- Page 4: Computability and Logic, Fifth Edit
- Page 10: CAMBRIDGE UNIVERSITY PRESS Cambridg
- Page 16: Contents Preface to the Fifth Editi
- Page 20: CONTENTS ix FURTHER TOPICS 19 Norma
- Page 26: xii PREFACE TO THE FIFTH EDITION on
- Page 32: Computability Theory
- Page 38: 4 ENUMERABILITY of the set being en
- Page 42: 6 ENUMERABILITY (second, fourth, si
- Page 46: 8 ENUMERABILITY (1, 1) —(1, 2) (1
- Page 50: 10 ENUMERABILITY 1.11 Example. Any
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12 ENUMERABILITY of base 10, we cou
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14 ENUMERABILITY If the intersectio
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2 Diagonalization In the preceding
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18 DIAGONALIZATION 1 2 3 4 s 1 s 1
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20 DIAGONALIZATION single infinite
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22 DIAGONALIZATION of sets. Strike
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24 TURING COMPUTABILITY out the com
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26 TURING COMPUTABILITY stroke, S 0
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28 TURING COMPUTABILITY 3.2 Example
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30 TURING COMPUTABILITY Example 3.4
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32 TURING COMPUTABILITY containing
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34 TURING COMPUTABILITY Problems 3.
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36 UNCOMPUTABILITY We now also want
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38 UNCOMPUTABILITY position with ou
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40 UNCOMPUTABILITY to say itself, d
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42 UNCOMPUTABILITY machine eventual
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44 UNCOMPUTABILITY if p(b) ≤ p(a)
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46 ABACUS COMPUTABILITY Historicall
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48 ABACUS COMPUTABILITY Figure 5-2.
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50 ABACUS COMPUTABILITY We remove a
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52 ABACUS COMPUTABILITY Figure 5-7.
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54 ABACUS COMPUTABILITY If it is sc
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56 ABACUS COMPUTABILITY Figure 5-12
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58 ABACUS COMPUTABILITY Three diffe
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60 ABACUS COMPUTABILITY Figure 5-16
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62 ABACUS COMPUTABILITY 5.7 Show th
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64 RECURSIVE FUNCTIONS (including z
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66 RECURSIVE FUNCTIONS Similarly, f
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68 RECURSIVE FUNCTIONS To put these
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70 RECURSIVE FUNCTIONS Proofs Examp
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72 RECURSIVE FUNCTIONS 6.3 Show tha
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74 RECURSIVE SETS AND RELATIONS dec
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76 RECURSIVE SETS AND RELATIONS whi
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78 RECURSIVE SETS AND RELATIONS Pro
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80 RECURSIVE SETS AND RELATIONS whe
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82 RECURSIVE SETS AND RELATIONS (b)
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84 RECURSIVE SETS AND RELATIONS 7.2
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86 RECURSIVE SETS AND RELATIONS so
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8 Equivalent Definitions of Computa
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90 EQUIVALENT DEFINITIONS OF COMPUT
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92 EQUIVALENT DEFINITIONS OF COMPUT
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94 EQUIVALENT DEFINITIONS OF COMPUT
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96 EQUIVALENT DEFINITIONS OF COMPUT
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98 EQUIVALENT DEFINITIONS OF COMPUT
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9 APrécis of First-Order Logic: Sy
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9.1. FIRST-ORDER LOGIC 103 The nonl
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9.1. FIRST-ORDER LOGIC 105 (For the
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9.2. SYNTAX 107 Thus the more or le
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9.2. SYNTAX 109 notation as the off
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9.2. SYNTAX 111 given formula to be
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PROBLEMS 113 9.3 Consider a languag
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10.1. SEMANTICS 115 The atomic sent
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10.1. SEMANTICS 117 yields two. So
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10.2. METALOGICAL NOTIONS 119 is tr
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10.2. METALOGICAL NOTIONS 121 (the
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PROBLEMS 123 can be said in a first
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PROBLEMS 125 10.14 Show that the fo
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11.1. LOGIC AND TURING MACHINES 127
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11.1. LOGIC AND TURING MACHINES 129
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11.1. LOGIC AND TURING MACHINES 131
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11.2. LOGIC AND PRIMITIVE RECURSIVE
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PROBLEMS 135 the decision problem f
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12 Models A model of a set of sente
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12.1. THE SIZE AND NUMBER OF MODELS
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12.1. THE SIZE AND NUMBER OF MODELS
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12.2. EQUIVALENCE RELATIONS 143 Nam
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12.2. EQUIVALENCE RELATIONS 145 b i
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12.3. THE LÖWENHEIM-SKOLEM AND COM
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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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14.3. OTHER PROOF PROCEDURES AND HI
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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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16.1. ARITHMETICAL DEFINABILITY 203
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16.1. ARITHMETICAL DEFINABILITY 205
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16.2. MINIMAL ARITHMETIC AND REPRES
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16.2. MINIMAL ARITHMETIC AND REPRES
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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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23.2. ARITHMETICAL DEFINABILITY AND
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23.2. ARITHMETICAL DEFINABILITY AND
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23.2. 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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25.2. OPERATIONS IN NONSTANDARD MOD
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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