Sets and Transfinite Algebra

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).

• 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.