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 Goedel ), 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 Goedel'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.

This book is a collection of articles, some introductory, some extended surveys, and some containing previously unpublished research, on a range of topics linking infinite permutation group theory and model theory. Topics covered include: oligomorphic permutation groups and omega-categorical structures; totally categorical structures and covers; automorphism groups of recursively saturated structures; Jordan groups; Hrushovski's constructions of pseudoplanes; permutation groups of finite Morley rank; applications of permutation group theory to models of set theory without the axiom of choice. There are introductory chapters by the editors on general model theory and permutation theory, recursively saturated structures, and on groups of finite Morley rank. The book is almost self-contained, and should be useful to both a beginning postgraduate student meeting the subject for the first time, and to an active researcher from either of the two main fields looking for an overview of the subject.

This book covers the basic theory of matrices and vector spaces. The book's three main parts cover (I) matrices, vector spaces, bases, and dimension; (II) inner products, bilinear and sesquilinear forms over vector spaces; (III) linear transformations, eigenvalues and eigenvectors, diagonalization, and Jordan normal form. An introduction to fields and polynomials over fields is also provided, and examples and applications are provided throughout. The approach throughout is rigorous, but without being unnecessarily abstract. In particular, this book would be suitable reading for a student with no prior exposure to abstract algebra. Although intended as a 'second course', the book is completely self-contained and all the material usually given in a 'first course' in presented fully in Part I, so the book provides a useful guide to the entire theory of vector spaces as usually studied in an undergraduate degree. Abstract methods are illustrated with concrete examples throughout, and more detailed examples highlight applications of linear algebra to analysis, geometry, differential equations, relativity and quantum mechanics. As such, the book provides a valuable introduction to a wide variety of mathematical methods.

'One Man and His God' describes the activities of a Liverpool Pastor who, in addition to running a church, reached out to violent Teddy Boys, delinquent girls and faced atheists and agnostics at Liverpool's Speakers Corner at the Pier Head. Exposing his faith to the questions and ridicule of unbelievers not only won converts, but caused him to re-examine his own beliefs and discard some of them. His own pilgrimage of faith continued as he helped to lead a joint Christian Aid/Army mercy dash across the Sahara where thousands were dying through drought and disease. I found it impossible, Richard writes, to tell a starving mother holding a dying baby that God loved her. Visiting India he witnessed remarkable development projects run by evangelicals, Hindus and Muslims. In the West Bank and Gaza he sympathised with the Israelis and agonised with the Palestinians. These visits helped to prune and re-shape his faith. The total inspiration of the Bible, the exclusiveness of the Evangelical Gospel and the Doctrine of everlasting punishment were all examined and laid aside as he moved forward in his quest to follow Christ. Illustrations and anecdotes abound in this book which is written to make you think and perhaps raise a smile. The authors sense of humour is evident as he describes his preachments and magic shows that served to spread the Gospel. Richard believes in the dynamic power of the 'whole Gospel' and feels he has found a no-nonsense faith that equips one for life today.