Lectures on the Hyperreals

Based on the author's lectures, this book introduces nonstandard analysis as a radically different way of viewing mathematical concepts and constructions, a source of new ideas, objects and proofs and a wellspring of powerful new principles of reasoning.

This is an introduction to nonstandard analysis based on a course of lectures given several times by the author. It is suitable for use as a text at the beginning graduate or upper undergraduate level, 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; a source of new ideas, objects and proofs; and a wellspring of powerful new principles of reasoning (transfer, overflow, saturation, enlargement, hyperfinite approximation etc.).

Zusammenfassung

R. Goldblatt

Lectures on the Hyperreals

An Introduction to Nonstandard Analysis

"Suitable for a graduate course . . . could be covered in an advanced undergraduate course . . . The author's ideas on how to achieve both intelligibility and rigor . . . will be useful reading for anyone intending to teach nonstandard analysis."-AMERICAN MATHEMATICAL SOCIETY

Inhalt
I Foundations.- 1 What Are the Hyperreals?.- 2 Large Sets.- 3 Ultrapower Construction of the Hyperreals.- 4 The Transfer Principle.- 5 Hyperreals Great and Small.- II Basic Analysis.- 6 Convergence of Sequences and Series.- 7 Continuous Functions.- 8 Differentiation.- 9 The Riemann Integral.- 10 Topology of the Reals.- III Internal and External Entities.- 11 Internal and External Sets.- 12 Internal Functions and Hyperfinite Sets.- IV Nonstandard Frameworks.- 13 Universes and Frameworks.- 14 The Existence of Nonstandard Entities.- 15 Permanence, Comprehensiveness, Saturation.- V Applications.- 16 Loeb Measure.- 17 Ramsey Theory.- 18 Completion by Enlargement.- 19 Hyperfinite Approximation.- 20 Books on Nonstandard Analysis.