Computability and Logic

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CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521877527 © George S. Boolos, John P. Burgess, Richard C. Jeffrey 1974, 1980, 1990, 2002, 2007 This publication is in copyright. Subject to statutory exception <strong>and</strong> to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2007 ISBN-13 978-0-511-36668-0 eBook (EBL) ISBN-10 0-511-36668-X eBook (EBL) ISBN-13 978-0-521-87752-7 hardback ISBN-10 0-521-87752-0 hardback ISBN-13 978-0-521-70146-4 paperback ISBN-10 0-521-70146-5 paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, <strong>and</strong> does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
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Page 8: Computability and Logic Fifth Editi
Page 16: Contents Preface to the Fifth Editi
Page 20: CONTENTS ix FURTHER TOPICS 19 Norma
Page 26: xii PREFACE TO THE FIFTH EDITION on
Page 32: Computability Theory
Page 38: 4 ENUMERABILITY of the set being en
Page 42: 6 ENUMERABILITY (second, fourth, si
Page 46: 8 ENUMERABILITY (1, 1) —(1, 2) (1
Page 50: 10 ENUMERABILITY 1.11 Example. Any
Page 54: 12 ENUMERABILITY of base 10, we cou
Page 58: 14 ENUMERABILITY If the intersectio
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2 Diagonalization In the preceding
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18 DIAGONALIZATION 1 2 3 4 s 1 s 1
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20 DIAGONALIZATION single infinite
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22 DIAGONALIZATION of sets. Strike
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24 TURING COMPUTABILITY out the com
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26 TURING COMPUTABILITY stroke, S 0
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28 TURING COMPUTABILITY 3.2 Example
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30 TURING COMPUTABILITY Example 3.4
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32 TURING COMPUTABILITY containing
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34 TURING COMPUTABILITY Problems 3.
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36 UNCOMPUTABILITY We now also want
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38 UNCOMPUTABILITY position with ou
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40 UNCOMPUTABILITY to say itself, d
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42 UNCOMPUTABILITY machine eventual
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44 UNCOMPUTABILITY if p(b) ≤ p(a)
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46 ABACUS COMPUTABILITY Historicall
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48 ABACUS COMPUTABILITY Figure 5-2.
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50 ABACUS COMPUTABILITY We remove a
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52 ABACUS COMPUTABILITY Figure 5-7.
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54 ABACUS COMPUTABILITY If it is sc
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56 ABACUS COMPUTABILITY Figure 5-12
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58 ABACUS COMPUTABILITY Three diffe
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60 ABACUS COMPUTABILITY Figure 5-16
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62 ABACUS COMPUTABILITY 5.7 Show th
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64 RECURSIVE FUNCTIONS (including z
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66 RECURSIVE FUNCTIONS Similarly, f
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68 RECURSIVE FUNCTIONS To put these
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70 RECURSIVE FUNCTIONS Proofs Examp
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72 RECURSIVE FUNCTIONS 6.3 Show tha
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74 RECURSIVE SETS AND RELATIONS dec
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76 RECURSIVE SETS AND RELATIONS whi
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78 RECURSIVE SETS AND RELATIONS Pro
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80 RECURSIVE SETS AND RELATIONS whe
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82 RECURSIVE SETS AND RELATIONS (b)
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84 RECURSIVE SETS AND RELATIONS 7.2
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86 RECURSIVE SETS AND RELATIONS so
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8 Equivalent Definitions of Computa
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90 EQUIVALENT DEFINITIONS OF COMPUT
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9 APrécis of First-Order Logic: Sy
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9.1. FIRST-ORDER LOGIC 103 The nonl
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9.1. FIRST-ORDER LOGIC 105 (For the
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9.2. SYNTAX 107 Thus the more or le
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9.2. SYNTAX 109 notation as the off
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9.2. SYNTAX 111 given formula to be
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PROBLEMS 113 9.3 Consider a languag
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10.1. SEMANTICS 115 The atomic sent
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10.1. SEMANTICS 117 yields two. So
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10.2. METALOGICAL NOTIONS 119 is tr
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10.2. METALOGICAL NOTIONS 121 (the
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PROBLEMS 123 can be said in a first
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PROBLEMS 125 10.14 Show that the fo
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11.1. LOGIC AND TURING MACHINES 127
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11.2. LOGIC AND PRIMITIVE RECURSIVE
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PROBLEMS 135 the decision problem f
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12 Models A model of a set of sente
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12.1. THE SIZE AND NUMBER OF MODELS
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12.1. THE SIZE AND NUMBER OF MODELS
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12.2. EQUIVALENCE RELATIONS 143 Nam
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12.2. EQUIVALENCE RELATIONS 145 b i
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12.3. THE LÖWENHEIM-SKOLEM AND COM
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PROBLEMS 149 of completeness shows
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PROBLEMS 151 12.12 Show that if two
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13 The Existence of Models This cha
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13.1. OUTLINE OF THE PROOF 155 13.3
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13.3. THE SECOND STAGE OF THE PROOF
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13.3. THE SECOND STAGE OF THE PROOF
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13.4. THE THIRD STAGE OF THE PROOF
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13.5. NONENUMERABLE LANGUAGES 163 w
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PROBLEMS 165 13.11 Let A be a sente
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14.1. SEQUENT CALCULUS 167 equivale
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14.1. SEQUENT CALCULUS 169 We natur
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14.1. SEQUENT CALCULUS 171 conditio
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14.9 Example. Proper use of the two
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14.2. SOUNDNESS AND COMPLETENESS 17
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14.2. SOUNDNESS AND COMPLETENESS 17
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14.3. OTHER PROOF PROCEDURES AND HI
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PROBLEMS 185 unformalized mathemati
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15 Arithmetization In this chapter
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15.1. ARITHMETIZATION OF SYNTAX 189
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15.1. ARITHMETIZATION OF SYNTAX 191
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Table 15-2. Gödel numbers of symbo
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15.2. GÖDEL NUMBERS 195 Now s is t
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PROBLEMS 197 where the presence of
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16 Representability of Recursive Fu
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16.1. ARITHMETICAL DEFINABILITY 201
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16.2. MINIMAL ARITHMETIC AND REPRES
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Another example is the proof of the
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16.3. MATHEMATICAL INDUCTION 215 Am
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PROBLEMS 217 and then second z ′
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PROBLEMS 219 relation < 1 of Proble
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17.1. THE DIAGONAL LEMMA AND THE LI
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17.1. THE DIAGONAL LEMMA AND THE LI
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17.2. UNDECIDABLE SENTENCES 225 Suc
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17.3*. UNDECIDABLE SENTENCES WITHOU
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17.3*. UNDECIDABLE SENTENCES WITHOU
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PROBLEMS 231 every axiom of Q is a
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THE UNPROVABILITY OF CONSISTENCY 23
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THE UNPROVABILITY OF CONSISTENCY 23
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HISTORICAL REMARKS 237 From (2) and
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HISTORICAL REMARKS 239 In the cours
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19 Normal Forms A normal form theor
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19.1. DISJUNCTIVE AND PRENEX NORMAL
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19.2. SKOLEM NORMAL FORM 247 both s
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19.2. SKOLEM NORMAL FORM 249 take f
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19.2. SKOLEM NORMAL FORM 251 19.9 T
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19.3. HERBRAND’S THEOREM 253 enum
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19.4. ELIMINATING FUNCTION SYMBOLS
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19.4. ELIMINATING FUNCTION SYMBOLS
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PROBLEMS 259 the subset of T consis
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20.1. CRAIG’S THEOREM AND ITS PRO
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20.1. CRAIG’S THEOREM AND ITS PRO
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20.3. BETH’S DEFINABILITY THEOREM
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20.3. BETH’S DEFINABILITY THEOREM
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PROBLEMS 269 if it is preceded by
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21.1. SOLVABLE AND UNSOLVABLE DECIS
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21.2. MONADIC LOGIC 273 the forms o
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21.3. DYADIC LOGIC 275 to a quantif
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21.3. DYADIC LOGIC 277 We then clai
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22 Second-Order Logic Suppose that,
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SECOND-ORDER LOGIC 281 (The foregoi
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SECOND-ORDER LOGIC 283 22.6 Example
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PROBLEMS 285 Problems 22.1 Does it
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23.1. ARITHMETICAL DEFINABILITY AND
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24 Decidability of Arithmetic witho
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DECIDABILITY OF ARITHMETIC WITHOUT
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DECIDABILITY OF ARITHMETIC WITHOUT
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PROBLEMS 301 if there are rational
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25.1. ORDER IN NONSTANDARD MODELS 3
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25.1. ORDER IN NONSTANDARD MODELS 3
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25.2. OPERATIONS IN NONSTANDARD MOD
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25.3. NONSTANDARD MODELS OF ANALYSI
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25.3. NONSTANDARD MODELS OF ANALYSI
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PROBLEMS 317 (In general, there cou
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26 Ramsey’s Theorem Ramsey’s th
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26.1. RAMSEY’S THEOREM: FINITARY
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26.2. K ÖNIG’S LEMMA 323 called
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26.2. K ÖNIG’S LEMMA 325 The pre
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27 Modal Logic and Provability Moda
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27.1. MODAL LOGIC 329 There is a no
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27.1. MODAL LOGIC 331 deducible fro
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27.1. MODAL LOGIC 333 For Propositi
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27.2. THE LOGIC OF PROVABILITY 335
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27.3. THE FIXED POINT AND NORMAL FO
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27.3. THE FIXED POINT AND NORMAL FO
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Annotated Bibliography General Refe
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344 INDEX categorical theory, see d
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346 INDEX Gödel sentence, 225 Göd
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348 INDEX Presburger, Max, see arit
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350 INDEX valid sentence, 120, 327
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