Book:Paul Bernays/Axiomatic Set Theory/Second Edition
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Paul Bernays: Axiomatic Set Theory
Published $\text {1968}$, Dover Publications
- ISBN 0-486-66637-9
Subject Matter
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 $\text {I}$. The Frame of Logic and Class Theory
- 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 $\text {II}$. The Start of General Set Theory
- 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 $\text {III}$. Ordinals; Natural Numbers; Finite Sets
- Chapter $\text {IV}$. Transfinite Recursion
- 1. The General Recursion Theorem
- 2. The Schema of Transfinite Recursion
- 3. Generated Numeration
- Chapter $\text {IV}$. Transfinite Recursion
- Chapter $\text {V}$. Power; Order; Wellorder
- 1. Comparison of Powers
- 2. Order and Partial Order
- 3. Wellorder
- Chapter $\text {V}$. Power; Order; 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 $\text {VI}$. The Completing Axioms
- 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 $\text {VII}$. Analysis; Cardinal Arithmetic; 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
- Chapter $\text {VIII}$. Further Strengthening of the Axiom System
- Index of Authors (Part $\text {I}$)
- Index of Symbols (Part $\text {II}$)
- Predicates
- Functors and Operators
- Primitive Symbols
- Index of Symbols (Part $\text {II}$)
- Index of Matters (Part $\text {II}$)
- List of Axioms (Part $\text {II}$)
- Bibliography (Part $\text {I}$ and $\text {II}$)
Further Editions
- 1958: Paul Bernays: Axiomatic Set Theory