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Index
Cover Title Page Copyright Page Dedication Preface 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 *§ 6. Model theory: duality § 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
*§ 28. Functions, terms *§ 29. Equality *§ 30. Equality vs. equivalence, extensionality *§ 31. Descriptions
Part Ii. Mathematical Logic and the Foundations of Mathematics
Chapter IV. The Foundations of Mathematics
§ 32. Countable sets § 33. Cantor’s diagonal method § 34. Abstract sets § 35. The paradoxes § 36. Axiomatic thinking vs. intuitive thinking in mathematics § 37. Formal systems, metamathematics § 38. Formal number theory *§ 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) § 43. Applications to formal number theory: undecidability (Church) and incompleteness (Godel’s theorem) § 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) § 52. Godel’s completeness theorem, and the Lowenheim-Skolem theorem, in the predicate calculus with equality § 53. Skolem’s paradox and nonstandard models of arithmetic § 54. Gentzen’s theorem *§ 55. Permutability, Herbrand’s theorem § 56. Craig’s interpolation theorem § 57. Beth’s theorem on definability, Robinson’s consistency theorem
Bibliography Theorem and Lemma Numbers: Pages List of Postulates Symbols and Notations Index
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