# Book:Paul Bernays/Axiomatic Set Theory/Second Edition

## Paul Bernays: Axiomatic Set Theory

Published $\text {1968}$, Dover Publications

ISBN 0-486-66637-9

### Subject Matter

Axiomatic Set Theory

### Contents

Preface

PART $\text {I}$. HISTORICAL INTRODUCTION by Abraham Fraenkel:
$1$. 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 Restriction
$6$. Development of Set-Theory from the Axioms of Z
$7$. Remarks on the Axiom Systems of von Neumann, Bernays, Gödel

PART $\text {II}$. AXIOMATIC SET THEORY
Introduction
Chapter $\text {I}$. 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 $\text {II}$. 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 $\text {III}$. Ordinals; Natural Numbers; Finite Sets
1. Fundaments of the Theory of Ordinals
2. Existential Statements on Ordinals. Limit Numbers
3. Fundamentals of Number Theory
4. Iteration. Primitive Recursion
5. Finite Sets and Classes
Chapter $\text {IV}$. Transfinite Recursion
1. The General Recursion Theorem
2. The Schema of Transfinite Recursion
3. Generated Numeration
Chapter $\text {V}$. Power; Order; Wellorder
1. Comparison of Powers
2. Order and Partial Order
3. Wellorder
Chapter $\text {VI}$. 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 $\text {VII}$. 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 $\text {VIII}$. 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 $\text {I}$)
Index of Symbols (Part $\text {II}$)
Predicates
Functors and Operators
Primitive Symbols
Index of Matters (Part $\text {II}$)
List of Axioms (Part $\text {II}$)
Bibliography (Part $\text {I}$ and $\text {II}$)