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Given that I am going through Munkres's book on topology , I had to give a glance at the topics included in the first chapter like that of Axiom of choice, The maximum principle, the equivalence of the former and the later etc. Given all this I doubt that I know enough of set theory , or more precisely and suiting to my business , Lack a good deal of rigor in my ingredients. I wanted to know whether research is conducted on set theory as an independent branch. Is there any book that covers all about set theory, like the axioms, the axiom of choice and other advanced topics in it. I have heard about the Bourbaki book, but am helpless at getting any soft copy of that book.

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I don't think Bourbaki is where you should start learning set theory ^^ –  Olivier Bégassat Dec 23 '12 at 20:05
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Karel Hrbacek & Thomas Jech, Introduction to Set Theory is a good introductory text at the senior undergraduate or first-year graduate level. It may be a bit too advanced, but you won’t find a good treatment of all of the topics that you mentioned at a much lower level. –  Brian M. Scott Dec 23 '12 at 20:11
    
@BrianM.Scott Any such book having an available PDF version . Sir, How about your lecture notes? –  danny gotze Dec 23 '12 at 20:24
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Other quite comprehensive books are Jech's Set Theory and Kenneth Kunen's Set theory: introduction to independence proofs. But those may be a little too fast-paced for a beginner... I'm pretty sure all these books have pdf versions, if you know where to look. Out of curiosity, what is this thing you call maximum principle? –  tomasz Dec 23 '12 at 20:25
    
@tomasz: Those are definitely too hard for someone who’s having some trouble with Munkres. Presumably the maximum principle is the Hausdorff maximum principle. –  Brian M. Scott Dec 23 '12 at 20:26
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2 Answers

Here are four suggestions (two "entry level" books, and two just a notch up in difficulty):

  1. Herbert B. Enderton, The Elements of Set Theory (Academic Press, 1997) is particularly clear, and nicely marks 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. It is written with a certain lightness of touch and proofs are often presented in particularly well-signposted stages. The last couple of chapters or so perhaps do get a bit tougher, but overall this really is quite exemplary exposition.
  2. 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, it is quite attractively written (as set theory books go!), and is indeed very well-structured for independent reading. The coverage is very similar to Enderton’s, and either book makes a fine introduction.
  3. 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. Again this is a very nicely put together book, and recommended if you want to consolidate your understanding after one of the first two books by reading another presentation of the basics and want then to push on just a bit. (Jech is of course a major author on set theory, and Hrbacek once won a AMA prize for maths writing.)
  4. 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.

All these books are in print, though none are cheap: indeed, Enderton’s is quite absurdly expensive. But all are ‘musts’ for any university library and are widely available. I’d strongly advise reading one of the first pair and then one of the second pair. (And do this before tackling more advanced books like Kunen's or the Jech bible which go more a lot more quickly through the basics and then deal with more advanced topics including forcing and large cardinals.)

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I'd recommend "Naive Set Theory" by Halmos. It is a fun read, in a leisurely style, starts from the axioms and prove the Axiom of Choice.

Also, see this XKCD. http://xkcd.com/982/

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Halmos's book has the great merits of clarity and great brevity. Just 102 pages of text in fact. As a first taster it is fine: but it probably won't give you enough -- you won't even end up knowing what ZFC is. (And of course it doesn't prove the Axiom of Choice: like any set theory book it only explains what it is and gives some motivation.) –  Peter Smith Dec 23 '12 at 23:00
    
Halmos' book is intended to contain the bare bones of what you need to know of set theory in order to do mathematics (in non set theory and logic heavy areas obviously). From the introduction: “In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some and here it is; read it, absorb it, and forget it.” –  tharris Dec 24 '12 at 16:06
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