Computability and Logic

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Index abacus (machine), 45ff, simulation of Turing machine by, 51ff abacus-computable function, 46ff abbreviation, 108ff accent ( ′ ), 64 Ackermann, Wilhelm, see Ackermann function Ackermann function, 84f A-correct, 290 Addison, John, see Addison’s theorem Addison’s theorem, 286, 289ff addition, abacus computability of, 48f, laws of, 218, Turing computability of, 29f address of a register, 46 ampersand (&), 102 analysis, 312ff, axiomatic, 314, predicative, 315 analytical sets, relations, functions, 284 antidiagonal sequence, 18, set, 18 Arabic numerals, see decimal representation argument(s) of a function, 4 Aristotle, 272 arithmetic (true arithmetic), 150, 207, non-standard models of, 150f, 302ff, undecidability of, 222, without addition, 295, without multiplication, 295; see also P, Q, R arithmetical classes, 287 arithmetical completeness theorem, see Solovay completeness theorem arithmetical definability, see arithmetical sets, arithmetical classes arithmetical equivalence of formulas, 205 arithmetical Löwenheim–Skolem theorem, 305, 317f arithmetical sets and relations, 199, 286ff arithmetical soundness theorem, 335 arithmetization of syntax, 187ff arrow (→), 102, 107f, 327 associative laws of addition and multiplication, 218, of conjunction and disjunction, 245 atomic formula, 107 atomic term, 107 avoidable appeal to Church’s thesis, 83 axiom of choice (AC), 163, 248, 341 axiom of enumerability, 281, of induction, 214, 283, 314, of infinity, 282 axioms of GL, 333, of K, 328, of P, 214–215, of Q, 207f, of R, 216, of S5, 340 axiom scheme, 214 axiomatizable theory, 191, finitely, 191 back-and-forth argument, 345 bars, 262 Barwise, Jon, 341 base-b representation of numbers, 11; see also binary, decimal, duodecimal base step in proof by induction, 109, 212–213 basic functions, 64 Behmann, Heinrich, see Löwenheim–Behmann theorem Benacerraf, Paul, xii Bernays, Paul, 233 Berry’s paradox, 227 Bertrand’s postulate, see Chebyshev’s theorem beta function (β-function) lemma, 203 Beth, Ewart W., see Beth definability theorem Beth definability theorem, 265ff biconditional (↔), 102, 108 binary representation of numbers, 11, 21, 89 Boole, George, 272 box (□, □), 328, 332 box of an abacus, 46 bound variables, 111, relettering of, 124 bounded minimization and maximization, 77 bounded quantification, 76 branch, 323 Büchi, J. R., see Turing–Büchi proof busy beaver problem, see productivity canonical domains, 142, 147 Cantor, Georg, 239; see also back-and-forth argument, Cantor’s theorem, diagonalization method, zig-zag enumeration Cantor’s theorem, 16ff caret (∧), 328 343
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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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Page 670: 320 RAMSEY’S THEOREM Figure 26-1.
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Page 682: 326 RAMSEY’S THEOREM which tells
Page 686: 328 MODAL LOGIC AND PROVABILITY Wha
Page 690: 330 MODAL LOGIC AND PROVABILITY 27.
Page 694: 332 MODAL LOGIC AND PROVABILITY The
Page 698: 334 MODAL LOGIC AND PROVABILITY Fro
Page 702: 336 MODAL LOGIC AND PROVABILITY The
Page 706: 338 MODAL LOGIC AND PROVABILITY dis
Page 710: 340 MODAL LOGIC AND PROVABILITY GL
Page 718: 344 INDEX categorical theory, see d
Page 722: 346 INDEX Gödel sentence, 225 Göd
Page 726: 348 INDEX Presburger, Max, see arit
Page 730: 350 INDEX valid sentence, 120, 327
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