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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?

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up vote 8 down vote accepted

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.

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that's a terrific book, but does it really go into the proofs of Godel's Theorems, as user requests? – Gerry Myerson Jul 19 '11 at 3:52
@Gerry. Yes, I believe so, and my opinion is shared by others, e.g. in the MO thread: Pete Clark and Carl Mummert. – Bill Dubuque Jul 19 '11 at 4:03
In that same thread, Bill, Antonio Porreca writes, "No, Franzén doesn’t give full proofs." On pp. 3-4, Franzen writes, "In this book, no knowledge of logic or mathematics (beyond an acquaintance with school mathematics) will be assumed, on the basis of the view that a sound informal understanding of the theorem is attainable without a study of formal logic." – Gerry Myerson Jul 19 '11 at 4:22
@Gerry If a reader lacks sufficient background to fill in the details of proofs sketched in Franzén's book then they can find them in the paper of Smorynski that I cited above. – Bill Dubuque Jul 19 '11 at 4:37

I suggest the terrific book "An Introduction to Gödel's Theorems" written by Peter Smith

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That book is recommended in two of the answers to the MO question mentioned in Bill Dubuques's answer. – Gerry Myerson Jul 19 '11 at 12:57

A couple of good, standard texts that handle Godel's work are Mendelson, Introduction to Mathematical Logic, and Enderton, A Mathematical Introduction to Logic.

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I recommend Gödel's Incompleteness Theorems by Raymond Smullyan. He motivates well and his style is appropriate for a mathematician reader.

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Seconded here - I have a slight bias towards Smullyan, but while he certainly has his weaknesses I find his writing particularly crisp. – Steven Stadnicki Jul 21 '11 at 22:02

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

As we speak, Scott Aaronson made a wonderful post about this on his blog:

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