Mathematical Logic

Overview

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 ...

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Mathematical Logic

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Overview

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.

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Product Details

  • ISBN-13: 9780486425337
  • Publisher: Dover Publications
  • Publication date: 12/18/2002
  • Series: Dover Books on Mathematics Series
  • Edition description: Reprint
  • Pages: 434
  • Sales rank: 432,340
  • Product dimensions: 5.50 (w) x 8.50 (h) x 0.88 (d)

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

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