# Book:András Hajnal/Set Theory

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## András Hajnal and Peter Hamburger:

## András Hajnal and Peter Hamburger: *Set Theory*

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

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

### Subject Matter

### 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

## Source work progress

- 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $2$. Definition of Equivalence. The Concept of Cardinality. The Axiom of Choice: Definition $2.2$

- to be reviewed