Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics, Volume 102) by Kenneth Kunen (PDF)

Description

Studies in Logic and the Foundations of Mathematics, Volume 102: Set Theory: An Introduction to Independence Proofs offers an introduction to relative consistency proofs in axiomatic set theory, including combinatorics, sets, trees, and forcing. The book first tackles the foundations of set theory and infinitary combinatorics. Discussions focus on the Suslin problem, Martin’s axiom, almost disjoint and quasi-disjoint sets, trees, extensionality and comprehension, relations, functions, and well-ordering, ordinals, cardinals, and real numbers. The manuscript then ponders on well-founded sets and easy consistency proofs, including relativization, absoluteness, reflection theorems, properties of well-founded sets, and induction and recursion on well-founded relations. The publication examines constructible sets, forcing, and iterated forcing. Topics include Easton forcing, general iterated forcing, Cohen model, forcing with partial functions of larger cardinality, forcing with finite partial functions, and general extensions. The manuscript is a dependable source of information for mathematicians and researchers interested in set theory.

User’s Reviews

Editorial Reviews: Review …provides a good introduction to relative consistence proofs in axiomatic set theory. It can be recommended as a graduate text on the subject. — Zentralblatt für Mathematik

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Very clear exposition.

⭐When I was a student, Paul Cohen’s theorem about the independence of continuum-hypothesis was still a hot topic, which attracted interest not only of the experts in set theory and mathematical logic. But the theorem appeared to be almost inaccessible for outsiders. Cohen’s own exposition and T. Jech small Lecture Notes volume based on an alternative form of Cohen’s method due to D. Scott (the theory of Boolean-valued models) seemed to be too dense and hardly penetrable for an ordinary mathematician (by which I mean a mathematician without any serious training in mathematical logic). I managed to read the first two chapters of Cohen’s book (I still consider the second chapter to be the best introduction into the axiomatic set theory; the first one about mathematical logic is also terrific, but to a big extent assumes that you already know at least the basics), and a part of the third chapter (the reading was interrupted by external circumstances). Later I read T. Jech’s small volume, which left me in a fairly confused state: I understood the proofs, but had no idea why they worked and what ideas are the key ones.Then K. Kunen’s book appeared. The reading was a very smooth ride. The exposition is very careful and detailed. At the same time there is a miracle: you see what the key ideas are without being directly told so (i.e. without the usual recourse of authors of textbooks). You will see that the success of the forcing method depends not just on the idea of forcing formal statements to be true, but also on a subtle balance of nearly contradicting properties of appropriate partially ordered sets.The book does not attempt to be encyclopedic. It has a clear goal: to prove the independence of the continuum-hypothesis. I believe that having such a clear goal provides a huge motivation for the reader (especially in contrast with collections of introductory chapters, as it is often the case with textbooks). The result itself is one of crowning achievements of mathematics of the twenties century. According to Hugh Woodin, by now the method of proof (Cohen’s forcing method) is turned from a tool to prove a theorem into a basic framework for thinking about set theory.But Cohen’s theorem and Kunen’s exposition are not just for the future experts in the set theory. It seems that most of mathematicians are hardly aware of how beautiful the proof of Cohen’s theorem is from purely mathematical point of view. Usually the theorem is mentioned in connection with its presumed philosophical implications. Personally, I came to consider the related philosophical questions (like the nature of infinity, for example) as irrelevant, and to appreciate the pure beauty of the theorem and its proof. I would recommend this book not only to graduate students intending to work in the set theory, but to any mathematician interested in learning an exceptionally deep and beautiful theory not necessarily related to his or her research. Kunen’s book is only 330 pages long; one rarely gets to a so deep result in such a short book.Of course, it is highly desirable to have some previous exposure to mathematical logic and axiomatic set theory. In an ideal world this would be a part of education of every mathematician. Even in the real world it is not very difficult to get the necessary background, and doing this would be worthwhile in any case.

⭐This is the classic text for learning forcing, and has strongly influenced every generation of set theorists since its publication. I know people who have used it for a first course in set theory, but it is quite terse on these matters, and that isn’t really its purpose. After a quick look at basic set theory, Kunen establishes relevant combinatorics at some length, and applies important notions from logic to set theory. Then he carefully develops the theory of forcing. All of this is terrific. But he does not give many examples of different kinds of forcing, so the book needs to be supplemented in a semi-major way by the reader or the instructor. I understand that a serious revision is underway. I hope it addresses this concern.

⭐This is the most widely used textbook for graduate -level set theory, and with good reason. Kunen manages to cover all the essentials of set theory in a quick 300 pages–and he does so with exceptional clarity and depth.I purchased a copy of the book when it was first published in 1980; I was a graduate student at the time, studying set theory. The book was good, but perhaps a bit advanced for the independent study approach that I was taking.Some years later when I had the opportunity to teach a graduate-level course in set theory, using Kunen’s text, I realized that as an adjunct to a lecture-based course, this book is ideal. I have also found that it clarifies subtle points about set theory that most authors gloss over. For instance, his treatment makes very clear how to define the forcing relation in the ground model; why inaccessibles can’t be proven consistent with ZFC using ZFC alone; how transfinite recursion should be formally stated in the theory and how it is to be used formally; and what the different approaches to forcing are.The main topics in the book are constructibility (developed on the basis of an understanding of ordinal definability) and forcing, with a final chapter on iterated forcing. Loads of material can be found in the vast number of exercises which, especially in the later chapters, provide a quick survey of important results in the literature.Kunen’s style is both entertaining and precise. Every sentence has the extraordinary quality of literally bursting with information. One can easily go back to this book year after year and expect new layers of insight to unfold. Kunen demonstrates both his mastery of set theory and mastery of the language in this superb text.

⭐For a graduate course on set theory, I don’t think there is any serious competition for Kunen. It assumes you have already had a basic course and starts right in using the concepts of axiomatic set theory and the properties of cardinals and ordinals, although it does start with a terse recap of logic and ZFC. The whole book is narrowly focused on leading up to the technique of forcing to prove consistency results under ZFC. It mostly avoids large cardinals, the Axiom of Choice, descriptive set theory, and model theory.The style is fairly readable, but I thought it was a little too messy for clarity. Without the key points being highlighted, I tended to get lost in the details. The early chapters have few exercises and not many examples. The last chapter on forcing is the big payoff of the book and has lots of exercises, but still a shortage of examples.I don’t think it makes a good reference, because it is not concise and clear enough. The monograph by Jech (3rd edition!) is vastly better for that. Even if you are currently using Kunen for a class, it is worth referring to Jech to clarify points in Kunen.

⭐This is a good book to read if you want to understand the motivation behind Cohen’s forcing technique used to create models of set theory in which various propositions (such as the axiom of choice and the continuum hypothesis) can be forced to be true or forced to be false. When I read it I was struck by the use of Martin’s axiom to motivate forcing, which works very well. You should read Paul Cohen’s great monograph “Set Theory and the Continuum Hypothesis” of course, in which (in book form) forcing originated; but forcing developed rapidly in the 1960s and 70s, and Kunen’s book captures the developments until the 1980s. It is a great achievement, and does a brilliant job of categorising the different approaches to forcing. I would also recommend Smullyan and Fitting’s “Set Theory and the Continuum Problem”, which introduces modal models (ie models expressed in the logic of possibility and necessity) as a way of understanding forcing. I gave it 4 stars only because a lot has happened with forcing since the 1980s; but before reading Jech’s “Set Theory” you should read Kunen’s book.

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