Nonstandard models of arithmetic are of interest to mathematicians through the presence of infinite (or nonstandard) integers and the various properties they inherit from the finite integers. Since their introduction in the 1930s (by Skolem and Godel ), 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 have been kept to a minimum. A basic grounding in elementary model theory and a familiarity with the notions of recursive, primitive recursive, and r.e. sets will be sufficient. Consequently, the book should be suitable for postgraduate students coming to the subject for the first time and a variety of exercises of varying degrees of difficulty will help to further the reader's understanding.
Beginning with Godel's incompleteness theorem, the book covers the prime models, cofinal extensions, end extensions, Gaifman's construction of a definable type, Tennenbaum's theorem, Friedman's theorem and subsequent work on indicators, and culminates in a chapter on recursive saturation and resplendency. The work is designed for postgraduates and research workers specializing in mathematical logic.
'"Models of Peano arithmetic" is a book that should have been written many years ago ... 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 presentation will certainly satisfy all who ever wondered what
such a formula looks like.'
R. Kossak, Zentralblatt für Mathematik und ihre Grenzgebiete Mathematics Abstracts
'It is carefully written and therefore recommended to anyone who wants to learn about this theorem.'
A. Di Bucchianico, MWG, No. 4, April 1993
'Kaye's book has many obvious virtues. It covers a great number of topics which are of the utmost importance for metamathematics of PA. Thus, the book is an advanced monograph which offers a unified treatment of various aspects of the first-order PA ... the author has succeeded in producing a text which can also serve as a graduate textbook of metatheory of PA; the book is clearly written and contains several well-motivated exercises.'
Jan Wole'nski, Jagiellonian University, Poland
'Its well-written presentaiton is aimed at a general (mathematical)audience.'
P. Schmitt, Monatshefte fur Mathematik
Preface; Background; The standard model; Discretely ordered rings; Gödel incompleteness; The axioms of Peano arithmetic; Some number theory in Peano arithmetic; Models of Peano arithmetic; Collection; Prime models; Satisfaction; Subsystems of Peano arithmetic; Saturation; Initial segments; The standard system; Indicators; Recursive saturation; Suggestions for further reading; Bibliography; Index.
Series: Oxford Logic Guides
Number Of Pages: 302
Published: 31st January 1991
Publisher: Oxford University Press
Country of Publication: GB
Dimensions (cm): 23.9 x 16.3
Weight (kg): 0.65