Nonstandard analysis (seminar talk given as a student)

These notes complement a seminar talk I gave as a student at Technische Universität Darmstadt as part of the StuVo (Studentische Vortragsreihe), Dec 10, 2007. While this text is meant to be informal in nature, any corrections as well as suggestions are very welcome.

Abstract

We discuss some ways to add infinitesimals to our usual numbers, get acquainted with ultrafilters and how they can be used to construct nonstandard extensions, and then provide an axiomatic framework for nonstandard analysis. As basic working examples we present nonstandard characterizations of continuity and uniform continuity. We close with a short external look at the nonstandard integers and point out connections with \(p\)-adic integers.

Contents

1. Introduction
   1. Approaches to Getting Infinitesimals
   2. The Use of Infinitesimals
2. Diving In
   1. Limits and Ultrafilters
   2. A First Nonstandard Peak
   3. An Axiomatic Approach
   4. Basic Usage
   5. The Source of Nonstandard Strength
   6. Stronger Nonstandard Models
   7. Studying \({}^\ast \mathbb Z\) in its Own Right
3. Conclusions

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