# An Introduction to Gödel's Theorems

Cambridge University Press, 26 lug 2007
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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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### Sommario

 Decidability and enumerability 8 Axiomatized formal theories 17 Capturing numerical properties 28 The truths of arithmetic 37 Sufficiently strong arithmetics 43 Two formalized arithmetics 51 Firstorder Pea no Arithmetic 71 Primitive recursive functions 83
 Using the Diagonalization Lemma 175 Secondorder arithmetics 186 Incompleteness and Isaacsons conjecture 199 Godels Second Theorem for PA 212 The derivability conditions 222 Deriving the derivability conditions 232 Reflections 240 About the Second Theorem 252

 Capturing p r functions 99 Q is p r adequate 106 A very little about Principia 118 The arithmetization of syntax 124 PA is incomplete 138 Godels First Theorem 147 About the First Theorem 153 Strengthening the First Theorem 162 2O The Diagonalization Lemma 169
 pRecursive functions 265 3O Undecidability and incompleteness 277 Turing machines 287 Turing machines and recursiveness 298 Halting problems 305 The ChurchTuring Thesis 315 Looking back 342 Index 356 Copyright

### Informazioni sull'autore (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.