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I have good knowledge of Propositional and Predicate Logic. I want to begin set theory. I am looking for books that explain set theory principles using Logic. And also books which have lots of theorems with proofs on basic set theory?

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Main goal is to become good at Algebra. I was told that Logic is the foundation in mathematics and basics of set theory appear in all branches of mathematics.I am not sure how will set theory help me in algebra. – Ramki Oct 31 '11 at 8:26
@Asaf: Most of the above books directly go into set theory – Ramki Oct 31 '11 at 8:37
If you just want the basics of set theory, any introductory chapter would do. In most, if not all, modern books which develop a theory from the ground. – Asaf Karagila Oct 31 '11 at 8:41
And if you want set theory to use, then Halmos's Naive Set Theory will give you more than enough. – Arturo Magidin Oct 31 '11 at 16:17

A first course in logic by Shawn Hedman covers both logic and set theory. I personally like Kunen's development of elementary set theory, but that might be a bit terse. It certainly is based on logic. Jech's book contains a lot of material, but might be a bit inaccessible at first.

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I downloaded these 3 books. thanks for referring "A first course in logic by Shawn Hedman". Kunen's book assumes that one knows well concepts of axiomatic set theory and properties of cardinals and ordinals, although it does start with a terse recap of logic and ZFC. Jech book covers advances in set theory well. – Ramki Oct 31 '11 at 10:08

If you want something basic, slow paced, with detailed proofs and solutions of some exercises, then this could be a good choice.

Schaum's outline of theory and problems of set theory and related topics By Seymour Lipschutz

This book was also recommended in an answer to this related question: Set theory practice problems?

I should add that I only have this book in hand only a few times, mostly when I was looking for some exercises. But what I wrote above more-or-less characterizes the books from Schaum's Outlines series I've seen so far.

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thanks for pointing to other thread. It was really useful. – Ramki Nov 2 '11 at 5:50

The following book, which I got in 1996 or 1997, might be suitable:

Set Theory, Logic and their Limitations by Moshe Machover (1996)

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Axiomatic Set Theory by Patrick Suppes is a classic, logic-based introduction to formal set theory and it's in Dover, so it's very cheap. You'll also find a very good chapter on axiomatic set theory in Elliot Mendelson's Introduction To Mathematical Logic.

There's an interesting pedagogical debate in mathematics regarding set theory: Should it be taught with or without some grounding in mathematical logic? I learned it without logic from Enderton's classic and very readable Elements of Set Theory-but Enderton's treatment is confusing in regards to the more subtle aspects of it,such as expressing the precise difference in ZFC between elements and sets.Personally,I find formal set theory to be rather confusing without the basics of logic. Many of the formal axioms of set theory were devised specifically to overcome logical paradoxes that develop without them-so expressing them as predicates in logical arguments are probably the clearest way to express the more subtle aspects of it. Expressing some of these more difficult points in logical notation goes a long way towards clearing them up.

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the point you made: Good knowledge of logic avoids confusion in Set Theory was already made to me. so I went on to Logic first and now coming to Set theory. I have to look into book by Patrick Suppes – Ramki Nov 2 '11 at 5:59

Levy's book on Basic Set Theory" is also very good. For an easy going and clear introduction you may go for Goderi's Classic; and somewhat more formal Vaught's Introduction or Hanjal's set theory. Kunen's 1980 work and Jech 2006 monograph are for advanced courses.

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