Computability and Logic

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INDEX 347 logical consequence, see consequence logical equivalence, see equivalence, logical logical symbols, 102 Löwenheim, Leopold, see Löwenheim–Behmann theorem, Löwenheim–Skolem theorem Löwenheim–Behmann theorem, 270; see also monadic logic Löwenheim–Skolem theorem, 137, 147ff, arithmetical, 305, 317f, and second-order logic, 279, 282, strong, 251, upward, 163 lower domain, 312 lower inequality, 297 lower part, 312 Lyndon, Roger, see Lyndon interpolation theorem Lyndon interpolation theorem, 268 machine table, 26 Maltsev (Malcev), A. I., see compactness theorem mathematical induction, proof by, see induction, mathematical matrix, 246 max function, 34 maximal principle, 163 McAloon, Kenneth, see Tennenbaum–McAloon theorem Mellema, Paul, xiii metalanguage, 121 metalogical, 120 min function, 34 minimal arithmetic, see Q minimal system of modal logic, see K minimization, 60f, 70ff, bounded, 77 modal logic, 123, 327ff modalized, 336 models, 137ff, existence of, 153ff, number of, 139ff, size of, 137, in modal logic, 329ff; see also epsilon model, non-standard model, standard model modus ponens, 328 monadic predicates and monadic logic, 272ff monadic represention of numbers, 24, modified, 63f monotone function, 98 mop-up, 54ff Mostowski, Andrzej, 341 multiplication, abacus computability of, 49, laws of, 218, Turing computability of, 29ff N *, see standard model of arithmetic N **, see standard model of analysis name, see constant natural language, 122f; see also Hilbert’s thesis necessity (□), 328 necessitation, 328 negation (∼), 75, 102, 107, 327 negation normal form, 243f n-generic, 291 nonconstructive proof, 182, 237f nonlogical symbols, 103 nonstandard models, of analysis, 312ff, of arithmetic, 150f, 302ff normal form, for sentences and sets, 243, conjunctive, 244, disjunctive, 244, full conjunctive and disjunctive, 245, for letterless sentences, 336ff, negation, 243f, prenex, 246, Skolem, 247f normal form, for terms, 297 nullity problem, 132 numerals, see base-b representation, monadic or tally representation object language, 121 objectual quantifier, 117 official and unofficial notation, see abbreviation omega-consistency and -inconsistency (ω-consistency and -inconsistency), 217, 226 omega-model (ω-model), 313 one-to-one function, 14 onto function, 14 open formula, 103, 112 open term, 103 ordinal numbers, 210 ordinary language, see natural language overspill, 147, 309 P (Peano arithmetic), 214–215 Padoa, Alessandro, see Padoa’s method Padoa’s method, 267 pairing function (J), 8f, 71 paradox, Berry, 227, Epimenides or liar, 106, 227, Grelling or heterological, 227, Richard, 21f, Russell’s, 285, Skolem, 252f parentheses, 102, 109; see also abbreviation parenthesis lemma, 109 Paris, Jeffrey, see Paris–Harrington theorem Paris–Harrington theorem, 325, 341 partition, 143; see also Ramsey’s theorem parity, 29 partial function, 7, recursive function, 71 Peano, Giuseppi, see P Peano arithmetic, see P Pendelbury, Michael J., xiii pi () notation, 69 places, 103 positively semi-decidable, see semi-decidable positively semi-definable, see semi-definable possibility (♦), 328 power, see exponentiation predecessor function, 69 predicate or relation symbol, 103, dyadic, 271, 275ff, identity, 103, 255ff, monadic, 272ff predicative and impredicative, 315f prefix, 246 premiss of a rule, 169 prenex normal form, 246 Presburger arithmetic, see arithmetic without multiplication
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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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Page 716: Index abacus (machine), 45ff, simul
Page 720: INDEX 345 difference function ( .
Page 726: 348 INDEX Presburger, Max, see arit
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