I am attempting to write an expository paper on non-standard models of PA that is accesible to students taking an introductory graduate course in mathematical logic (covering Godel's incompleteness theorems, the diagonalization lemma, models, etc.). In this paper I want to give an explanation of some results such as Tennanbaum's Theorem (there does not exist a countable recursive model of PA that is not isomorphic to the standard model). By, "give an explanation of", I mean to actually work through an explanation of the proof, some of the techniques involved, and the general overlap between techniques used in the proofs of Tennanbaum's theorem, some theorems proven by Rosser on extensions of PA, Robinson's overspill lemma, etc. (Note: I want to avoid digressing into an explanation of forcing if possible).

My question is, what books or online resources do you know of that would be useful for me? That is, do you happen to know of surveys of these topics that are around on arXiv or JSTOR? I have been digging through the mathematical logic section of arXiv for papers and I found a few that are useful, but I thought that some mathematicians/logicians on MO might know of some papers that give a just survey of the introductory results regarding non-standard models of PA.

Thank you!

Models of Peano Arithmeticis good and accessible. – Ed Dean Mar 2 '12 at 0:41