Real Analysis Through Modern Infinitesimals provides a course on mathematical analysis based on Internal Set Theory (IST) introduced by Edward Nelson in 1977. After motivating IST through an ultrapower construction, the book provides a careful development of this theory representing each external class as a proper class. This foundational discussion, which is presented in the first two chapters, includes an account of the basic internal and external properties of the real number system as an entity within IST. In its remaining fourteen chapters, the book explores the consequences of the perspective offered by IST as a wide range of real analysis topics are surveyed. The topics thus developed begin with those usually discussed in an advanced undergraduate analysis course and gradually move to topics that are suitable for more advanced readers. This book may be used for reference, self-study, and as a source for advanced undergraduate or graduate courses.

'Nader Vakil has shown with his text that advanced calculus and much of related abstract analysis can be explained and simplified within the context of internal set theory.'

Peter Loeb Source: SIAM Review

'Real Analysis through Modern Infinitesimals intends to be used and to be useful. Nonstandard methods are deployed alongside standard methods. The emphasis is on bringing all tools to bear on questions of analysis. The exercises are interesting and abundant.'

James M. Henle and Michael G. Henle Source: MAA Reviews