# An Introduction to Gödel's Theorems

Cambridge University Press, Jul 26, 2007 - Mathematics
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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User Review  - Usman - Goodreads

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

#### Review: An Introduction to Gödel's Theorems

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

### Popular passages

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.