Computability and Logic

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346 INDEX Gödel sentence, 225 Gödel–Berry formula, 228 Gödel–Chaitin formula, 228 Gödel–Grelling formula, 227 Gödel–Rosser sentence, 226 Gödel-style proof of Church’s theorem, 126, 132ff Goldfarb, Warren, xiii graph principle, first, 82, second, 96 graph relation of a function, 75 greatest common divisor, 86 Grelling or heterological paradox, 227 Guenthner, Franz, 341 halting, of a Turing machine, 26, in standard configuration or position, 32, 91 halting function, 38f halting problem, 40 Hare, Caspar, xiii Harrington, Leo, see Paris–Harrington theorem Henkin, Leon, 285; see also Henkin axioms, Henkin sentence Henkin axioms, 162, 164 Henkin sentence, 235 Herbrand, Jacques, see Herbrand–Church theorem, Herbrand’s theorem Herbrand’s theorem, 253ff heterological or Grelling paradox, 227 Hilbert, David, 238; see also Hilbert’s thesis Hilbert’s thesis, 185 Hindu–Arabic numerals, see decimal representation homogeneous set, 320 horizontal section, 86 identifying nodes of a flow chart, 43 identity function(s), 5, 57, 64 identity of indiscernibles, 280 identity relation, 104, Whitehead–Russell definition of, 281 identity symbol, 103, elimination of, 255ff implication, logical, 101 implicit definability, 266 impredicative and predicative, 315f incompleteness of second-order logic, see Gödel completeness theorem, failure for second-order logic incompleteness theorems, see first incompleteness theorem, second incompleteness theorem inconsistent sentence or set, 148, theory, 191, in modal logic, 328 individual symbol, see constant individual variable, 278; see also variable induction, mathematical, proof by, 212– 213, complete, 213 induction axioms, 214, second-order, 283 induction hypothesis, 109 induction on complexity, proof by, 109ff induction scheme, 214 induction step in a proof, 109, 213 infinitary Ramsey’s theorem, 321 infinity, axiom of, 282 instance of a formula, 112 interpolant, 261 interpolation theorem, see Craig interpolation theorem, Lyndon interpolation theorem interpretation of a language, 102, 103f, in modal logic, 327 inverse function, 14 inversion lemma, 179f, 186 irrefutable, see consistent isomorphism, 139ff isomorphism lemma, 140 isomorphism type, 142 J, see pairing function junction, see conjunction, disjunction K (minimal system of modal logic), 328ff Kant’s theorem, 269 Kleene normal form theorem, 94 Kleene, Steven Cole, 341; see also Kleene normal form theorem, Kleene’s theorem Kleene’s theorem, 82 Kochen, Simon, xiii König’s lemma, 322ff Kreisel, Georg, see Tennenbaum–Kreisel theorem Kripke, Saul, xi; see also Kripke completeness theorem Kripke completeness theorems, 329ff L ∗ , see language of arithmetic L ∗∗ , see language of analysis Lagrange’s theorem, 204, 213 Lambek, Joachim, see abacus Lambek machine, see abacus language, 103, empty, 103, meta-, 121, natural, 122f, non-enumerable, 162f, object, 121, of analysis, 312, of arithmetic, 103 leapfrog routine, 31 least common multiple, 86 least-number principle, 214 left introduction rules, 170 left movement of a Turing machine, 26 left number, 89 Leibniz’s law, 280 length function, 80 letterless sentence, 336, normal form theorem for, 336ff level of a tree, 322f Lewis, David, xiii liar or Epimenides paradox, 106 linear order, 151f lines or steps, 168 Löb, M. H., see Löb’s theorem Löb’s theorem, 236 logarithm functions, 79
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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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Page 716: Index abacus (machine), 45ff, simul
Page 720: INDEX 345 difference function ( .
Page 726: 348 INDEX Presburger, Max, see arit
Page 730: 350 INDEX valid sentence, 120, 327
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