Computability and Logic

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17.2. UNDECIDABLE SENTENCES 225 Such a G T is called a Gödel sentence for T , and such an R T a Rosser sentence for T .ThusaGödel sentence ‘says of itself that’ it is unprovable, and a Rosser sentence ‘says of itself that’ if there is a witness to its provability, then there is an earlier witness to its disprovability. 17.8 Theorem. Let T be a consistent, axiomatizable extension of Q. Then a Rosser sentence for T is undecidable for T . Proof: Suppose the Rosser sentence R T is provable in T . Then there is some a that witnesses the provability of R T . Since T is consistent, ∼R T is not also provable, and so no m witnesses the disprovability of R T , and in particular, no m < n does so. It follows that the rudimentary sentence Prf T ( R T , n)&∼∃z
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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
Page 430: 200 REPRESENTABILITY OF RECURSIVE F
Page 470: 17 Indefinability, Undecidability,
Page 474: 222 INDEFINABILITY, UNDECIDABILITY,
Page 478: 224 INDEFINABILITY, UNDECIDABILITY,
Page 484: 17.3*. UNDECIDABLE SENTENCES WITHOU
Page 488: 17.3*. UNDECIDABLE SENTENCES WITHOU
Page 492: PROBLEMS 231 every axiom of Q is a
Page 496: THE UNPROVABILITY OF CONSISTENCY 23
Page 500: THE UNPROVABILITY OF CONSISTENCY 23
Page 504: HISTORICAL REMARKS 237 From (2) and
Page 508: HISTORICAL REMARKS 239 In the cours
Page 516: 19 Normal Forms A normal form theor
Page 520: 19.1. DISJUNCTIVE AND PRENEX NORMAL
Page 524: 19.2. SKOLEM NORMAL FORM 247 both s
Page 528: 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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