# Computability and Logic

Cambridge University Press, Mar 4, 2002 - Philosophy - 356 pages
This fourth edition of one of the classic logic textbooks has been thoroughly revised by John Burgess. The aim is to increase the pedagogical value of the book for the core market of students of philosophy and for students of mathematics and computer science as well. This book has become a classic because of its accessibility to students without a mathematical background, and because it covers not simply the staple topics of an intermediate logic course such as Godel's Incompleteness Theorems, but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has now enhanced the book by adding a selection of problems at the end of each chapter, and by reorganising and rewriting chapters to make them more independent of each other and thus to increase the range of options available to instructors as to what to cover and what to defer.

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### Contents

 Enumerability 3 12 Enumerable Sets 7 Diagonalization 16 Turing Computability 23 Uncomputability 35 42 The Productivity Function 40 Abacus Computability 45 52 Simulating Abacus Machines by Turing Machines 51
 152 Godel Numbers 192 153 More Godel Numbers 196 Representability of Recursive Functions 199 162 Minimal Arithmetic and Representability 207 163 Mathematical Induction 212 164 Robinson Arithmetic 215 Indefinability Undecidability Incompleteness 221 172 Undecidable Sentences 225

 53 The Scope of Abacus Computability 57 Recursive Functions 63 62 Minimization 70 Recursive Sets and Relations 73 72 Semirecursive Relations 80 73 Further Examples 83 Equivalent Definitions of Computability 88 82 Universal luring Machines 94 83 Recursively Enumerable Sets 96 A Precis of FirstOrder Logic Syntax 101 92 Syntax 106 A Precis of FirstOrder Logic Semantics 114 102 Metalogical Notions 119 The Undecidability of FirstOrder Logic 126 112 Logic and Primitive Recursive Functions 132 Models 137 122 Equivalence Relations 142 123 The LowenheimSkolem and Compactness Theorems 146 The Existence of Models 153 132 The First Stage of the Proof 156 133 The Second Stage of the Proof 157 134 The Third Stage of the Proof 160 135 Nonenumerable Languages 162 Proofs and Completeness 166 142 Soundness and Completeness 174 143 Other Proof Procedures and Hilberts Thesis 179 Arithmetization 187
 173 Undecidable Sentences without the Diagonal Lemma 227 The Unprovability of Consistency 233 Normal Forms 243 192 Skolem Normal Form 247 193 Herbrands Theorem 253 194 Eliminating Function Symbols and Identity 255 The Craig Interpolation Theorem 260 202 Robinsons Joint Consistency Theorem 264 203 Beths Definability Theorem 265 Monadic and Dyadic Logic 270 212 Monadic Logic 273 213 Dyadic Logic 275 SecondOrder Logic 279 Arithmetical Definability 286 232 Arithmetical Definability and Forcing 289 Decidability of Arithmetic without Multiplication 295 Nonstandard Models 302 252 Operations in Nonstandard Models 306 253 Nonstandard Models of Analysis 312 Ramseys Theorem 319 262 Konigs Lemma 322 Modal Logic and Provability 327 272 The Logic of Provability 334 273 The Fixed Point and Normal Form Theorems 337 Hints for Selected Problems 341 Index 349 Copyright