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Richard Kaye

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.

Amazon.com info
Hardcover 304 pages  
ISBN: 019853213X
Salesrank: 3846839
Weight:1.4 lbs
Published: 1991 Clarendon Press
Amazon price $181.20
Marketplace:New from $142.12:Used from $122.99
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Amazon.co.uk info
Hardcover 302 pages  
ISBN: 019853213X
Salesrank: 4293310
Weight:1.4 lbs
Published: 1991 Clarendon Press
Marketplace:New from £95.15:Used from £79.99
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Amazon.ca info
Hardcover 302 pages  
ISBN: 019853213X
Salesrank: 2543065
Weight:1.4 lbs
Published: 1993 Clarendon Press
Amazon price CDN$ 236.17
Marketplace:New from CDN$ 204.36:Used from CDN$ 220.84
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Product Description
Non-standard models of arithmetic are of interest to mathematicians through the presence of infinite integers and the various properties they inherit from the finite integers. Since their introduction in the 1930s, they have come to play an important role in model theory, and in combinatorics through independence results such as the Paris-Harrington theorem. This book is an introduction to these developments, and stresses the interplay between the first-order theory, recursion-theoretic aspects, and the structural properties of these models. Prerequisites for an understanding of the text have been kept to a minimum, these being a basic grounding in elementary model theory and a familiarity with the notions of recursive, primitive recursive, and r.e. sets. Consequently, the book is suitable for postgraduate students coming to the subject for the first time, and a number of exercises of varying degrees of difficulty will help to further the reader's understanding.