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Mathematical Logic

Stephen Cole Kleene
Publisher: 
Dover Publications
Publication Date: 
2002
Number of Pages: 
416
Format: 
Paperback
Price: 
24.95
ISBN: 
9780486425337
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Steven Deckelman
, on
05/23/2016
]

This is a Dover reprint of Stephen Cole Kleene’s classic book, published originally by John Wiley and Sons in 1967. A pdf scan of the original 1967 version is available on line at the Internet Archive. This presumably means the book is in the public domain.

Kleene (1909–1994), intended this book to be an introduction to mathematical logic at the undergraduate level, in contrast to his earlier and more celebrated Introduction to Metamathematics (1952), which was intended as a graduate text. The older book is often considered to be the better of the two. Indeed, this book contains many references to the older book and some of its parts are actually abridged versions of the older book.

The book is about 400 pages and divided into two parts. The first part introduces elementary mathematical logic, that is, propositional and predicate calculus. The second part delves into logic and the foundations of mathematics (computability, decidablity, formal number theory, Turing machines, incompleteness, completeness etc.). There are many exercises and an extensive bibliography.

Since its appearance fifty years ago there have been myriad other introduction to logic texts published, many of which might be considered more accessible to undergraduates, such as Herbert Enderton’s A Mathematical Introduction to Logic. Nevertheless I find Kleene’s book to still be quite lucid introduction that might work well as a text for honors students.

When a book enters the pantheon of historical classics is not always clear, but certainly the author’s stature as one of the great logicians of the twentieth century makes the book’s exposition interesting in and of itself.


Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.

 

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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