A monograph containing a historical introduction by A. A. Fraenkel to the original Zermelo-Fraenkel form of set-theoretic axiomatics, and Paul Bernays’ independent presentation of a formal system of axiomatic set theory. No special knowledge of set thory and its axiomatics is required. With indexes of authors, symbols and matters, a list of axioms and an extensive bibliography.

Table of Contents for Axiomatic Set Theory

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, GÖDEI"
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)

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