Graduate Level Mathematical Logic Textbooks I'm currently a first year mathematics graduate student, and am at an institution which does not have any work being done in mathematical logic, or any logicians on the staff. I've taken a mathematical logic course with the textbook by Leary, 'A Friendly Introduction to Mathematical Logic, 2nd. ed.' (I actually took the class with the author at his university), and am comfortable with it's contents. 
I am looking for recommendations of graduate mathematical logic textbooks that would reflect the work and content done in a graduate logic course, so that I may see more advanced model theory and proof theory and get a better feel for whether or not these are topics that I would more enjoy studying.
 A: It's hard to find a "one-size-fit-all" book which fits this description, especially since the literature on logic is getting quite advanced. Back in the days, Joseph Shoenfield's Mathematical Logic was (and still is, in some places) a standard reference, and it does contain a fair bit of model theory (up to Ryll-Nardzewski's categoricity theorem, though do note that there are a lot of important results and definitions in the exercises), some proof theory (say, Herbrand's theorem and Gödel's consistency proof, though, sadly, no cut-elimination or Gentzen style consistency proofs), some recursion theory and some set-theory (including one of the first textbook presentations of forcing). 
A more up-to-date book in the same style (and quite a mammoth) is Hinman's Fundamentals of Mathematical Logic, which also focuses on almost everything except proof-theory (the model-theoretic part is quite advanced for a general introduction, getting until Morley's theorem).
One option that does include a bit more of proof theory (and a much less of everything else) is Dirk van Dalen's Logic and Structure, which has some very basic model theory (ultraproducts, model completeness) and has a chapter on normalization for natural deduction systems. It's also considerably shorter than the other, so it may be a nice option. In fact, my recommendation would be to skim van Dalen (which, if I remember correctly, is not much more difficult than Leary) to see if you enjoy what you find there. If so, then you can perhaps focus on more specialized books, such as Poizat's A Course in Model Theory or Pohler's Proof Theory.
A: I haven't looked through the book by Leary that you mentioned, so I don't know how it compares to the following:


*

*Manin

*Chiswell & Hodges

*Mundici
A: Disclaimer: I haven't taken a graduate level logic course.
Peter Smith a retired professor, who used to teach logic at the University of Cambridge put up a guide here with a list of books: http://www.logicmatters.net/tyl/
S. C. Kleene's Introduction to Metamathematics got reviewed by Michael Beeson when it got republished and as I recall, Dr. Beeson said that the book still had relevance for graduate students as a starting point.  It is also the first book on Peter Smith's list.  The most recent review of the book on Amazaon also says: "This 1952 book by Stephen Cole Kleene (1909-1994) is essential for anyone who wants to understand mathematical logic at the graduate level."  It looks like it has over 900 citations on CiteSeer (I don't know if that is large for citations in logic), is still getting cited at present. 
