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I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?

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When you say "more examples", do you mean more examples of the early syntactical stuff, or more thorough lists of specific theories to which general results apply? For the latter, the book by Donald Monk is good, although its notation takes getting used to. The chapters on decidable and undecidable theories include many concrete examples. For the former, you should think about upper-level undergraduate books. Most graduate-level books in logic (and other parts of mathematics) have very few worked examples of basic theorems. They assume you will work out examples on your own at that level. –  Carl Mummert Sep 10 '10 at 12:03
    
I think what I mean is more introduction about the intuition from which the theory was generalized. –  ciciplus Sep 14 '10 at 3:47
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10 Answers

For my work in this area, I refer to :

  • Richard Epstein "Classical Mathematical Logic"
  • Wolfgang Rautenberg "A Concise Introduction to Mathematical Logic"
  • Jon Barwise "Handbook of Mathematical Logic"
  • Jean Heijenoort "From Frege to Gödel"
  • We Li "Mathematical Logic"

Rautenberg has a lot of examples, exercise, but is very heavy going (at least for me). Epstein is fairly recent and very well laid out. Barwise is the most comprehensive for when you need to deep dive.

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Ebbinghaus, Flum and Thomas. Mathematical Logic (Amazon)

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I attended a course on mathematical logic where a similar book by Ebbinghaus in German language was used. I can only recommend it. His style is not what some might call "easy", but it is very clear and with an attention to detail, which in its extent may be uncommon even in introductory books in this field. –  knuton Nov 25 '10 at 23:24
    
I warmly recommend the latest German edition. The English edition has received some devastating reviews‌​, which makes me unsure whether it really matches the qualities of the German text (including such niceties as worked out solutions to the sometimes challenging exercises). –  Thomas Klimpel Oct 13 '13 at 22:53
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Shoenfield's "Mathematical Logic". The notation is a bit dated, but the exercises are great.

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+! for this great classic. All it needs is some examples. For some crazy reason, authors thought graduate textbooks didn't need examples in those days,which still puzzles me........... –  Mathemagician1234 Oct 31 '11 at 19:25
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François G. Dorais and others made some great recommendations to me some time ago over on MathOverflow. They're fairly high-level (not exactly introductory courses) but they're good reads.

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Yuri Manin, A course in mathematical logic.

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There's a new addition of this book, now entitled A Course in Mathematical Logic for Mathematicians. –  Quinn Culver Aug 22 '12 at 22:26
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A book that should be read by everyone in mathematics regardless of level is Wolfe's A Tour Through Mathematical Logic. It's simply a compulsory read,I couldn't put it down. It gives a broad overview of mathematical logic and set theory along with it's history and is absolutely beautifully written. That's the best place for anyone to begin.

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I'd love to know why this was downvoted- it would be amusing to hear my fan club's excuse. –  Mathemagician1234 Jul 11 '12 at 16:58
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Cori, Lascar, Pelletier, Mathematical Logic : A course with exercises -- Part I and Part II. Especially the second one. Amazon link

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I suggest to read:

  • The incompleteness phenomenon, by H. Judah and M. Goldstern.
  • There is a very good on-line course notes by L. van den Dries: http://www.math.uiuc.edu/~vddries/410notes/main.dvi
  • A Mathematical Introduction to Logic by H. Enderton.
  • Mathematical logic, by H.-D. Ebbinghaus, J. Flum and W. Thomas.
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I recently started studying from An Introduction to Mathematical Logic and Type Theory: To Truthe through Proof by Peter B. Andrews. It's great at my level of mathematical knowledge. Perhaps this is more introductory than you are looking for.

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