Lectures on the hyperreals: an introduction to nonstandard analysis

An introduction to nonstandard analysis based on a course given by the author. It is suitable for beginning graduates or upper undergraduates, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions. It is a source of new ideas, objects and proofs, and a wealth of powerful new principles of reasoning. The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line. Highlights include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set-theoretic approach to enlargements than is usual.

What are the Hyperreals?- Large Sets.- Ultrapower Construction of the Hyperreals.- The Transfer Principle.- Hyperreals Great and Small.- Convergence of Sequences and Series.- Continuous Functions.- Differentiation.- The Riemann Integral.- Topology of the Reals.- Internal and External Sets.- Internal Functions and Hyperfinite Sets.- Universes and Frameworks.- The Existence of Nonstandard Entities.- Permanence, Comprehensiveness, Saturation.- Loeb Measure.- Ramsey Theory.- Completion by Enlargement.- Hyperfinite Approximation.- Books on Nonstandard Analysis.

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