# An Introduction to Gödel's Theorems

Cambridge University Press, Feb 21, 2013 - Mathematics - 388 pages
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.

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I have read the first edition of this title and have found it to be one of the best, if not _the_ best introductions to the topic (possibly second only to Boolos et. al. "Computability and Logic". I am looking forward to the 2nd Edition.
Please note that at present (October 2013) the 'View Ebook' link on the page for the 2nd edition takes you to the purchase page for the *1st* edition ebook. Caveat Emptor.
Google, when this was pointed out to them, suggested I contact Peter Smith to get the issue corrected.

### Contents

 Functions and enumerations 8 Effective computability 14 Effectively axiomatized theories 25 Capturing numerical properties 36 The truths of arithmetic 46 Induction 56 What Q can prove 71 Firstorder Peano Arithmetic 90
 Speedup 201 Incompleteness and lsaacsons Thesis 219 Godels Second Theorem For PA 233 On the unprovability of consistency 239 Generalizing the Second Theorem 245 Lobs Theorem and other matters 252 Deriving the derivability conditions 258 The Second Theorem Hilbert minds and machines 272

 l5 LA can express every p r function 113 A very little about Principia 130 The arithmetization of syntax 136 2O Arithmetization in more detail 144 PA is incomplete 152 Godels First Theorem 161 About the First Theorem 167 The Diagonalization Lemma 177 Rossers proof 185 Broadening the scope 191 Tarskis Theorem 197
 u Recu rsive fu nctions 285 Q is recursively adequate 297 Turing machines 310 Turing machines and recursiveness 321 Halting and incompleteness 328 The ChurchTu ring Thesis 338 Proving the Thesis? 348 Looking back 367 Index 383 Copyright