Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text. It begins with an elementary but thorough overview of mathematical logic of first order. The treatment extends beyond a single method of formulating logic to offer instruction in a variety of techniques: model theory (truth tables), Hilbert-type proof theory, and proof theory handled through derived rules.The second part supplements the previously discussed material and introduces some of the newer ideas and the more profound results of twentieth-century logical research. Subsequent chapters explore the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. The author, Stephen Cole Kleene, was Cyrus C. MacDuffee Professor of Mathematics at the University of Wisconsin, Madison. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. COVER TITLE COPYRIGHT 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. 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 BACK COVER
this Reprint Of Kleene's 1967 Text Provides An Elementary But Thorough Treatment Of Mathematical Logic Of The First Order. Kleene (formerly: Mathematics, University Of Wisconsin, Madison) Covers A Variety Of Methods And Techniques Including Model Theory (truth Tables), Hilbert-type Proof Theory, And Proof Theory Through Derived Rules. Subsequent Chapters Cover Topics Formal Number Theory, Godel's Completeness Theorem, Genzen's Theorem, Skolem's Paradox, And Nonstandard Models Of Arithmetic. Annotation (c)2003 Book News, Inc., Portland, Or
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.
