unsolvable Problems The Kleene-Mostowski Hierarchy: Recursively Enumerable Sets Other Notions of Computability 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 ines Diagrams Partial Recursive Functions: Unsolvable Problems The Kleene-Mostowski Hierarchy: Recursively Enumerable Sets Other Notions of Computability 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.