An Introduction to Gödel's Theorems
2nd Edition
Part of Cambridge Introductions to Philosophy
- Author: Peter Smith, University of Cambridge
- Date Published: February 2013
- availability: Out of stock in print form with no current plan to reprint
- format: Paperback
- isbn: 9781107606753
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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.
Read more- 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
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: February 2013
- format: Paperback
- isbn: 9781107606753
- length: 402 pages
- dimensions: 246 x 175 x 18 mm
- weight: 0.79kg
- availability: Out of stock in print form with no current plan to reprint
Table of Contents
Preface
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.
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