Book:András Hajnal/Set Theory

András Hajnal and Peter Hamburger: Set Theory

Published $\text {1999}$, London Mathematical Society

ISBN 0-521-59667-X (translated by Attila Máté)

Subject Matter

• Set Theory

Contents

Preface (Piscataway, New Jersey, December 1998)

Part I. Introduction to set theory

Introduction

1. Notation, conventions

2. Definition of equivalence. The concept of cardinality. The Axiom of Choice

3. Countable cardinal, continuum cardinal

4. Comparison of cardinals

5. Operations with sets and cardinals

6. Examples

7. Ordered sets. Order types. Ordinals

8. Properties of wellordered ses. Good sets. The ordinal operation

9. Transfinite induction and recursion. Some consequences of the Axiom of Choice, the Wellordering Theorem

10. Definition of the cardinality operation. Properties of cardinalities. The cofinality operation

11. Properties of the power operation

Hints for solving problems marked with ${}^*$ in Part I

Appendix. An axiomatic development of set theory

Introduction

A1. The Zermelo-Fraenkel axiom system of set theory

A2. Definition of concepts; extension of the language

A3. A sketch of the development. Metatheorems

A4. A sketch of the development. Definitions of simple operations and properties (continued)

A5. A sketch of the development. Basic theorems, the introduction of $\omega$ and $\R$ (continued)

A6. The ZFC axiom system. A weakening of the Axiom of Choice. Remarks on the theorems of Sections 2-7

A7. The role of the Axiom of Regularity

A8. Proofs of relative consistency. The method of interpretation

A9. Proofs of relative consistency. The method of models

Part II. Topics in combinatorial set theory

12. Stationary sets

13. $\Delta$-systems

14. Ramsey's Theorem and its generalizations. Partition calculus

15. Inaccessible cardinals. Mahlo cardinals

16. Measurable cardinals

17. Real-valued measurable cardinals, saturated ideals

18. Weakly compact and Ramsey cardinals

19. Set mappings

20. The square-bracket symbol. Strengthenings of the Ramsey counterexamples

21. Properties of the power operation. Results on the singular cardinal problem

22. Powers of single cardinals. Shelah's Theorem

Hints for solving problems of Part II

Bibliography

List of symbols

Name index

Subject index

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