Computability and Logic

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22 DIAGONALIZATION of sets. Strike them out to obtain an irredundant enumeration of all sets of positive integers that have definitions in English. Now consider the set of positive integers defined by the condition that a positive integer n is to belong to the set if and only if it does not belong to the nth set in the irredundant enumeration just described. This set does not appear in that enumeration. For it cannot appear at the nth place for any n, since there is a positive integer, namely n itself, that belongs to this set if and only if it does not belong to the nth set in the enumeration. Since this set does not appear in our enumeration, it cannot have a definition in English. And yet it does have a definition in English, and in fact we have just given such a definition in the preceding paragraph.
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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
Page 24: Preface to the Fifth Edition The or
Page 28: PREFACE TO THE FIFTH EDITION xiii T
Page 36: 1 Enumerability Our ultimate goal w
Page 40: 1.1. ENUMERABILITY 5 we might have
Page 44: 1.2. ENUMERABLE SETS 7 member of A,
Page 48: 1.2. ENUMERABLE SETS 9 The pair (m,
Page 52: 1.2. ENUMERABLE SETS 11 a pair G(n)
Page 56: 1.2. ENUMERABLE SETS 13 on Example
Page 60: PROBLEMS 15 the set of all positive
Page 64: DIAGONALIZATION 17 supposition must
Page 68: DIAGONALIZATION 19 has that much ti
Page 72: PROBLEMS 21 2.4 Show that the set o
Page 78: 24 TURING COMPUTABILITY out the com
Page 82: 26 TURING COMPUTABILITY stroke, S 0
Page 86: 28 TURING COMPUTABILITY 3.2 Example
Page 90: 30 TURING COMPUTABILITY Example 3.4
Page 94: 32 TURING COMPUTABILITY containing
Page 98: 34 TURING COMPUTABILITY Problems 3.
Page 102: 36 UNCOMPUTABILITY We now also want
Page 106: 38 UNCOMPUTABILITY position with ou
Page 110: 40 UNCOMPUTABILITY to say itself, d
Page 114: 42 UNCOMPUTABILITY machine eventual
Page 118: 44 UNCOMPUTABILITY if p(b) ≤ p(a)
Page 122: 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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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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11.2. 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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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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