Table of 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 (Gödel's theorem)
44. Applications to formal number theory: consistency proofs (Gödel'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. Gödel's completeness theorem: introduction
49. Gödel's completeness theorem: the basic discovery
50. "Gödel's completeness theorem with a Gentzen-type formal system, the Löwenheim-Skolem theorem"
51. Gödel's completeness theorem (with a Hilbert-type formal system)
52. "Gödel's completeness theorem, and the Löwenheim-Skolem theorem, in the predicate calculus with equality"
53. Skolen'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
