Introduction to Mathematical Logic

One The Propositional Calculus.- 1. Propositional Connectives. Truth Tables.- 2. Tautologies.- 3. Adequate Sets of Connectives.- 4. An Axiom System for the Propositional Calculus.- 5. Independence. Many-Valued Logics.- 6. Other Axiomatizations.- Two Quantification Theory.- 1. Quantifiers.- 2. Interpretations. Satisfiability and Truth. Models.- 3. First-Order Theories.- 4. Properties of First-Order Theories.- 5. Additional Metatheorems and Derived Rules.- 6. Rule C.- 7. Completeness Theorems.- 8. First-Order Theories with Equality.- 9. Definitions of New Function Letters and Individual Constants.- 10. Prenex Normal Forms.- 11. Isomorphism of Interpretations. Categoricity of Theories.- 12. Generalized First-Order Theories. Completeness and Decidability.- 13. Elementary Equivalence. Elementary Extensions.- 14. Ultrapowers. Nonstandard Analysis.- 15. Semantic Trees.- Three Formal Number Theory.- 1. Axiom System.- 2. Number-Theoretic Functions and Relations.- 3. Primitive Recursive and Recursive Functions.- 4. Arithmetization. Gödel Numbers.- 5. The Fixed Point Theorem. Gödel's Incompleteness Theorem.- 6. Recursive Undecidability. Church's Theorem.- Four Axiomatic Set Theory.- 1. An Axiom System.- 2. Ordinal Numbers.- 3. Equinumerosity. Finite And Denumerable Sets.- 4. Hartogs' Theorem. Initial Ordinals. Ordinal Arithmetic.- 5. The Axiom of Choice. The Axiom of Regularity.- 6. Other Axiomatizations of Set Theory.- Five Effective Computability.- 1. Algorithms. Turing Machines.- 2. Diagrams.- 3. Partial Recursive Functions. Unsolvable Problems.- 4. The Kleene-Mostowski Hierarchy. Recursively Enumerable Sets.- 5. Other Notions of Effective Computability.- 6. Decision Problems.- Answers to Selected Exercises.- Notation.