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The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to be written on NIP theories, is an introduction to the subject that will appeal to anyone interested in model theory: graduate students and researchers in the field, as well as those in nearby areas such as combinatorics and algebraic geometry. Without dwelling on any one particular topic, it covers all of the basic notions and gives the reader the tools needed to pursue research in this area. An effort has been made in each chapter to give a concise and elegant path to the main results and to stress the most useful ideas. Particular emphasis is put on honest definitions, handling of indiscernible sequences and measures. The relevant material from other fields of mathematics is made accessible to the logician.

Model theory. --- Independence (Mathematics) --- Logic, Symbolic and mathematical.

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Starting with a simple formulation accessible to all mathematicians, this second edition is designed to provide a thorough introduction to nonstandard analysis. Nonstandard analysis is now a well-developed, powerful instrument for solving open problems in almost all disciplines of mathematics; it is often used as a ‘secret weapon’ by those who know the technique. This book illuminates the subject with some of the most striking applications in analysis, topology, functional analysis, probability and stochastic analysis, as well as applications in economics and combinatorial number theory. The first chapter is designed to facilitate the beginner in learning this technique by starting with calculus and basic real analysis. The second chapter provides the reader with the most important tools of nonstandard analysis: the transfer principle, Keisler’s internal definition principle, the spill-over principle, and saturation. The remaining chapters of the book study different fields for applications; each begins with a gentle introduction before then exploring solutions to open problems. All chapters within this second edition have been reworked and updated, with several completely new chapters on compactifications and number theory. Nonstandard Analysis for the Working Mathematician will be accessible to both experts and non-experts, and will ultimately provide many new and helpful insights into the enterprise of mathematics.

Mathematics. --- Functional Analysis. --- Measure and Integration. --- Operator Theory. --- Game Theory, Economics, Social and Behav. Sciences. --- Mathematical Logic and Foundations. --- Number Theory. --- Functional analysis. --- Operator theory. --- Logic, Symbolic and mathematical. --- Number theory. --- Mathématiques --- Analyse fonctionnelle --- Théorie des opérateurs --- Logique symbolique et mathématique --- Théorie des nombres --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Nonstandard mathematical analysis. --- Analysis, Nonstandard mathematical --- Mathematical analysis, Nonstandard --- Non-standard analysis --- Nonstandard analysis --- Measure theory. --- Game theory. --- Mathematical logic. --- Model theory --- Number study --- Numbers, Theory of --- Algebra --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Functional analysis --- Math --- Science --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Games, Theory of --- Theory of games --- Mathematical models --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)