From Book News, Inc.Burris (U. of Waterloo, Canada) divides Kevin Compton's ideas into two parts, one on density in number systems, and the other on the application of number theoretic density results to obtain logical limit laws. The first part, on additive number systems, is suitable as an advanced undergraduate special topics reading text. The second, on multiplicative number systems, demonstrates the challenge and reward of becoming reasonably comfortable with Dirichlet series.Book News, Inc.R, Portland, ORProduct Description:This book shows how a study of generating series (power series in the additive case and Dirichlet series in the multiplicative case), combined with structure theorems for the finite models of a sentence, lead to general and powerful results on limit laws, including $0 - 1$ laws. The book is unique in its approach to giving a combined treatment of topics from additive as well as from multiplicative number theory, in the setting of abstract number systems, emphasizing the remarkable parallels in the two subjects. Much evidence is collected to support the thesis that local results in additive systems lift to global results in multiplicative systems. All necessary material is given to understand thoroughly the method of Compton for proving logical limit laws, including a full treatment of Ehrenfeucht-Fraissé games, the Feferman-Vaught Theorem, and Skolem's quantifier elimination for finite Boolean algebras. An intriguing aspect of the book is to see so many interesting tools from elementary mathematics pull together to answer the question: What is the probability that a randomly chosen structure has a given property? Prerequisites are undergraduate analysis and some exposure to abstract systems.

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