Overview

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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9781009737845

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Author Information

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).

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