Axiomatic Set Theory

Since the beginning of the twentieth century, set theory, which began with Euclid's Elements and was revived in the nineteenth century, has become increasingly important in almost all areas of mathematics and logic. In Part I of this excellent monograph, A. A. Fraenkel presents an introduction to the original Zermelo-Fraenkel form of set-theoretic axiomatics and a history of its subsequent development.In Part II Paul Bernays offers an independent presentation of a formal system of axiomatic set theory, covering such topics as the frame of logic and class theory, general set theory, transfinite recursion, completing axioms, cardinal arithmetic, and strengthening of the axiom system.The book is directed to the reader who has some acquaintance with problems of axiomatics and with the standard methods of mathematical logic. No special knowledge of set theory and its axiomatics is presupposed. Readers will find the critical apparatus at the back of the book especially helpful. It includes indexes of authors, symbols and matters, a list of axioms, and an extensive bibliography.

Table of Contents:
PREFACE PART I. HISTORICAL INTRODUCTION 1. INTRODUCTORY REMARKS 2. ZERMELO'S SYSTEM. EQUALITY AND EXTENSIONALITY 3. "CONSTRUCTIVE" AXIOMS OF "GENERAL" SET THEORY" 4. THE AXIOM OF CHOICE 5. AXIOMS OF INFINITY AND OF RESTRICTION 6. DEVELOPMENT OF SET-THEORY FROM THE AXIOMS OF Z 7. "REMARKS ON THE AXIOM SYSTEMS OF VON NEUMANN, BERNAYS, GODEI" PART II. AXIOMATIC SET THEORY INTRODUCTION CHAPTER I. THE FRAME OF LOGIC AND CLASS THEORY 1. Predicate Calculus; Class Terms and Descriptions; Explicit Definitions 2. Equality and Extensionality. Application to Descriptions 3. Class Formalism. Class Operations 4. Functionality and Mappings CHAPTER II. THE START OF GENERAL SET THEORY 1. The Axioms of General Set Theory 2. Aussonderungstheorem. Intersection 3. Sum Theorem. Theorem of Replacement 4. Functional Sets. One-to-one Correspondences CHAPTER III. ORDINALS; NATURAL NUMBERS; FINITE SETS 1. Fundaments of the Theory of Ordinals 2. Existential Statements on Ordinals. Limit Numbers 3. Fundaments of Number Theory 4. Iteration. Primitive Recursion 5. Finite Sets and Classes CHAPTER IV. TRANSFINITE RECURSION 1. The General Recursion Theorem 2. The Schema of Transfinite Recursion 3. Generated Numeration CHAPTER V. POWER; ORDER; WELLORDER 1. Comparison of Powers 2. Order and Partial Order 3. Wellorder CHAPTER VI. THE COMPLETING AXIOMS 1. The Potency Axiom 2. The Axiom of Choice 3. The Numeration Theorem. First Concepts of Cardinal Arithmetic 4. Zorn's Lemma and Related Principles 5. Axiom of Infinity. Denumerability CHAPTER VII. ANALYSIS; CARDINAL ARITHMETIC; ABSTRACT THEORIES 1. Theory of Real Numbers 2. Some Topics of Ordinal Arithmetic 3. Cardinal Operations 4. Formal Laws on Cardinals 5. Abstract Theories CHAPTER VIII. FURTHER STRENGTHENING OF THE AXIOM SYSTEM 1. A Strengthening of the Axiom of Choice 2. The Fundierungsaxiom 3. A one-to-one Correspondence between the Class of Ordinals and the Class of all Sets INDEX OF AUTHORS (PART I) INDEX OF SYMBOLS (PART II) Predicates Functors and Operators Primitive Symbols INDEX OF MATTERS (PART II) LIST OF ATOMS (PART II) BIBLIOGRAPHY (PART I AND II)