Computability and Logic

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348 INDEX Presburger, Max, see arithmetic without multiplication preservation upwards and downwards, 164f prime decomposition, 13 primitive recursion, 58f, 67 primitive recursive functions, 67, 132ff, real numbers, 86, sets or relations, 73 print operation of a Turing machine, 26 probability measure, 301 product, see multiplication, pi notation productivity, of a Turing machine, 42 projection functions, 64 proof by contradiction, 170, 238 proof by induction, see induction proof procedure, 166ff proof theory, 179 provability logic, 387; see also GL provability predicate, 234, 335 provable (⊢), 191 power, see exponentiation Putnam, Hilary, xiii Q (minimal arithmetic), 207ff Qin Jiushao (Ch’in Chiu-shao), see Chinese remainder theorem quadruples of a Turing machine, 26 quantifier, 102, bounded, 76, existential, 102, 107, universal, 102, 107 Quine, Willard Van Orman, xiii quotient function, 12, 61 R (Robinson arithmetic), 216ff Rado, Tibor, see productivity Ramsey, Frank Plumpton, see Ramsey’s theorem Ramsey’s theorem, 319ff, glorified, 325, infinitary, 321 random access, 46 range of a function, 7 recursion, see primitive recursion recursion equations, 67 recursive function, 61, 71, set or relation, 73 recursively enumerable sets, 96ff; see also semi-recursive sets recursively inseparable sets, 98 reduct, 247 reflexive relation, 143 refutation, refutability, 167ff; see also inconsistent registers of an abacus, see box regular function or relation, 71 Reidhaar-Olson, Lisa, 336 relation, 73, 104 relation symbol or predicate, 103 relation variable, 279 relatively prime, 86 relativized quantifiers, 296 relettering bound variables, 124 remainder function, 12, 61 representable, 207ff Richard’s paradox, 21f right introduction rules, 170 right movement of a Turing machine, 26 right number, 89 Robinson, Abraham, see Robinson’s joint consistency theorem Robinson arithmetic, see R Robinson, Raphael, 341; see also R Robinson’s joint consistency theorem, 264f Rosser, J. Barkley, see Rosser sentence Rosser sentence, 225 rudimentary formula, 204; see also existentialrudimentary, universal-rudimentary rudimentary function, 206 rule of inference, 169 Russell, Bertrand, see Russell’s paradox, Whitehead–Russell definition of identity Russell’s paradox, 285 S5, 340 Sambin, Giovanni, 336; see also De Jongh–Sambin theorem Santa Claus, 235 satisfaction of a formula, 117ff satisfaction properties, 153f satisfiable sentence or set, 120 Scanlon, T. M., xiii scanned symbol, 25 scheme, see axiom scheme Schur’s theorem, 325 scoring function, 40f second graph principle, 96 second incompleteness theorem, 232ff second-order logic, 279ff section of a relation, see horizontal section, vertical section secures (⇒), 167f Segerberg, Krister, see Segerberg completeness theorem Segerberg completeness theorem, 334 semantics of first-order logic, 114ff, distinguished from syntax, 106 semi-decidable set or relation, effectively, 80, recursively, see semi-recursive set or relation semi-definable set or relation, 218 semi-recursive set or relation, 80ff; see also recursively enumerable sentence or closed formula, of first-order logic, 103, 112, of modal logic, 328, of second-order logic, 280 sentence letter, 103, 107, 114, 327 sentential logic, 301, 327 sequents and sequent calculus, 166ff Shoenfield, J. R., 341 sigma () notation, 69 signature of an equivalence relation, 143, 151 size-n set, 320 Skolem, Thoralf, see Löwenheim–Skolem theorem, Skolem normal form
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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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5.3. 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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8.2. 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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10 APrécis of First-Order Logic: S
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11 The Undecidability of First-Orde
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128 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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17 Indefinability, Undecidability,
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222 INDEFINABILITY, UNDECIDABILITY,
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18 The Unprovability of Consistency
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234 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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258 NORMAL FORMS if P A holds of a
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20 The Craig Interpolation Theorem
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262 THE CRAIG INTERPOLATION THEOREM
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21 Monadic and Dyadic Logic We have
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272 MONADIC AND DYADIC LOGIC associ
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274 MONADIC AND DYADIC LOGIC Our in
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276 MONADIC AND DYADIC LOGIC For th
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278 MONADIC AND DYADIC LOGIC Svu is
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280 SECOND-ORDER LOGIC just sets, o
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282 SECOND-ORDER LOGIC is true. But
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284 SECOND-ORDER LOGIC Proof: A fir
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23 Arithmetical Definability Tarski
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288 ARITHMETICAL DEFINABILITY Proof
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290 ARITHMETICAL DEFINABILITY 23.4
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292 ARITHMETICAL DEFINABILITY Case
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294 ARITHMETICAL DEFINABILITY A-cor
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296 DECIDABILITY OF ARITHMETIC WITH
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25 Nonstandard Models By a model of
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304 NONSTANDARD MODELS Note that a
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306 NONSTANDARD MODELS the proof of
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308 NONSTANDARD MODELS Before givin
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310 NONSTANDARD MODELS in P, but to
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312 NONSTANDARD MODELS And suppose
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314 NONSTANDARD MODELS The axiomati
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316 NONSTANDARD MODELS But while th
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318 NONSTANDARD MODELS for the cons
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320 RAMSEY’S THEOREM Figure 26-1.
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