Models of Peano arithmetic. (English) Zbl 0744.03037

Oxford Logic Guides. 15. Oxford: Clarendon Press. x, 292 p. (1991).
“Models of Peano arithmetic” is a book that should have been written many years ago. The topic covers a very broad spectrum of results, some dealing directly with most fundamental issues, and it has links to all major fields in mathematical logic. However, the subject has never had a standard introductory text. For many technical reasons potential authors have found the task of writing such a text rather difficult, leaving this interesting area of research without the presentation it deserves. Kaye’s book fills this gap in literature remarkably well. The book consists of 16 sections. The first five are devoted to an elegant exposition of Gödel’s first and second incompleteness theorems. Although this is not one of the covered topics, this part can be recommended as an excellent introduction to more advanced explorations of weak fragments of arithmetic. In the next three sections classical results about cofinal and end extensions of models are discussed. Section 9 brings the very detailed work which is necessary to define in PA a \(\Sigma_ 1\) universal truth formula. The presentation will certainly satisfy all who ever wondered what such a formula looks like. Once the tedious work is done, Kaye proceeds with presentation of other classical themes: Tennenbaum’s theorem, Friedman’s theorem on initial segments and the arithmetized completeness theorem, Scott sets and standard systems are also discussed in this part. Section 14 contains material on indicators. Indicators, once fashionable tools for obtaining independence results, are now considered obsolete by some. It is good that this topic has been included here, the method of indicators is an original, specifically arithmetical tool, quite worth studying for its own sake. In this section we are also given a proof of the famous Paris-Harrington independence result. The book concludes with a section on recursive saturation, where the highlights are the results of Kotlarksi, Krajewski and Lachlan on full satisfaction classes. The text is supplemented with a good deal of exercises at various levels, and is accessible to a beginner with some basic knowledge in model theory. Many important topics that fall under the category of “Models of arithmetic” are not, and could not have been, covered by Kaye’s book. Most of them however are mentioned in the appendix with suggestions for further reading. One can hope now that texts covering these other areas will appear soon.


03C62 Models of arithmetic and set theory
03F30 First-order arithmetic and fragments
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations