I would like to learn more about the proofs of Gödel's incompleteness theorems. I have read and am rereading Gödel's proof by Nagel, Newman, and Hofstadter. I like it very much, but I would like something a bit more substantial (but not too substantial). More specifically, I am a professor of mathematics, but not a logician. (If it helps, I am a probabilist.) I would like to read something that is aimed more at my level and is preferably self-contained. But this is purely for my own curiosity, so I do not want to learn a whole new field to understand it. Does anyone have any suggestions?
Highly recommended: Torkel Franzén. Gödel's Theorem: An Incomplete Guide to its Use and Abuse. Wellesley, Massachusetts: A K Peters, Ltd., 2005. x + 172 pp. ISBN 1-56881-238-8.
Some reviews: NAMS (Panu Raatikainen), Phil. Math. review (Stewart Shapiro) and R. Zach.
For a more concise technical treatment see Smorynski's 47 page exposition in the Handbook of Mathematical Logic (edited by J. Barwise). Smorynski is an expert in the field and a gifted expositor. I recall being highly influenced by many of his expositions as a student (some less technical than the cited paper). Note also that the 1165 page Handbook is an excellent general reference on logic for a mathematician.
See also this Related MO question.
I suggest the terrific book "An Introduction to Gödel's Theorems" written by Peter Smith http://www.amazon.com/Introduction-Theorems-Cambridge-Introductions-Philosophy/dp/0521674530
A couple of good, standard texts that handle Godel's work are Mendelson, Introduction to Mathematical Logic, and Enderton, A Mathematical Introduction to Logic.
I recommend Gödel's Incompleteness Theorems by Raymond Smullyan. He motivates well and his style is appropriate for a mathematician reader.
It is also possible to proof the incompleteness theorem utilizing computability theory, by reducing it to the halting problem. I think this method is easier to understand, since it just relies on some basic knowledge of the undecidability of the halting problem.
Here is a reference http://www.logicmatters.net/resources/pdfs/PartIII.pdf
As we speak, Scott Aaronson made a wonderful post about this on his blog: http://www.scottaaronson.com/blog/?p=710