Textbooks on set theory
Solution 1:
(Excerpted from an earlier version of a study guide to logic texts more generally -- you will find the latest version here: http://www.logicmatters.net/students/tyl/)
Mere lists are fairly uninteresting and unhelpful. So let's be a bit more selective!
We should certainly distinguish books covering the elements of set theory – the beginnings that anyone really ought to know about – from those that take on advanced topics such as ‘large cardinals’, proofs using forcing, etc.
On the elements, two excellent standard ‘entry level’ treatments are
Herbert B. Enderton, The Elements of Set Theory (Academic Press, 1997) is particularly clear in marking off the informal development of the theory of sets, cardinals, ordinals etc. (guided by the conception of sets as constructed in a cumulative hierarchy) and the formal axiomatization of ZFC. It is also particularly good and non-confusing about what is involved in (apparent) talk of classes which are too big to be sets – something that can mystify beginners.
Derek Goldrei, Classic Set Theory (Chapman & Hall/CRC 1996) is written by a staff tutor at the Open University in the UK and has the subtitle ‘For guided independent study’. It is as you might expect extremely clear, and is indeed very well-structured for independent reading.
Still starting from scratch, and initially only half a notch up in sophistication, we find two more really nice books (also widely enough used to be considered "standard", whatever exactly that means):
Karel Hrbacek and Thomas Jech, Introduction to Set Theory (Marcel Dekker, 3rd edition 1999). This goes a bit further than Enderton or Goldrei (more so in the 3rd edition than earlier ones). The final chapter gives a remarkably accessible glimpse ahead towards large cardinal axioms and independence proofs.
Yiannis Moschovakis, Notes on Set Theory (Springer, 2nd edition 2006). A slightly more individual path through the material than the previously books mentioned, again with glimpses ahead and again attractively written.
My next recommendation might come as a bit of surprise, as it is something of a ‘blast from the past’: but don’t ignore old classics: they can have a lot to teach us even if we have read the modern books:
- Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy, Foundations of Set-Theory (North- Holland, 2nd edition 1973). This puts the development of our canonical ZFC set theory into some context, and also discusses alternative approaches. It really is attractively readable. I’m not an enthusiast for history for history’s sake: but it is very much worth knowing the stories that unfold here.
One intriguing feature of that last book is that it doesn’t at all emphasize the ‘cumulative hierarchy’ – the picture of the universe of sets as built up in a hierarchy of levels, each level containing all the sets at previous levels plus new ones (so the levels are cumulative). This picture – nowadays familiar to every beginner – comes to the foreground again in
- Michael Potter, Set Theory and Its Philosophy (OUP, 2004). For mathematicians concerned with foundational issues this surely is – at some stage – a ‘must read’, a unique blend of mathematical exposition (mostly about the level of Enderton, with a few glimpses beyond) and extensive conceptual commentary. Potter is presenting not straight ZFC but a very attractive variant due to Dana Scott whose axioms more directly encapsulate the idea of the cumulative hierarchy of sets.
Turning now to advanced topics Two books that choose themselves as classics are
Kenneth Kunen, Set Theory (North Holland, 1980), particularly for independence proofs.
Thomas Jech, Set Theory: The Third Millenium Edition (Springer 2003), for everything.
And then there are some wonderful advanced books with narrower focus (like Bell's on Set Theory: Boolean Valued Models and Independence Proofs). But this is already long enough and in fact, if you can cope with Jech's bible, you'll be able to find your own way around the copious literature!
Solution 2:
I used the book Set Theory by Andras Hajnal and Peter Hamburger and got the impression (since I was taking the class during a program in Hungary) that it was a common book there. It has a good introduction to naive set theory and a lot of more advanced topics in combinatorial set theory as well.
A link to the book is here.
Solution 3:
Here are some books not included in your list.
Kunen has completely rewritten his text Set Theory: An Introduction to Independence Proofs. See Amazon. It contains a lot of new material.
Holz, Steffens, Weitz, Introduction to Cardinal Arithmetic. The first chapter (about 100 pages) of this book is a very good introduction to set theory. One of the best I have ever seen.
Just and Weese have a two volume introduction published by the AMS. The second volume is a very good second course if you like their conversational style.
Drake, Set Theory, An Introduction to Large Cardinals. Contains introductory material as well as some advanced topics.
Drake, Singh, Intermediate Set Theory. If I recall correctly, this book contains a detailed development of set theory and constructibility.
There is a new Dover edition of Smullyan, Fitting, Set Theory and the Continuum Problem. This book has a non-standard approach to different topics.
The new Dover edition of Lévy's Basic Set Theory contains an errata not available in the old version.
Schimmerling's new book, A Course on Set Theory, looks like a nice and compact introduction.
Henle, An Outline of Set Theory is a problem-oriented text. It has a section on Goodstein's theorem.
Five classic texts still relevant today:
Sierpiński, Cardinal and Ordinal Numbers. A very rich collection of results on ordinal and cardinal arithmetic.
Kuratowski, Mostowski, Set Theory. The old bible of set theory?
Heinz Bachmann, Transfinite Zahlen. Unfortunately, I don't know any German. As far as I understand, this book contains some results not found in Sierpinski's book.
Cohen, Set Theory and the Continuum Hypothesis. I guess the last chapter on forcing is quite dated. But the previous chapters are insightful.
Erdős, Hajnal, Máté, Rado, Combinatorial Set Theory: Partition Relations for Cardinals. This is a more specialized book with a quick review of basics. It looks like this book is still published: Amazon link.
There are some online texts available as well. I remember seeing notes by Steve Jackson, J. Donald Monk and Sy Friedman among others.
Solution 4:
P.R.Halmos: Naive Set Theory (1974)
An excellent "Outline of the elements of naive set theory" as the author himself describes the book. The purpose of the book is to equip the beginning student of advanced mathematics with the necessary minimum of set theory "with minimum of philosophic discourse and logical formalism". This means the following: the axioms of set theory are stated and discussed in the most gentle way, followed by an account of all classical topics treated under the label of set theory in the same manner. This is a great place to start for anyone.
Suppes: Axiomatic Set Theory (1960)
If after reading Halmos, you develop an appetite for a more formal treatment (of ZF theory), Suppes (1960) is a great companion to continue this journey with (as recommended in Halmos (1960)). Be prepared for some logical formalism and great historical references.
Both books are available from Dover, and it is not a bad idea to get the original printing by van Nostrand of either. Have fun!
Solution 5:
Frank Drake's book, "Set Theory: An Introduction to Large Cardinals" (already mentioned by Ali Kare) deserves your attention even if you're not particularly interested in large cardinals. Despite its subtitle, it contains very nice presentations of a lot of general set-theoretic background before getting to the large cardinals.