Computability and Logic

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24 Decidability of Arithmetic without Multiplication Arithmetic is not decidable: the set V of code numbers of sentences of the language L of arithmetic that are true in the st<strong>and</strong>ard interpretation is not recursive (nor even arithmetical). But for some sublanguages L* of L, if we consider the elements of V that are code numbers of sentences of L*, then the set V* of such elements is recursive: arithmetic without some of the symbols of its language is decidable. A striking case is Presburger arithmetic, or arithmetic without multiplication. The present chapter is entirely devoted to proving its decidability. We have used (true) arithmetic to mean the set of sentences of the language of arithmetic L ={0,
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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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Page 570: 21 Monadic and Dyadic Logic We have
Page 574: 272 MONADIC AND DYADIC LOGIC associ
Page 578: 274 MONADIC AND DYADIC LOGIC Our in
Page 582: 276 MONADIC AND DYADIC LOGIC For th
Page 586: 278 MONADIC AND DYADIC LOGIC Svu is
Page 590: 280 SECOND-ORDER LOGIC just sets, o
Page 594: 282 SECOND-ORDER LOGIC is true. But
Page 598: 284 SECOND-ORDER LOGIC Proof: A fir
Page 602: 23 Arithmetical Definability Tarski
Page 606: 288 ARITHMETICAL DEFINABILITY Proof
Page 610: 290 ARITHMETICAL DEFINABILITY 23.4
Page 614: 292 ARITHMETICAL DEFINABILITY Case
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Page 624: DECIDABILITY OF ARITHMETIC WITHOUT
Page 628: DECIDABILITY OF ARITHMETIC WITHOUT
Page 632: PROBLEMS 301 if there are rational
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Page 668: 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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