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I have taken two introductory courses on logic. One was an undergraduate level and the second one was at the graduate level. Both used a set of notes written by the instructor. I'm thinking about graduate school in the field and would like to spend my next summer reading more about logic.

What I'm looking at is a good list of rigorous graduate level textbooks on the most important topics in logic that would get one started and which would provide a good foundation for graduate level studies. I have the whole summer and my senior year to prepare.

Which books would you recommend? My plan at this point would be to read a rigorous graduate level textbook on basic logic and then one on model theory. The ones that I've looked at are:

Manin, A Course in Mathematical Logic for Mathematicians

Marker, Model Theory: An Introduction

Would this be a nice place to start or are there better options?

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  • $\begingroup$ Form my, the introduction to Set theory was J.D. Monk "Introduction to set theory", about logic I like very much P. Johnstone, "Notes on Logic and Set Theory " together Elliot Mendelson's "Introduction To Mathematical Logic"... YUry MAnin Book is a hard book for me (readable, but slowly..) $\endgroup$
    – Buschi Sergio
    Commented Apr 30, 2012 at 20:31
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    $\begingroup$ I second Andreas's recommendation for Kunen's book. Marker's book on model theory is good, but beware of the errors:homepages.math.uic.edu/~marker/mt-errors.html. $\endgroup$
    – Zach N
    Commented Apr 30, 2012 at 23:24

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For set theory, I recommend Drake's "Set theory: An introduction to large cardinals" (which also contains a good treatment of various other parts of set theory, before concentrating on large cardinals) and Kunen's "Set theory: An introduction to independence proofs." The "bible" of set theory is Jech's "Set Theory" but I think it may be easier to learn from Drake and Kunen first. [Do you get the impression that set theorists are not very creative when it comes to book titles?]

For logic in general, Shoenfield's "Mathematical Logic" covers an amazing amount of material in rather few pages. Most of it is great (if you do the exercises, into which a lot of material has been squeezed) but I wouldn't recommend his approach to constructibility and forcing --- use Kunen instead.

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