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Computability and Logic Fifth Editi
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For SALLY and AIGLI and EDITH
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viii CONTENTS BASIC METALOGIC 9 APr
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Preface to the Fifth Edition The or
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PREFACE TO THE FIFTH EDITION xiii T
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1 Enumerability Our ultimate goal w
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1.1. ENUMERABILITY 5 we might have
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1.2. ENUMERABLE SETS 7 member of A,
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1.2. ENUMERABLE SETS 9 The pair (m,
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1.2. ENUMERABLE SETS 11 a pair G(n)
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1.2. ENUMERABLE SETS 13 on Example
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PROBLEMS 15 the set of all positive
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DIAGONALIZATION 17 supposition must
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DIAGONALIZATION 19 has that much ti
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PROBLEMS 21 2.4 Show that the set o
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3 Turing Computability A function i
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TURING COMPUTABILITY 25 carried out
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TURING COMPUTABILITY 27 in general
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TURING COMPUTABILITY 29 It will the
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TURING COMPUTABILITY 31 But if ther
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TURING COMPUTABILITY 33 A numerical
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4 Uncomputability In the preceding
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4.1. THE HALTING PROBLEM 37 machine
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4.1. THE HALTING PROBLEM 39 Turing
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4.2. THE PRODUCTIVITY FUNCTION 41 g
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4.2. THE PRODUCTIVITY FUNCTION 43 F
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5 Abacus Computability Showing that
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5.1. ABACUS MACHINES 47 be thought
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5.1. ABACUS MACHINES 49 Figure 5-4.
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5.2. SIMULATING ABACUS MACHINES BY
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5.2. SIMULATING ABACUS MACHINES BY
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5.2. SIMULATING ABACUS MACHINES BY
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5.3. THE SCOPE OF ABACUS COMPUTABIL
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5.3. THE SCOPE OF ABACUS COMPUTABIL
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PROBLEMS 61 Figure 5-17. Minimizati
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6 Recursive Functions The intuitive
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One might indicate this in shorthan
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6.1. PRIMITIVE RECURSIVE FUNCTIONS
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6.1. PRIMITIVE RECURSIVE FUNCTIONS
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PROBLEMS 71 The total function f is
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7 Recursive Sets and Relations In t
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7.1. RECURSIVE RELATIONS 75 Given a
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7.1. RECURSIVE RELATIONS 77 Proof:
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7.1. RECURSIVE RELATIONS 79 So the
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answer. Thus if S is a semidecidabl
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7.3. FURTHER EXAMPLES 83 Proof: Sup
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7.3. FURTHER EXAMPLES 85 after whic
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PROBLEMS 87 7.9 Let f (n) bethenth
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8.1. CODING TURING COMPUTATIONS 89
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8.1. CODING TURING COMPUTATIONS 91
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8.1. CODING TURING COMPUTATIONS 93
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8.2. UNIVERSAL TURING MACHINES 95 P
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PROBLEMS 97 relation of the restric
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Basic Metalogic
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102 APRÉCIS OF FIRST-ORDER LOGIC:
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104 APRÉCIS OF FIRST-ORDER LOGIC:
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106 APRÉCIS OF FIRST-ORDER LOGIC:
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108 APRÉCIS OF FIRST-ORDER LOGIC:
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110 APRÉCIS OF FIRST-ORDER LOGIC:
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112 APRÉCIS OF FIRST-ORDER LOGIC:
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10 APrécis of First-Order Logic: S
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116 APRÉCIS OF FIRST-ORDER LOGIC:
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118 APRÉCIS OF FIRST-ORDER LOGIC:
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120 APRÉCIS OF FIRST-ORDER LOGIC:
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122 APRÉCIS OF FIRST-ORDER LOGIC:
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124 APRÉCIS OF FIRST-ORDER LOGIC:
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11 The Undecidability of First-Orde
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128 THE UNDECIDABILITY OF FIRST-ORD
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130 THE UNDECIDABILITY OF FIRST-ORD
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132 THE UNDECIDABILITY OF FIRST-ORD
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134 THE UNDECIDABILITY OF FIRST-ORD
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136 THE UNDECIDABILITY OF FIRST-ORD
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138 MODELS There are actually sever
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140 MODELS If we also let 0 denote
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142 MODELS But this is simply the c
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144 MODELS the number of equivalenc
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146 MODELS elements are isolated, a
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148 MODELS (b) Any set of sentences
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150 MODELS vertices of a square, an
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152 MODELS Such a model A will cons
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154 THE EXISTENCE OF MODELS (S7) If
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156 THE EXISTENCE OF MODELS 13.6 Le
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158 THE EXISTENCE OF MODELS (E3) If
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160 THE EXISTENCE OF MODELS which i
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162 THE EXISTENCE OF MODELS And ind
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164 THE EXISTENCE OF MODELS Problem
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14 Proofs and Completeness Introduc
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168 PROOFS AND COMPLETENESS true.
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170 PROOFS AND COMPLETENESS Table 1
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172 PROOFS AND COMPLETENESS is beca
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174 PROOFS AND COMPLETENESS 14.13 E
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176 PROOFS AND COMPLETENESS new int
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178 PROOFS AND COMPLETENESS whether
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180 PROOFS AND COMPLETENESS possibl
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182 PROOFS AND COMPLETENESS Proofs:
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184 PROOFS AND COMPLETENESS In addi
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186 PROOFS AND COMPLETENESS Unless
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188 ARITHMETIZATION in which, as el
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190 ARITHMETIZATION 15.5 Corollary
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192 ARITHMETIZATION 15.7 Corollary.
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194 ARITHMETIZATION This means that
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196 ARITHMETIZATION For the first c
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198 ARITHMETIZATION 15.2 Let Ɣ be
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200 REPRESENTABILITY OF RECURSIVE F
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202 REPRESENTABILITY OF RECURSIVE F
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204 REPRESENTABILITY OF RECURSIVE F
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206 REPRESENTABILITY OF RECURSIVE F
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208 REPRESENTABILITY OF RECURSIVE F
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210 REPRESENTABILITY OF RECURSIVE F
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212 REPRESENTABILITY OF RECURSIVE F
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214 REPRESENTABILITY OF RECURSIVE F
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216 REPRESENTABILITY OF RECURSIVE F
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218 REPRESENTABILITY OF RECURSIVE F
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17 Indefinability, Undecidability,
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222 INDEFINABILITY, UNDECIDABILITY,
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224 INDEFINABILITY, UNDECIDABILITY,
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226 INDEFINABILITY, UNDECIDABILITY,
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228 INDEFINABILITY, UNDECIDABILITY,
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230 INDEFINABILITY, UNDECIDABILITY,
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18 The Unprovability of Consistency
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234 THE UNPROVABILITY OF CONSISTENC
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236 THE UNPROVABILITY OF CONSISTENC
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238 THE UNPROVABILITY OF CONSISTENC
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Further Topics
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244 NORMAL FORMS Proof: The proof i
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246 NORMAL FORMS 19.3 Theorem (Full
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248 NORMAL FORMS For (1) and (3.1)
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250 NORMAL FORMS can be the domains
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252 NORMAL FORMS as ‘set (of natu
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254 NORMAL FORMS Now if Ɣ is satis
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256 NORMAL FORMS in a sense to be m
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- Page 550: 20 The Craig Interpolation Theorem
- Page 554: 262 THE CRAIG INTERPOLATION THEOREM
- Page 558: 264 THE CRAIG INTERPOLATION THEOREM
- Page 562: 266 THE CRAIG INTERPOLATION THEOREM
- Page 566: 268 THE CRAIG INTERPOLATION THEOREM
- Page 570: 21 Monadic and Dyadic Logic We have
- Page 574: 272 MONADIC AND DYADIC LOGIC associ
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- Page 582: 276 MONADIC AND DYADIC LOGIC For th
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- Page 620: 24 Decidability of Arithmetic witho
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25.2. OPERATIONS IN NONSTANDARD MOD
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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