Graduate level elementary logic books

I've done two courses on Logic during my Bachelor course, but they were very basic. Now I'm going to start by PhD, and I'm interested in learning "real Logic". Could you please provide some references for logic, starting from the foundations, but in a "graduate level-approach"?

Giving some examples of what I want: When I first studied one-variable Calculus, my professor used Spivak's books and various exercises from some Analysis books. When I got into multi-variable calculus, the professor spent a month or so talking about general topology, and did most things in the context of Banach spaces (and translating the results for $\mathbb{R}^n$). Also, my Analysis courses were given using Rudin's "Introduction to Mathematical Analysis". I believe both Spivak and Rudin's books make "graduate level-approaches to undergraduate subjects", and that's what I'm looking for logic.

Also, if there are some references for more advanced subjects, like Model Theory, they could come in handy.

Thank you.

Solution 1:

There's an expansive Teach Yourself Logic self-study Guide to the literature available (with supplementary materials) at http://www.logicmatters.net/tyl which should be helpful because it covers quite a number of texts giving some indication of their level/approach, and telling you more about what they cover.

The author's recommendations are mostly not-too-idiosyncratic, I'm informed ....

Solution 2:

Maybe Boolos and Jeffrey's Computability and Logic is what you're looking for?

From the Preface: "...for the student...who has mastered the material ordinarily covered in a first course in logic and who wishes to advance his or her acquaintance with the subject. The aim of the book is to present the fundamental theoretical results about logic, and to cover certain other metatheoretical results whose proofs are not readily obtainable elsewhere."

Edit: But I (ahem) think Peter Smith probably knows whereof he speaks, and his recommendation should be your first lead to follow up.