Logic, computers, and sets
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New York
Chelsea
1970
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Datensatz im Suchindex
| _version_ | 1819765882654556160 |
|---|---|
| adam_text | CONTENTS
Preface Hi
PART ONE GENERAL SKETCHES
CHAPTER I. The Axiomatic Method 1
§ 1. Geometry and axiomatic systems 1
§ 2. The problem of adequacy 5
§ 3. The problem of evidence 8
§ 4. A very elementary system L 14
§ 5. The theory of non negative integers 19
§ 6. Godel s theorems 23
§ 7. Formal theories as applied elementary logics 28
CHAPTER II. Eighty Years of Foundational Studies 34
§ 1. Analysis, reduction and formalization 34
§ 2. Anthropologism 39
§ 3. Finitism 41
§ 4. Intuitionism 43
§ 5. Predicativism: standard results on number as being 44
§6. Predicativism: predicative analysis and beyond 46
§7. Platonism 48
§ 8. Logic in the narrower sense 53
§ 9. Applications 56
CHAPTER III. On Formalization 57
§ 1. Systematization 57
§ 2. Communication 58
§ 3. Clarity and consolidation 59
§4. Rigour 60
§ 5. Approximation to intuition 61
§ 6. Application to philosophy 63
§ 7. Too many digits 64
§ 8. Ideal language 65
§ 9. How artificial a language? 66
§ 10. The paradoxes 67
v
CHAPTER IV. The Axiomatization of Arithmetic 68
§ 1. Introduction 68
§ 2. Grassmann s calculus 69
§3. Dedekind s letter 73
§ 4. Dedekind s essay 74
§ 5. Adequacy of Dedekind s characterization 77
§6. Dedekind and Frege 79
CHAPTER V. Computation 82
§ 1. The concept of computability 82
§ 2. General recursive functions 89
§ 3. The Friedberg Mucnik theorem 93
§ 4. Metamathematics 97
§ 5. Symbolic logic and calculating machines 100
§ 6. The control of errors in calculating machines 107
PART TWO CALCULATING MACHINES
CHAPTER VI. A Variant to Turing s Theory of Calculating
Machines 127
§ 1. Introduction 127
§ 2. The basic machine B 128
§ 3. All recursive functions are 5 computable 133
§ 4. Basic instructions 144
§ 5. Universal Turing machines 150
§ 6. Theorem proving machines 154
CHAPTER VII. Universal Turing Machines: An Exercise in Coding 160
CHAPTER VIII. The Logic of Automata (with A. W. Burks) .... 175
§ 1. Introduction 175
§ 2. Automata and nets 175
§ 3. Transition matrices and matrix form nets 202
§ 4. Cycles, nets, and quantifiers 214
CHAPTER IX. Toward Mechanical Mathematics 224
§ 1. Introduction 224
§ 2. The propositional calculus (system P) 229
§ 3. Program /: the propositional calculus P 231
vi
§ 4. Program //: selecting theorems in the prepositional calculus .. 234
§ 5. Completeness and consistency of the system P and P, 236
§ 6. The system Pe: the propositional calculus with equality 237
§ 7. Preliminaries to the predicate calculus 238
§ 8. The system Qp and the AE predicate calculus 240
§ 9. Program /// 243
§ 10. Systems Qq and Qr alternative formulations of the AE
predicate calculus 245
§ 11. System Q: the whole predicate calculus with equality 248
§ 12. Conclusions 253
Appendices I—VII
CHAPTER X. Circuit Synthesis by Solving Sequential Boolean
Equations 269
§ 1. Summary of problems and results 269
§ 2. Sequential Boolean functional and equations 270
§ 3. The method of sequential tables 272
§ 4. Deterministic solutions 274
§5. Related problems 279
§ 6. An effective criterion of general solvability 281
§ 7. A sufficient condition for effective solvability 286
§ 8. An effective criterion of effective solvability 290
§ 9. The normal form (5) of sequential Boolean equations 294
§ 10. Apparently richer languages 299
§ 11. Turing machines and growing automata 301
PART THREE FORMAL NUMBER THEORY
CHAPTER XI. The Predicate Calculus 307
§ 1. The propositional calculus 307
§ 2. Formulations of the predicate calculus 309
§ 3. Completeness of the predicate calculus 317
CHAPTER XII. Many Sorted Predicate Calculi 322
§ 1. One sorted and many sorted theories 322
§ 2. The many sorted elementary logics Ln 326
§ 3. The theorem (I) and the completeness of L» 328
§ 4. Proof of the theorem (IV) 329
vii
CHAPTER XIII. The Arithmetization of Metamathematics 334
§ 1. Godel numbering 334
§ 2. Recursive functions and the system Z 342
§ 3. Bernays lemma 345
§ 4. Arithmetic translations of axiom systems 352
CHAPTER XIV. Ackeemann s Consistency Proof 362
§ 1. The system Za 362
§ 2. Proof of finiteness 366
§ 3. Estimates of the substituents 370
§ 4. Interpretation of nonfinitist proofs 372
CHAPTER XV. Partial Systems of Number Theory 376
§ 1. Skolem s non standard model for number theory 376
§ 2. Some applications of formalized consistency proofs 379
PART FOUR IMPREDICATIVE SET THEORY
CHAPTER XVI. Different Axiom Systems 383
§ 1. The paradoxes 383
§ 2. Zerrnelo s set theory 388
§ 3. The Bernays set theory 394
§ 4. The theory of types, negative types, and new foundations .. 402
§ 5. A formal system of logic 415
§ 6. The systems of Ackermann and Frege 423
CHAPTER XVII. Relative Strength and Reducibility 432
§ 1. Relation between P and Q 432
§ 2. Finite axiomatization 436
§ 3. Finite sets and natural numbers 439
CHAPTER XVIII. Truth Definitions and Consistency Proofs 443
§ 1. Introduction 443
§ 2. A truth definition for Zermelo set theory 445
§ 3. Remarks on the construction of truth definitions in general .. 455
§ 4. Consistency proofs via truth definitions 459
§5. Relativity of number theory and in particular of induction .... 466
§ 6. Explanatory remarks 473
viii
CHAPTER XIX. Between Number Theory and Set Theory 478
§ 1. General set theory 480
§ 2. Predicative set theory 489
§ 3. Impredicative collections and ^ consistency 497
CHAPTER XX. Some Partial Systems 507
§ 1. Some formal details on class axioms 507
§ 2. A new tlieory of element and number 515
§ 3. Set theoretical basis for real numbers 525
§ 4. Functions of real variables 532
PART FIVE PREDICATIVE SET THEORY
CHAPTER XXI. Certain Predicates Defined by Induction Schemata 535
CHAPTER XXII. Undecidable Sentences Suggested by Semantic
Paradoxes 546
§ 1. Introduction 546
§ 2. Preliminaries 547
§ 3. Conditions which the set theory is to satisfy 549
§ 4. The Epimenides paradox 552
§ 5. The Richard paradox 554
§ 6. Final remarks 557
CHAPTER XXIII. The Formalization of Mathematics 559
§ 1. Original sin of the formal logician 559
§ 2. Historical perspective 559
§3. What is a set? 561
§ 4. The indenumerable and the impredicative 562
§ 5. The limitations upon formalization 564
§ 6. A constructive theory 565
§ 7. The denumerability of all sets 567
§8. Consistency and adequacy 569
§ 9. The axiom of reducibility 574
§ 10. The vicious circle principle 576
§ 11. Predicative sets and constructive ordinals 578
§ 12. Concluding remarks 581
CHAPTER XXIV. Some Formal Details on Predicative Set Theories 585
§ 1. The underlying logic 585
ix
§ 2. The axioms of the theory 2 589
§ 3. Preliminary considerations 593
§ 4. The theory of non negative integers 597
§ 5. The enumerability of all sets 601
§ 6. Consequences of the enumerations 606
§ 7. The theory of real numbers 608
§ 8. Intuitive models 611
§ 9. Proofs of consistency 614
§ 10. The system R 619
CHAPTER XXV. Ordinal Numbers and Predicative Set Theory .... 624
§ 1. Systems of notation for ordinal numbers 625
§ 2. Strongly effective systems 627
§ 3. The Church Kleene class B and a new class C 632
§ 4. Partial Herbrand recursive functions 637
§ 5. Predicative set theory 639
§ 6. Two tentative definitions of predicative sets 646
§ 7. System H: the hyperarithmetic set theory 648
x
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| indexdate | 2024-12-23T19:46:30Z |
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| title | Logic, computers, and sets |
| title_alt | Survey of mathematical logic |
| title_auth | Logic, computers, and sets |
| title_exact_search | Logic, computers, and sets |
| title_full | Logic, computers, and sets Hao Wang |
| title_fullStr | Logic, computers, and sets Hao Wang |
| title_full_unstemmed | Logic, computers, and sets Hao Wang |
| title_short | Logic, computers, and sets |
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