Book:András Hajnal/Set Theory

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András Hajnal and Peter Hamburger: Set Theory

Published $1999$, London Mathematical Society

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

Subject Matter


Preface (Piscataway, New Jersey, December 1998)
Part I. Introduction to set theory
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
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
List of symbols
Name index
Subject index