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 Gentzentype formal system, the LöwenheimSkolem theorem" 

51. Gödel's completeness theorem (with a Hilberttype formal system) 

52. "Gödel's completeness theorem, and the LöwenheimSkolem 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 

