Computability and Logic

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PROBLEMS 125 10.14 Show that the following pairs are equivalent: (a) ∀xF(x)&∀yG(y) and ∀u(F(u)&G(u)). (b) ∀xF(x) ∨∀yG(y) and ∀u∀v(F(u) ∨ G(v)). (c) ∃xF(x)&∃yG(y) and ∃u∃v(F(u)&G(v)). (d) ∃xF(x) ∨∃yG(y) and ∃u(F(u) ∨ G(u)). [In (a), it is to be understood that u may be a variable not occurring free in ∀xF(x) or∀yG(y); in particular, if x and y are the same variable, u may be that same variable. In (b) it is to be understood that u and v may be any distinct variables not occurring free in ∀xF(x) ∨∀yG(y); in particular, if x does not occur in free in ∀yG(y) and y does not occur free in ∀xF(x), then u may be x and y may be v. Analogously for (d) and (c).]
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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
Page 228: Basic Metalogic
Page 234: 102 APRÉCIS OF FIRST-ORDER LOGIC:
Page 258: 10 APrécis of First-Order Logic: S
Page 284: 11.1. LOGIC AND TURING MACHINES 127
Page 296: 11.2. LOGIC AND PRIMITIVE RECURSIVE
Page 300: PROBLEMS 135 the decision problem f
Page 304: 12 Models A model of a set of sente
Page 308: 12.1. THE SIZE AND NUMBER OF MODELS
Page 312: 12.1. THE SIZE AND NUMBER OF MODELS
Page 316: 12.2. EQUIVALENCE RELATIONS 143 Nam
Page 320: 12.2. EQUIVALENCE RELATIONS 145 b i
Page 324: 12.3. THE LÖWENHEIM-SKOLEM AND COM
Page 328: PROBLEMS 149 of completeness shows
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PROBLEMS 151 12.12 Show that if two
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13 The Existence of Models This cha
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13.1. OUTLINE OF THE PROOF 155 13.3
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13.3. THE SECOND STAGE OF THE PROOF
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13.3. THE SECOND STAGE OF THE PROOF
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13.4. THE THIRD STAGE OF THE PROOF
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13.5. NONENUMERABLE LANGUAGES 163 w
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PROBLEMS 165 13.11 Let A be a sente
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14.1. SEQUENT CALCULUS 167 equivale
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14.1. SEQUENT CALCULUS 169 We natur
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14.1. SEQUENT CALCULUS 171 conditio
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14.9 Example. Proper use of the two
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14.2. SOUNDNESS AND COMPLETENESS 17
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14.2. SOUNDNESS AND COMPLETENESS 17
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14.3. OTHER PROOF PROCEDURES AND HI
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PROBLEMS 185 unformalized mathemati
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15 Arithmetization In this chapter
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15.1. ARITHMETIZATION OF SYNTAX 189
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15.1. ARITHMETIZATION OF SYNTAX 191
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Table 15-2. Gödel numbers of symbo
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15.2. GÖDEL NUMBERS 195 Now s is t
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PROBLEMS 197 where the presence of
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16 Representability of Recursive Fu
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16.1. ARITHMETICAL DEFINABILITY 201
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16.2. MINIMAL ARITHMETIC AND REPRES
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Another example is the proof of the
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16.3. MATHEMATICAL INDUCTION 215 Am
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PROBLEMS 217 and then second z ′
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PROBLEMS 219 relation < 1 of Proble
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17.1. THE DIAGONAL LEMMA AND THE LI
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17.1. THE DIAGONAL LEMMA AND THE LI
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17.2. UNDECIDABLE SENTENCES 225 Suc
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17.3*. UNDECIDABLE SENTENCES WITHOU
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17.3*. UNDECIDABLE SENTENCES WITHOU
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PROBLEMS 231 every axiom of Q is a
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THE UNPROVABILITY OF CONSISTENCY 23
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THE UNPROVABILITY OF CONSISTENCY 23
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HISTORICAL REMARKS 237 From (2) and
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HISTORICAL REMARKS 239 In the cours
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19 Normal Forms A normal form theor
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19.1. DISJUNCTIVE AND PRENEX NORMAL
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19.2. SKOLEM NORMAL FORM 247 both s
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19.2. SKOLEM NORMAL FORM 249 take f
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19.2. SKOLEM NORMAL FORM 251 19.9 T
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19.3. HERBRAND’S THEOREM 253 enum
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19.4. ELIMINATING FUNCTION SYMBOLS
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19.4. ELIMINATING FUNCTION SYMBOLS
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PROBLEMS 259 the subset of T consis
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20.1. CRAIG’S THEOREM AND ITS PRO
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20.1. CRAIG’S THEOREM AND ITS PRO
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20.3. BETH’S DEFINABILITY THEOREM
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20.3. BETH’S DEFINABILITY THEOREM
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PROBLEMS 269 if it is preceded by
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21.1. SOLVABLE AND UNSOLVABLE DECIS
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21.2. MONADIC LOGIC 273 the forms o
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21.3. DYADIC LOGIC 275 to a quantif
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21.3. DYADIC LOGIC 277 We then clai
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22 Second-Order Logic Suppose that,
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SECOND-ORDER LOGIC 281 (The foregoi
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SECOND-ORDER LOGIC 283 22.6 Example
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PROBLEMS 285 Problems 22.1 Does it
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23.1. ARITHMETICAL DEFINABILITY AND
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24 Decidability of Arithmetic witho
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DECIDABILITY OF ARITHMETIC WITHOUT
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DECIDABILITY OF ARITHMETIC WITHOUT
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PROBLEMS 301 if there are rational
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25.1. ORDER IN NONSTANDARD MODELS 3
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25.1. ORDER IN NONSTANDARD MODELS 3
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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
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