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An Introduction to Gödel's Theorems

2nd Edition

$34.99 (X)


Part of Cambridge Introductions to Philosophy

  • Date Published: May 2013
  • availability: In stock
  • format: Paperback
  • isbn: 9781107606753

$34.99 (X)

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About the Authors
  • In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book – extensively rewritten for its second edition – will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.

    • Includes a wide coverage of related formal results, not all easily available in other textbooks
    • Provides optional material for follow-up work, to develop the learning of the interested teacher/student
    • Contains historical and philosophical commentary
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    Reviews & endorsements

    "Smith breathes new life into the work of Kurt Godel in this second edition … Recommended. Upper-division undergraduates through professionals."
    R. L. Pour, Choice

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    Product details

    • Edition: 2nd Edition
    • Date Published: May 2013
    • format: Paperback
    • isbn: 9781107606753
    • length: 402 pages
    • dimensions: 246 x 175 x 18 mm
    • weight: 0.79kg
    • availability: In stock
  • Table of Contents

    1. What Gödel's theorems say
    2. Functions and enumerations
    3. Effective computability
    4. Effectively axiomatized theories
    5. Capturing numerical properties
    6. The truths of arithmetic
    7. Sufficiently strong arithmetics
    8. Interlude: taking stock
    9. Induction
    10. Two formalized arithmetics
    11. What Q can prove
    12. I∆o, an arithmetic with induction
    13. First-order Peano arithmetic
    14. Primitive recursive functions
    15. LA can express every p.r. function
    16. Capturing functions
    17. Q is p.r. adequate
    18. Interlude: a very little about Principia
    19. The arithmetization of syntax
    20. Arithmetization in more detail
    21. PA is incomplete
    22. Gödel's First Theorem
    23. Interlude: about the First Theorem
    24. The Diagonalization Lemma
    25. Rosser's proof
    26. Broadening the scope
    27. Tarski's Theorem
    28. Speed-up
    29. Second-order arithmetics
    30. Interlude: incompleteness and Isaacson's thesis
    31. Gödel's Second Theorem for PA
    32. On the 'unprovability of consistency'
    33. Generalizing the Second Theorem
    34. Löb's Theorem and other matters
    35. Deriving the derivability conditions
    36. 'The best and most general version'
    37. Interlude: the Second Theorem, Hilbert, minds and machines
    38. μ-Recursive functions
    39. Q is recursively adequate
    40. Undecidability and incompleteness
    41. Turing machines
    42. Turing machines and recursiveness
    43. Halting and incompleteness
    44. The Church–Turing thesis
    45. Proving the thesis?
    46. Looking back.

  • Author

    Peter Smith, University of Cambridge
    Peter Smith was formerly Senior Lecturer in Philosophy at the University of Cambridge. His books include Explaining Chaos (1998) and An Introduction to Formal Logic (2003) and he is also a former editor of the journal Analysis.

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