Verfügbarkeit: Logic, computers, and sets

Logic, computers, and sets:

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Bibliographische Detailangaben
1. Verfasser:	Wang, Hao (VerfasserIn)
Format:	Buch
Sprache:	Undetermined
Veröffentlicht:	New York Chelsea 1970
Ausgabe:	Repr.
Schlagworte:	Menge Mathematische Logik
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	Früher u.d.T.: Wang, Hao: A survey of mathematical logic
Beschreibung:	X, 651 S.

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Datensatz im Suchindex

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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
adam_txt	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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spellingShingle	Wang, Hao Wang, Hao Logic, computers, and sets Menge (DE-588)4038613-2 gnd Mathematische Logik (DE-588)4037951-6 gnd
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title	Logic, computers, and sets
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title_auth	Logic, computers, and sets
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title_exact_search_txtP	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
title_sort	logic computers and sets
topic	Menge (DE-588)4038613-2 gnd Mathematische Logik (DE-588)4037951-6 gnd
topic_facet	Menge Mathematische Logik
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015111066&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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