Nonstandard Analysis - A Simplified Approach

TL;DR

This paper presents a simplified approach to nonstandard analysis by deriving many real analysis notions using minimal logical tools and ultrapower structures, making the theory more accessible.

Contribution

It introduces a streamlined method for developing nonstandard analysis with only two basic atomic formulas and applies it to ultrapower structures generated by free ultrafilters.

Findings

Many real analysis concepts are obtained using minimal logical complexity

Almost all characteristics hold for ultrapower structures generated by free ultrafilters

The approach simplifies the foundational framework of nonstandard analysis

Abstract

In this monograph, nonstandard characteristics for many notions from real analysis are obtained and applied. However, only two simple types of atomic formula are used and almost all of the characteristics are shown to hold for a simple ultrapower styled structure generated by any free ultrafilter over the natural numbers.