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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.

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logicmatters.net/tyl –  symplectomorphic May 24 at 17:17

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up vote 7 down vote accepted

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 ....

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Expansive, but not expensive! :-) –  Asaf Karagila May 24 at 17:34

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.

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I think this book is not quite at the graduate level, although it is a very good book. –  Carl Mummert May 25 at 12:45
    
Probably a fair comment. The book was assigned in one of my undergraduate courses. The OP is looking for something "elementary" and at the same time somewhat advanced; perhaps if I'd majored in math, his examples would have given me a clearer idea what exactly he was looking for. Anyway, I thought Boolos and Jeffrey might be worth a look, but I did lead with the word "maybe." :) Another title that comes to mind, assigned in a grad/undergrad cross-listed course I took later, is Enderton's Mathematical Intro to Logic. –  StumpyLeg May 25 at 19:11

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