CONTENTS
PART I. ELEMENTARY MATHEMATICAL LOGIC
CHAPTER I. THE PROPOSITIONAL CALCULUS
§ 1. Linguistic considerations: formulas
§ 2. Model theory: truth tables, validity
§ 3. Model theory: the substitution rule, a collection of valid formulas
§ 4. Model theory: implication and equivalence
§ 5. Model theory: chains of equivalences
§ 7. Model theory: valid consequence
*§ 8. Model theory: condensed truth tables
§ 9. Proof theory: provability and deducibility
§ 10. Proof theory: the deduction theorem
§ 11. Proof theory: consistency, introduction and elimination rules
§ 12. Proof theory: completeness
§ 13. Proof theory: use of derived rules
*§ 14. Applications to ordinary language: analysis of arguments
*§ 15. Applications to ordinary language: incompletely stated arguments
CHAPTER II. THE PREDICATE CALCULUS
§ 16. Linguistic considerations: formulas, free and bound occurrences of variables
§ 17. Model theory: domains, validity
§ 18. Model theory: basic results on validity
*§ 19. Model theory: further results on validity
§ 20. Model theory: valid consequence
§ 21. Proof theory: provability and deducibility
§ 22. Proof theory: the deduction theorem
§ 23. Proof theory: consistency, introduction and elimination rules
§ 24. Proof theory: replacement, chains of equivalences
§ 25. Proof theory: alterations of quantifiers, prenex form
*§ 26. Applications to ordinary language: sets, Aristotelian categorical forms
*§ 27. Applications to ordinary language: more on translating words into symbols
CHAPTER III. THE PREDICATE CALCULUS WITH EQUALITY
*§ 30. Equality vs. equivalence, extensionality
PART II. MATHEMATICAL LOGIC
AND THE FOUNDATIONS OF MATHEMATICS
CHAPTER IV. THE FOUNDATIONS OF MATHEMATICS
§ 33. Cantor’s diagonal method
§ 36. Axiomatic thinking vs. intuitive thinking in mathematics
§ 37. Formal systems, metamathematics
*§ 39. Some other formal systems
CHAPTER V. COMPUTABILITY AND DECIDABILITY
§ 40. Decision and computation procedures
§ 41. Turing machines, Church’s thesis
§ 42. Church’s theorem (via Turing machines)
§ 44. Applications to formal number theory: consistency proofs (Godel’s second theorem)
*§ 45. Application to the predicate calculus (Church, Turing)
*§ 46. Degrees of unsolvability (Post), hierarchies (Kleene, Mostowski).
*§ 47. Undecidability and incompleteness using only simple consistency (Rosser)
CHAPTER VI. THE PREDICATE CALCULUS (ADDITIONAL TOPICS)
§ 48. Godel’s completeness theorem: introduction
§ 49. Godel’s completeness theorem: the basic discovery
§ 50. Godel’s completeness theorem with a Gentzen-type formal system, the Lowenheim-Skolem theorem
§ 51. Godel’s completeness theorem (with a Hilbert-type formal system)
§ 53. Skolem’s paradox and nonstandard models of arithmetic
*§ 55. Permutability, Herbrand’s theorem
§ 56. Craig’s interpolation theorem
§ 57. Beth’s theorem on definability, Robinson’s consistency theorem