Product Description
Although this book deals with basic set theory (in general, it stops short of areas where model-theoretic methods are used) on a rather advanced level, it does it at an unhurried pace. This enables the author to pay close attention to interesting and important aspects of the topic that might otherwise be skipped over. Written for upper-level undergraduate and graduate students, the book is divided into two parts. The first covers pure set theory, including the basic notions, order and well-foundedness, cardinal numbers, the ordinals, and the axiom of choice and some of its consequences. The second part deals with applications and advanced topics, among them a review of point set topology, the real spaces, Boolean algebras, and infinite combinatorics and large cardinals. A helpful appendix deals with eliminability and conservation theorems, while numerous exercises supply additional information on the subject matter and help students test their grasp of the material. Slightly revised republication of the edition published by Springer-Verlag, Berlin and New York, 1979. New preface. Introduction. Appendix. Bibliography. Index of Notation. Index. 20 figures.
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Customer Reviews (1)

The basics and some advanced topics beyond that
A solid foundation in set theory is one of the essential preconditions of being a good, working mathematician. I have taught nearly every course in the undergraduate mathematics curriculum and the basic concepts of set theory crop up on a regular basis. I have also taught nearly every course in the undergraduate computer science curriculum and those same ideas also pop up on a regular basis. Just last week when I was using finite automata to describe language recognizers it was necessary to remind the class about some basic concepts of set theory.
This book extends the idea of basic set theory beyond what most people would have as a definition. Most of the books in basic set theory place too much emphasis on the basic and not enough on the more advanced topics that students need for future study. Levy thoroughly covers the advanced topics, providing an overview of all of the areas of set theory that graduate students in mathematics need. Point set topology, real spaces, Boolean algebras, and infinite combinatorics are all covered in the second, more advanced part of the book. The coverage throughout is complete albeit succinct. It consists largely of a series of proposition - proof statements with few, if any examples.
This is not a book that a beginner can use to learn set theory, it requires a high level of mathematical maturity before it can be understood. Unfortunately, there are no exercises collected at the end of the chapters, so I hesitate to recommend it for advanced undergraduate or graduate courses in the field. In coverage, it is acceptable, but my personal belief is that all math textbooks should contain exercises at the end of the chapters. It can give you a solid, advanced foundation in set theory, but you have to work hard, slowly and be able to practice on your own without additional problems to work through. It is very well suited for a role as an advanced reference work.
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Product Description
This book is a novel introduction to set theory, including axiomatic development, consistency, and independence results. It is self-contained and covers all the set theory that a mathematician should know. It also presents, for the first time in a textbook, the double inductionand superinduction principles, and Cowen's theorem. The book will interest students and researchers in logic and set theory. ... Read more

Customer Reviews (3)

The symbols ARE missing
Perhaps there were different printing runs, but as the first reviewer experienced, my copy lacked the symbols for membership, subset, quantifiers, etc. It really does make the book useless. I too would like to have a correctly printed copy.
Update. I returned the book, explaining that the symbols were missing and got a replacement yesterday, Sept. 29. Unfortunately the replacement also lacks the symbols. The reviewer who had symbols is in Europe. Perhaps the US printing is messed up. I am going to go to a book store where I can look at the pages before I buy the book.

A fair review of an original book on set theory
Apparently some people had bought a printing of this Dover edition without maths symbols. In my copy of this Dover edition all maths symbols are here. Maybe there are different print runs and the first is without maths symbol. Maybe I have bought (with Amazon.fr) a second printing where the problem has been corrected.

Turning to the content this book is on independence proofs in set theory.

The first part, probably written by Smullyan (his witty style is easily recognized), begins with an exposition of the basics of set theory (with an axiomatics based on the concepts of sets and classes) and then exposes the consistency proofs of Gôdel of the axiom of choice and of the generalized continuum hypothesis. It is very readable, if not always as precise as it should be necessary.

The second part is the most original. It exposes the method of forcing with the help of a modicum of modal logic (all the necessary concepts of modal logic are explained), then proceed to prove the Independence of the Axiom of Choice and of the Continuum Hypothesis. The use of modal logic helps very much to understand the forcing method. But I must say that some points at the beginning of the exposition contain errors or are not exposed with all necessary details. And many points are left as exercices (from easy to difficult) to the reader. This is why I give only 3 stars to the book.

But overall, it is a very easy and readable introduction to the independence proofs in set theory, only to be compared with Kunen's "Set Theory".

Unreadable due to missing symbols
I just purchased the book.It is unreadable as many of the mathematical symbols are missing.The introduction makes it sound very interesting. ... Read more

Product Description
Mathematics of Fuzzy Sets: Logic, Topology and MeasureTheory is a major attempt to provide much-needed coherence for themathematics of fuzzy sets. Much of this book is new material requiredto standardize this mathematics, making this volume a reference toolwith broad appeal as well as a platform for future research. Fourteenchapters are organized into three parts: mathematical logic andfoundations (Chapters 1--2), general topology (Chapters3--10), and measure and probability theory (Chapters11--14). Chapter 1 deals with non-classical logics and their syntactic andsemantic foundations. Chapter 2 details the lattice-theoreticfoundations of image and preimage powerset operators. Chapters 3 and 4lay down the axiomatic and categorical foundations of general topologyusing lattice-valued mappings as a fundamental tool. Chapter 3 focuseson the fixed-basis case, including a convergence theory demonstratingthe utility of the underlying axioms. Chapter 4 focuses on the moregeneral variable-basis case, providing a categorical unification oflocales, fixed-basis topological spaces, and variable-basiscompactifications. Chapter 5 relates lattice-valued topologies to probabilistictopological spaces and fuzzy neighborhood spaces. Chapter 6investigates the important role of separation axioms in lattice-valuedtopology from the perspective of space embedding and mapping extensionproblems, while Chapter 7 examines separation axioms from theperspective of Stone-Cech-compactification and Stone-representationtheorems. Chapters 8 and 9 introduce the most important concepts andproperties of uniformities, including the covering and entourageapproaches and the basic theory of precompact or complete [0,1]-valueduniform spaces. Chapter 10 sets out the algebraic, topological, anduniform structures of the fundamentally important fuzzy real line andfuzzy unit interval. Chapter 11 lays the foundations of generalized measure theory andrepresentation by Markov kernels. Chapter 12 develops the importanttheory of conditioning operators with applications to measure-freeconditioning. Chapter 13 presents elements of pseudo-analysis withapplications to the Hamilton--Jacobi equation and optimizationproblems. Chapter 14 surveys briefly the fundamentals of fuzzy randomvariables which are [0,1]-valued interpretations of random sets. ... Read more

Product Description
Now available in paperback, this monograph is a self-contained exposition of the main results and methods of descriptive set theory. It develops all the necessary background material from logic and recursion theory, and treats both classical descriptive set theory and the effective theory developed by logicians. ... Read more

Customer Reviews (1)

The finest book of its type
Introduction to Axiomatic Set Theory by E.J. Lemmon is by far the very best presentation to be found.Either as a tutorial, a reference, or as a foundational model upon which to build further, this book will provide the systematic throughness and clarity that will almost certainly make your intellectual journey a joyful one.PLEASE BRING THIS BOOK BACK INTO PRINT !!! ... Read more

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Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Now available in this two volume set, it contains all the important information both volumes provides. Considered to be a complete benefit for all researchers and academics in theoretical computer science, logicians and philosophers who study the foundations of mathematics, and those working in differential geometry and continuum physics. ... Read more

Customer Reviews (1)

An easy call
Very simply: if you want to know a very great deal about topos theory, buy this book. I mean, seriously, if you plan to make real work on topos theory a part of your life, then grit your teeth and come up with the money. If you do not want to know a very great deal about it, do not buy this book. You can use it at the library as a reference.

If you merely want a professional understanding of what topos theory is, then read Johnstone's earlier TOPOS THEORY. That far shorter book gives a better overview. My Amazon review of it discusses others on the subject. Most are more accessible than Johnstone's books and go more into particular aspects of the theory.

This book is a reference on all the methods, and the latest results, in topos theory. If you want the definition of "split opfibration", it is here, along with some 80 pages of background, examples, and motivation. Johnstone does an heroic job of unifying the terminology and organizing the theorems.

More than that, Johnstone has written down an expert, encyclopedic view of the subject today. It is rare for a top mathematical researcher to give so deep an account of their field. It is rare for anyone to even work out such an explicit, coherent, extensive account of the whole. Not everyone will agree with his view. Some would like to see much less of such logical topics as "allegories", others would like to see the logic more formalized from the start. But Johnstone builds a case for his choices: partly implicit in his success at explaining things this way, and partly by explicit reasons.

If you want to know that much about the subject then you want to immerse yourself in this book. ... Read more

Product Description
Logic forms the basis of mathematics, and is hence a fundamental part of any mathematics course. It is a major element in theoretical computer science and has undergone a huge revival with the every- growing importance of computer science. This text is based on a course to undergraduates and provides a clear and accessible introduction to mathematical logic. The concept of model provides the underlying theme, giving the text a theoretical coherence whilst still covering a wide area of logic. The foundations having been laid in Part I, this book starts with recursion theory, a topic essential for the complete scientist. Then follows Godel's incompleteness theorems and axiomatic set theory. Chapter 8 provides an introduction to model theory. There are examples throughout each section, and varied selection of exercises at the end. Answers to the exercises are given in the appendix. ... Read more

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High Quality Content by WIKIPEDIA articles! In universal algebra and in model theory, a structure consists of a set along with a collection of finitary functions and relations which are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields, vector spaces and lattices. The term (universal) algebra is used for structures with no relation symbols. Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. ... Read more

Product Description
This introduction to modern set theory covers all aspects of its two main general areas: classical set theory including large cardinals, infinitary combinatorics, desriptive set theory, and independence proofs starting with Goedel's proof around 1938 followed by Cohen's proof in 1963, whereby Cohen's method of forcing probably had a greater influence on mathematics. The author's primary emphasis is on forcing and large cardinals (on which he has collected an enormous amount of material which had previously been available only through scattered journal articles or private communication) but there is a substantial discussion of descriptive set theory and infinitary combinatorics as well. The author's presentation is very well-organized and carefully worked out and has become a standard reference. ... Read more

Customer Reviews (5)

Clear and comprehensive
The author tried to cover everything a set theorist should master plus a representative selection of topics of current interest. That makes for a lot of ground to cover, but Jech did a great job. The writing is very well organized and clear. Every short chapter has many exercises, often with hints. There are extensive sections on applications of forcing. The indexes are really good.

There has to be a down side, of course. In order to squeeze so much in, he had to be brief. There is little context provided, especially in Part I: Basic Set Theory. There are rarely any examples and only the main facts are covered. That is all part of an understandable compromise, but I have a serious complaint (my only one) about the references. He gives detailed historical references in each chapter, but no references to further reading. He could have done it with hardly any use of space and it would have been very helpful.

Because of the brevity, it is a bit hard to learn from, but it makes a great secondary reference. For example, its explanations are often clearer and more direct than in Kunen and with more detailed proofs. It you are going to have any more exposure to set theory than an introductory course, you will probably want to buy a copy. (BTW, the 2e was just a corrected reprint; 3e is a complete rewrite.)

A good but unreadable book
It's really a good book for researchers in set theory. But it is NOT an introduction for students who want to know what is set theory. You will feel you are so stupid if you read this book without any set theoretical background. I recommand Kunnen's book for those people who are interested in set theory but have no any (or only a little) set theory knowledge.

This edition has been completely revised!
Just wanted to point out that all the reviews here dated before Feb 2003 are referring to older editions.The new one has been totally revised (no laundry list of corrections at the end) and also expanded -- lots of material from the last 25 years of set theory research is now included. Most notable among these is material on proper forcing and pcf theory.(There is even a section on my research interest, mutually stationary sets, and this is a notion which was just published for the first time 2 years ago!)The book is still just as informative and readable as the previous editions.

EDIT: I still agree with everything I wrote above; nearly 6 years later, I still read portions of this book almost every day (and I'm not even doing set theory per se professionally anymore).However I should state for the record that the book is RIDDLED with typos and minor errors.So, be prepared to read critically.

A classic
This book is a wonderful reference volume for set theory. It contains a clear and readable explaination of all the things a set theorist needs to know. I have only one complaint: "revised edition" simply meansthat a 20-odd page errata has been appended....

My favorite book on set theory.
In 1979, I was a first-year graduate student in mathematics. One summerday, I was looking in the math section of Stanford bookstore and saw thisthick green volume with the simple title Set Theory (by Thomas Jech). Icouldn't help pulling the tome off the shelf. I flipped through the pagesin awe. This book had everything about mathematics that I had always wantedto know.

After about an hour, I reluctantly looked at the price and itwas just too much; I had to put it back on the shelf. But for the nextmonth, that book was all I could think about. I finally went back andbought it.

Two years later after hooking up with my adviser and embarkingon research in set theory, I started working through Jech's book startingon page 1. It took me 2 years to work through the entire book, and for muchof that time I had the opportunity to present what I was learning inseminars.

That book is a real treasure. I don't think I've spent as muchtime poring over any other book. I think the presentation of material isfantastic and the coverage is thorough (or it was at the time I studiedit--probably his recently updated work also has this attribute).

I wouldrecommend this book (or rather the most recent edition of it) to anyserious graduate student specializing in set theory.

Two areas where Ineeded supplementary study were in his approaches to the constructibleuniverse and to forcing. These are important areas, and Jech does a finejob in his approach, but certain approaches other than his have become moreof a standard, and any serious researcher will have to become familiar withthese standards. Jech uses Boolean algebras (primarily) in his developmentof forcing (and his development is excellent) whereas by now, the usualapproach is with partial orders. Also, Jech develops L as a transitivemodel that is closed under "Godel operations"--a perfectly validapproach. These days, though, the formula-based approach is more common inthe literature.

Nonetheless, Jech's wide variety of forcing applications,his in-depth treatment of large cardinals, and his compact surveys ofsaturated ideals and descriptive set theory make his work really anoutstanding contribution. ... Read more