Book:András Hajnal/Set Theory

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



Source work progress
* : $2$. Definition of Equivalence. The Concept of Cardinality. The Axiom of Choice: Definition $2.2$
 * to be reviewed