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# StuVo talk on nonstandard analysis

These are complementary notes for a StuVo talk I gave at Technische Universität Darmstadt, 10 December 2007. While this text is supposed 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