This two-part book offers a rigorous yet accessible exploration of set theory and transfinite algebra, with a particular emphasis on the axiom of choice and its applications. Part I presents an informal axiomatic introduction to the foundations of set theory, including a detailed treatment of the axiom of choice and its equivalents, suitable for advanced undergraduates. Part II, aimed at graduate students and professional mathematicians, treats selected topics in transfinite algebra where the axiom of choice, in one form or another, is useful or even indispensable. The text features self-contained chapters for flexible use, and includes material rarely found in the literature, such as Tarski's work on complete lattices, Hamel's solution to Cauchy's functional equation, and Artin's resolution of Hilbert's 17th problem. Over 140 exercises, with full solutions provided in the Appendix, support active engagement and deeper understanding, making this a valuable resource for both independent study and course preparation.
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A two-part exploration of set theory and transfinite algebra, emphasizing the axiom of choice and its applications.
Introduction; Part I. Set Theory:
1. The axioms of set theory;
2.
Correspondences, mappings, and quotient sets;
3. Ordered sets;
4. Around the
axiom of choice;
5. Cardinals and ordinals;
6. First-order logic and the
axioms of set theory revisited;
7. Some excursions; Part II. Topics in
Transfinite Algebra:
8. Group and ring structures on non-empty sets;
9.
Orderable abelian groups and fields;
10. Subdirect decomposition of algebras;
11. Dependence relations, rank functions, and closure operators;
12.
Semisimple and injective modules;
13. The Jacobson radical of a ring;
14.
Artin's solution of Hilbert's 17th problem; Appendix. Solutions to exercises;
References; Index.
Thomas Müller, formerly of Queen Mary & Westfield College (London), is affiliated with the University of Vienna as a Professorial Research Fellow in the Department of Mathematics. His main research interests are in the areas of algebra, combinatorics, and geometry. He is the co-author of 'A Universal Construction for Groups Acting Freely on Real Trees' (2012).
