
Richard Kaye
Models of Peano Arithmetic
The first part of the book gives rigorous proofs of Gödel's incompleteness theorem and related results. Kaye then gets on to the properties of nonstandard integers, such as Tennenbaum's theorem which says that arithmetic on nonstandard integers is not recursive  i.e. you can't do calculations in the way you are used to with other mathematical systems. You might think that this made it impossible to study the properties of nonstandard integers, but that is what Kaye does in the rest of the book, deriving results such as Friedman's theorem  every countable nonstandard model is isomorphic to a proper initial segment of itself.
Kaye says that some of the book should be understood by an advanced undergraduate student. I have my doubts about this, but the book is well laid out, and will give a flavour of the subject even to those who can't follow the details. Also it's probably the best book if you want to study nonstandard integers in depth.