Computability and Logic

Recommendations

Info

150 MODELS vertices of a square, <strong>and</strong> the denotation of Q is the relation between vertices of being diagonally opposite. Show that U <strong>and</strong> V are isomorphic. 12.6 Consider a language with just one nonlogical symbol, a two-place predicate
Page 2:
This page intentionally left blank
Page 8:
Computability and Logic Fifth Editi
Page 12:
For SALLY and AIGLI and EDITH
Page 18:
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 76:
3 Turing Computability A function i
Page 80:
TURING COMPUTABILITY 25 carried out
Page 84:
TURING COMPUTABILITY 27 in general
Page 88:
TURING COMPUTABILITY 29 It will the
Page 92:
TURING COMPUTABILITY 31 But if ther
Page 96:
TURING COMPUTABILITY 33 A numerical
Page 100:
4 Uncomputability In the preceding
Page 104:
4.1. THE HALTING PROBLEM 37 machine
Page 108:
4.1. THE HALTING PROBLEM 39 Turing
Page 112:
4.2. THE PRODUCTIVITY FUNCTION 41 g
Page 116:
4.2. THE PRODUCTIVITY FUNCTION 43 F
Page 120:
5 Abacus Computability Showing that
Page 124:
5.1. ABACUS MACHINES 47 be thought
Page 128:
5.1. ABACUS MACHINES 49 Figure 5-4.
Page 132:
5.2. SIMULATING ABACUS MACHINES BY
Page 136:
Page 140:
Page 144:
5.3. THE SCOPE OF ABACUS COMPUTABIL
Page 148:
5.3. THE SCOPE OF ABACUS COMPUTABIL
Page 152:
PROBLEMS 61 Figure 5-17. Minimizati
Page 156:
6 Recursive Functions The intuitive
Page 160:
One might indicate this in shorthan
Page 164:
6.1. PRIMITIVE RECURSIVE FUNCTIONS
Page 168:
6.1. PRIMITIVE RECURSIVE FUNCTIONS
Page 172:
PROBLEMS 71 The total function f is
Page 176:
7 Recursive Sets and Relations In t
Page 180:
7.1. RECURSIVE RELATIONS 75 Given a
Page 184:
7.1. RECURSIVE RELATIONS 77 Proof:
Page 188:
7.1. RECURSIVE RELATIONS 79 So the
Page 192:
answer. Thus if S is a semidecidabl
Page 196:
7.3. FURTHER EXAMPLES 83 Proof: Sup
Page 200:
7.3. FURTHER EXAMPLES 85 after whic
Page 204:
PROBLEMS 87 7.9 Let f (n) bethenth
Page 208:
8.1. CODING TURING COMPUTATIONS 89
Page 212:
Page 216:
Page 220:
8.2. UNIVERSAL TURING MACHINES 95 P
Page 224:
PROBLEMS 97 relation of the restric
Page 228:
Basic Metalogic
Page 234:
102 APRÉCIS OF FIRST-ORDER LOGIC:
Page 238:
Page 242:
Page 246:
Page 250:
Page 254:
Page 258:
10 APrécis of First-Order Logic: S
Page 262:
Page 266:
Page 270:
Page 274:
Page 278: 124 APRÉCIS OF FIRST-ORDER LOGIC:
Page 282: 11 The Undecidability of First-Orde
Page 286: 128 THE UNDECIDABILITY OF FIRST-ORD
Page 306: 138 MODELS There are actually sever
Page 310: 140 MODELS If we also let 0 denote
Page 314: 142 MODELS But this is simply the c
Page 318: 144 MODELS the number of equivalenc
Page 322: 146 MODELS elements are isolated, a
Page 326: 148 MODELS (b) Any set of sentences
Page 332: PROBLEMS 151 12.12 Show that if two
Page 336: 13 The Existence of Models This cha
Page 340: 13.1. OUTLINE OF THE PROOF 155 13.3
Page 344: 13.3. THE SECOND STAGE OF THE PROOF
Page 348: 13.3. THE SECOND STAGE OF THE PROOF
Page 352: 13.4. THE THIRD STAGE OF THE PROOF
Page 356: 13.5. NONENUMERABLE LANGUAGES 163 w
Page 360: PROBLEMS 165 13.11 Let A be a sente
Page 364: 14.1. SEQUENT CALCULUS 167 equivale
Page 368: 14.1. SEQUENT CALCULUS 169 We natur
Page 372: 14.1. SEQUENT CALCULUS 171 conditio
Page 376: 14.9 Example. Proper use of the two
Page 380:
14.2. SOUNDNESS AND COMPLETENESS 17
Page 384:
14.2. SOUNDNESS AND COMPLETENESS 17
Page 388:
14.3. OTHER PROOF PROCEDURES AND HI
Page 392:
Page 396:
Page 400:
PROBLEMS 185 unformalized mathemati
Page 404:
15 Arithmetization In this chapter
Page 408:
15.1. ARITHMETIZATION OF SYNTAX 189
Page 412:
15.1. ARITHMETIZATION OF SYNTAX 191
Page 416:
Table 15-2. Gödel numbers of symbo
Page 420:
15.2. GÖDEL NUMBERS 195 Now s is t
Page 424:
PROBLEMS 197 where the presence of
Page 428:
16 Representability of Recursive Fu
Page 432:
16.1. ARITHMETICAL DEFINABILITY 201
Page 436:
Page 440:
Page 444:
16.2. MINIMAL ARITHMETIC AND REPRES
Page 448:
Page 452:
Page 456:
Another example is the proof of the
Page 460:
16.3. MATHEMATICAL INDUCTION 215 Am
Page 464:
PROBLEMS 217 and then second z ′
Page 468:
PROBLEMS 219 relation < 1 of Proble
Page 472:
17.1. THE DIAGONAL LEMMA AND THE LI
Page 476:
17.1. THE DIAGONAL LEMMA AND THE LI
Page 480:
17.2. UNDECIDABLE SENTENCES 225 Suc
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
Page 532:
19.2. SKOLEM NORMAL FORM 251 19.9 T
Page 536:
19.3. HERBRAND’S THEOREM 253 enum
Page 540:
19.4. ELIMINATING FUNCTION SYMBOLS
Page 544:
19.4. ELIMINATING FUNCTION SYMBOLS
Page 548:
PROBLEMS 259 the subset of T consis
Page 552:
20.1. CRAIG’S THEOREM AND ITS PRO
Page 556:
20.1. CRAIG’S THEOREM AND ITS PRO
Page 560:
20.3. BETH’S DEFINABILITY THEOREM
Page 564:
20.3. BETH’S DEFINABILITY THEOREM
Page 568:
PROBLEMS 269 if it is preceded by
Page 572:
21.1. SOLVABLE AND UNSOLVABLE DECIS
Page 576:
21.2. MONADIC LOGIC 273 the forms o
Page 580:
21.3. DYADIC LOGIC 275 to a quantif
Page 584:
21.3. DYADIC LOGIC 277 We then clai
Page 588:
22 Second-Order Logic Suppose that,
Page 592:
SECOND-ORDER LOGIC 281 (The foregoi
Page 596:
SECOND-ORDER LOGIC 283 22.6 Example
Page 600:
PROBLEMS 285 Problems 22.1 Does it
Page 604:
23.1. ARITHMETICAL DEFINABILITY AND
Page 608:
Page 612:
Page 616:
Page 620:
24 Decidability of Arithmetic witho
Page 624:
DECIDABILITY OF ARITHMETIC WITHOUT
Page 628:
DECIDABILITY OF ARITHMETIC WITHOUT
Page 632:
PROBLEMS 301 if there are rational
Page 636:
25.1. ORDER IN NONSTANDARD MODELS 3
Page 640:
25.1. ORDER IN NONSTANDARD MODELS 3
Page 644:
25.2. OPERATIONS IN NONSTANDARD MOD
Page 648:
Page 652:
Page 656:
25.3. NONSTANDARD MODELS OF ANALYSI
Page 660:
25.3. NONSTANDARD MODELS OF ANALYSI
Page 664:
PROBLEMS 317 (In general, there cou
Page 668:
26 Ramsey’s Theorem Ramsey’s th
Page 672:
26.1. RAMSEY’S THEOREM: FINITARY
Page 676:
26.2. K ÖNIG’S LEMMA 323 called
Page 680:
26.2. K ÖNIG’S LEMMA 325 The pre
Page 684:
27 Modal Logic and Provability Moda
Page 688:
27.1. MODAL LOGIC 329 There is a no
Page 692:
27.1. MODAL LOGIC 331 deducible fro
Page 696:
27.1. MODAL LOGIC 333 For Propositi
Page 700:
27.2. THE LOGIC OF PROVABILITY 335
Page 704:
27.3. THE FIXED POINT AND NORMAL FO
Page 708:
27.3. THE FIXED POINT AND NORMAL FO
Page 712:
Annotated Bibliography General Refe
Page 718:
344 INDEX categorical theory, see d
Page 722:
346 INDEX Gödel sentence, 225 Göd
Page 726:
348 INDEX Presburger, Max, see arit
Page 730:
350 INDEX valid sentence, 120, 327
show all

Computability and Logic

Create successful ePaper yourself

Delete template?

Save as template?