A Book of Set Theory

by Charles C. Pinter

Suitable for upper-level undergraduates, this accessible approach to set theory poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. Starting with a repetition of the familiar arguments of elementary set theory, the level of abstract thinking gradually rises for a progressive increase in complexity.
A historical introduction presents a brief account of the growth of set theory, with special emphasis on problems that led to the development of the various systems of axiomatic set theory. Subsequent chapters explore classes and sets, functions, relations, partially ordered classes, and the axiom of choice. Other subjects include natural and cardinal numbers, finite and infinite sets, the arithmetic of ordinal numbers, transfinite recursion, and selected topics in the theory of ordinals and cardinals. This updated edition features new material by author Charles C. Pinter.

Genres: Mathematics, Nonfiction, Logic

256 pages, Paperback

First published September 18, 2013

Reviews

[WarpDrive · Rating 5 out of 5]

This is an excellent textbook perfectly suitable for upper-level undergraduates.
Very informative, comprehensive and generally accurate, this books starts with the basic conceptual apparatus of set theory, which is then progressively developed into an array of ever more sophisticated topics, resolutely getting into post-graduate territory in the final sections of the book (when more advanced topics such as inaccessible ordinals, the relationship between the cumulative hierarchy V and the constructible universe L, elements of model theory, and issues of consistency and independence, are all treated by the author).

Well written, clear and succinct, not infested with too many typos (will I ever find an advanced mathematics book that is completely typo-free?), this is one of the best books on this utterly fascinating subject that I have found so far. The presentation of some critical concepts is also enriched with relevant historical background and interesting philosophical insights.

I also appreciated that almost all theorems are provided with a comprehensive proof (in some cases, though, some intermediate steps are not fully explained, therefore some "interpretation" or further analysis is required), and that the succession of topics is always logically coherent, and it follows a very natural progression.
The way the ordinals are treated is absolutely first-class, a brilliant examples of conceptual lucidity and intelligibility. I also particularly enjoyed the chapter where the conceptual equivalence Axiom of Choice -> Hausdorff's Maximal Principle -> Zorn's Lemma -> Well-ordering Theorem -> Axiom of Choice is explained by the author.

An excellent choice for anybody who is interested in pursuing a good level of understanding (at upper-level undergraduate depth) of this critically important subject that represents one of the conceptual foundations of modern mathematics. Yes, set theory can be a subtle, slippery realm that can quickly becomes highly technical when more sophisticated topics are studied, but this is a fundamental subject matter, representing an absolute "must-know" if really you want to comprehend the ultimate conceptual underpinning of the beautiful world of mathematics - no meaningful philosophy of mathematics can be discussed without a prior decent level of understanding of set theory.

4.5 stars, rounded up to 5 - a must-have book, a very enjoyable read, a very solid introduction to be also kept for future reference.

[Arttu · Rating 5 out of 5]

This book offers a comprehensive journey through axiomatic set theory, showing how much of mathematics can be built from a remarkably small list of axioms. After an intriguing historical introduction, it covers all the elementary topics of the subject: sets and proper classes, relations, and ways in which relations order classes. Even a "simple" topic such as the natural numbers is reconstructed formally before the author discusses larger structures and the fundamental notions of cardinal and ordinal numbers. With this machinery in place, the final chapter returns to the historical thread by sketching theorems on the consistency and independence of the Axiom of Choice and the Continuum Hypothesis.

I enjoyed the book because it offers a great deal to readers with no previous background in axiomatic set theory. The author's engaging style continually places new concepts in a broader mathematical context. The text not only strengthened my intuition for set theory – so important throughout mathematics – but also deepened my understanding of mathematics as a language and of the limits of that language. For these reasons I can see myself returning to the book to refresh my grasp of the fundamentals. For now I give it five stars, subject to revision after a second reading.

[Hyunrae Cho]

Some good examples but not rigorous.

[Roberto Carrasco Betancourth · Rating 5 out of 5]

Amazing book.