Provability, Computability and Reflection (eBook)

Front Cover 1
Foundations of Set Theory 4
Copyright Page 5
Contents 6
Preface 9
CHAPTER I. THE ANTINOMIES 12
§ 1. HISTORICAL INTRODUCTION 12
§ 2. LOGICAL ANTINOMIES 16
§ 3. SEMANTICAL ANTINOMIES 20
§ 4. GENERAL REMARKS 22
§ 5. THE THREE CRISES 25
§ 6. BIBLIOGRAPHICAL REMARKS 27
CHAPTER II. AXIOMATIC FOUNDATIONS OF SET THEORY. THE AXIOM OF CHOICE 30
§ 1. INTRODUCTION 30
§ 2. THE PRIMITIVE RELATION. EQUALITY AND EXTENSIONALITY 38
§ 3. “CONSTRUCTIVE” AXIOMS OF GENERAL SET THEORY 44
§ 4. THE AXIOM OF CHOICE 55
§ 5. AXIOMS OF INFINITY AND OF RESTRICTION 92
§ 6. THE AXIOM SYSTEM OF VON NEUMANN. THE SET-THEORETICAL RELATIVISM (Paradox of Skolem) 107
§ 7. THE AXIOM SYSTEMS OF BERNAYS AND GÖDEL. RELATIVE CONSISTENCY PROOFS 121
§ 8. DERIVATION OF SET THEORY FROM THE AXIOMS 135
CHAPTER III. TYPE-THEORETICAL APPROACHES 147
§ 1. THE IDEAL CALCULUS 147
§ 2. THE GENERAL THEORY OF CLASSES 151
§ 3. QUINE’S NEW FOUNDATIONS 155
§ 4. QUINE’S MATHEMATICAL LOGIC 157
§ 5. THE HIERARCHY OF LANGUAGES AND THE RAMIFIED CLASS CALCULUS 161
§ 6. WANG’S SYSTEME S 164
§ 7. LORENZEN’S OPERATIONIST SYSTEM 169
§ 8. THE LOGICISTIC THESIS 171
§ 9. TYPES, CATEGORIES, AND SORTS 179
§ 10. IMPREDICATIVE CONCEPT FORMATION 185
§ 11. SET THEORIES BASED UPON NON-STANDARD LOGICS 193
CHAPTER IV. INTUITIONISTIC CONCEPTIONS OF MATHEMATICS 207
§ 1. HISTORICAL INTRODUCTION. THE ABYSS BETWEEN DISCRETENESS AND CONTINUITY 207
§ 2. THE CONSTRUCTIVE CHARACTER OF MATHEMATICS. MATHEMATICS AND LANGUAGE 218
§ 3. THE PRINCIPLE OF THE EXCLUDED MIDDLE 226
§ 4. MATHEMATICS AND LOGIC. LOGICAL CALCULUS 238
§ 5. THE PRIMORDIAL INTUITION OF INTEGERS. FREE-CHOICE SEQUENCES AND BROUWER’S CONCEPT OF SET 255
§ 6. MATHEMATICS AS TRIMMED ACCORDING TO THE INTUITIONISTIC ATTITUDE 264
CHAPTER V. METAMATHEMATICAL AND SEMANTICAL APPROACHES 276
§ 1. THE HILBERT PROGRAM 276
§ 2. FORMAL SYSTEMS, LOGISTIC SYSTEMS, AND FORMALIZED THEORIES 281
§ 3. INTERPRETATIONS AND MODELS 291
§ 4. CONSISTENCY, COMPLETENESS, CATEGORICALNESS, AND INDEPENDENCE 296
§ 5. DECIDABILITY AND RECURSIVENESS. ARITHMETIZATION OF SYNTAX 308
§ 6. THE LIMITATIVE THEOREMS OF GÖDEL, TARSKI, CHURCH AND THEIR GENERALIZATIONS 314
§ 7. THE METAMATHEMATICS AND SEMANTICS OF SET THEORY 327
§ 8. PHILOSOPHICAL REMARKS 343
BIBLIOGRAPHY 360
INDEX OF AUTHORS 417
INDEX OF TERMS AND SYMBOLS 424
ERRATA 427