Includes bibliographical references (p. 371-386) and index.
Undergraduate students with no prior instruction in mathematical logic will benefit from this multi-part text. Part I offers an elementary but thorough overview of mathematical logic of 1st order. Part II introduces some of the newer ideas and the more profound results of logical research in the 20th century. 1967 edition.
PART I. ELEMENTARY MATHEMATICAL LOGIC
CHAPTER I. THE PROPOSITIONAL CALCULUS
and#160; 1. Linguistic considerations: formulas
and#160; 2. "Model theory: truth tables,validity "
and#160; 3. "Model theory: the substitution rule, a collection of valid formulas"
and#160; 4. Model theory: implication and equivalence
and#160; 5. Model theory: chains of equivalences
and#160; 6. Model theory: duality
and#160; 7. Model theory: valid consequence
and#160; 8. Model theory: condensed truth tables
and#160; 9. Proof theory: provability and deducibility
and#160; 10. Proof theory: the deduction theorem
and#160; 11. "Proof theory: consistency, introduction and elimination rules"
and#160; 12. Proof theory: completeness
and#160; 13. Proof theory: use of derived rules
and#160; 14. Applications to ordinary language: analysis of arguments
and#160; 15. Applications to ordinary language: incompletely stated arguments
CHAPTER II. THE PREDICATE CALCULUS
and#160; 16. "Linguistic considerations: formulas, free and bound occurrences of variables"
and#160; 17. "Model theory: domains, validity"
and#160; 18. Model theory: basic results on validity
and#160; 19. Model theory: further results on validity
and#160; 20. Model theory: valid consequence
and#160; 21. Proof theory: provability and deducibility
and#160; 22. Proof theory: the deduction theorem
and#160; 23. "Proof theory: consistency, introduction and elimination rules"
and#160; 24. "Proof theory: replacement, chains of equivalences"
and#160; 25. "Proof theory: alterations of quantifiers, prenex form"
and#160; 26. "Applications to ordinary language: sets, Aristotelian categorical forms"
and#160; 27. Applications to ordinary language: more on translating words into symbols
CHAPTER III. THE PREDICATE CALCULUS WITH EQUALITY
and#160; 28. "Functions, terms"
and#160; 29. Equality
and#160; 30. "Equality vs. equivalence, extensionality"
and#160; 31. Descriptions
PART II. MATHEMATICAL LOGIC AND THE FOUNDATIONS OF MATHEMATICS
CHAPTER IV. THE FOUNDATIONS OF MATHEMATICS
and#160; 32. Countable sets
and#160; 33. Cantor's diagonal method
and#160; 34. Abstract sets
and#160; 35. The paradoxes
and#160; 36. Axiomatic thinking vs. intuitive thinking in mathematics
and#160; 37. "Formal systems, metamathematics"
and#160; 38. Formal number theory
and#160; 39. Some other formal systems
CHAPTER V. COMPUTABILITY AND DECIDABILITY
and#160; 40. Decision and computation procedures
and#160; 41. "Turing machines, Church's thesis"
and#160; 42. Church's theorem (via Turing machines)
and#160; 43. Applications to formal number theory: undecidability (Church) and incompleteness (Gand#246;del's theorem)
and#160; 44. Applications to formal number theory: consistency proofs (Gand#246;del's second theorem)
and#160; 45. "Application to the predicate calculus (Church, Turing)"
and#160; 46. "Degrees of unsolvability (Post), hierarchies (Kleene, Mostowski)."
and#160; 47. Undecidability and incompleteness using only simple consistency (Rosser)
CHAPTER VI. THE PREDICATE CALCULUS (ADDITIONAL TOPICS)
and#160; 48. Gand#246;del's completeness theorem: introduction
and#160; 49. Gand#246;del's completeness theorem: the basic discovery
and#160; 50. "Gand#246;del's completeness theorem with a Gentzen-type formal system, the Land#246;wenheim-Skolem theorem"
and#160; 51. Gand#246;del's completeness theorem (with a Hilbert-type formal system)
and#160; 52. "Gand#246;del's completeness theorem, and the Land#246;wenheim-Skolem theorem, in the predicate calculus with equality"
and#160; 53. Skolen's paradox and nonstandard models of arithmetic
and#160; 54. Gentzen's theorem
and#160; 55. "Permutability, Herbrand's theorem"
and#160; 56. Craig's interpolation theorem
and#160; 57. "Beth's theorem on definability, Robinson's consistency theorem"
BIBLIOGRAPHY
THEOREM AND LEMMA NUMBERS: PAGES
LIST OF POSTULATES
SYMBOLS AND NOTATIONS
INDEX