Models of Peano Arithmetic

Goedel's incompleteness theorem tells us that no finite set of axioms can capture the essence of the integers - there will always be non-standard models of such a set of axioms. In Models of Peano Arithmetic Richard Kaye takes a look at such nonstandard models. The book is not for the fainthearted though - there's no gentle introduction, from the start it is full of highly abstract symbols. The book is aimed at aimed at postgraduates starting research into the subject, and assumes some previous experience of model theory.

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 non-standard integers, such as Tennenbaum's theorem which says that arithmetic on non-standard 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 non-standard integers, but that is what Kaye does in the rest of the book, deriving results such as Friedman's theorem - every countable non-standard 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 non-standard integers in depth.