In recent years Leon Harkleroad has mostly concentrated on mathematical aspects of music, but he still enjoys revisiting his old stomping grounds of mathematical logic. Real Analysis through Modern Infinitesimals contains a large (four-digit) number of exercises, both incorporated into the text and gathered into separate sections. Indeed, three pages later in the Preface, Vakil says that “nfinitesimals are served as a side dish” and makes a point of mentioning that he introduces, e.g., continuity and the Riemann integral in delta-epsilon, rather than nonstandard, terms. However, Vakil actually presents the ultrapower construction first and then uses it as motivation for internal set theory.Īlthough the Preface states that “Our goal here is to explore the applications of modern infinitesimals in studying the central topics of real analysis,” the book treats nonstandard techniques as useful additions to the usual toolkit, rather than insisting on viewing everything exclusively through a nonstandard lens. This axiomatic approach allows one to work with a hyperreal system without having to deal with its explicit construction in terms of ultrapowers and the like. The book approaches nonstandard analysis through Edward Nelson’s internal set theory. The second part, Elements of Abstract Analysis, covers more advanced topics - for example, Banach spaces, the Daniell integral, the Baire Category Theorem, and integral operators. In what follows, we ma- nipulate sequences of real numbers. Having infinitesimals, it is not 'standard.' Nor is it 'nonstandard,' however, as this term now has a well-defmed meaning. The first part of the book, Elements of Real Analysis, addresses single-variable calculus - limits, continuity, derivatives, integrals, sequences, and series. I call the system 'non-nonstandard analysis' to draw at- tention to its misfit nature. Real Analysis through Modern Infinitesimals adds to the modest number of textbooks on nonstandard analysis that are aimed at an audience of upper-division undergraduates to graduates.
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