An Introduction to Gödel's Theorems (Google eBook)

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Cambridge University Press, Jul 26, 2007 - Mathematics
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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 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.
  

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Review: An Introduction to Gödel's Theorems

User Review  - Usman - Goodreads

Very Bstract, dense, but if you stick to it, you will reap the benefits !! Read full review

Contents

1 What Godels Theorems say
1
2 Decidability and enumerability
8
3 Axiomatized formal theories
17
4 Capturing numerical properties
28
5 The truths of arithmetic
37
6 Sufficiently strong arithmetics
43
Taking stock
47
8 Two formalized arithmetics
51
21 Using the Diagonalization Lemma
175
22 Secondorder arithmetics
186
Incompleteness and
199
24 Gbdels Second Theorem for PA
212
25 The derivability conditions
222
26 Deriving the derivability conditions
232
27 Reflections
240
About the Second Theorem
252

9 What Q can prove
58
93 Adding g to Q
61
10 Firstorder Peano Arithmetic
71
11 Primitive recursive functions
83
expat 1H s 4min A
98
12 Capturing pr functions
99
13 Q is pr adequate
106
A very little about Principia
118
15 The arithmetization of syntax
124
16 PA is incomplete
138
17 G6dels First Theorem
147
About the First Theorem
153
19 Strengthening the First Theorem
162
20 The Diagonalization Lemma
169
29 uRecursive functions
265
30 Undecidability and incompleteness
277
31 Turing machines
287
32 Turing machines and recursiveness
298
BEBE19
301
BBB
302
33 Halting problems
305
34 The ChurchTuring Thesis
315
35 Proving the Thesis?
324
36 Looking back
342
Further reading
344
Bibliography
346
Copyright

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Page 3 - following it up right back to the primitive truths. If, in carrying out this process, we come only on general logical laws and on definitions, then the truth is an analytic one.

About the author (2007)

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

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