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Classic Set Theory: For Guided Independent Study (Chapman & Hall Mathematics S) 1st Edition
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Unlike most of the set theory books, Derek's book starts from "every day mathematics"(arithmetics, real analysis or algebra) and gradually lead you to see the connection between "every day mathematics" and abstract set theory. He did not put you in the "empty sky". On the contrary, this book enhances your set theory knowledge from a practical mathematics point of view and by the way deepen your mathematics knowledge from a set theoretical point of view.
My suggestion: Grab it and read it from the first page to the end. It must be the most fascinating experience you will find in set theory!
This book now stands, along with Smullyan's "Set Theory and the Continuum Problem", in my "personal bible" for this part of maths.
1)Suberb organization of the material. Ideas are built gradually without logical gaps or regressions. Definitions and theorems are clearly stated. "What follows from what" is always transparent. Even the choice of paragraphs is so well thought, that one can easilly assign a title in each of them for quick reference later on.
2)Each subject is clearly introduced within its historical and logical context. Each theorem (and even exercise) is motivated for its importance and its merits in the global picture of Set theory.
3)The logic and intuition behind the proofs is given (as well as the proof itself...) in a well organized and not unecessarily wordy manner.
4) There are exercises within the main text (which, as usual, are well motivated for their importance) with solutions folowing right after. In this way, one may develop skills and understanding, without getting frustrated or spending too much time. There are also exercises in the end of each section which are interesting and not too difficult.
5) There are comments aside of the main text, which range from ideas concerning a proof to historical remarks or recommendations to the reader. In this way, the main text remains clean of tangencies, but never dry.
I could continue praising this book, but let me cut it short by saying just this: it is one of those proper (i.e. rigorous) math textbooks that invite you to read each following chapter and to turn each page to see what's next. Having finished it I feel I have a pretty firm understatnding of the basics.
I only wish that Goldrei could write a second book on specialized topics (say, similar to the topics covered in Devlin's "The Joy of Sets", or Moschovakis' book), with the same energy and enthousiasm that wrote this one.