This book is a total rewrite of the author's Set Theory: An Introduction to Independence Proofs, first published in 1980. Both books are intended for
readers who have studied axiomatic set theory and who want to learn the basics of the independence proofs, starting with the independence of the Continuum Hypothesis (CH). There are two reasons why this rewrite was needed. First, there has been a lot of set theory discovered over the past thirty years. Not all of this impacts on these two books, which just cover the basics of the subject, but some of the discoveries need to be reflected in the text.
For example, the first book could essentially culminate with the consistency
of Martin's Axiom (MA) with -,CH, but now even an elementary text in set
theory must give some discussion of results independent of MA + -, CH, and at least mention the Proper Forcing Axiom.
Second, model-theoretic methods have become increasingly prevalent in mathematical arguments. Working in an axiomatic set theory (such as ZFC) ,
one can formalize the syntax of logic and then prove theorems about model theory, and in particular about models of set theory. This idea was well-known to logicians going back to the 1940s, but even in the 1980s, there were still many researchers in set theory and its applications who did not know much logic. So, pains were taken to avoid model-theoretic methods in the first book. But now, anyone doing research in the more set-theoretic areas of topology and analysis has learned basic model theory and knows how to apply model theoretic techniques to set-theoretic problems. So, the present book describes these techniques in Chapter I, as part of a review of basic logic, and then in later chapters applies these techniques to prove mathematical theorems and independence results.
One spinoff of this rewrite is that Gödel's proof of the consistency of the Generalized Continuum Hypothesis (GCH) has been downgraded from a whole chapter in the original text to one section in this text (part of Chapter II, "Easy Consistency Proofs"). The reason is that once one is familiar with the subject of models of set theory (which historically grew out of Gödel's proof), the construction of his model for GCH seems like almost the obvious thing to do.
《A Book of Set Theory》by Charles Pinter
这是我读过的最好的集合论的书。不仅这一本，Pinter的所有数学书（标题都是以“A Book of XX”开头）都值得读。
《Foundations of Mathematical Analysis》by J. K. Truss
或者《An.introduction.to.Analysis》by 作者: Arlen Brown / Carl Pearcy
第二本《Foundations of Mathematical Analysis》by Truss, J, K.