on April 4, 2000
The only reason I won't say it's THE BEST introduction to set theory is that I haven't read ALL such introductions. I am (obviously) a student of logic and I worked my way through the whole book a few years ago. It is an insightful development of set theory, both as a foundation for mathematics and a distinctive mathematical discipline in its own right. Set theory can be developed from a "naive" or an "axiomatic" perspective. The naive approach simply asks the reader to accept arguments about sets on the basis of informed intuition, whereas the axiomatic approach relies on showing how mathematical proofs can be formalized as deductions from a precise axiom system. Enderton's book deftly combines both approaches ; axiomatic considerations are isolated from the rest of the text and identified by a stripe running down the side of the page. Those who are not interested in axioms can avoid dealing with them almost entirely, but enthusiasts of formal rigor (like me!) won't be disappointed either. The axioms, which comprise a system known as Zermelo Fraenkel set theory with Choice, are introduced as needed in the overall development (so Replacement Axioms aren't mentioned until page 179). The text develops relations and functions as well as natural and real number systems, and then goes on to cardinals, orderings, and ordinals. I particularly enjoyed Enderton's well-motivated exposition of ordinals, which clearly shows how these numbers measure the lengths of well-orderings. His treatment of cardinals, transfinite induction, and the Axiom of Choice, is enlightening as well. A final chapter, which includes cofinality and inaccessible cardinals, should whet the student's appetite for further study in set theory. I have a hard time thinking of anything negative to say about this book. Perhaps it would be better if its nicely annotated bibliography were a bit more extensive. If you wanna learn set theory, buy this book!
on February 23, 2010
This is a solid book about set theory. The presentation is rigorous. Yet, unlike many textbooks, Enderton presented the material in a approachable way: the author explains the reason behind various conventions, discuss about the various mathematical and philosophical implications of the theory and the reading feels continuous (at not like a big collection of unconnected theorems). For people experienced with set theory, the first few chapters might be a bit slow, but on the other hand novices will have the chance to get used to the abstractness of sets.
However, some topics are divided among various chapters (e.g. axioms of choice is introduced in chapter 3, then presented in detail in chapter 6, but the discussion isn't complete until the second half of chapter 7). This was done so that harder parts of some topics come only after the reader has acquired a good foundation with set theory. The problem is that these topics are therefore slightly hard to follow and feel incoherent. Also, the book doesn't go really far. The discussion about the universe of set, for example, could have been longer.
Still, I consider this book to be very good and would definitely suggest it to any person who want to learn the basis of set theory
on June 17, 2004
Perhaps because it is a Foundations book -- in my mathematics training it always seemed that the people who did the best job of motivating and explaining (or at least making you feel you understood) the material were Foundations people -- but this book has a presentation polished to the point where the closest genre of mathematics text in level of polish would be intro calculus books, where the problems theorems and proofs have been worked over for many many many years. Here, however, the material is in great part relatively recent - probably the closest to contemporary stuff you can see as an undergraduate -- in Real Analysis, by contrast, you may well just be coming out of the 19th century by graduate school. This polish, I have discovered in later years, facilitates use of this book for self-study and it is a wonderful text for providing rapid refreshment of important concepts. I have over the years referred back to it on a number of occassions and have always been pleasantly reminded what a wonderful book it is.
This is a very nice book and the best introduction to the material I have seen (although, given the number of intro books I have seen on the topic, this may not be a strong statement).